Topology in Quantum (Field) Theory Falk Bruckmann U Regensburg, Summer term 2016 Version Wednesday 20 th July, 2016, 10:48 content: solitonic objects stabilized by topology from kinks in 1d to instantons in 4d mathematical description: homotopy/winding numbers, fibre bundles, moduli space fermion zero modes and index theorem physical mechanisms: tunnelling, charge and flux quantization, Borel resummation and beyond, anomalies, (towards) confinement Reviews/Books/Lecture notes [Raj] R. Rajaraman, “Solitons And Instantons. An Introduction To Solitons And Instantons In Quantum Field Theory,” North-holland (1982) [still the classical book] [Col] S. Coleman, “Aspects of Symmetry,” Cambridge University Press (1985) [in particular the sections ‘Classical lumps and their quantum descendants’ and ‘The uses of instantons’, very nice physical intuitions] [Nak] M. Nakahara, “Geometry, topology and physics,” Bristol, UK: Hilger (1990) Graduate student series in physics [nice account of mathematics needed] [Vac] T. Vachaspati, “Kinks and domain walls: An introduction to classical and quantum solitons,” Cambridge University Press (2010) [Zin] J. Zinn-Justin, “Path Integrals in Quantum Mechanics,” Oxford University Press (2005) [Kle] H. Kleinert, “Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets,” World Scientific (2004) [tHo] G. ’t Hooft, “Monopoles, instantons and confinement,” Saalburg lecture notes, arxiv.org: hep-th/0010225. 0-1
Transcript
Topology in Quantum (Field) Theory
Falk BruckmannU Regensburg, Summer term 2016
Version Wednesday 20th July, 2016, 10:48
content: solitonic objects stabilized by topology from kinks in 1d to instantons in 4d
mathematical description: homotopy/winding numbers, fibre bundles, moduli space
[Raj] R. Rajaraman, “Solitons And Instantons. An Introduction To Solitons And InstantonsIn Quantum Field Theory,” North-holland (1982)
[still the classical book]
[Col] S. Coleman, “Aspects of Symmetry,” Cambridge University Press (1985)
[in particular the sections ‘Classical lumps and their quantum descendants’ and ‘Theuses of instantons’, very nice physical intuitions]
[Nak] M. Nakahara, “Geometry, topology and physics,” Bristol, UK: Hilger (1990) Graduatestudent series in physics
[nice account of mathematics needed]
[Vac] T. Vachaspati, “Kinks and domain walls: An introduction to classical and quantumsolitons,” Cambridge University Press (2010)
[Zin] J. Zinn-Justin, “Path Integrals in Quantum Mechanics,” Oxford University Press (2005)
[Kle] H. Kleinert, “Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, andFinancial Markets,” World Scientific (2004)
[tHo] G. ’t Hooft, “Monopoles, instantons and confinement,” Saalburg lecture notes,arxiv.org: hep-th/0010225.
0-1
[Ton] D. Tong, “TASI lectures on solitons: Instantons, monopoles, vortices and kinks,”arxiv.org: hep-th/0509216.
[‘inverse order’: from 4d down to 0d, rather advanced: including string points of view]
[Wei] E.J. Weinberg, “Classical solutions in quantum field theory,” Cambridge UniversityPress (2012)
[Shi] M. Shifman, “Advanced topics in quantum field theory: A lecture course,” CambridgeUniversity Press (2012)
for 4d Yang-Mills instantons (typically including low-dimensional motivation, too):
[SchShu] T. Schafer and E. V. Shuryak, “Instantons in QCD,” Rev. Mod. Phys. 70 (1998)323, arxiv.org: hep-ph/9610451
[the experts]
[Dia1] D. Diakonov, “Instantons at work,” Prog. Part. Nucl. Phys. 51 (2003) 173, arxiv.org:hep-ph/0212026.
[dito]
[Dia2] D. Diakonov, “Instantons, solitons, monopoles ...”, Lecture Series @ RU Bochum(2008), http://www.tp2.ruhr-uni-bochum.de/lehre/skripte/qcd/solitons/
0-2
1 Motivation
some things are not similar to others at all, i.e. cannot be deformed into each other by smalldeformations/perturbations
two linked loops cannot be separated in 3d without cutting them (but in 4d)
one can rope (lasso) a cow, but not peel an orange without cutting it
these situations are described by a ‘topological’/‘winding’ number taking integers only
in physics such numbers stabilize solitonic objects, which are solutions of nonlinear differential(wave) equations
1.1 Experimental Physics
the first ‘soliton’:
‘The soliton phenomenon was first described in 1834 by John Scott Russell (1808-1882)who observed a solitary wave in the Union Canal in Scotland. He reproduced thephenomenon in a wave tank and named it the ”Wave of Translation”.’ [wikip.: soliton]
‘ ... a rounded, smooth and well-defined heap of water, which continued its course alongthe channel apparently without change of form or diminution of speed.’
such solitons are typically stabilized by an integer number like above and thereforecannot be deformed into vacuum
1.2 Mathematics
the functions e−1/x or e−1/x2 have a Taylor series identical to zero
(since all derivatives are of the form limx→0
1x#e−1/x = lim
y→∞y#e−y = 0)
‘there are some phenomena which are impossible to understand by perturbation theory,regardless of how many orders of perturbation theory we use’ [wikip.: nonperturbative]= nonperturbative
1.3 Theoretical Physics
tunnelling:
probability ∼ exp(− length
√2m(potential height− energy)/~
), nonperturbative in ~
(quantum) statistics:
partition sum = exp(−free energy or potential/T ) , looks nonpert. in temperature T
related by analogy between partition sum and QM path integral
1-1
2 Kinks
first in the mexican hat potential in 1d Quantum mechanics
2.1 Tunnelling, mexican hat potential, inverted potential
start by potential barrier in 1d QM: V = V0 for −v < x < v, zero elsewhere
wave functions for E < V0:
ψ(x) =
1 exp(ikx) +R exp(−ikx) for x < −v# exp(κx) + #′ exp(−κx) for − v < x < v
T exp(ikx) + 0 for v < x
(2.1)
where 1 and 0 have been chosen for convenience, and k =√
2mE/~, κ =√
2m(V0 − E)/~
the four unknowns – the reflection amplitude R, the transmission amplitude T plus theprefactors # and #′ – are fixed by demanding ψ to be continuous and differentiable at±v
for high barriers, the transmission probability
|T | → . . . exp(− 4 v
√2m(V0 − E)︸ ︷︷ ︸
for . . . ~
/~)
(2.2)
decreases exponentially with the width v and the square root of the effective heightV0 − E of the barrier
this is measured in units of ~, in which |T | is nonperturbative
T can also be read off from the probability current j = ~m Im (ψ∗∂xψ)
for nonconstant potentials: Wentzel-Kramers-Brillouin formula
T ∼ exp(− 2
∫turning points
dx√
2m(V (x)− E)/~)
(2.3)
from the classical p2
2m = E − V note that√
2mE = ~k is the momentum in the
classically allowed region, whereas in the classically forbidden region√
2m(V − E) =√2m(−)(E − V ) can be interpreted as the momentum in the inverted potential
moreover,∫dx p =
∫dt px looks like an action (since H = px− L and H = E = 0)
also note that the sign of p2 ∼ x2 can also be changed by t → iτ ‘going to Euclideantime’ (see below)
let’s look at this for a concrete example
2-1
the mexican hat/Higgs potential/bottom of wine bottle:
V (x) =λ
4!(x2 − v2)2 , λ > 0 (2.4)
nonnegative, zero at the minima x = ±v, the curvature around which gives the springconstant k
from now on let’s put the mass in the QM problem to 1 (all quantites in mass units,i.e. with the corresponding power of mass)
then k = ω2, the frequency of the oscillation
in QFT the minima are vacua and the curvature around them gives a mass
we keep calling it ω
V =1
2V ′′(±v)︸ ︷︷ ︸
= ω2
(x∓ v)2 +O((x∓ v)3,4) , ω2 =λ
3v2 (2.5)
will be the mass of a scalar particle, the Higgs
(for those who know the Higgs mechanism: mass ∼ Higgs vev. × coupling X)
the cubic and quartic term are interactions
expectation for the lowest states in large barriers:
approximately given by the two ground states from harmonic oscillators around minima,E0 = 1
2 ~ω, split by tunnelling
‘for a quantum system the ground state cannot be degenerate’ [Zin]
for the tunnelling between −v and v one has to set E = 0 and – as discussed before –look at the inverted potential V = −V
2.2 The kink solution
indeed in V , there exists a classical solution rolling down from x = −v via the minimumx = 0 up to x = +v
plus the backward solution
of course, this takes infinite time: x(t = −∞) = ∓v, x(t = +∞) = ±v
keep in mind that particles could also stay at ∓v forever
the equation of motion to be solved is mx = −dVdx = dV
dx , but we immediately make useof energy conservation
1
2x2 + V (x) = E = 0 (recall m = 1) (2.6)
which lowers the degree of the differential equation from two down to one (we’ll see asimilar trick at work later in more complicated systems)
2-2
it is still nonlinear (linear diff. eqs. do not support interesting solutions)
the explicit solutions can be obtained by separation of variables:
x2 = −2V (2.7)
dx
dt= ±
√2λ
4!(v2 − x2) (2.8)∫
dx
v2 − x2= ±
∫dt
√2λ
4!(2.9)
there is one integration constant, that we put on the time side∫ x
0
dx′
v2 − x′2= ±
∫ t
t0
dt′√
2λ
4!, t = t0 x = 0 (2.10)
(|x| < v)1
vartanh
x
v= ±
√λ
12(t− t0) now use (2.5) (2.11)
artanhx
v= ± ω
2(t− t0) (2.12)
x(t) = ± v tanh(ω
2(t− t0)
)(2.13)
-10 -5 5 10Ω t
-1.0
-0.5
0.5
1.0xv
Fig. 1: A kink at t0 = 3 and an antikink at t0 = −4.
interpretation
indeed moving from ∓v to ±v in infinite time
upper sign: kink1, lower sign: antikink
the approach to ±v is exponential in time, since tanh(ω2 t)t→±∞−→ ±(1− e−ωt)
at t = t0 the (anti)kinks visit x = 0, the maximum of V , also called ‘false vacuum’
we have thus obtained two families of solutions with the (anti)kink time t0 ∈ R being afree parameter, a so-called ‘collective cordinate’; it is of course related to the invarianceof the system under time translations leading to energy conservation
1deutsch: der/die Kink [Seemannssprache/nordeutsch] = unerwunschter Knoten im Tau bzw. Knick inStahltrosse [Duden, wikipedia]
2-3
at t = t0 the slope is ω/2 · v, such that in linear approximation x = ± v is reached aftera time difference of 2/ω
actually xv = f(ωt) is a universal function up to the shift in the argument
both kinetic and potential2 energy of the solutions read
λv4
4!
1
cosh4(ω2 (t− t0)
) (2.14)
i.e. these objects are localised at t = t0 with decay time 2/ω (see above): ‘tunnellingevents’; instantaneous → ‘instant-ons’
integrating this over time yields √λ
27v3 =
ω3
λ(2.15)
if perturbation theory is performed at fixed curvature ω and in small λ, then theseobjects have large energies/are heavy
it is clear that such solutions exists whenever minima of same height (in V , maxima inV ) exist
another very important such case follows now
2V or |V |, as |x| ≤ v
2-4
3 Solitons
3.1 The sine-Gordon model in (quantum) mechanics . . .
potential with cosine (physicists’ humor: Klein-Gordon eq.→ ‘sine-Gordon’ eq. as eom.)
V (x) = a(1− cos 2π
x
v
)(a, v > 0) (3.1)
minima at x = Z v of height zero again, in between V positive
curvature:
V ′′(Z v) = ω2 = a(2π
v
)2(3.2)
classical solutions interpolating (rolling) between kv and (k ± 1)v are typically calledsolitons/antisolitons (for the nomenclature see below)
explicit solution, obtained as above:
x(t) = ± v
π/2arctan exp
(ω(t− t0)
)(3.3)
recall that arctan(0, 1,∞) = 0, π/4, π/23
-10 -5 5 10Ω t
-1.0
-0.5
0.5
1.0xv
Fig. 2: A soliton at t0 = 3 and an antisoliton at t0 = −4 (t0 is where the maximum of thepotential at a half-integer multiple of v is visited, as indicated).
approaching to 0 and ±v like exp(−ω|t|) again4
kinetic and potential energy
2a
cosh2(ω(t− t0)
) (3.4)
localised in |t− t0| . 1/ω again
integrates to 2a/ω ∼ ω3/λ again, cf. Eq. (2.15) with coupling constant λ ∼ a/v4 (fromV (iv)(Z v))
3and for the kink running backwards in time that arctan(1/x) = π/2− arctan(x)4since arctan(exp(t))
t→−∞−→ exp(t) and arctan(exp(t))t→∞−→ π/2− exp(−t)
3-1
3.2 The Bogomolnyi trick
revisit Hamilton’s principle of least action, still for the inverted potential; the action
S =
∫dt(1
2x2 − V
)(3.5)
yields the classical equations of motion, when S is stationary, δS = 0, under variationsof the path x(t), keeping x(tinitial) and x(tfinal) fixed
here comes an interesting trick/view on that [Bogomolnyi]
by virtue of a ‘superpotential’ W (x), provided V ≥ 0(dWdx
)2= 2V (3.6)
we can rewrite the action into a square and a total derivative
S =
∫dt(1
2[x∓
√2V ]2 ± x
dW/dx︷ ︸︸ ︷√2V︸ ︷︷ ︸
dW/dt
)check=
∫dt(1
2[x+ 0 + 2 |V |︸︷︷︸
=V=−V
])
(3.7)
=
∫dtnonnegative±W (x(t))
∣∣t=∞t=−∞ (3.8)
where we assumed a monotonically rising W (x): dW/dx = +√
2V (x)
we obtain a bound
S ≥ ±W (x(t))∣∣t=∞t=−∞ (3.9)
which is the same for all paths with same x(t = ±∞), as required in Hamilton’s principle
note that in classical mechanics, this is a way to define/obtain classical solutions,whereas in Feynman’s path integral for quantum mechanics all of these paths (‘con-figurations’) are integrated over with their particular actions
this surface term can be seen as a ‘topological’ contribution to the action, that is in-variant under deformations of the path at any finite time (‘in the bulk’)
the (anti)kink/soliton obeys x = ±√
2V , cf. Eq. (2.7), and make the square vanish
these objects are thus absolute minima of the action obeying the equality in the bound(3.9)
once again: other paths evolving from the same initial to the same final x (thus sameW (x(±∞))) must have actions higher than that of the soliton, which is the ‘minimalamount for a transition’
through this trick, the differential equation to be solved is of first order only
seen before, will apply to higher dimensions later
3-2
in some more detail:√
2V = 2√a∣∣ sinπx
v
∣∣ (note half angle compared to V in(3.1)) (3.10)
for a soliton from x = 0 to x = v we can neglect the absolute value and get W fromintegration
W (x)x∈[0,v]
=2√a v
π
(1− cosπ
x
v
), W (v)−W (0) =
4√a v
π(3.11)
which can also be obtained from the action of the soliton directly integrating its kineticenergy and potential over t giving ω3/λ, cf. below Eq. (3.4)
what about the other bound in Eq. (3.9)? for the soliton considered here, it would giveS > −4
√av/π, which is of course weaker
antisoliton: now (x < 0) the lower equality is obeyed and the lower bound is stronger
results in the same minimal action for all paths evolving like the antisolition (why shouldx = (0→ v) be cheaper than x = (0→ −v)?)
no such bound for vacua x = (v → v) and x = (−v → −v)
3.3 Multisolitons I
there are infinitely many minima in the SG model, but so far we have discussed onlysingle transitions from k v to (k ± 1)v
from the Bogomolnyi trick we immediately get the corresponding multiple of the minimalaction for such paths
(since by construction W is monotonically increasing, i.e. looks piecewise like the re-stricted W above, therefore W (2v)−W (v) = W (v)−W (0) etc.)
the rolling particle picture, however, is delicate
we have to push the particle a little more to overcome the hill, but then it escapes allthe way to infinity
we’ll come back to this issue later
of course, all this is irrelevant for the mexican hat: after a kink the particle is in theright minimum = on the right hill, thus only an antikink could follow
this is even trickier and we’ll come back to that as well
3.4 . . . and in (quantum) field theory
classical, in 1+1 dimensions; first an experimental approach
chain of pendulums/pendula, angles ϕ with index a
realizes cosine potential∑
a V (ϕa) and kinetic energies ∝ ϕa2
3-3
neighbors coupled through springs or just by torsion of the rubber band in between:energy cost ∝ (ϕa − ϕa+1)2
in the continuum limit of that system we arrive at a field theory
change of variables: ϕa(t)→ ϕ(t, x)
(and rescaling such that kinetic and potential energy become those used below)
ϕ is a ‘scalar’ field depending on a position x (here in 1d) and time
energy
H =
∫dx[12ϕ2 +
c2
2(∂ϕ
∂x)2 + V (ϕ)
](3.12)
[. . .] is the corresponding energy density
the parameter c is given by the particular coupling (spring constant)
which dimension? velocity, namely of the waves appearing without potential
in a relativistic theory, thanks to the required Poincare invariance, c can only be thevelocity of light
we set c = 1 (meaning space and time have the same dimension)
actually a Hamiltonian should be written in terms of fields (like positions) and momenta
in a field theory they are functions of x as well: pa(t) → Π(t, x) and the energy startswith Π2/2 (again m = 1)
the Lagrange picture is nicer in relativistic theories
with continuum counterpart of the common rule H =∑
a paϕa − L where pa = ∂L∂ϕa
weobtain immediately:
L =
∫dx[ 1
2(∂tϕ)2 − 1
2(∂xϕ)2 − V (ϕ)︸ ︷︷ ︸L
](3.13)
again, the integrand L = L(t, x) is the corresponding Lagrange density
while the Hamiltonian is the sum of kinetic and potential energy, the Lagrangian is thedifference
the first two terms are indeed of the relativistic form
throughout the lecture the sum convention for indices appearing twice is applied
considering the action S =∫dtL, also the integration becomes symmetric in (t, x)5
5and the factor√| det g| also present in a relativistic action is just a constant
3-4
the equations of motion6:
∂µ∂L
∂(∂µϕ)− ∂L∂ϕ
= 0 ∂2t ϕ− ∂2
xϕ+∂V
∂ϕ= 0 (3.15)
3.5 Moving solitons
where are the soliton solutions?
a first trivial connection: uniform excitations ϕ(t, 6x) meaning back to mechanics
soliton = being at the bottom at t0 with just the right energy to climb up with timeand reach the top at infinite future
note that the soliton discussed for QM tunnelling moves in V = −V , but for SG this isjust V up to a shift in argument, i.e. we can take over the solution and its properties
by analogy between t and x in e.o.m. above:
there exists a solution ϕ(6 t, x), where now the transition is done as a function of x
now indeed in V , from relative sign in e.o.m.
so preparing the chain such that the pendulums are down at both ends7 and up inbetween and preparing the angles carefully following the analytic solution, the chainshould not change in time (stabilized by a subtle interplay between x-gradient (springs)and potential (gravity))
static solution, from (3.3) with x(t)→ ϕ(x)
ϕ(x) = ± v
π/2arctan exp
(ω(x− x0)
)(3.16)
now x0 is the spatial position of the soliton
from boosting this we’ll now obtain solitons moving at a constant speed
2d Lorentz transformation/boost Λ:
Λ = γ
(1 −β−β 1
), β =
v
c= v ∈ (−1, 1) , γ =
1√1− β2
∈ [1,∞) (3.17)
need more details (in 2d)?
coordinates transform as (t′
x′
)= Λ
(tx
)(3.18)
and a scalar field like our ϕ does not pick up an additional factor
6the resulting sine-Gordon equation is sometimes written as ∂yzϕ = sinϕ; one obtains that by putting a = 1,v = 2π and using ‘light cone coordinates’ y = (x+ t)/2, z = (x− t)/2 (basically rotating the metric fromthird to first Pauli matrix)
7at infinity, so this can probably be only a gedankenexperiment
3-5
new moving solutions ϕmov are obtained as
ϕmov(t, x) = ϕ(t′, x′)static= ϕ(x′) = ϕ
( x− βt√1− β2
)(3.19)
rescaling the constant x0 in the static solution (3.16) we obtain
ϕmov(t, x) = ± v
π/2arctan exp
( ω√1− β2
(x− x0 − βt))
(3.20)
best characterised by following the point, where ϕmov = ± v/2 – as before – or moregenerally by the isolines of same value of ϕmov, cf. Fig. 3
Fig. 3: Profiles of moving solitons with velocity β = 1/10 (left) and β = 9/10 (right). Thecontours shown in the bottom panels refer to the same values of the field in both plots.Of course, solutions with negative β-values move towards the left, whereas antisolitonschange with x from zero to negative (in general smaller) values of the field.
obviously, the latter means x−βt = const., which is nothing but a motion with constantvelocity β (actually β · c)
tilted lines in Fig. 3, experimentally: a soliton walking along the chain
like for massive particles in special relativity, this motion is time-like (= inside thelightcone/connected to a static wordline)
3-6
in this spirit, the region of order 1/ω, in which the transition is almost completed,might be viewed as the extension of the particle (this particle is pretty sharp as thetails extending beyond this region are exponentially small with distance)
this region also gets length contracted; the easiest way to see this is the gradient ofϕmov (wrt. x), which has a factor 1/
√1− β2 that gets larger, when β comes close to
±1, cf. Fig. 3 right panels
last but not least the energy is also that of a relativistic particle
with (3.12) and (3.19) we compute the dependence on β:
E =
∫dx[12
(∂tϕmov(t, x))2 +1
2(∂xϕmov(t, x))2 + V (ϕmov(t, x))
](3.21)
=
∫dx′(∂x′∂x
)−1 [12
(∂x′ϕ(x′))2(∂x′
∂t
)2+(∂x′∂x
)2+ V (ϕ(x′))]
(3.22)
eom. of ϕ=
∫dx′ γ−1 1
2(∂x′ϕ(x′))2
γ2 + γ2β2 + 1︸ ︷︷ ︸
γ2(β2+1/γ2)=γ2
(3.23)
= γ · β-indep. = γ · E|β=0 recall γ =1√
1− β2≥ 1 (3.24)
E is the relativistic mass, bigger or equal than the rest mass E|β=0 (the mass in therest frame)
3.6 Topological current
define the ‘topological’ current
jµ =1
vεµν∂νϕ (3.25)
it is conserved, ∂µjµ = 0, due to antisymmetry of ε
not using any symmetry of the system = not a Noether current
corresponding charge:
Q =
∫dx j0 =
1
v
∫dx ∂xϕ =
1
vϕ∣∣∣x=∞
x=−∞=
1
v(k+ − k−)v ∈ Z (3.26)
the values of ϕ at x = ±∞ cannot be moved away from the discrete vacua k±v, unlessthe potential energy becomes infinite (
∫dx V (ϕ)︸ ︷︷ ︸
→6=0
=∞, kinetic and gradient energies arepositive and thus cannot help)
the topological charge is discrete and – therefore – does not change with time, ∂tQ = 0
for the mexican hat potential the space of vacua ± v is formally a Z2 group, thereforesome authors speak of ‘Z2 kinks’ [Vac]
3-7
3.7 Multisolitons II
task for students: soliton-soliton, soliton-antisoliton, breather etc.
3.8 Nomenclature
here I try to collect precise notions for the objects we are dealing with, but be aware that theliterature is not consistent on that
definitions: let ε(t, ~x) be the energy density of the field theory at hand (the zero-zerocomponent of the energy-momentum tensor)
for relativistic scalar theories ε = ϕ2/2 + (∂~xϕ)2/2 + V (ϕ), in the SG model see (3.12)and the calculation following (3.21)
a solution of the classical field e.o.m. is dissipative, if its maximum decays in time[Col]
limt→∞
max~x
ε(t, ~x) = 0 (3.27)
“Most of the simple field theories with which we are familiar have the property that allof their non-singular solutions of finite total energy are dissipative. This is the case forMaxwell’s equations, the Klein-Gordon equation, etc. However, . . . ” [Col, Sec. 6.1]
impossible e.g. in the linear Klein-Gordon eq. ∂2t ϕ− ∂2
xϕ+m2ϕ = 0 (from V = m2
2 ϕ2)
– being linear, plane waves with (angular) frequency ω and wave number ~k solve itprovided the dispersion relation ω2 = ~k2 +m2 = ω2(~k) holds
– consider a wave packet sharply peaked around central wave number k0
– then the phase moves with the phase velocity ω(k0)/k0 and the wave packet’senvelope moves with group velocity ω′(k0)
– in addition, the wave packet’s width increases in time with ω′′(k0) (which is nonzerounless ω(k) is linear) since wave packets with slightly different k0’s differ in groupvelocity by this quantity making the packet wider
– more precise: saddle point/Gaussian approximation around k0 (since Fourier am-plitude is sharply peaked around it) leads to an expansion of iω(k)t . . .
in nonlinear wave equations different wave number components talk to each other
‘lumps’ = nondissipative (and regular and of finite energy) [Col]
‘solitary wave’ = localised energy density of the form (and regular) [Raj]
ε(t, x) = ε(~x− ~v t) (3.28)
moves at velovity ~v
a breather is thus not a soliton, but a lump
3-8
Galileian or Lorentz invariance can be used to boost a static sol. to such a moving one
‘soliton’ = solitary wave with an additional requirement on the scattering of several ofthem [Raj], which is the fascinating property of these objects
if (the energy of) a field configuration in the infinite past separates into the superpositionof N single solitary waves,
ε(t, x)t→−∞−→
N∑i=1
ε0(~x− ~vit− ~ai) (3.29)
then, to be a soliton, it should assume the same form in the infinite future, up todisplacements ~δi:
ε(t, x)t→+∞−→
N∑i=1
ε0(~x− ~vit− ~ai + ~δi) (3.30)
meaning that shape and velocity of the solitons are restored, after the scattering event(in which all kinds of things might happen)
want to reinterprete the displacements by ’time delays’ ..− ~vit+ ~δi?= ..− ~vi(t− t0,i)
in higher dimensions only if ~vi ∝ ~δi as vectors
the SG solitons obey this scattering condition (cf. the exact soliton-soliton and soliton-antisoliton solutions)
the mexican hat kink is a solitary wave, but it can only be followed by an antikink, forthe time evolution of such system see [Vac] and references therein
some other equations that admit solitons in 1+1 dimensions:
both are nonrelativistic; they emerge from nonlinear optics and water waves, respectively
the soliton of KdV has the familiar cosh in the denominator and the typical argumentx − v(t − t0) (with fine-tuned factors), the many-soliton solutions of KdV are knownand obey the soliton scattering property above
3-9
4 Fluctuations and zero modes
4.1 Introduction
still kinematics, on the way to dynamics: tunnelling
back to 1d, motion in inverted potential
motivation
(anti)kink and (anti)solitons minimize the action in inverted potential, i.e. minimize the‘Euclidean action’ SE
the Euclidean path integral, t→ −iτ (see later), has integrands exp(−SE)/~
kinks etc. with minimal SE are thus saddle points in this integral, sharper upon ~→ 0
⇒ need to explore paths around kink with same initial and final positions x(τ = ∓∞)
motivation in the original action: there is an imaginary unit in front of S in the pathintegral; classical solutions are thus points of stationary phase; they dominate becausenear other paths the phases tend to cancel
what we know already
first order variations around classical solutions vanish thanks to Hamilton’s principle
kinks etc. are absolute minima of the action thanks to Bogomolnyi bound
⇒ paths near such solutions should have higher actions
not quite! changing the time t0 does not change the action – called a flat direction –and this should give rise to an exact zero mode in the fluctuation operator
zero modes should occur for all parameters (‘moduli’) with which the action does notvary
4.2 Fluctuation operator
derivation: functions of vectors, (Hessian) matrix of second derivatives in expansion
f(~x) = f(~x0) + first order +1
2
∑i,j
(~x− ~x0)i∂2f
∂xi∂xj
∣∣∣~x=~x0
(~x− ~x0)j + . . . (4.1)
continuous version for functionals:
SE [x(τ)] = SE [xcl(τ)] + 0 +
∫dτdτ ′ (x− xcl)(τ)
1
2
δ2SEδx(τ)δx(τ ′)
∣∣∣x(τ)=xcl(τ)
(x− xcl)(τ′) + . . .
(4.2)
where xcl(τ) denotes the classical path
4-1
for our actions SE =∫dτ[− 1
2 x∂2τx+ V (x(τ))
]δSE
δx(τ)δx(τ ′)
∣∣∣x=xcl
=[− ∂2
τ + V ′′(xcl(τ))]︸ ︷︷ ︸
fluctuation operator
δ(τ − τ ′) (4.3)
where ′ is the derivative wrt. x
the delta function removes one integration in (4.2), thus the second order variationbecomes the expectation value of the fluctuation operator in the ‘wave function’ (x −xcl)(τ)
a Schrodinger problem (~2/2m = 1) in τ with as potential the second derivative ofV evaluated on the classical solution xcl(τ) ← eventually a (particular) function of τ
on the real line since τ ∈ (−∞,∞) and no variations at the ends, (x−xcl)(τ = ∓∞) = 0
eigenvalue equation: [− ∂2
τ + V ′′(xcl(τ))]ψn(τ) = ω2
nψn(τ) (4.4)
‘linearized’ ‘stability equation’, ψn are ‘normal modes’
for the kink with V (x) from (2.4) and xcl(τ) from (2.13)
V ′′(xkink(τ)) = ω2(1− 3/2
cosh2(ω(τ − τ0)/2)
)(4.5)
for the SG soliton with V from (3.1) and xcl(τ) from (3.3)
V ′′(xSG soliton(τ)) = ω2(1− 2
cosh2(ω(τ − τ0))
)(4.6)
these are famous Poschl-Teller potentials (shifted by τ0)
the continuous spectra of them start at the asymptotic values of the potentials, ω2
4.3 Zero modes
explicitly: the two Poschl-Teller potentials above possess zero eigenvalues8
as mentioned already, these are related to the fact that changing τ0 in the solutionsdoes not change the action and thus render the fluctuation zero incl. its second orderpart evaluated here
in full generality
if for some parameter µ
S[xcl(τ ;µ)] = const. (4.7)
8and the kink one has another discrete eigenvalue between zero and the continuous spectrum
4-2
then the vanishing second order can be expressed as
∂xcl
∂µ
δ2S
δx2cl
∂xcl
∂µ= 0 (4.8)
therefore the flat directions should also give the wave function to zero eigenvalue in thefluctuation operator:
ψ0(τ) ∼ ∂xcl(τ ;µ)
∂µ(4.9)
for normalizability and normalization wait a minute
for µ = τ0 just shifting τ we obtain
ψ0(τ) ∼ ∂xcl(τ)
∂τ0∼ ∂xcl(τ)
∂τ(4.10)
check that the solutions’ derivatives indeed give the zero modes of the two PT potentials!
eventually the normalizability/square integrability (otherwise it wouldn’t be an eigen-mode!):∫
dτ(∂xcl(τ)
∂τ
)2 Bog.=
∫dτ[1
2
(∂xcl(τ)
∂τ
)2+ V (xcl(τ))
]= solutions’ action (mass) <∞
(4.11)
4-3
5 Tunnelling
back to our original motivation and the first physical application of solitons
1d, QM, first kinks in mexican hat, later for SG
to extract energies, we use the propagator and its late time behavior
(lattice field theories: masses calculated numerically in the same fashion)
the Euclidean propagator/kernel of Euclidean time (−it/~ → −τ) evolution operatorin position space and energy representation reads:
〈xf |e−Hτ |xi〉 ≡ K(xf , xi, τ) =∑n
ψ∗n(xf ) e−Enτ ψn(xi) (5.1)
just insert complete energy basis (nothing to do with Euclidean yet), note order of‘initial’ and ‘final’
now path integral with Euclidean action
K(xf , xi, τ) =
x(τ/2)=xf∫x(−τ/2)=xi
Dx exp(−SE [x]) (5.2)
just repeat derivation of path integral . . .
Dx means integration over all paths x(τ) with the given boundary condition and
SE =
∫dτ(1
2x2 + V (x)
)(5.3)
now the dot means differentiation wrt. τ
the times in the boundary conditions are perhaps unusual but can be chosen at will aslong as the difference is τ ; will be best suited for our purposes
for large τ the leading exponential gives the ground state energy with information onthe ground state wave function; the next exponential the first excited states etc.
we are especially interested in the energy splitting E1 − E0 caused by tunnelling
saddle point approximation = expand around maxima in weight = class. solutions xcl
5-1
to second order:
K(v,±v, τ) ≈x(τ/2)=v∫
x(−τ/2)=±v
Dx exp(− SE [xcl] + 0 +
1
2
∫dτ (x− xcl)(τ)
[− ∂2
τ + V ′′(xcl(τ)](x− xcl)(τ)
)(5.5)
?!=
e−SE [xcl]√det(− ∂2
τ + V ′′(xcl(τ))) (5.6)
where the second line would be the result of a gaussian integration9 having shifted theintegration measure by the ‘constant’ xcl(τ)
5.1 Zero mode treatment
but we know the fluctuation operator has a zero mode, in the direction of which theweight does not decay (which would render the determinant zero): we have to treatthat direction differently
in addition, once we will collect all saddle point contributions, we would effectivelyintegrate over the moduli parameter τ0 and thus overcount this flat direction
using that the Hermitian fluctuation operator provides a complete, orthonormal andreal basis, we project out the corresponding component of the fluctuation (x − xcl)(τ)explicitly:
(x− xcl)(τ) =∞∑n=0
cn ψn(τ) (5.7)
∫dτ (x− xcl)
(− ∂2
τ + V ′′(xcl))(x− xcl) =
∞∑n=0
λnc2n λn ≥ 0 , cn real (5.8)
∫Dx =
∫ ∞∏n=0
dcn (prop. factor neglected) (5.9)
no Jacobian in the last line, cf. [Zin]
indeed the integration yields the determinant, when excluding the zero mode component:∫ ∞∏n=1
dcn e−λnc2n/2 ∼
∞∏n=1
1√λn
=1√
det′(− ∂2
τ + V ′′(xcl(τ))) (5.10)
where det′ stands for the determinant without zero eigenvalues
goal: fix c0, but integrate over τ0 instead
from the decomposition (5.7), acting with∫dτ ψ0(τ) :
c0 =
∫dτ (x(τ)− xcl(τ ; τ0))ψ0(τ ; τ0) ≡ c0(τ0) (5.11)
9which other path integrals could we calculate . . .
5-2
where we also know (4.10), (4.11)
ψ0 =1√Scl
∂xcl
∂τ, Scl ≡ SE(xcl) (5.12)
introducing an identity into the integral [Faddeev-Popov technique, for gauge fixing]
from the delta function property
δ(f(τ)) =∑i
δ(τ − τi)|f ′(τi)|
, where f(τi) = 0 (5.13)
with one zero
|f ′(τi)| δ(f(τ)) = δ(τ − τi) (5.14)
we obtain the useful identity ∫dτ0 |c′0(τ0)| δ(c0(τ0)) = 1 (5.15)
in words: c0 fixed (in the full integrand), τ0 integrated instead, additional factor: Jaco-bian , with more parameters than τ0: Jacobian = metric in moduli space
the Jacobian is calculable from (5.11) and (5.12):
J ≡ ∂τ0c0(τ0) =1√Scl
∫dτ ∂τ0
((x− xcl)
∂xcl
∂τ
)(5.16)
=1√Scl
∫dτ (−∂τ )
((x− xcl)
∂xcl
∂τ
)+
1√Scl
∫dτ
∂x
∂τ
∂xcl
∂τ(5.17)
= boundary term +1√Scl
∫dτ[∂xcl
∂τ+O(cn)
]∂xcl
∂τ(5.18)
=1√Scl
∫dτ xcl
2 =√Scl (5.19)
where we have used that linear = odd terms in the cn vanish upon Gaussian integration
finally:∫Dx exp(−SE [x]) =
∫ ∏n=1
dcn
∫dc0 . . . =
e−SE [xcl]√det′
(. . .) ∫ dc0
∫dτ0 J δ(c0)︸ ︷︷ ︸∫dτ0 J
. . . (5.20)
5.2 Multiple kinks and the final result
our approximation so far is based on the fact that the kink is a classical solution with oneof the required boundary conditions, x(−τ/2) = −v and x(−τ/2) = v, when τ → ∞ (thisargument is why we have chosen boundary conditions at ±τ/2)
5-3
since the (anti)kink is exponentially localised, it could also serve as a very good approximatesaddle point for large, but finite τ
even more: we will also include sequences of alternating kinks and antikinks as approx-imate saddle points of the path integral
meaningful approximation when they form a dilute enough gas ← to be checked a posteriori
we have to slightly modify the quantities computed so far for one kink
let n be the total number of kinks and anikinks together
actually nothing should depend on “anti”, apart from boundary conditions
action of n (anti)kinks: Sn ' nScl
meaning no interactions
time integration, same Jacobian
τ/2∫−τ/2
dτ1J
τ/2∫τ1
dτ2J . . .
τ/2∫τn−1
dτnJ = Jnτn
n!(5.21)
(cf. time-ordered exponentials) used that nothing else depends on τ ’s when no interac-tions are taken into account
actually for τi − τi−1 < 1/ω there are strong overlap effects, but this range should benegligible for a dilute gas
determinants (with zero modes excluded): more complicated
intuitively: in the Schrodinger equation −∂2τ +V ′′(xmultiple kinks(τ)) the particle sees the
free case most of the time, on top of that local Poschl-Teller deformations 1/ cosh2(#(τ−τ..)) from n kinks
⇒ factorization
1√det′ S′′n
=Kn√
detS′′n=0
= Kne−ω/2·τ (5.22)
into the nth power of some K to be calculated and the soliton-free case
the latter has a constant fluctuation potential V ′′(xcl = ±v) = ω2 and no zero modesneed to be excluded
of course this is nothing but the path integral with the harmonic potential V = ω2
2 x2
entering the Euclidean action (no zeroth order exponential since SE [xcl] = SE [x ≡±v] = 0) and from the energy representation we conclude that the late time behaviorof the Euclidean propagator is given in terms of the lowest energy ω/2,
1√detS′′n=0
= Kharm(v, v, τ)τ→∞−→ e−ω/2·τ (5.23)
let’s agree on this form for now and come back to determinants later (they are divergentanyhow)
5-4
finally put everything together:
K(v,±v, τ) = #∑
n even/odd
e−nScl√Scl
n τn
n!Kne−ω/2·τ (5.24)
exponential, actually cosh and sinh
(of exponential e−Scl and other stuff)
= # e−ω/2·τe√SclKe
−Scl ·τ ± e−√SclKe
−Scl ·τ
(5.25)
identification with (5.4): the first term (smallest exponent) is ∝ e−E0τ etc.
E0,1 =ω
2∓√SclK e−Scl (5.26)
the split of energies is exponential in the kink action
moreover, the wave functions ψ0,1 are even/odd: since e.g. the prefactor of the first termis ψ∗0(v)ψ0(±v) and it does not change sign in (5.25)
this is to be expected in QM for symmetric potentials
to see that this effect is nonperturbative, reintroduce ~ in the path integral weightsexp(−S[x]/~), which immediately propagates to ~ω/2 (harmonic oscillator of course)and exp(−Scl/~) in the energy difference
E1 − E0 ∼√SclK e−Scl/~ (5.27)
since from the Bogomolnyi bound Scl =∫ v−vdτ
√2V (x), the factor exp(−Scl/~) is noth-
ing but the WKB factor of (2.2) for turning points ±v at E = 0
here it comes about by taking all multiple kinks into account (which exponentiatede−Scl)
note: individual energies receive perturbative corrections O(~2) etc. that are bigger thanthe term kept here, but for the energy splitting the leading term is the derived one
general remark: the value of the action S does not matter in classical physics, whereonly δS = 0 is used to derive eom.s; in quantum physics S/~ enters
validity: check of diluteness
in the multiple kink sum
K ∝∑n
(Wτ)n
n!, ∆E = 2W (5.28)
the dominant contribution comes from the terms where n ≈ Wτ (for small n thenumerator still grows faster than the denominator, for large n suppression from Stirling’sformula: n! ≈ nn)
density there: nτ ≈W , thus on average kinks separated by 1/W , while their size is 1/ω
dilute iff W ω, i.e. ∆E E0, that is if the barriers are large with fixed perturbativeground state, i.e. fixed curvature at the bottom
5-5
in other words√Scl e
−Scl ω/K = #, i.e. kinks have large action
both in units of ~ and the coupling λ, since S~ ∼
ω3
~λ 1 (see the classical actions (2.15)and below (3.4))
therefore: small ~ = semiclassical regime small λ = perturbative regime
still nonperturbative effect (!)
5.3 Related results
5.3.1 Bands from tunnelling
in the SG model
physical expectation: periodic potential → energy bands
in the sum over classical trajectories: superpose solitons and antisolitons with any order
these new aspects will appear to be related
consider K(Nv, 0, τ) which means any combination of n solitons and n′ antisolitons aslong as n− n′ = N (recall K(xf , xi, τ))
in order to freely sum over n and n′ (which are otherwise constrained), write this with aLagrange multiplier: δn−n′−N,0 =
why not 1(n+n′)! as it would be the case for treating n+ n′ solitons?
this is just one possibility to achieve n−n′ = N transitions out of(n+n′
n
)(= number of
possibilities to choose the solitons’ ordinary numbers), the product of both is 1n!n′!
this has a neat interpretation:
– the lowest sector is a band characterised by a continuous quantum number θ
– the energies Eθ are cosines in θ around the central energy ω/2, still the groundstate energy around the valleys
the lowest energy is assumed for θ = 0 (maximal cos θ)
5-6
it is twice as far from the central energy as E0 with two minima: many vacua lowerthe energy further, since the wave function has more room to spread
dito for highest energy Eθ=π vs. E1
– the last factor contains the periodicity of the corresponding wave functions, sayN = 1:
ψ∗θ(v)ψθ(0) = exp(iθ) (5.32)
the lowest state with θ = 0 is strictly periodic as expected
recall that periodic potentials are solved by Bloch waves ψn,k(x) = exp(ikx)φn,k(x)where φ is strictly periodic when shifted by one lattice spacing, here v, such thatψ is periodic up to a phase exp(ikv)
in our case the band index is n = 0 and kv can be identified with θ; the latteris related to the energy by the dynamics of the particular periodic system, cf. theDirac-delta comb and Eθ above
“Hearteningly, this is just the right answer.” [Col]
5.3.2 Decay of metastable states from tunnelling
as a model for the decay of a metastable state we consider a potential with one minimum ofvalue 0 at x and again the curvature there shall be ω plus a branch unbounded from below,say V turns negative for x ≥ v
example (check yourself): V = −ω2
2v x2(x− v)
a state prepared at x = 0 will eventually decay with some probability/rate
the quantitative value is important since if the corresponding decay time exceeds the ageof the universe, the state is practically stable; for instance, in some grand unified modelsthe proton becomes unstable, but with very large life time (still experiments with manyprotons give lower bounds on the life time as large as O(1030) years (depending on thedecay channel))
the inverted potential has a classical solution starting and ending – again in infinitepast/future – at x = 0 and bounce off at x = v at some Euclidean time τ0
this so-called bounce solution shares many properties with (anti)kinks and (anti)solitons,but also displays one new aspect, that will be shown to be related to decay
in the potential above: x(τ) = v/ cosh2(ω/2 · (τ − τ0))
these bounces are localised around their times and we can and will superpose them forthe propagator starting and ending at the metastable x = 0:
K(0, 0, τ) =∑n (all)
(Wτ)n
n!e−ωτ/2 (5.33)
5-7
where K in W came from (the ratio of) the determinant of the fluctuation operator inthe classical background without zero modes, cf. (5.22) with n = 1,
K ∼ 1√det′
(− ∂2
τ + V ′′(xcl(τ))) (5.34)
for our example
V ′′(xbounce(τ)) = ω2(1− 3
cosh2(ωτ/2)
)(5.35)
which is indeed a Poschl-Teller potential again, but now with a negative mode [liter-ature or one of the tasks]
what happened? recall that we know the zero mode to be rooted in the independenceof τ0 and this remains true here
ψ0(τ) ∼ ∂xcl
∂τ0∼ ∂xcl
∂τ(5.36)
but xcl(τ) for the bounce is a bump and thus its derivative crosses zero at τ = τ0 (anddecays to zero away from it being normalizable)
check this for our example . . .
invoke QM folklore: a mode having one node is the first excited state and implies theground state being below it
the existence of the negative mode is thus generic
formally the determinant is negative, its square root is purely imaginary and so are K,W and eventually
E0 =ω
2− i Γ
2(5.37)
from resonance theory: Γ is the decay width of the metastable state prepared at x = 0
qualitatively: time evolution e−iE0t ∼ e−Γt/2
as before, the multi-bounce contribution is the leading one not for E0 itself, but for itsimaginary part Γ, and it is nonperturbative
for nice and detailed discussions of this effect see [Wei, Col, Zin]
Intermezzo: Multisolitons in SG
recall that in the SG model with parameters v and ω, the one-(anti)soliton
ϕ(t, x) = ± v
π/2arctan exp
(ωγ(x− βt)
)(5.38)
is moving at velocity β with the relativistic (contraction) factor γ = 1/√
1− β2 and that onecould (equivalently) shift space and time by x0 and t0, respectively
here we discuss three more solutions known analytically ← related to integrability of SGmeaning it has an infinite number of (independent) conserved charges
5-8
Two-soliton
we analyse (with fixed sign and without further space or time shifts)
ϕSS =v
π/2arctan
β sinh(ωγ · x)
cosh(ωγ · βt)(5.39)
for simplicity we will use positive ‘velocities’ β’s (the precise meaning will become clearin a moment)
note the similarities in prefactor and arctan to the one-soliton above, its exp has ‘dou-bled’ into sinh and cosh
for the profiles see Fig. 4
Fig. 4: Profile of the two-soliton, Eq. (5.39).
it is clear that in x it performs a transition from −v to v and thus has topological chargeQ = 2 (cf. (3.26))
this is a conserved charge and can be read off at any time slice
−ϕSS is an antisoliton-antisoliton with Q = −2
we argued from the particle picture in Sec. 3.3 that a static two-soliton cannot exist
in fact in the limit β → 0 (to get rid of time-dependence), the solution vanishes10
we further conjecture that in the infinite past it separates into two separated solitonsperforming single transitions −v → 0 and 0→ v
with velocities β and −β, such they approach each other
by symmetry in t this will then also be the case for the infinite future obeying thescattering requirement of the soliton definition, Sec. 3.8
we will verify the decomposition into such ‘constituent solitons’ (and potential displace-ments) explicitly for the next solution
10for any fixed (t, x): the ϕ = 0 area in the middle eats up everything
5-9
the energy of this solution can be calculated to be 2γ times that of a single soliton
(integrate energy density as in (3.21), at any time slice, t = 0 is convenient)
just the sum of two moving solitons, as could easily be calculated at infinite past/futureassuming the decomposition
bigger than just the sum of two static solitons, which is the Bogomolnyi bound, butonly applicable to static solutions, here the kinetic energy (∂tϕ)2/2 is added
from the contour plots of ϕ (or plots of energy density below) one can further see thatthese ‘constituents’ recoil/‘never get in touch with one another’ and thus are repulsive
how to obtain different velocities? (= tilt the graph?)
observe that the center of mass of this two-soliton is zero, it is at rest
we can always boost it with some β′ and consequently it will be further squeezed by γ′
(follow the steps in Sec. 3.5, just Eq. (3.19) has to be taken in full glory, as ϕSS doesdepend on t′) which then should further enlarge its energy
this indeed yields two solitons at different velocities which – due to different γ contrac-tions – is tantamount to scattering two solitons of ‘different shape’
the velocities ‘go through’ the scattering event, in that respect ‘repelling solitons nottouching each other’ is misleading
Soliton-antisoliton
here we analyse
ϕSA =v
π/2arctan
sinh(ωγ · βt)β cosh(ωγ · x)
(5.40)
note the similarity to the one- and two-soliton above, but somehow t and x have inter-changed their roles (and the explicit β has moved to the denominator)
for the profiles see Fig. 5, indeed almost a rotated two-soliton
we conjecture that this constitutes a soliton-antisoliton configuration
obviously the charge vanishes, Q = 0
the energy is that of ϕSS : same decompositon in the infinite past/future and antisolitonvs. soliton does not matter
in the Q = 0 sector there is no lower bound on the energy, but our configuration hasbeen prepared to have more than two ‘units’ of energy and keeps it
let’s decompose it into its ‘constituents’ analytically
we only need to reduce sinh and cosh to single exponentials, valid in asymptotic regions
let |t| and |x| be large (of course ω is the scale to compare to, with factor γ and γβ)
for simplicity let x by largely negative, then cosh(ωγx)→ exp(−ωγx), 1/2 will cancel
5-10
Fig. 5: Profile of the soliton-antisoliton, Eq. (5.40).
for t we have to distinguish positive and negative: sinh(ωγβt)→ sign(t)·exp(ωγsign(t)βt)
we also write the factor 1/β as an exponential and use the arctan being an odd functionto arrive at
ϕSA|t|,|x|→∞−→ sign(t)
v
π/2arctan exp
(ωγ[x− (−sign(t)β) · t] + log 1/β
)(x < 0) (5.41)
we can immediately identify for negative t an antisoliton with velocity β and for positivet a soliton with velocity −β, which agrees with the figure
moreover, log 1/β can be turned into x-displacements in the sense of the soliton defini-tion (3.30); here we prefer to introduce time shifts:
t < 0 : . . . ωγ[x− β(t−
δt︷ ︸︸ ︷log 1/β
ωγβ)] . . . (x < 0) (5.42)
t > 0 : . . . ωγ[x+ β(t− − log 1/β
ωγβ︸ ︷︷ ︸−δt
)] . . . (x < 0) (5.43)
since β < 1, the quantity δt is positive
interpretation, also follow the ϕ/v = ±1/2 curves in Fig. 6:
positive/negative time shift in past/future: (anti)solitons have started later, but arriveearlier
in between a higher velocity (interpolate between the curves in the figure): attractive
ϕSS can be shown to have the opposite time shifts, which confirms it being repulsive
note, however, that this is not reflected in the classical energies, being the same for ϕSSand ϕSA
5-11
Fig. 6: Soliton-antisoliton contours ϕ/v = −3/4,−1/2,−1/4, 1/4, 1/2, 3/4 (ϕ = 0 at t = 0) inthe exact solution (left) and extrapolating their asymptotic behavior to t = 0 (right).
Breather
here we analyse
ϕB =v
π/2arctan
sin(ωγ′ · βt)β cosh(ωγ′ · x)
, γ′ =1√
1 + β2∈ (0, 1] (5.44)
formally it appears from the soliton-antisolition solution by changing the velocity toimaginary (only possible, iff ϕ stays real and it does, from continuing sinh(#β)/β), γ′
will have a different meaning
so it should be somehow connected to the soliton-antisoliton solution, which is attractive
obviously this solution is periodic in time, with period: T = 2πω
√1+β2
β
in the limit β → 0 the period diverges, i.e. towards unbounded motion
in the limit of large β the period is bounded from below by 2πω
for the profiles see Fig. 7
roughly identifying the constituents by where ϕ increases or decreases, one can see thatthey oscillate around each other
we interprete this as soliton-antisoliton bound state, called ‘breather’ or ‘doublet’
analogy in attractive Coulomb potential: Kepler ellipses instead of open orbits
the energy is also continued from that of ϕSA, it is γ′ times that of two static solitonsand thus smaller than that of ϕSA
(at β = 0 they match as γ′ = 1 = γ)
the ‘constituents’ are never separated and can thus never be properly defined
does not fit the definitions of solitons nor solitary wave (was: energy density(~x− ~vt))
5-12
Fig. 7: Profile of the breather, Eq. (5.44).
but it is nondissipative (verify in energy density) = a lump
the breather plays a role in the action-angle variable representation of the SG model
there is an equivalence of the SG model to a fermionic Thirring model
(we may come back to these points later in the lecture)
5-13
Fig. 8: Energy densities of the three solutions discussed in this section: guess which is which.
5-14
5.4 After-math: one-dimensional determinants
in the determinant of a Schrodinger type operator, det (−∂2τ + V ), there are two sources of
infinities: the eigenvalues are not bounded from above – ultraviolet divergence – and theirnumber is continuous – infrared divergence
therefore we need two ‘regularizations’
the UV divergence can be circumvented through division by the determinant of anothersuch operator
det(−∂2τ + V )
det(−∂2τ + V )
' det−∂2
τ + V
−∂2τ + V
(5.45)
mathematically, this relation holds for operators with a finite number of eigenvalues(matrices), but otherwise the second expression defines the first one (being ∞/∞)
physically one expects that eigenvalues/high excitations are insensitive to the particularproperties of the potential and thus universal cancelling in the ratio above
in the end we will compute the denominator determinant in some ‘absolute’ sense
the IR divergence can be circumvented through forcing the system into an interval ofsize L with Dirichlet boundary conditions = ‘particle in a box’, or periodic boundaryconditions (recall the corresponding discrete QM spectra for the free case)
in the end L→∞
5.4.1 ‘Relative’
we use phase shifts, another computation uses a quasi-zero mode of the Schrodingeroperator [Gel’fand, Yaglom 1960], see also
‘to compute the determinant .. we actually do not have to know any of its eigenvalues’
for asymptotically flat potentials, say of same height V (x → ±∞) = ω2 (as in thefluctuation operators we will analyse), the wave functions become plane waves of theform discussed for the barrier (2.1) with reflection and transmission coefficients
the wave number k is related to the eigenvalues (energies) λ as k =√λ− ω2 when√
2m/~ has been set to unity
we will use the plane wave form in the finite volume, too, assuming L being larger thanthe typical interaction length
fortunately, the Poschl-Teller potentials we need in the fluctuation operators are solvedto great detail
via supersymmetric QM one can relate them to the free case . . .
5-15
they are reflectionless (!) and for the kink potential
V = ω2(1− 3/2
cosh2(ωτ/2)
)(5.46)
the transmission reads
T =ω − ik−ω − ik
ω/2− ik−ω/2− ik
=: eiδ, R = 0⇒ ψ(x) =
1T
eikx (5.47)
δ(k) = − 2[
arctan(k
ω) + arctan(
2k
ω)] . . . phase shift (5.48)
for periodic boundary conditions ψk(−L/2) = ψk(L/2) one obviously needs
kL+ δ(k) = 2πn (5.49)
which in particle/lattice physics context is known as Luscher’s formula
obviously this (nonalgebraic) equation for k has a discrete set of solutions kn
for the free case of course:
k(0)n L = 2πn (5.50)
for the calculation we use that for L→∞ one can expand around the free case:
kn = k(0)n + ∆kn : k(0)
n L + ∆knL︸ ︷︷ ︸small · large
+ δ(k(0)n ) + ∆knδ
′(k(0)n ) +O(∆k2
n)︸ ︷︷ ︸small
= 2πn
furthermore, we need to know that the nonzero discrete eigenvalue is 34 ω
2 [Kle, (17.55)]
for the determinant ratio we use for V the free case at the asymptotic ω2, for which nozero mode subtraction is needed:
logdet′(−∂2
τ + V )
det(−∂2τ + ω2)
= log34ω
2∏n λ(kn)∏
n λ(k(0)n )
= log(3
4ω2) +
∑n
logk2n + ω2
k(0)2n + ω2
(5.51)
logdet′(−∂2
τ + V )
det(−∂2τ + ω2)
− log(3
4ω2) =
∑n
log
(1 +
2k(0)n ∆kn
k(0)2n + ω2
)=∑n
2k(0)n ∆kn
k(0)2n + ω2
(5.52)
above= − 1
L
∞∑n=−∞
2k(0)n
k(0)2n + ω2
δ(k(0)n )
Riemann= − 1
2π
∫ ∞−∞
dk2k
k2 + ω2δ(k) (5.53)
=1
π
∫ ∞0dk log(k2 + ω2)
∂δ(k)
∂k(integration by parts) (5.54)
a comparison with the last term in the first line shows, that ∂δ(k)/∂k is the density ofstates (with the density of states from the free case subtracted)
rescaling the integration variable to the dimensionless x = k/ω:
logdet′(−∂2
τ + V )
det(−∂2τ + ω2)
− log(3
4ω2) =
1
π
∫ ∞0dx log(ω2(x2 + 1))(−2)
( 1
1 + x2+
2
1 + (2x)2
)=
1
π(−2π) log(3ω2) = log(
1
(3ω2)2) (5.55)
5-16
the final result is:(det′(−∂2
τ + V )
det(−∂2τ + ω2)
)−1/2
=
(3/4 · ω2
9ω4
)−1/2
=√
12ω cf. [HK, 17.129] (5.56)
5.4.2 ‘Absolute’
for the determinant ratio of the previous subsection and for Eq. (5.22) we still have toconfirm the free case result, to wit 1/
√det(−∂2 + ω2) ∼ exp(−ω/2 · τ)
we require the wave functions to vanish at −τ/2 and τ/2 = Dirichlet conditions, fromboundary conditions of the path integral, ‘particle in a box’
I removed τ from the second derivative above as the role of τ is to denote the extent ofthe box
eigenfunctions such11 that −∂2 can be substituted by (k πτ )2 with k = 1, 2, . . .∞
taking the logarithm we need to compute a diverging sum
s = log det(−∂2 + ω2) = log∞∏k=1
((π k/τ)2 + ω2
)(5.57)
=1
2
∞∑k=−∞
log((∆ · k)2 + ω2
)− 1
2logω2 ∆ = π/τ (5.58)
poor man’s trick: write
∂s
∂ω= ω
∞∑k=−∞
1
(∆ · k)2 + ω2− 1
ω(5.59)
now this sum converges, and for large τ meaning small ∆ we invoke the Riemann sum12
∆∞∑
k=−∞f(∆ · k)→
∞∫−∞
dx f(x) to obtain:
∂s
∂ω→ ω
∆
∫ ∞−∞
dx
x2 + ω2− 1
ω=π
∆− 1
ω= τ − 1
ω(5.60)
s→ ωτ − log(ω) (5.61)
1√det(−∂2 + ω2)
= exp(−s2
)→ exp(−ω2τ) · τ -indep. (τ →∞) (5.62)
where the second term on the right hand side is actually smaller than the first one forlarge τ ’s
this is the expected result, but where has the singularity gone?
11cosine of odd, sine of even multiples of the lowest frequency12actually the sum in (5.59) can be computed exactly yielding τ coth(ωτ), which contains the integral and
exponentially small (in τ) corrections; in the thermodynamic context, τ ∼ 1/T , these are Matsubara sums
5-17
the step from the first to the second line is valid up to adding ω-independent terms,which enter the final result as ω-independent factors; it is those kind of singularitiesthat we have removed by the trick
we have interchanged derivative and infinite sum violating the mathematical conditionsfor such a manipulation to yield the same result
rewrite the logarithms and define (interchanging derivative and infinite sum again!)
log(k2 + ω2) = − ∂
∂z
∣∣∣z=0
(k2 + ω2)−z (5.63)∑k
log(k2 + ω2) := − ∂
∂z
∣∣∣z=0
∑k
(k2 + ω2)−z (5.64)
the sum on the right hand side is a well-defined function of z (and ω) for large enoughreal parts of z and can be continued analytically to the origin, where it gives a finiteresult
∗ some related fun: Riemann’s zeta function
ζ(z) :=
∞∑k=1
k−z (5.65)
a sum like above, converges for Re z > 1 and is regular around the origin, where e.g.
ζ(0) = −1
2“=”1 + 1 + 1 + . . . (5.66)
how to continue? write k−z as an integral with a Gamma function (’Mellin transform’),after which the k-sum can be done . . .
ζ is very interesting for number theory and statistics: it has a simple pole at z = 1and trivial zeros at z = −2,−4, . . . plus infinitely many nontrivial zeros in the strip0 < Re z < 1, conjectured13 to be at Re z = 1/2 and at kind of random Im z etc.
13B. Riemann (1859): “Hiervon ware allerdings ein strenger Beweis zu wunschen; ich habe indess die Auf-suchung desselben nach einigen fluchtigen vergeblichen Versuchen vorlaufig bei Seite gelassen, da er fur dennachsten Zweck meiner Untersuchung entbehrlich schien.” checked numerically for 1010 solutions, unsolvedClay Millenium Problem
5-18
6 Fermions in solitonic backgrounds I
(‘solitons in the presence of Fermi fields’ (!) [Raj])
assume the bosonic part of the action be mexican hat with a static14 (anti)kink solutionϕ(x) = ± v tanh(ω2x) , Eq. (2.13) with t→ x, t0 = 0
couple massless fermions via a Yukawa interaction with a coupling constant, say g > 0:
Lint = g ψ(t, x)ϕ(x)ψ(t, x) (6.1)
plus conventional kinetic term for the fermions:
Lf = ψ(t, x)(iγµ∂µ − m︸︷︷︸
0
)ψ(t, x) (6.2)
(recognize the Dirac equation as the eom. ∂L/∂ψ = 0) where
ψ := ψ†γ0 , γµγν + γνγµ!
= 2ηµν1 (6.3)
ϕ acts like a ‘space-dependent mass’
the trivial vacua ϕ ≡ ±v give a fermion mass of gv (Higgs effect: coupling × vev)
the sign of the mass term is irrelevant as there are always fermion solutions of bothof energy (Dirac sea etc.), e.g. particle at rest solutions of the Dirac equation: ψ(x) ∼exp(±imt), or relativistic relation E2 −~k2 = m2 derived from the squared Dirac = KGequation
the kink solution ϕ(x) approaches that asymptotically, but vanishes at the kink location
symmetry of the entire system: ψ → γ5ψ thus ψ → −ψγ5 (since ψ = ψ†γ0) plusϕ→ −ϕ
the latter is a symmetry of the bosonic Lagrangian, too (and relates kink and antikink)
a mass term with fixed ϕ = v is known to break this ‘chiral symmetry’
the minimal representation of the γ-matrices depends on the dimension, in 2 dimensionsone can use the Pauli matrices with potential factors of i (check anticommutator above):
γ0 = σ1 =
(0 11 0
), γ1 = iσ2 =
(0 1−1 0
), γ5 := γ0γ1 = −σ3 =
(−1 00 1
)(6.4)
spinors in 2 dimensions are thus two-component vectors
the last γ-matrix has been named in analogy to 4 dimensions; it obeys (γ5)2 = 12 andcan be used to define chiral projectors (12 ± γ5)/2; since in our representation γ5 isdiagonal, this is a chiral or Weyl basis
14again, boosts are possible, provided the fermions are multiplied by the corresponding factor
6-1
note that γ0 and γ5 are hermitian, while γ1 is antihermitian (in Euclidean space all γµ’scan be chosen hermitian)
6.1 Zero modes
as the background kink is static, we factorize a trivial time-dependence
ψ(t, x) = e−iωtχ(x) (6.5)
ω is related to the energy, can also be seen from the Hamiltonian density (cf. Sec. 3.4):
H = Πψ − L Π =∂L∂ψ
= ψiγ0 (6.6)
= −ψ(iγ1∂x + gϕ
)ψ (6.7)
(that the time-derivative disappears is typical for fermions), further, using the Diracequation:
H = −ψ(−iγ0∂t)ψ = ωψγ0ψ = ωψ†ψ (= ωχ†χ) (6.8)
χ is a two-spinor obeying (from the Dirac equation):
−(iγ1∂x + gφ(x)
)χ(x) = γ0ωχ(x) (6.9)
−(iγ5∂x + γ0gφ(x)
)︸ ︷︷ ︸=: D
χ(x) = ωχ(x) (6.10)
the differential operator D is hermitian (since i∂x is) and thus has real eigenvalues ω
symmetry: since D is made of γ5 and γ0, it anticommutes with γ1, and the latter relateseigenfunctions of opposite ω:
the mirror modes are also normalized in the same way:
χ†−ωχ−ω = χ†ω((γ1)†γ1
)︸ ︷︷ ︸−(γ1)2 = 12
χω (6.14)
the situation for ω = 0 is special since the second eigenfunction might be not linearlyindependent (indeed it will not, since the first one will be an eigenfunction to γ1)
on this subspace, D commutes with γ1
D, γ1 = 0D=0⇒ [D, γ1] = 0 (6.15)
6-2
and thus γ1 can be measured15 together with D, with eigenvalues ±i:
γ1sσ = iσsσ σ ∈ ±1 (6.16)
factorizing χω=0(x) = hσ(x) · sσ, the zero eigenvalue equation (6.9) for the scalar func-tions hσ in these two spin sectors becomes:(
− σ∂x + gϕ(x))hσ(x) = 0 (6.17)
with the obvious solution (note first order only):
hσ(x) = exp(σg
∫ x
ϕ)
(6.18)
the existence of such fermionic zero modes seems to go back to Dashen, Harlacher,Neveu, Phys.Rev. D 10 (1974) 4130
since h−1(x) = 1h+1(x) , only one of these could be square-integrable
for the kink we can obtain the exact solution:
hσ(x) = cosh(ωx
2
) 2gvω·σ
(6.19)
which is square-integrable for σ = −1 (since g > 0, exp. decay) only
likewise, for the antikink (ϕ changes sign and cosh turns into 1/cosh) we obtain asquare-integrable solution for σ = +1 only, with the same x-dependence
consistent with the fact that ϕ → −ϕ goes along with ψ → γ5ψ, in fact since γ5
anticommutes with γ1, it relates eigenmodes of the latter with opposite eigenvalue σ inthe way we discussed above
in the vacua ϕ ≡ ±v no such mode exists (technically because functions exp(±vx) blow
up at one of the two x-ends), plane waves have ω =
√~k2 + (gv)2, |ω| ≥ gv
in general potentials with two minima, say v− < 0 and v+ > 0 the solitons will approachconstants with different signs as x→ ±∞ and this (soliton goes through zero!) enablessquare-integrability
the existence of fermionic modes with zero energy seems to be tightly connected to thetopological charge being nonzero and their spinor character seems to be sensitive tothe sign of the topological charge
this will be the content of index theorems
needless to say that if the (anti)kink is localised at x0 then so will its zero mode
you should have heard the word ‘zero mode’ before, namely in the bosonic fluctuationoperator around the kink; are these related??
we can turn the first order differential equation (6.17) with σ = −1 into a second orderequation with any differential operator from the left
(. . . ∂x + . . .)︸ ︷︷ ︸B
(∂x + gϕ(x))︸ ︷︷ ︸A
h−1(x) = 0 (6.20)
15if you like such a statement for a hermitian operator, then repeat everything for iγ1
6-3
to ensure that no first derivatives remain, that the zero mode is the lowest one etc.take B = A† = −∂x + gϕ(x) (i∂x is hermitian), which can only possess nonnegativeeigenvalues as 〈ψ|A†A|ψ〉 =‖ A|ψ〉 ‖2≥ 0 (and zero exactly if A|ψ〉 = 0 as for h−1)
the product A†A is a ‘supersymmetric’ (SUSY) Schrodinger operator16
this has sech(ωx/2)2 in it like the bosnoc potential V ′′mex. hat(ϕkink(x)) in (4.5) and –‘magically’ – agrees with it for a particular choice of the coupling g
(SUSY tells you the couplings in the bosonic and the fermionic sectors are fine-tuned)
Vg=ω/v(x) = ω2(1− 3/2
cosh2(ωx/2)
)(6.22)
indeed with that g we obtain h−1(x) = cosh(ωx/2)−2 which you can check is the explicitbosonic zero mode
∗ SUSY comes with a huge amount of beautiful theory, which we cannot present here
e.g. there exists an extension of ordinary space with Grassmann variables called ‘super-space’, in which SUSY becomes just a generalized translation invariance
moreover, generalized derivatives exists that provide SUSY building blocks, i.e. ensurecertain kinetic and potential terms to be SUSY
a SUSY Lagrangian reads [Shi, Eq. (71.1)]
L =1
2(∂µϕ)2 − W ′(ϕ)2
2+ ψ
[iγµ∂µ +W ′′(ϕ)
]ψ (6.23)
where W ′ = ∂ϕW and ψ is actually a Majorana (in some representation real) spinor,which is important for the matching of the bosonic and fermionic degrees of freedom
we recognize our superpotential from the Bogomolnyi bound, Eq. (3.6),
W ′(ϕ) =√
2V (ϕ)mex. hat
=
√2λ
4!(ϕ2 − v2) (6.24)
which now also determines the Yukawa coupling:
W ′′(ϕ) =
√λ
3ϕ =
ω
vϕ = gϕ
∣∣g=ω/v
(cf. (6.22)) (6.25)
this way of coupling bosons and feermions is thus prefered by SUSY, and as a conse-quence the one-loop determinants (of the fluctuation operators) cancel including thezero modes
for more details see [Shi]
16did you realize in QM: knowing one eigenvalue and its eigenmode fixes the potential in the Schrodingerequation, for a zero mode V = ψ′′0 /ψ0 and indeed h′′−1/h−1 gives g2ϕ2 − gϕ′ = Vg
6-4
6.2 Fermion number 1/2
some strange consequence of the zero mode [Jackiw, Rebbi, Phys.Rev. D 13 (1976) 3398]
canonical quantization of fermions
for free fermions through decomposition into Fourier modes
later more general: eigenmodes of the Dirac equation (for the correct time evolution)
the coefficients are promoted to operators (bq, d†q) for ψ(x) and (b†q, dq) for ˆψ(x)
q labels the eigenmodes (in the free case spin and momentum)
the parts with d(†) belong to negative energy solutions/antifermions/holes . . .
all bq and dq annihilate the vacuum: bq|0〉 = 0 = dq|0〉
b†q creates particles (e.g. electrons), d†q creates antiparticles (e.g. positrons) with oppositecharge (see below)
the use of anticommutators results in the Pauli principle that identical particles cannotbe created: (b†q)2|0〉 = 1
2b†q, b†q|0〉 = 0
energy from Hamiltonian
the first guess (cf. Eq. (6.8))∫d3x ψ†i∂tψ =
∫∑q
E(q)(b†q bq − dqd†q
)(6.28)
does not vanish on the vacuum
instead one ‘normal-orders’ all quantities: bring d† to the left of d – such that allquantities vanish on the vacuum – neglecting the anticommutator met in between: aconstant that is only a redefinition of the zero point of energy (this vacuum is called‘(filled) Dirac sea’ etc.)
H =
∫d3x : ψ†i∂tψ : =
∫∑q
E(q)(b†q bq + d†qdq
)(6.29)
the second term has changed sign, such that there is nothing unphysical about the anti-particles created by d†: no negative energy (just opposite charge: see below); lookingback it was essential to use the anti commutator to arrive at this plus sign
6-5
conserved fermion current and charge
classically, from U(1) invariance ψ → eiαψ and ψ → ψe−iα (like in QM or complexscalar field theory) the Noether current
jµ = ψγµψ (6.30)
is conserved, ∂µjµ = 0 (act on both ψ and ψ and use Dirac equation or observe that
jµ = ∂L/∂(∂µψ) and use eom. noting that ∂L/∂ψ = 0)
in the corresponding charge, b†q bq and dqd†q come with the same sign, but normal ordering
changes that to
Q =
∫d3x : ψ†γ0ψ : =
∑q
(b†q bq − d†qdq
)(6.31)
again the vacuum has no expectation value = is neutral, but b†q and d†q create stateswith opposite charges
in the kink background include the zero mode χ0:
ψ(x) = aχ0(x) +∑q
(bq, d
†q
)(6.32)
and ψ† with a†
a and a† are also promoted to operators with a, a† = 1; all other anticommutators,
also including b(†)q and d
(†)q , vanish
these zero mode components do not add to the energy (6.29) and the vacuum is degen-erate, actually two-fold:
let |kink〉 be the state annihilated by all bq, dq and also by a, which is not mandatory
a|kink〉 = 0 (6.33)
the other state obtained from it by applying a† is called |kink′〉
once again, |kink〉 and |kink′〉 are of same energy, what about the charge?
the naive charge a†a is already normal-ordered, because a is unpaired in the decompo-sition (6.32) (no ..† in ψ)
this would give 0 on |kink〉 and 1 on |kink′〉, this asymmetry is physically unreasonable
the better choice is
Q = a†a− 1
2=
1
2(a†a− aa†) (6.36)
6-6
which can also be shown to give zero charge on the trivial vacua [Shi]
now this gives a fractional and opposite fermion number on the kink states:
Q|kink〉 = −1
2|kink〉 , Q|kink′〉 =
1
2|kink′〉 (6.37)
[Shi]: this fractional charge would be measured locally in an experiment, the “missing1/2” is further away in the system, e.g. on an antikink (see the application below)
6.3 Kinks and zero modes in polyacetylene
QFT theory in an organic polymer! [Su, Schrieffer, Heeger, Phys.Rev.Lett. 42 (1979) 42]
see e.g. Jackiw’s Dirac medal lecture hep-th/9903255, we also follow 0709.3248
now vacuum = Fermi surface between completely filled valence band and completely emptycondution band; the former are equivalent to the anti-particles = positrons
in a sense the Dirac sea (‘for particle physics . . . an unphysical construct’ [Jackiw]) has becomereality
polyacetylene = long chain of carbon atoms C with one hydrogen atom attached to each
⇒ each C ‘still has three arms’ = can built up a single and a double bond to its neighbors
⇒ single and double bonds alternate (‘conjugated chain’), but in two configurations Aand B which are obtained from each other by a shift or applying a reflection (parity),‘right-left symmetry broken by Peierls instability’, see Fig. 9
A and B are two degenerate ground states
(‘microscopic’) Hamiltonian σ3i∂x +σ1ϕ(x) like our D from (6.10) with mirror energies
ϕ(x) is a phonon field, it measures the displacement of the carbon atoms from anequidistant lattice
the two components are not spin, but right and left moving electrons in a linear approx-imation
idealisation: continuum and infinite volume
in the lowest state the double-bonded C’s are closer to each other and measured fromone lattice site a uniform positive and negative shift occurs for A and B, respectively
correspondingly, ϕ = ±v (our trivial vacua), and the electron spectrum has a gap (mass)
what is a kink here?
= a defect where two single bonds meet, see fourth line in Fig. 9
left of it say A with ϕ(x→ −∞) = −v, right of it B with ϕ(x→ +∞) = +v
a stable excited state, also called domain wall
6-7
O
A
B
B+S
B+2S
+v
−v
Fig. 9: from Rao, Sahu, Panigrahi, ‘Fermion Number Fractionization’, arxiv: 0709.3248
that comes with a new localised electron state (at the kink position X) in the middleof the energy gap, the fermion zero mode
topological: invariant under small deformations
(high doping⇒many solitions⇒ band⇒ conducting polymer [Nobel prize in Chemistry2000])
what is fractional fermion number here?
agrees with el. charge
consider kink-antikink, see fifth line in Fig. 9, well separated
one bond less, shared between the states
but one bond means two electrons . . .
with net spin 0, so a domain wall acquires a charge e without spin (!)
another condensed matter example of fractional charge: quantum Hall effect
6-8
7 Towards solitons in higher dimensions
7.1 A first encounter of vortices
consider a bosonic field ϕ in a potential V (ϕ) with minima at V = 0 (can always shift thepotential to achieve that) in d spatial dimensions and with finite energy
the energy ∫ddx(1
2(∂tϕ)2 +
1
2(∂iϕ)2 + V (ϕ)
)∂i =
∂
∂xi, i = 1, . . . , d (7.1)
is finite, only if for |~x| → ∞ the field ϕ approaches a minimum of V
the boundary of space, |~x| → ∞ is a hypersphere, ∂Rd = Sd−1 for d ≥ 2, and thusconnected, or just two points ∂R = ±∞ = Z2, and thus disconnected
(a) discrete minima as for the kink:
by continuity, ϕ has to stay in one minimum at the entire boundary sphere
this looks too trivial (but topology in the O(3) model is of this kind, see (7.19))
(b) continuous manifold of minima/vacuum manifold:
e.g. consider a vector field ~ϕ ∈ RN : ‘internal space’/‘color space’
and let the potential depend only on its modulus, V (~ϕ) = V (|~ϕ|)
then |~ϕ| has to approach a value that minimizes V , but the direction of ~ϕ can still varyon a hypersphere in internal space
this yields a mapping:
~ϕ∞ : Sd−1︸ ︷︷ ︸= ∂Rd
→ SN−1︸ ︷︷ ︸⊂ RN
(7.2)
and this leads to the concept of homotopies, which will be covered soon
for the moment, let’s rely on intuition and analyse the simplest cases
(i) space is two-dimensional
(ii) ϕ has two components
the corresponding mappings are
(i) ϕ∞ : S1 → SN−1 (7.3)
(ii) ϕ∞ : Sd−1 → S1 (7.4)
the first one can be thought of as putting a circle/lasso on a sphere, the lasso can snapto a point and this is ‘topologically trivial’, unless N = 2 as well
the second one is not so easy to grasp: map a higher sphere onto a lower one; from thetables: also trivial unless d = 2 (can be nontrivial: S3 → S2)
7-1
then
ϕ∞ : S1 → S1 (7.5)
is the way one circle can be wrapped onto another circle, cf. Fig. 10
Fig. 10: Visualization of topologically different mappings S1 → S1, guess the windingnumbers.
such mapping are characterised by an integer winding number, that is topological inthe sense of being invariant under small deformations of the mapping
to arrive at this situtation consider a two-component or complex scalar field ϕ in 1 + 3dimensions, that is say static and independent of the third component x3: ‘line object’
for simplicity, let the potential be minimal at |~ϕ| = 1
let (ρ, φ) be the polar coordinates in (x1, x2)
moreover, a two-component vector can be mapped to a complex ϕ = ϕ1 + iϕ2 !
configurations with a nontrivial winding number Q in ϕ∞ could behave as
~ϕρ→∞−→
(cos(Qφ)sin(Qφ)
), ϕ
ρ→∞−→ exp(iQφ) (7.6)
the winding could also be performed with a varying velocity in the angle φ, but thisshould lead to additional energy
the configuration looks like a two-dimensional hedgehog, the following one can be ob-tained by a global rotation:
~ϕρ→∞−→
(− sin(Qφ)
cos(Qφ)
)(7.7)
this is tangential to the radius vector and represents a vortex
a finite integrated potential energy can be arranged by having |~ϕ| approach 1 fast enough
the kinetic energy from the winding, however, is infinite:
(~∇~ϕ)2 = (∂ρ~ϕ)2 +1
ρ2(∂φϕ)2 = positive +
Q2
ρ2(7.8)∫
d2x1
2(~∇~ϕ)2 ≥ 2π
∫ ∞ρ0
dρ ρQ2
2ρ2∼ log ρ
∣∣∞ρ0
=∞ (7.9)
7-2
(a bit sloppy: lower bound ρ0 not specified)
the energy is not finite as required, although this divergence is ‘weak’ (logarithmic)
present for any configuration with winding number
later we will present a way to avoid it
what about continuing ϕ to the ‘bulk’ = finite ρ
the conservation of the winding number seems to lead to a singularity
well, ~ϕ is not constrained to the vacuum manifold anymore and thus can change itsmodulus, to avoid the singularity a zero |~ϕ| = 0 is needed at least at one point
the existence of such a zero is another topological invariant, whereas the point where ithappens is not, and there could be more points, the windings of which partially cancel,only the net winding is fixed at the boundary to be Q
there is not much room to directly generalize kinks/solitons to higher dimensions! (like wetried above)
assume we have a functional (Euclidean action or energy) for scalar fields in d dimensionsof the usual form :
I[ϕ] =
∫ddx
1
2(∂iϕ)2 + V (ϕ)
= Ikin[ϕ] + Ipot[ϕ] i = 1, . . . , d (7.10)
(one more derivative than in (7.1))
for the kinks in mexican hat or the solitons in SG we started in 1+1d, but these arestatic configurations and thus minimize the potential energy
∫d1x(∂xϕ)2/2 + V (ϕ),
thus d = 1
equivalently, these are solutions (‘instantons’) in 1d QM with inverted potential
all we will discuss now applies to vector fields ~ϕ and constrained fields as well
a solution, that is also stable, should obey:
δI = 0 (7.11)
δ2I ≥ 0 (7.12)
consider a particular variation, a rescaling of the coordinate (‘conformal trafo’):
ϕλ(x) = ϕ(λx) (7.13)
then the x-measure provides a factor λ−d and
Ikin[ϕλ] =λ2−dIkin[
≡ ϕ︷︸︸︷ϕ1 ] (7.14)
Ipot[ϕλ] =λ−dIpot[ϕ1] (7.15)
7-3
being a solution means:
0!
=δI
δλ
∣∣∣λ=1
= (2− d)Ikin[ϕ] + (−d)Ipot[ϕ] (7.16)
while stability needs:
0!≤ δ2I
δλ2
∣∣∣λ=1
= . . . = 2(2− d)Ikin[ϕ] (7.17)
⇒ these solutions only exist in d ≤ 2
in d = 2: Ipot[ϕ] = 0, having shifted V such that V ≥0, this means the trivial V =0 (*)
⇒ such solitons are restricted to one spatial dimension
(or the kinetic part vanishes: constant field = trivial vacuum solution)
no no-go theorem without ways around it:
⇒ other – e.g. gauge – theories (will help the vortex)
⇒ (*) and constrained fields: equality δ2I/δλ2 = 0 means conformal trafo yields a flatdirection and the object’s size is a collective coordinate (assume ϕ(x; size) = ϕ(x/size))
7.3 Definition of O(N) sigma models
let ~ϕ ∈ RN be normalized to 1 and consider simply the kinetic ‘innocent looking’ [Raj]term
L =1
2(∂µ~ϕ)2, |~ϕ| = 1 meaning ϕaϕa = 1 , a = 1, . . . N (7.18)
already this action gives rise to nonlinear equations of motion (think of solving theconstraint by writing the field and the action in terms of angles)
due to its symmetry, these models are called nonlinear17 O(N) sigma models
this system is most interesting in two dimensions, µ = 1, 2 (which may be thought ofas two spatial coordinates (static solutions) or one spatial coordinate plus Eucl. time)
the main effects are asymptotic freedom and a dynamically generated mass gap; notethat the theory does not contain a dimensionful constant , the dimensionful mass isgenerated by quantum effects
for the O(3) model as well as the CP (N −1) generalizations of it, one also has topologyand instantons (but not in O(N ≥ 4) models, see in a minute)
all like in 4d Yang-Mills theory
another motivation obviously comes from (classical) spins
17one can obtain it from linear sigma models having no constraint, but a potential being minimal at |~ϕ| = 1:perform the limit of the potential becoming infinitely narrow there
7-4
for a finite action, the field must not vary at infinity, otherwise vortex divergence (7.9),
~ϕρ→∞−→ ~ϕ0 (7.19)
now the ‘point infinity’ can be added to R arriving at a sphere S2, like the Riemannsphere compactifies the complex space to C ∪ ∞ (this is not an S1 ⊂ R2 at fixed ρ)
together with the normalization of the vector ~ϕ to an (N − 1)-sphere in internal space,the configurations are mappings
~ϕ : S2 → SN−1 (7.20)
this is again a homotopy (group), ‘one level higher’ than the vortex’ S1 → S..
trivial unless S2 → S2, thus N = 3: the O(3) configurations are characterised by thecorresponding integer winding number (wrapping two-spheres on two-spheres, perhapsless easy to imagine, see next section)
note that due to the presence of the gradient term in the action, we always assumecontinuous and differentiable ~ϕ’s as classical solutions; which configurations dominatethe path integral is a difficult question
outlook: the configuration space will split into those labeled by winding number Q
in each sector a Bogomolnyi bound S ≥ # ·Q will apply, and the instantons as solutionsof the classical eom. will be those with minimal action obeying the equality and a firstorder differential equation only (still nonlinear and partial)
two-dimensional locations as collective coordinates etc.
especially for the Bogomolnyi bound we need formulas to calculate winding numbers Q
we will give an (incomplete) account of the necessary mathematics in the next section
7-5
8 Homotopy groups
in this section we will give some mathematical definitions of what ‘continuous deformations’are, how one classifies (classes of) mappings that cannot be continuously deformed into eachother and how the corresponding winding numbers can be computed (which then enter Bo-gomolnyi bounds etc.)
we will not go into the mathematical definitions of the spaces Y that are probed by thesemappings; [Nak], which we follow, for instance considers topological spaces, in which onlyneighborhood is defined – this is the real mathematical topology – and consequently contin-uous mappings
most of the time we will consider smooth (differentiable) spaces (manifolds = locally RN , butnot necessarily globally), like spheres, tori, Lie groups etc., and consequently have smoothmappings in mind
often we need intervals as domains for our arguments, take them normalised to I = [0, 1], youmay think of s ∈ I as a time progressing from 0 to 1 (while an image moves in Y )
8.1 Pathes in classes and group structure
a path γ is simply a mapping into such a space, γ : I → Y
initial and end point are what one expects: y0 = γ(0), y1 = γ(1), of course y0,1 ∈ Y
and closed pathes are called loops: y1 = y0 being base point
a trivial loop stays at its base point all the time γtriv(s) ≡ y0 ∀s
to be continuously deformable into each other is expressed by being homotopic ∼ andis defined by the existence of an interpolation such that ‘everything remains smooth’ orjust continuous
γ ∼ γ′, if there exists F : I× I → Y continuous with F (s, 0) = γ(s) and F (s, 1) = γ′(s),of course for all s
intermediate pathes come in as F (s, t 6= 0, 1) = γt(s), visualize this!
it is not enough that all γt(s) are continuous in s, we also need continuity in t, otherwisewe ould ‘jump over holes’
for closed pathes one also requires the interpolation to have the same base point:F (0, t) = F (1, t) = y0 for all t
intuitively: if a loop does not enclose a whole, then it is homotopic to the trivial loop(plot the image of F )
and should actually be identified with the trivial loop, this is done by noting that
∼ is an equivalence relation, meaning
reflectivity: γ ∼ γ
8-1
symmetry: γ ∼ γ′ γ′ ∼ γ
transitivity: if γ ∼ γ′ and γ′ ∼ γ′′, then γ ∼ γ′′
not hard to write down F ’s to prove all three properties
consequently, we can define the equivalence class of a loop by [γ], called homotopyclass
traversing two loops after each other is called the product γ ∗ γ′ of them
just shrink the time intervals (‘double the velocity’) to define:
(γ ∗ γ′)(s) =
γ(2s) for s ∈ [0, 1/2]
γ′(2s− 1) for s ∈ [1/2, 1](8.1)
(could also be defined for nonclosed pathes as long as end point of one path is the initialpoint of the other)
immediately the question appears, whether the order in this product matters; this willbe answered by the group structure on the loops with this product
actually on the equivalence classes, since one can show that the product can be definedon them: [γ] ∗ [γ′] := [γ ∗ γ′] is independent of the representatives and thus well-defined
the inverse path γ−1 is defined by traversing backwards: γ−1(s) = γ(1− s), and canalso be extended to the equivalence classes: [γ]−1 := [γ−1]
now the equivalence classes of loops together with the product ∗ form a group
we need to proof that (0) the product stays within the space (X) plus the group axioms:
(i) unit element: [γtriv]
(ii) inverse elements: [γ]−1
(iii) associativity of ∗
come up with proofs yourself or visit [Nak]
note that we do need the equivalence, just on loops γ−1 ∗ γ 6= γtriv
8.2 First homotopy group
this group of loop classes is named first homotopy group or fundamental group π1(Y, y0)
‘there does not exist a routine procedure to compute the fundamental groups’ [Nak]
some important properties and examples:
of course the group depends on the space Y probed, but also on the base point y0
actually not in most cases: if the space Y is path (also called arcwise) connected, i.e.if for any two points there exists a path between them, then the fundamental group isindependent of the base point
8-2
(to be fully correct, π1(Y, y0) is isomorphic to π1(Y, y1); isomorphic means a bijectivemapping respecting the group structure as well)
π1’s see the same structures/holes after having walked to the other base point and back
π1 of ‘similar spaces’
spaces can be ‘identified’ via mappings:
YfgY ′ (8.2)
e.g. for two manifolds to be diffeomorphic, f must be bijective (invertible and map ontothe entire Y ′), g is its inverse and both must be differentiable
for two top. spaces to be homeomorphic, the same conditions, but f and f−1 must becontinuous only
to be of same homotopy type (homotopy equivalent) is a weaker condition: it onlyneeds fg ∼ identityY ′ and gf ∼ identityY
this gives an equivalence relation among spaces, main result for us:
if Y and Y ′ are of same homotopy type, then π1(Y, y0) ∼= π1(Y ′, f(y0))
diffeomorphic/homeomorphic spaces are also of same homotopy type and thus the fun-damental groups agree for them, too
example: R is of same homotopy type as a point and thus has the same π1 = e,the trivial group consisting of just the unit element; these spaces are certainly notdiffeomorphic/homeomorphic (different dimension!)
similarly, the annulus and the circle have the same π1 (the latter is a ‘deformationretract’ of the former [Nak]) and so does the punctured plane R2\0
in that sense π1 measures the presence of holes, a space with trivial π1 is ‘simplyconnected’
example: π1(S1) = Z
we have guessed this in the vortex section 7.1, ‘easily understood even by children’ [Nak]
the proof is ‘not too obvious’ [Nak], it uses a one-to-one correspondence between theloops γ : S1 → S1 and functions γ : R→ R with γ(φ+ 2π) = γ(φ) + 2πn for all φ
γ ‘unwinds’ γ and has angles as argument and value
n ∈ Z is the degree or winding number
in the loop product the angles add up and thus the fundamental group is Z with additionas group multiplication
computing the degree
8-3
as an integral:
deg γ = n =1
2π
(γ(2π)− γ(0)
)=
1
2π
∫ 2π
0dφ
∂γ
∂φ(8.3)
helpful for physics: Bogomolnyi bound etc.
read 2π as volume of sphere and integral over the pullback of the volume form under γ
defined in differential geometry and generalizable to higher degrees, see the next section
as a sum over preimages:
fix some point on the image circle α being regular = finite number of preimages underγ and nonzero derivative there, then
deg γ =∑
i : γ(φi) = α
sign∂γ
∂φ
∣∣∣φ = φi
(8.4)
sums the orientation of the mapping γ when passing α (independent on α)
e.g. a degree of 5 can be obtained by 7 preimages with positive derivative and 2 withnegative derivative (opposite orientation), but ‘at least’ with 5 preimages with positivederivative (or nonregular points)
π1 of product spaces
is the direct sum over the individual π1’s [Nak, Sec. 4.3.1]
wikipedia writes a product: π1(Y × Y ′) ∼= π1(Y ) × π1(Y ′) ‘The fundamental groupfunctor [from spaces to fundamental groups] takes products to products.’
π1(TN ) = Z⊕ . . .⊕ Z (N times), meaning N separate winding numbers being additive
cylinder: the expected single winding number π1(S1 × R) = π1(S1)⊕ e = Z
in this context one also writes the trivial goup as 0 (with addition as group multiplica-tion)
more examples
spheres: π1(SN ) = 0 for all N ≥ 2
Lie groups as manifolds: π1(SU(N)) = 0 for all N
π1(SO(N)) = Z2 for N ≥ 3 and π1(SO(2)) = π1(U(1)) = Z
torus and Klein bottle from presentation of groups
start from generators xk ∈ X, which are letters of
words: w = xi1k1xi2k2. . . xinkn (8.5)
8-4
where n is finite and all ij ∈ Z, i.e. including inverse powers
let the words be reduced = no adjacent x’s are the same, otherwise reducing it by addingthe powers
these carry a group structure: the product is obtained by juxtaposing words and, ifnecessary, reduce the result
a word with no letters is the empty word denoted by 1, it acts as the unit element
reversing the order and all powers in a word gives the inverse word (as for matrices)
this is called the free group generated by X
example: one element X = x
the words are . . . , x−2, x−1, 1, x1, x2, . . . and multipying them means adding the power:this gives the group Z
but in general these groups can be non-abelian
constraints are written in the form like words
xi1k1xi2k2. . . xinkn = 1 (8.6)
and together with the generators this yields the presentation of a group
example: one element x constrained by xm = 1 gives the cyclic group Zm = x : xm = 1
torus and Klein bottle are obtained from rectangles by identifying the edges in a par-ticular way, parallel or antiparallel, see Fig. 11:
Fig. 11: The arrows on the outer rectangle visualize the identifications to arrive at the 2-torus(left) and the Klein bottle (right). Note that as a result all corners are identified.Also depicted are the noncontractible closed loops x and y, which are constrained bythe inner loop being contractible.
due to these identifications, x and y are noncontractible closed loops and thus generatorsof the π1’s
but there are also constraints by observing that particular combinations of them can beshrunk to a point and thus are trivial (put a rectangular into the bulk and blow it up)
for the torus: x y x−1 y−1 = 1 (visualize on a doughnut!) meaning x y = y x
⇒ every word can be ordered to xiyj which generates two independent Z’s
for the Klein bottle: x y x−1 y = 1
the words cannot be simplified further than replacing y−1 = x y x−1 everywhere
π1(Klein bottle) = x, y : x y x−1 y = 1 is nonabelian
for the ‘figure eight’ (also called 2-bouquet) we have just x and y without constraints
π1 = x, y : ∅, again nonabelian
8-5
real projective space RPN
the set of lines through the origin in RN+1: N -dim. manifold (with the correspondingneighborhood and smoothness)
equivalently: sphere SN with antipodal points ±~x identified: RPN = SN/Z2
for those who know that: rotation group SO(3) from the ‘covering group’18 SU(2) byidentifying ± group elements19 (−12 ∼ 12 is enough) which are indeed antipodal pointson the sphere S3 representing SU(2), thus SO(3) ' RP 3
a path x connecting the base point with its antipodal is now noncontractible (all ofthese pathes fall into one equivalence class)
[x] ∗ [x] is trivial again [Nak, Fig. 4.16]
π1(RPN ) = Z2 for all N ≥ 2 (later π1 for quotient of groups)
π1 and the first homology group
consider ‘objects’ embedded into the space called simplexes of various dimension: points,lines/edges, triangles with interiors etc.
homology groups detect objects without boundary that are not boundaries of anotherobject of one dimension higher20
group structure: define an integer or (simply real) ‘occupation number’ for every object(free abelian group generated by this objects)
another topological invariant of spaces H1 = closed 1-simplexes (‘lines’) that are notboundary of 2-simplexes
looks like π1 but always abelian: must divide out the commutator from the latter [Nak]
8.3 Higher homotopy groups
the even lower homotopy is an exception: the zeroth homotopy group π0(Y ) is just the numberof connected components of Y and not a group
higher homotopy groups can be constructed by mapping spheres or tori etc.
an n-sphere can be contructed by taking the n-cube In = I × I × . . .× I and identifying allpoints on the boundary ∂In ← at least one of the si’s is 0 or 1, Sn = In/∂In
in analogy to pathes we therefore define n-loops with base points y0 as mappings
γ : In → Y with γ(∂In) = y0 (8.7)
18the (for connected Lie groups unique) group with the same Lie algebra that is simply connected19popular: a spin rotated by SU(2) becomes the original one “after a 4π rotation”; adjoint representation with
Pauli matrices σa: g ∈ SU(2) and R(g) ∈ SO(3) are related by ~x2 = R(g)(~x1) ~x2~σ = g · ~x1~σ · g†, thenlook at g = −12
20the dimensions of these spaces added up with alternating signs gives the Euler characteristic
8-6
two such loops are homotopic to each other again, if an interpolation F with an addi-tional argument exists such that . . .
y0
s2
s1
γ
γ
γ γ ’ γ ’ γ’γ
’
γ
Fig. 12: The product on higher loops: the loops are traversed after one another in s1 (left),but letting the images ‘swim in y0’ (second from left) this is equivalent to defining theproduct with s2 (third from left). All these mappings are equivalent and – continuingthis process – so is the product in the other order (right).
this figure also explains why this is equivalent to gluing in any other s
for the group structure we still need γ−1, again define it by going backwards in s1
we get a group called the nth homotopy group πn(Y )
it has many properties in common with π1, but one important feature is different:
higher homotopy groups are always abelian, as Fig. 12 ‘proves’
examples:
spheres [wikipedia]:
πn(SN ) =
0 for n < N
Z for n = N
‘depends’ for n > N
(8.8)
e.g.: π4(S2) = Z2, π10(S4) = Z24 × Z3
π3(S2) = Z: Hopf number = linking number of loop like preimages
groups [Nak]:
π2(SU(N)) = 0, but π3(SU(N)) = Z
(knowing SU(2) ' S3, this is reduced to π2,3(S3) from above)
π2(SO(N)) = 0, but π3(SO(N ≥ 3)) = Z apart from π3(SO(4)) = Z⊕ Z
the degree (as in previous section) of a map ϕ between same-dimensional spaces
again at regular points α: finite number of preimages and on all of them nonzerodeterminant of the (Jacobian) matrix of partial derivatives, then
deg ϕ =∑
i : ϕ(xi) = α
sign det∂ϕ
∂x
∣∣∣x = xi
(8.9)
8-7
(possibly use local coordinates x in ‘charts’)
for image spheres SN and preimage spaces closed and oriented (and of dim. N), wehave that two mappings are homotopic precisely if their degrees coincide [Dubrovin,Fomenko, Novikov, Modern Geometry II, $ 13]21
or as integral over pullback of volume form
for S2 → S2, when the preimage sphere is the compactification of R2 3 (x1, x2) and theimage sphere is embedded in R3:
deg ϕ =
∫R2
d2x q(x) , q(x) =1
8πεabcϕ
a∂iϕb∂jϕ
cεij , (ϕa)2 = 1 (8.10)
(with sum convention in i, j, a, b, c and trivial metric in them meaning whether anindex is a subscript or superscript does not matter)
triple [Spat] product of ϕ, ∂xϕ and ∂yϕ (this way it can be discretized . . . )
the integrand q is a total derivative (divergence, locally) and thus the degree is invariantunder small deformations
to see that introduce angles for ϕ:
ϕ1,2 = sin θcossin
φ, ϕ3 = cos θ (8.11)
recall the S2 volume form in these angles is sin θdθdφ and q reflects that (‘pull-back’)
q(x) =1
4πεij sin θ ∂iθ ∂jφ = ∂i
[1
4πεij(±1− cos θ)∂jφ
](8.12)
valid around θ = 0 and θ = π, respectively, where φ is not well-defined, but where theprefactor vanishes
recall that (θ, φ) can be taken on several times (for various (x1, x2)) and Q counts thatby integration
21From the degree one can also derive the Fundamental Theorem of Algebra that any complex polynomial hasat least one root.
8-8
9 Instantons in sigma models
9.1 Bogomolnyi bound in O(3)
recall the action and its density (Eucl. space):
S =
∫d2x s(x) s =
1
2∂iϕ
a∂iϕa (9.1)
the degree is called topological charge Q with the topological density q from (8.10):
Q =
∫d2x q(x) 4πq =
1
2εabcϕ
a∂iϕb∂jϕ
cεij (9.2)
now try to guess a bound on the action as some multiple of Q like in (3.7)
= action density as a square plus a term being proportional to the top. density
here we go:
(∂iϕa ± εabc ϕb∂jϕcεij)2 = (∂iϕ
a)2 ± 2 εabcϕb∂iϕ
a︸ ︷︷ ︸−εabcϕa∂iϕb
∂jϕcεij + (εabc ϕ
b∂jϕcεij)
2︸ ︷︷ ︸(∂iϕ
a)2
(9.3)
square = 4s∓ 16πq (9.4)
hence:
S ≥ 4π|Q| (9.5)
thus to have a particular topological charge sector, a minimal amount of gradients andthus action is needed
“solitons stabilised by topology”
9.2 Instanton solutions in O(3)
again: classical solutions minimize (extremize) the action, which follows from the equality inthe bound (9.5), the latter has beautiful consequences
for the Bogomolnyi equality the square has to vanish everywhere:
∂iϕa = ∓εabc ϕb∂jϕcεij (9.6)
what do these equations mean? stereographic coordinates22:
ω1,2 =ϕ1,2
1− ϕ3(9.7)
(watch out for other conventions with a factor 2 in this expression)
22for the inverse transformation use |ω|2 =1−ϕ2
3(1−ϕ3)2
= 1+ϕ31−ϕ3
, ϕ3 = |ω|2−1
|ω|2+1etc.
9-1
ϕ3 = −1 (south pole) amounts to ω = 0
ϕ3 = 1 (north pole) amounts to ω =∞, should naturally be included
now Eqs. (9.6) become:
∂1ω1 = ± ∂2ω2, ∂1ω2 = ∓ ∂2ω1 (9.8)
further, upon complexifications ω = ω1 + iω2, z = x1 + ix2 this becomes
(∂1 ± i∂2)ω = 0 (9.9)
defining
∂ =1
2(∂1 − i∂2) , ∂∗ =
1
2(∂1 + i∂2) = −(∂)† , (9.10)
such that ∂z = 1, ∂z∗ = 0 etc., these are nothing but the Cauchy-Riemann conditions:
∂∗ω = 0 (9.11)
∂ω = 0 (9.12)
and easily solved by meromorphic or antimeromorphic functions:
ω =ω(z) for Q > 0 (9.13)
ω =ω(z∗) for Q < 0 (9.14)
remember that poles in ω are allowed, including those at z =∞
in terms of ω, the densities reads: [Fateev, Frolov, Shvarts, Nucl.Phys. B154 (1979) 1]
s = 4|∂ω|2 + |∂∗ω|2
(1 + |ω|2)2(9.15)
q =1
π
|∂ω|2 − |∂∗ω|2
(1 + |ω|2)2(9.16)
which reveals the nonlinearity of the sigma model, see also [Shi, (29.1)]
one can see again that s = 4πq iff ∂∗ω = 0 and s = −4πq iff ∂ω = 0
example 1: simply a linear function
ω =1
λ(z − z0) =
0 (ϕ on south pole) for z = z0
|ω| = 1 (ϕ3 = 0, equator) for |z − z0| = λ
∞ (ϕ on north pole) for z =∞(9.17)
Belavin-Polyakov monopole [JETP Lett. 22 (1975) 245] or (baby) skyrmion
expect Q = 1 from the covering argument: for every image there is exactly one preimage
9-2
compared to the vacuum – where ϕ is say at the north pole everywhere23 – this config-uration is a ‘defect’ that cannot be removed (what to do with the south pole??) anddisorders the ‘spins’ ϕ
this costs energy/action, but has more entropy (possibilities) than the vacuum
with the ‘other orientation’ z → z∗ one should obtain a Q = −1 solution (X)
λ has length dimension (since ϕ and ω are dim.less)
whereas z0 = (x1)0 + i(x2)0 looks like a two-dimensional location (X)
indeed the action = topological density is localised at z0, it reads:
q =1
π
|λ|2
(|λ|2 + |z − z0|2)2(9.18)
and one can see that |λ| simply rescales the solution, since in dimensionless units:
|λ|2q =1
π
1
(1 + ( |z−z0||λ| )2)2(9.19)
the total topological charge
Q =
∫d2x q(x) = 1 (9.20)
is independent on λ and z0 which therefore are collective coordinates
the former we have already anticipated by Derrick’s argument at the end of Sec. 7.2
the phase of λ rotates the solution
example 2: inverse of ex. 1
ω = λ1
z − z0=
∞ for z = z0
0 for z =∞(9.21)
poles and zeros interchanged, still Q = 1, even same density q
example 3: both zero and pole at finite z
ω =z − zz − z
=
0 for z = z
∞ for z = z(9.22)
constituents at z and z?
no, same density as above around the center of mass (z + z)/2 with size λ = |z − z|/2
example 4: generalize the above to rational functions
ω =
|Q|∑i=1
λiz − z0(i)
(9.23)
23this would break the O(3) symmetry spontaneously down to the O(2) around the prefered direction, whichin two dimensions contradicts the Mermin-Wagner theorem valid in the full quantum theory
9-3
charge Q solutions, since Q singularities (ϕ on north pole) and holomorphic (Q > 0; forQ < 0 replace z by z∗ again), zeros (ϕ on south pole): at z =∞, of Qth order
actually this is the general charge Q solution [Shi]
it has 4Q real parameters/moduli: Q two-dim. locations, Q sizes and Q phases
example 5: instantons on a cylinder R× S1
ω = exp(2πQz/β) , Q ∈ Z (9.24)
obeys periodic bc.s under z → z + iβ, x2 → x2 + β, meaning nonzero temperature forthe (bosonic) ω or ϕ
the topological charge in a β-strip is Q indeed, with the topological density q ∼1/ cosh2(2πx1/β) being localised in space and static in ‘time’
(only one location: not the most general solution)
instanton-antiinstantons: add ω’s, approximate solutions when dilute
9.3 Definition of CP (N − 1) models
“The CPN−1 system does its best to imitate QCD.” [Actor, Fortschr.Phys. 33 (1985) 6, 333]
but can also be realized experimentally in cold atoms [C. Laflamme et al., Annals Phys. 370(2016) 117]
confusion starts already with the name, since called CP (N) by roughly half of the papers (weare in line with [Shi], [Dia1], but not with [Raj])
let n be a complex N -dim. vector, n = (n1, . . . , nN )T , normalized to 1: n†n = 1
= complex projective space, analogous to the real one mentioned before
again two space-time dimensions
Lagrangian, coupling set to unity
L = (Din)†(Din), Di = ∂i + iAi (9.25)
includes a gauge field which is nondynamical, since no field strength involved
auxiliary to write L in a convenient form quadratic in n
A can be replaced by the solution of its equation of motion:
L =− n†D2i n (since D†i = −Di) (9.26)
0!
=∂L∂Ai
= −2n†Din · i = −2i(n†∂in+ iAin†n) (9.27)
Ai = in†∂in (9.28)
9-4
this gauge field is actually real24 and abelian (a number); plugging it back yields
L = (∂in)†(∂in) + (n†∂in)2 (9.29)
note the nontrivial mixing of kinetic and n4 terms: (∂in∗a)(δab − nan∗b)(∂inb)
equation of motion for n via a Lagrange multiplier for the constraint
one may forget Ai from now on, but one may also use the more convenient Lagrangian(9.25) with Ai an abbreviation for in†∂in
symmetries:
local U(1):
na → eiΛ(x)na ∀a , Ai → Ai − ∂iΛ . . . gauge trafo X (9.30)
diagonal part of:
global U(N) 3 G:
na → Gabnb , Ai → Ai (9.31)
the lowest nontrivial model is CP (1), i.e. N = 2, with a two-dimensional vector n, thatis normalized and in which one phase can be gauged away by (9.30)⇒ 2 real parameters
can be mapped onto the O(3) model via
ϕA = (n†)a(σA)abnb (9.32)
with the Pauli matrices σA
CP (N−1) models are generalizations of O(3) that keep topology, to be seen in a minute
in contrast to O(N), for which the finite action requirement in Sec. 7.3 leads to trivialmappings π2(SN )
9.4 Topology and instantons in CP (N − 1)
asymptotic behavior from finite action again [Raj, (4.112)]:
|Din| → 0 (9.33)
∀a : ∂ina + iAina → 0 (9.34)
Ai → i∂inana
= i∂i|na||na|︸ ︷︷ ︸
imag!
= 0
− ∂i arg na︸ ︷︷ ︸real
!= a-indep.
(9.35)
∂i|na| → 0, na → na(x)ei arg na(x) (9.36)
arg na → arg na,(0)(x) + γ(x) (9.37)
~n→ ~n(0)ei γ(x) (9.38)
24 since (n†∂in)∗ = (n†∂in)† = (∂in)†n = n†(−∂i)n
9-5
argue the other way round: n winds as γ does, but a (circular) Ai → −∂iγ compensatesthat in the covariant derivative (∂i + iAi)n
this winding at the asymptotic circle gives rise to a winding number S1 → S1:
remarkably simple and a total derivative (and similar to Yang-Mills instantons, seelater)
consider as a meromorphic function – similar to what was done in O(3) – a polynomialof degree k in each component wa, i.e. the highest degree over all components shall be k
top. charge from winding at spatial ∞:
wa =k∑j=1
ca,jzj |z|→∞−→ ca,kz
k (9.53)
|w| |z|→∞−→√∑
a
|ca,k|2 · |z|k (9.54)
nT =wT
|w||z|→∞−→
(c1,k, . . . , cN,k)√∑a |ca,k|2︸ ︷︷ ︸nT(0)
zk
|z|k︸︷︷︸= eiφk : Q = k
(9.55)
N · |Q| complex coefficients
⇒ real dim. of moduli space: 2 ·N · |Q| (partly gauged away!?)
cf. instantons in 4d YM with Nc colors, i.e. gauge group SU(Nc)
real dim. of moduli space: 4 ·Nc · |Q|
SU(Nc = 2) should be compared to CP (1) ∼= O(3)
can use local U(1) symmetry to make n1 real, and multiplications of w to set w1 = 1 (ifnonzero)
then in CP (1) ∼= O(3) we have w = (1, ω) i.e. just one (anti)meromorphic function ω
compatible with the definitions of ω from ϕ (9.7), ϕ from n (9.32) and n from w (9.48)
there n(x) is a complex M ×N matrix (M < N) normalized to 1M
the CP (N − 1) model is the simplest Grassmann model with M = 1 or in turn theGrassmann model is like an M -flavor CP (N − 1) model (with the flavors interacting ina particular way)
similar action and topology and (anti)meromorphic (anti)instantons
9-7
more analogies to gauge theories:
representation on a space-time lattice exists, but obscures topology (continuum concept)
top. charge Q can be made part of the action with a parameter θ = theta-angle; samesolitons (q is a total derivative and does not contribute to the e.o.m.), but particularsuperpositions in path integral/ensemble; violates CP and P invariance (in the case ofgauge theories experiments put very strong constraints, i.e. θ tiny; why?)
9.5 Fermions in solitonic backgrounds II: CP (N − 1)
one more analogy to QCD, and to kinks from before
complex structure will play a prominent role again
with gauge fields the coupling to fermions is straightforward
Lf + Lint = ψγµ(∂µ + iAµ)ψ (9.56)
where – as in the kink background – we have considered zero mass, and in Euclideanspace there is no imag. unit in the action
the Euclidean gamma matrices obeying γµγν + γνγµ = 2ηµν1 can again be chosen asPauli matrices γ1,2 = σ1,2 and the third one, γ5 = −iγ1γ2 = σ3, is diagonal again, withthe imag. factor chosen to make it hermitian
since the Dirac operators is proportional to γµ (and not 1 as for the mass), it anticom-mutes with γ5
again, on zero modes the Dirac operator commutes with γ5 that thus can be measured
γ5 has eigenvalues ±1 for spinors with only an upper/lower component denoted by l(eft)and r(ight): chirality
moreover, in complex language
γµDµ =
(0 D1 − iD2
D1 + iD2 0
)=
(0 DD∗ 0
)antiherm. X (9.57)
for zero modes we hus have to solve
D(∗)ψl(r) = 0 (9.58)
use the complex A’s when solution, say positive Q: A = i∂ log |w|, A∗ = −i∂∗ log |w|(9.50), then we need:
recall that the general solution for w with charge Q is a polynomial (vector) of degree Q,such that asymptotically |w| → |z|Q
9-8
finally normalizability
to avoid singularities χ should be an entire function (|w|±1 cannot cancel that: it has zand z∗)
it should thus have a Taylor series in nonnegative powers of z(∗)
this alone cannot be square integrable, but the factor |w|−1 (only that sign) helps
the χ-series has to truncate before that of w, i.e. at the order Q− 1 ← Q coefficients
check asymptotics: in the case of sigma models one requires normalizability on thesphere:
∫d2x
1+x2· |ψ|2 <∞, such that a decay |ψ|2 → (|z|Q−1/|z|Q)2 = 1/x2 is ok even in
2d [D’Adda, Di Vecchia, Luscher, Nucl. Phys. B152 (1979) 125]
check bulk: problems from |w|−1? |w| should not vanish, since we want na = wa|w| (w is
a complex vector)
result (still Q > 0): we have Q independent normalizable zero modes in ψr
solutions with negative Q’s: same functional form, but some sign changes, such thatthe zero modes sit in the other chirality sector, i.e. has the opposite γ5-eigenvalue
realization of the index theorem
nr − nl = Q (9.61)
where nl,r are the number of left/right handed zero modes (conventions: depends onthe def. of γ5 etc.)
explicitly shown on solutions of the eom., but as a topological invariant (integer!) italso holds in their vicinity and thus in the entire Q-sectors
it is the asymptotic winding that determines both, top. charge and fermion normaliz-ability, without reference to the complex structure used for solutions
9-9
10 Vortices
lines stabilized by winding of a scalar field at infinity, we already know that:
as it stands the vortex energy diverges, see Sec. 7.1
a gauge fields can help, see sigma models in Sec. 9.4
let’s first discuss such systems (mostly following Ryder’s textbook)
10.1 Spontaneous symmetry breaking in Higgs models
10.1.1 Global symmetries
just a scalar field with mexican hat potential
V (~ϕ) =λ
2(|~ϕ|2 − v2)2 (10.1)
with ~ϕ a vector with N components, say real (the complex case is pretty analogous)
theory (= Lagrangian incl. kinetic term) invariant under global rotations in O(N)
each |~ϕ| = v is a vacuum, choose a direction: ϕvac = (v, 0, . . . , 0)T
consider fluctuations around ϕvac, if they cost energy to quadratic order, then massive
here: fluctuations in the first component rescale the absolute value and thus go out ofthe minimum valley = massive, all other flucutations stay in the valley = massless =Goldstone modes: N − 1
symmetry of the vacuum lower = spontaneous symm. breaking
rotations that keep the vacuum = around an axis, group O(N − 1) (embedded in thebigger rotation matrix with upper left entry = 1)
Goldstone theorem: if the symmetry breaking pattern is G→ H, then massless modesexist for all generators inG/H (coset space; “inG but not inH”), number = dim(G)−dim(H)
just group theory: indep. of representation of the group and shape of the potential
with dim(O(N)) = N(N − 1)/2, we indeed get N − 1 massless modes
value of the mass:
λ
2((v + η)2 − v2)2 ⊃ λ
2(2v η)2 !
=m2
2η2 , m = 2
√λ v (10.2)
(self)coupling times vacuum expectation value vev
10-1
10.1.2 Local symmetries with gauge fields
abelian Higgs model: ϕ with one complex component, Minkowski space ηµν = diag(1,−1,−1,−1)(later static anyway)
L = (Dµϕ)∗(Dµϕ)− V (|ϕ|)− 1
4FµνF
µν , (10.3)
Dµ = ∂µ − ieAµ , FµνFµν = − 2 ~E2︸︷︷︸
~F0.2, kin.
+ 2 ~B2︸︷︷︸pot.
(10.4)
remark on conventions for kinetic and mass terms (free case without gauge fields):
ϕ∗(−∂µ∂µ −m2)ϕ vs. real ϕi−∂µ∂µ−m2
2 ϕi, both give the correct relativistic dispersionrelation pµp
µ−m2 = 0, E2−~p 2 = m2, the former when treating ϕ and ϕ∗ as independent
local U(1) symmetry, “scalar QED”25, gauge transformations:
ϕ→ g ϕ g = eiΛ (10.5)
Aµ → Aµ +1
e∂µΛ (10.6)
Dµϕ→ Dµϕ (10.7)
Fµν → Fµν (10.8)
can be used to gauge ϕ to real
masses and interactions from expansions around vev:
ϕ = v +η1 + iη2√
2, v ∈ R (10.9)
√2 to match the complex-real conventions mentioned above, expanded Lagrangian:
L =1
2(∂µη1)(∂µη1)− 1
4FµνF
µν . . . kin. and pot. terms (10.10)
− λv2η21 + (ev)2AµA
µ . . . mass terms (10.11)
+ η2-terms . . . eliminated by gauge freedom∗ (10.12)
+ cubic and quartic terms . . . interactions (10.13)
∗: new ϕ real again, “physical” or “unitary” gauge
vector and scalar field massive = Higgs mechanism:
mϕ =√
2√λ v (10.14)
mA =√
2 ev (10.15)
both of the form coupling times vev
indeed η1 runs up the hill
25generalization to nonabelian gauge fields: Georgi-Glashow (SU(2)adj. ' SO(3)) and Weinberg-Salam(SU(2)× U(1)→ U(1)′, electroweak)
10-2
the in-valley component η2 – which used to be massless Goldstone boson and could notbe gauged away in the previous model with global symmetry only – now points in agauge direction: the gauge fields has “eaten up” its mass and has become massive
counting of degrees of freedom: a massive gauge field has one more degree of freedomthan a massless one, the longitudinal polarization26
in the electroweak symmetry breaking the W (and Z) bosons are of this kind
a term prop. to AµAµ in a fundamental theory would break gauge invariance, this is
why the Higgs mechanism is needed
these masses will play a role for the vortex solutions we are after now
10.2 Equations of motion, vortex solution, Bogomolnyi bound
eom. = generalized Klein-Gordon and Maxwell equations:
−DµDµϕ− λ(|ϕ|2 − v2)ϕ = 0 (10.16)
∂νFµν = jµ ≡ ie(ϕ∗Dµϕ− ϕDµϕ∗) (10.17)
the structure of the current looks similar to that in quantum mechanics (from δL/δAµ)
it is conserved, ∂µjµ = 0, consistent with ∂µ∂νFµν = 0 (symmetric times antisymmetric)
look for solution with special properties:
– static ∂0 = 0
– A0 = 0 , thus ~E = 0 and no kinetic terms ⇒ H = −Lpot
– x3-indep., ∂3 = 0, and A3 = 0 ⇒ B1,2 = 0, B3 ≡ B
– finite energy per length in x3 ⇒ |ϕ| → v asymptotically in (x1, x2)
unit charge = unit winding number [Nielsen-Olesen 73]: symmetric ansatz, again inpolar coordinates (ρ, φ) :
ϕ = eiφϕ(ρ) ϕ(ρ)→ v (10.18)
Aρ = 0 (10.19)
Aφ = A(ρ) (10.20)
note that ϕ(ρ = 0) = 0, otherwise the scalar field is ambiguous at the origin
better ϕ = 0 must occur ‘somewhere’ since of course the whole configuration can beshifted in (x1, x2), which is a collective coordinate
this is the ‘false vacuum’ (maximal potential), but stabilized by topology: the correctvacuum around it winds such that the false vacum is trapped
applications in phase transitions of the early universe: cosmic strings [Kibble 1976], butstrongly constrained by cosmic microwave background
26not cancelling against the time-like component anymore [Ryder]
10-3
recall that vortices are governed by the first homotopy group π1 of some internal man-ifold – actually G/H from the symmetry breaking pattern – and that π1 is the onlyhomotopy group, that could be nonabelian
so one might come up with physical systems where the superposition/fusion of vorticesis ruled by vortex quantum numbers that are not additive but follow a more complicatedgroup multiplication (e.g. Z2: a double vortex is like no vortex)
back to equations of motion: become a coupled system of two nonlinear ordinary dif-ferential equations of second order in ϕ(ρ) and A(ρ)
numerical solution: ϕ(ρ) increases monotonically from 0 to v, A(ρ) and thus also B(ρ)decrease monotonically from some value to 0
asymptotics for large ρ:
ϕ(ρ)→ v + # exp(− mϕ√2ρ) (10.21)
A(ρ)→ 1
e
1
ρ+ #′
1√ρ
exp(−mAρ) (10.22)
B(ρ)→ 0 + #′′1√ρ
exp(−mAρ) (B =1
ρ∂ρ(ρAφ)) (10.23)
the masses govern the exponential corrections to the leading behavior
the circular Aφ has the correct radius dependence (see (9.40)) to cancel the φ-gradientof ϕ; there is no δ(0) in B as one could expect from that form of A (e.g. [1/ρ2] = [δ(2d)]),since this form is only valid asymptotically and smoothed out near the origin
energy per length, with i = 1, 2:
E =
∫d2x
[|Diϕ|2 + V (ϕ) +
1
2B2]
(10.24)
Bogomolnyi bound in the case e2 = λ (the physics of this case will be discussed later):
E =
∫d2x
[|∂iϕ± e εijAjϕ|2 +
1
2
B ±
√λ(|ϕ|2 − v2)
2 ± ev2B]
(10.25)
check that this reproduces all terms needed: in particular B|ϕ|2 from two terms, forwhich we need e2 = λ
note that the first term is not a covariant derivative
obviously:
E ≥ ev2|Φmag| = 2πev2|Q| (10.26)
where the equality yields solutions of the e.o.m.
these are easier to solve since only first order in ϕ(ρ) and A(ρ) (B ∼ ∂A), still numericsneeded
10-4
10.3 Vortices in superconductors
first a qualitative picture:
Cooper pair density ϕ, ‘condensate’ = macroscopic quantum effect
ϕ nonzero or zero distinguishes superconducting and normal state
ϕ nonzero → zero at boundaries, within the ‘coherence length’ ξ ← inverse of our mϕ,actually ξ =
√2/mϕ
in external magnetic fields B:
B expelled, if small, by persistent currents (no resistance!) = Meissner-Ochsenfeld effect= perfect diamagnetism
within the ‘penetration depth’ Λ [London2] ← inverse of our mA
large B: mechanism breaks down, how: energy balance
elements of the theory:
Cooper pair is charged by −2e thus coupling to Maxwell fields: ϕ complex
Cooper pair is spinless since electrons of opposite spin: ϕ scalar (not so in e.g. superfluidhelium)
nonzero vacuum value of ϕ achieved by a potential of mexican hat type
actually the coefficient in front of |ϕ|2 is T -dependent: above Tc positive such thatvacuum is just the normal state ϕ = 0
underlying theory: Bardeen-Cooper-Schrieffer, then leading order in |ϕ|2 and gradients
ϕ is the order parameter of a second order phase transition [Ginzburg-Landau] since itvanishes continuously (for a first order transition and its metastable state one wouldneed two disconnected vacua where as a function of e.g. temperature one takes overfrom the other)
nonrelativistic coupling to gauge fields: 12m∗ |Diϕ|2 where m∗ ' 2melectron
magnetic field costs (free) energy: ~B2/2
⇒ nonrelativistic abelian Higgs model with qualitatively the same vortex solutions
vortices, large B and types of superconductors:
beyond a critical B, the magnetic field cannot be expelled from the SC anymore, butenters
how? layer of transition, recall the two typical length in scalar and magnetic (gauge)field, ξ and Λ, respectively
large ξ: B counteracted by currents = costs energy, but ϕ not yet in its vacuum = notmuch energy gained ⇒ positive surface energy
10-5
type I superconductors
found first (elementary superconductors)
superconductivity breaks down suddenly: first order transition in B
Ginzburg-Landau parameter: κ := Λξ =
mϕ√2mA
= 1√2
√λe . . . dimensionless
in κ the vev v has dropped out
in most materials κ is approximatedly T -independent
type I for κ < 1/√
2 = κcrit
note that exactly there λ = e2, which is where the Bogomolnyi bound is at work
type II superconductors, negative surface energy = ’likes surfaces’
B enters in many small segments ⇒ vortices
quantized magn. flux: see below
a critical B where this starts and another one where also vortices eventually give roomfor the normal state, second order transitions!?
these vortices repel (= type II prefer a large total surface, in contrast to type I wherevortices attract and thus collapse to a giant normal domain)
⇒ vortex lattice [Abrikosov 1957, paper delayed by Landau who did not believe in it(!)]
triangular [Abrikosov himself made a mistake and predicted the slightly more energycostly square lattice]
observed in experiments: by neutron diffraction, later by decoration
10.4 Aharonov-Bohm effect and flux quantization
1959, measured 1960
a (cylindrical) solenoid with magnetic field B inside only
⇒ no effect on particles around it in classical theory
embed it into a double-slit electron experiment with its typical interference pattern
⇒ the latter is shifted in the presence of B
again: never having experienced a force from B
but there must be a circular gauge field Aφ ∝ B/r (or ∝ Φ/r) due to Stokes’ theorem
a free quantum particle with charge q on its way picks up a phase factor exp(iq∫d~r ~A)
(nonabelian theories: Wilson line)
from the wave function being proportional to exp(i∫d~r ~p) and the canonical momentum
becoming ~p+ q ~A
10-6
two electrons moving through different slits and interfering at the screen differ by aphase q
∮d~r ~A = qΦmag
this is what causes the interference pattern to change
the quantization of the flux of vortices in superconductors is now straightforward
the wave function ϕ of the doubly charged Cooper pairs around a vortex must be single-valued, therefore
qΦ = 2πZ ⇒ Φ = Φ0Z with Φ0 =2π
q
SC=
2π
2e~ =
h
2e(10.27)
where in the last expressions we have reintroduced Planck’s constant appearing in themomentum and thus in all quantum mechanical phase factors
the experimental value is Φ0 = 2.067833758(46) · 10−15 Wb (Weber) [wikipedia]
10-7
11 Magnetic Monopoles
11.1 Dirac’s idea in due ‘form’
the Maxwell equations in vacuum27
components relativistic forms
div ~B = 0
curl ~E + ~B = 0∂µε
µνρσFρσ = 0 dF = 0 homogeneous
div ~E = ρ
curl ~B − ~E = ~j∂µF
µν = jν ∗d ∗F = j inhomogeneous
are already pretty nice on the left hand side in treating electric and magnetic fields onequal footing, but not so on the right hand side
more symmetric, if magnetic charges ρmag exist [Dirac 1931]
say on the electric side ~ = 0 which means ρ = 0 from current conservation ∂νjν = 0
consistent with ∂ν∂µFµν = 0 (symm. times antisymm.)
so try
div ~B = ρmag (11.1)
with a static ρmag (no jmag), which in the simplest case is a point source
but the homogeneous equations were the prerequisites for introducing a gauge potential
components relativistic forms~B = curl ~A
~E = −grad φ− ~AFµν = ∂µAν − ∂νAµ F = dA if homogeneous
since it follows that ∂µεµνρσFρσ ∼ ∂µ∂ρεµ.ρ. = 0 or simply dF = d2A = 028
thus at the location of the monopole the gauge field must be singular somehow
why gauge fields at all? coupling matter, for which ~B exerts a force, but, as we learnedfrom Aharonov-Bohm, also A has an influence
let’s sit on a sphere around the magnetic charge, so no singularity there
27differential forms in a nutshell: F is an (antisymmetric) two-form 12Fµνdxµ∧dxν = 1
2Fµν(dxµdxν−(µ ν));
d the correct antisymmetrised differential operator arriving at a next higher form and d2 = 0, meaning,depending on what it acts, div curl = 0 or curl grad = 0; the Hodge star takes n-forms to dim(manifold)−nforms with the ε-tensor whose indices are partially up and partially down and for which one needs ametric, e.g. ∗1 is the volume form and (∗)2 = ±1; finally j = jµdxµ is a one form (for which one cannotantisymmetrise)
28A is a one-form, whose gauge freedom is adding a gradient A → A + 1edΛ (Aµ → Aµ + 1
e∂µΛ) giving the
same F = dA, again since d2Λ = 0
11-1
even then the existence of a gauge field is precluded by Stokes’ theorem: consider aclosed path C on the sphere and the two sector/caps S(1,2) with C as boundary, then∮
Cd~r ~A =
∫S(1)
d2~x ~B = −∫S(2)
d2~x ~B (11.2)
where the sign comes from the orientation of C as the boundary of one S(.)
the two caps add up to the entire sphere and we obtain
0 =
∫S(1)∪S(2)
d2~x ~B = Φmag (11.3)
the mathematical question behind:
on such a sphere we still have the homogeneous Maxwell equation dB = 0
this is a closed 2-form on S2 (actually every differential form (antisymmetric unlike themetric) on a two-dim. manifould is closed, since there is no 3-form)
is this form also exact, B = dA, with A a 1-form?
more sophisticated: B ∈ kernel dr, but also B ∈ image dr−1 with r = 2?
the rth, here 2nd, ‘de Rham cohomology’ Hr=2(M = S2) answers exactly this question
this is very tightly connected29 to ‘homology’, the question whether there are closedobjects (simplexes) on that manifold that are not boundaries of another object of onedimension higher; see the end of Sec. 8.2 for the connection to homotopy
indeed H2(S2) is nontrivial, there exist such forms that are closed but not exact: thevolume form! if it was exact as well, the argument above would give zero volume
upshot: even away from the magn. charge there is no unique regular gauge field
to be concrete, assume further a radial field, which from a point source is Coulombic:
~B = qm~r
r3, div ~B = 4πqm δ
(3)(x) (11.4)
from ~r/r3 = −~∇(1/r) and the 3d Greens function ~∇2(1/r) = ∆(1/r) = −4πδ(3)(x)
the choice of the constant is as ambiguous as the systems of units (CGS etc.) inelectrodynamics
the magnetic flux is
Φmag =
∮d2~x · ~B = 4πqm (11.5)
29de Rham’s theorem for compact manifolds, idea: duality by integrating an n-form over an n-simplex
11-2
11.2 The Dirac string and charge quantisation
let’s write down gauge fields, that are ‘almost regular’; in spherical coordinates:
Ar = 0 (tangential suffices, see integral for Stokes) (11.6)
Aθ = 0 (11.7)
Aφ =qm
r
±1− cos θ
sin θ(11.8)
the magnetic field is what we want:
curl ~A = ~eθ(∂Ar∂..
,∂(rAφ)
∂r
)+ ~eφ
(∂Ar∂..
,∂Aθ∂..
)+
~err sin θ
(∂(sin θAφ)
∂θ− ∂Aθ
∂..
)(11.9)
= ~erqm
r2
∂(±1− cos θ)
sin θ ∂θ= qm
~err2
(11.10)
so what’s wrong? and why ±1?
this gauge field is not regular everywhere, because φ isn’t! check the z-axis!
with +1 it is regular at cos θ = 1, so ok at positive z’s, but not at negative z’s
with −1 it is not regular at positive z’s
(with everything else than ±1 it is not regular at both parts of the z-axis)
closer inspection of the singularity
+1 at negative z’s and near the z-axis, θ ≈ π ⇒ +1 − cos θ ≈ 2 and r sin θ = ρ :=√x2 + y2, thus
Aφ =2qm
ρ(11.11)
this is a circular gauge field with the correct 1/distance decay in 2d that gives a winding(see CP (N − 1)-eq. (9.40) and vortex-eq. (10.22)) and thus a magnetic flux, but hereon an infinitesimal disc ⇒ B must be singular (+ smooth Coulomb part (11.4) notcontributing at inf. small circles)
indeed in coordinates perpendicular to the z-axis30
Ax =2qm
ρ2(−y) = −2qm ∂y log ρ , Ay =
2qm
ρ2x = 2qm ∂x log ρ (11.12)
and with the 2d Greens function ∆ log ρ = 2π δ(2)(x), we finally arrive at
Bz = ∂xAy − ∂yAx = 4πqm δ(2)(x)θ(−z) (11.13)
this object, carrying a localized singular B, is called the Dirac string
30in polar coordinates formally A = grad (2φ), but φ is not single valued
11-3
the B-field points towards the location of the monopole with flux∫(−d2(x, y))Bz =
−4πqm (in the vicinity of the z-axis we can use this measure) exactly compensatingthat of the monopole, Eq. (11.5)
equivalent: compensating the source since div B = 4πqmδ(2)(x)∂zθHeaviside(−z) =
−4πqmδ(3)(x)
this is the invariant statement of the Dirac string, while its geometry is irrelevant
with the −1 gauge field, for instance, we get the same influx at the positive z-axis
but the Dirac string could also be curved
consequences for a charged quantum particle
wave function must be single valued, as for the superconductor in Sec. 10.4
(go far away from the monopole such that we can speak of a free particle)
we can immediately take over the quantisation condition = ‘Dirac string invisible’ whenencircling it:
2πZ = qΦ = 4πq qm , q qm =Z2
(11.14)
for the electric charge q of every particle
a single monopole in the universe would thus require all electric charges to be quantised!in multiples of 1/(2qm)
(in turn the charges qm of monopoles should be quantised as well)
11.3 Wu-Yang construction
live with regular but only local gauge fields, here two, giving the same field strength, thusrelated by a gauge transformation
use the +1 gauge field on the northern hemisphere (of the sphere), where it is regular,plus a little bit south of the equator
and the −1 gauge field on the southern hemisphere plus a little bit north of the equator
on the equator (the argument can be extended to a strip around it)
A(±)φ =
qm
r
±1− 0
1(11.15)
A(+)φ = A
(−)φ +
2qm
r(11.16)
A(+) = A(−) + grad (2qmφ) (11.17)
this has the form of a gauge transformation, e.g. Eq. (10.6) with charges e→ q
A(+) = A(−) +1
qgrad (2qqmφ) = A(−) − i 1
eΩ−1grad Ω, Ω = exp
(i · 2qqmφ
)(11.18)
11-4
this is unique – as it must – under the same quantization condition 2qqm ∈ Z
11.4 Spontaneous symmetry breaking in a nonabelian gauge-Higgs model
for nonabelian & nonsingular magnetic monopoles a la ’t Hooft-Polyakov
model: Yang-Mills with gauge group SU(2) (for simplicity) and Higgs in the adjointrepresentation (see below) [Georgi-Glashow 1972]
close to electroweak with gauge group SU(2)× U(1) [Weinberg-Salam 1967/68]
Lagrangian:
L = trDµϕDµϕ− V (ϕ)− 1
2trFµνF
µν (11.19)
nonabelian field strength and gauge field in the Lie algebra of SU(2):
Fµν = F aµνσa2, Aµ = Aaµ
σa2
(11.20)
where half the Pauli matrices are the generators of SU(2) with commutation relationand normalization
[σa2,σb2
] = iεabcσc2, tr
σa2
σb2
=δab2
(11.21)
e.g. 12 trFµνF
µν = 14 (F aµν)2
the field strength in terms of Aµ contains a new term
Fµν = ∂µAν − ∂νAµ − ig[Aµ, Aν ] , F aµν = ∂µAaν − ∂νAaµ + g εabcA
bµA
cν (11.22)
with coupling constant g
as a result nonabelian gauge bosons can interact among themselves via three- and four-vertices (U(1)-photons cannot) and even the pure gauge theory is nontrivial: asymptoticfreedom etc.
ϕ could be a complex two-vector and in the covariant derivative Dµ the gauge field couldact on it by multiplication: ϕ ‘in the fundamental representation’, holds for quarks inSU(3) gauge theory
here: ϕ is also a two-by-two matrix expandable in Pauli matrices (lives in the SU(2)algebra, too) and the gauge field acts on it by commutation:
ϕ = ϕaσa2, ϕa ∈ R , Dµϕ = ∂µϕ− ig[Aµ, ϕ] (11.23)
(Dµϕ)a = ∂µϕa + gεabcA
bµϕ
c “ = Dµϕa ” (11.24)
in other words, a real three-vector ‘in the adjoint representation’, written as multipli-cation (index µ dropped, upper and lower index have no meaning):
εabcAbϕc
!= (Aadjϕ)a =(Aadj)acϕ
c (11.25)
(Aadj)ac!
= Ab(T adjb )ac ⇒ (T adj
b )ac = εabc (11.26)
11-5
possible for every Lie algebra with ε replaced by ‘structure constants’ fabc expandingthe commutator in generators as in Eq. (11.21) above
mexican hat potential
V (ϕ) =λ
8
(2 trϕ2 − v2
)2=λ
8
((ϕa)2 − v2
)2(11.27)
leads to symmetry breaking
local gauge transformations with Ω = exp(inaσa/2) an element of the gauge groupSU(2):
Aµ → ΩAµΩ−1 − i
g(∂µΩ)Ω−1 , ϕ→ ΩϕΩ−1 (11.28)
as a consequence
Fµν → ΩFµνΩ−1 , Dµϕ→ Ω(Dµϕ)Ω−1 (11.29)
and all terms in the action are gauge invariant
actually the symmetry of the Lagrangian is SO(3) = SU(2)/Z2 since Ω = −12 has noeffect on ϕ (and Aµ)
for Goldstone theorem: SU(2) and SO(3) – as one of its representations – have thesame number of generators, namely three
vacua at the bottom of the potential:
(ϕa)2 = v2 (11.30)
invariant under which Ω’s in (11.28) above? rotations around that axis: SO(2) ≡ U(1)
‘stabiliser subgroup’ in the context of Lie groups acting on manifolds [Nak]
e.g. diagonal ϕ = ϕ3 σ32 , Ω should commute with it ⇒ Ω = exp(in3σ3/2) diagonal, too
one generator
symmetry breaking SO(3)→ SO(2) yields two massive gauge fields
perpendicular to ϕ (eaten up those Goldstone bosons)
and one massive scalar field (Higgs)
masses similar to the abelian case: coupling times v
+ one massless gauge field along ϕ
11.5 Magnetic monopoles a la ’t Hooft-Polyakov
static and A0 = 0 ⇒ no electric field, but could be added
11-6
finite energy needs |ϕ| :=√
(ϕa)2 → v, normalised vector field:
na =ϕa
|ϕ|: S2 = ∂R3 → S2
color (11.31)
has an integer winding number/degree, see (8.10)
deg n =1
8π
∫S2∞
εabcna∂in
b∂jncεij k d
2σk ∈ Z (11.32)
for general symmetries G and H of theory and vacuum, the winding is in π2(G/H), forwhich more theory exists
symmetric ansatz: hedgehog
na → xa
r=: xa (11.33)
covers the sphere once ⇒ deg n = 1
note the mixing of color and spatial directions, ‘spin from isospin’
the full ansatz reads
ϕa = xaϕ(r) , ϕ(r)→ v (11.34)
for finite energy in addition Diϕ→ 0
with ∂i(1/|ϕ|)→ 0 this becomes Di n→ 0, thus as before Ai has to compensate for thewinding of n
perpendicular to radius could be enough, indeed
−gεabcAbi xc → ∂ixa =
1
r(δia − xixa) (11.35)
multiplying the rhs. with xi gives zero, and a sufficient condition for the lhs. to vanishis xiAbi = 0, this suggests:
with A ∝ 1/r guess what the magnetic field strength is!
Coulombic 1/r2, but there is color . . .
Bai =
1
2εijkF
ajk e.g. B3 = F12 = −F21 (11.38)
=1
2εijk(2∂jA
ak + gεabcA
biA
cj
)= . . . . . . (11.39)
= δia(− A(r)
r−A′(r)
)+ xixa
(− A(r)
r+A′(r) + gA2(r)
)(11.40)
11-7
what other symmetric (i, a)-structures could one have . . . , asymptotically:
Bai → −
xi
gr2xa (11.41)
indeed far field of a point charge (no dipole or higher components) with qm = −1/g inEq. (11.4) and color along the scalar field
the magnetic charge seems to be related to the degree of n like:
−qmg = deg n ∈ Z (11.42)
up to a factor 2 this is the same quantisation of electric and magnetic charges as forDirac’s monopole
here no singularity: ϕ(origin) = 0, i.e. in false vacuum, and A(origin) = 0 see below
that this holds can be shown with ’t Hooft’s field strength tensor:
Gµν := F aµνna − 1
gεabc n
aDµnbDνn
c (11.43)
= ∂µ(Aaνna)− ∂ν(Aaµn
a)− 1
gεabc n
a∂µnb∂νn
c (11.44)
note that the first line contains gauve invariant terms, whereas the second does not
for the ‘radial gauge’ we employed in the ansatz the terms Dµnb and Aaνn
a vanish andlooking at (µ, ν) = (j, k) and multiplying by εijk/2 and one gets Ba
i na in the first line
and −4π/g times the density of deg n in the second, thus Φ ∼ deg n
Bogomolnyi bound: only in the BPS limit [Bogomolnyi-Prasad-Sommerfield], where thepotential vanishes, λ→ 0, but its consequence |ϕ| → v is kept
energy:
E =
∫d3x tr
[( ~Dϕ)2 + ~B2
]=
∫d3x tr
[( ~Dϕ∓ ~B)2 ± 2 ~B ~Dϕ
](11.45)
≥ 2∣∣∫ d3x tr( ~B ~Dϕ)
∣∣ = 2∣∣∫ d3x tr( ~D( ~Bϕ)− ( ~D ~B)ϕ)
∣∣ (11.46)
= 2∣∣∫ d3x ~∂ tr( ~Bϕ)
∣∣ = 2∣∣∫S2∞
d2~σ tr( ~Bϕ)∣∣ = 2v
∣∣∫S2∞
d2~σ tr( ~Bn)∣∣ (11.47)
= v∣∣∫S2∞
d2~σ ~Bana∣∣ = 4πv |qm| (11.48)
where we used the Leibniz rule for the covariant derivative (in the adjoint) and later theBianchi identity Dµε
µνρσFρσ = 0 (from writing the field strength in terms of a gaugefield, as in ED without monopoles) with ν = 0 and that the trace of a commutatorvanishes
the monopole solutions obey the first order equation ~Dϕ = ± ~B, which is well-studiedsince integrable
11-8
what is so special about the BPS limit? mass of the Higgs vanishes ⇒ only algebraicdecay “long-range”, actually Coulombic (see below)
for (anti)monopole combination this yields a second interaction of same nature as theelectromagnetic one, in some cases it compensates the former
for unit charge, the solution for ϕ(r) and A(r) is known in terms of hyperbolic functions
with ξ = gvr:
ϕa =xa
g
H(ξ)
rH(ξ) = ξ coth ξ − 1→
O(ξ2) for ξ → 0
ξ − 1 for ξ →∞(11.49)
Aai =εaib x
b
g
1−K(ξ)
rK(ξ) =
ξ
sinh ξ→
1 +O(ξ2) for ξ → 0
exp(−ξ) for ξ →∞(11.50)
and has the expected asymptotics A→ exp(−evr) (perp. to ϕ, the A along ϕ is massless)and ϕ→ v +O(1/r)
can one abelianise the monopole?
to extract the particle content of a gauge-Higgs model, one typically gauges the field ϕin some fixed direction (see Sec. 10.1.2), say ϕ ∝ σ3; ‘unitary gauge’
this is impossible for the ’t Hooft-Polyakov monopole, as it would change the windingnumber of n from 1 to 0: one cannot comb the hedgehog
let’s look back at the ’t Hooft field strength tensor: the first line is gauge invariant andstill provides the Coulombic εijkGjk ∝ xi/r2 on an asymptotic two-sphere
let’s gauge na = (0, 0, 1)T on a large cap of that sphere, such that the winding, e.g.na = (0, 0,−1)T , is pushed into the complementary cap
then the third term in the second line vanishes almost everywhere, and the first twoterms are written in terms of A3
i , this field can be shown to be the gauge field of theDirac monopole (up to a factor 2) being ok on that certain cap and providing theCoulomb field
however, on the complement the abelian field aν = Aaνna (which is regular since ϕ stays
away from zero asymptotically) has to give the opposite magnetic field (see the Stokesargument (11.2)), e.g. pushing that cap to be the total sphere one sees the Dirac string
thus the total magnetic charge resides in the winding of n
there are no such monopoles in the electroweak sector of the Standard model
the breaking is SU(2)isospin×U(1)Y → U(1)electromagn, but the latter U(1) is embeddednontrivially as reflected by the Weinberg angle, see Ryder’s textbook for more details
in Grand Unified Theories the Standard model is embedded in a larger group like SU(5)and monopoles are in principle possible, but have never been detected
note that from BPS the mass is proportional to vg ∼
mAg2
, where mA ∼ gv is the mass
of the massive gauge boson (‘W’), and thus monopoles are even heavier than the latter(if the coupling is mall)
11-9
12 Instantons
pure gauge = Yang-Mills theory in 4d with Euclidean time
action density in various forms:
S =
∫d4xL , L =
1
2trFµνFµν = tr( ~E,2 + ~B2) = trF ∧ ∗F (12.1)
=1
2( ~E2
a + ~B2a) , if F = FaTa with trTaTb =
1
2δab (12.2)
in the second line we expanded in generators of the Lie algebra of say an SU(N) group,a = 1, . . . , N
without ∗ in the first line this would be the antisymmetric product FµνεµνρσFρσ, see Q below
for the field strength in terms of Aµ (and the coupling) and gauge transformations see Sec. 11.4
12.1 Selfduality and topology
dual field strength:
Fµν =1
2εµνρσFρσ, forms: F = ∗F (12.3)
interchanges E and B, e.g.:
E1 = F01 = F23 = B1 (12.4)
and has two more trivial, but useful properties:
˜F = F , (Fµν)2 = (Fµν)2 (12.5)
Bogomolnyi bound:
S =
∫d4x
(1
4tr(Fµν ∓ Fµν)2 ± 1
2FµνFµν
)(12.6)
≥ 8π2
g2|Q| (12.7)
where:
Q =
∫d4x
g2
16π2trFµνFµν︸ ︷︷ ︸
∼ tr ~E ~B ∼ trF ∧ F
∈ Z (12.8)
is the instanton number or topological charge
equality for (anti) selfdual field strength Fµν = ±Fµν , see below
12-1
Chern-Simons current:
g2
16π2trFµνFµν = ∂µKµ where Kµ =
g2
16π2εµνρσ
(Aaν∂ρA
aσ +
g
3εabcA
aνA
bρA
cσ
)(12.9)
note that Kµ is not a gauge invarirant quantity
thus the instanton number is determined by the behavior at the boundary, what is it?
finite action: Fµνr→∞−→ 0, where r =
√x2µ is the 4d radius
A = 0 or better a gauge transformations thereof, ‘pure gauge’:
second order e.o.m.s DµFµν = 0 (without sources, cf. Maxwell eq.s in Sec. 11.1) followfrom the Bianchi identity DµεµνρσFρσ ∼ DµFµν = 0 following from F (A)
from now on specialise to SU(2) for a while
spherical solutions: spherically symmetric pure gauge
Ω =1
r
(x012 + ixiσi) ∈ SU(2) (12.13)
(every SU(2) element exp(inaσa/2) can be written in this way)
unit instanton number guaranteed
with Ω−1 = Ω† = (x012 − ixiσi)/r one gets from (12.10):
Aaµ →1
gηaµν
2xνr2
(12.14)
12-2
with the ‘t Hooft tensor
ηa0ν = −δaν = −ην0 , ηaij = εija (12.15)
again mixing space and color
ansatz in the whole R4 (so far boundary)
Aaµ =1
gηaµν2xν ·A(r) ⇒ A(r) =
1
r2 + ρ2for any ρ ∈ (0,∞) (12.16)
F aµν = −1
gηaµν
4ρ2
(r2 + ρ2)2(12.17)
trF 2µν =
#
g2
ρ4
(r2 + ρ2)4(12.18)
‘instanton in regular gauge’: [Belavin, Polyakov, Schwartz, Tyupkin 1975]
– Q = 1
– A(r)→ 1r2Xand regular at origin
– F selfdual, since η is
– action density has the expected 1/g2 dependence (Bog. bound S ≥ 8π2/g2 · 1)
and is localised in space and time (event) with ρ a size parameter (such that integralremains one)
the latter is rooted in the fact that the theory has no dimensionful parameter(coupling dim.less in 4d) just like CP (N − 1) models in 2d
– this and the location of the instanton gives 5 moduli
– F → 1r4
faster than 1r2
expected from A
multipole expansion in 4d: no monopole, F → 1r4
amounts to a dipole field
anti-instanton Ω(Q=−1) = Ω−1
in the calculation of the Chern-Simons current tr (∂Ω−1 ·Ω)3 use ∂Ω−1 ·Ω = −Ω−1 ·∂Ωtr=
−∂Ω · Ω−1 = − old term, i..e three minus signs)
‘instanton in singular gauge’
– same action density, but F and A differ by a gauge transformation (that is singularat origin and infinity)
Aaµ =1
gη aµν2xν ·
ρ2
r2(r2 + ρ2)→ 1
r3(η similar to η) (12.19)
F → 1
r4(12.20)
– A decays stronger (more natural given the dipole F ), such that zero winding num-ber at the infinite sphere
(recall that the Chern-Simons current is not gauge invariant)
12-3
– the winding now resides in a small S3 around the origin, where A is singular
– when one compactifies space to an S4 , these two gauge fields are valid on northernand southern patch, as for the Dirac monopole
and on the ‘equator’ (fixed r) are related by a gauge transformation (x012 ±ixaσa)/r that carries the winding and is singular at the origin and infinity
the topology on e.g. the 4-torus is different
solutions of higher charge: ansatz [Corrigan, Fairlie, ’t Hooft, Wilczek]
Q lumps of top. = action density at locations yk with sizes ρk, when well separated,otherwise nonlinear overlap effects
5Q moduli, but 3Q relative color orientations are missing (all have the same ηaµν)
dimension of moduli space is known to be 8Q − 3 (the −3 come from global gaugerotations)
in general gauge groups 4NcQ moduli (looks like Nc constituents per charge and their4 space time locations)
Q = 2 is known completely thanks to a similar ansatz (Jackiw, Nohl, Rebbi)
Atiyah-Drinfeld-Hitchin-Manin formalism
all (!) instantons from a nonlinear algebraic problem (not simple)
on S1×R3 (nonzero temperature!) one can generate instantons by chains of R4 instan-tons along x0 = infinite sum in Π with locations shifted by (β,~0) [Harrington, Shepard1978; later generalised by Kraan, van Baal and Lee, Lu 1998]
in the large ρ-limit and after some gauge transformation this becomes the ’t Hooft-Polyakov monopole(!) [Rossi; in the generalisation even N such monopoles]
what is the relation? in this limit a static configuration: Ei = ∂iA0− ig[Ai, A0] = DiA0
call A0 = Φ (commutator thus in the adjoint representation)
action per length: S =
∫d3x tr
((DiΦ)2 +B2
i
)is that of the gauge-Higgs model without
V , i.e. in BPS limit
12-4
in fact the selfduality becomes DiΦ = ±Bi
12.3 Fermions in solitonic backgrounds III
exist chiral zero modes and an index theorem nr − nl = Q
usual steps: massless fermions couple via covariant derivative ⇒ γ5 anticommutes andon zero modes commutes with Dirac operator ⇒ γ5 can be chosen diagonal with ±12:eigenmodes with γ5 = +1 are those with upper two components etc. ⇒ for Q = ±1instantons only one sector has a normalisable zero mode⇒ ψ consists of a color structuretimes ψ(r) = ρ/(r2 + ρ2)3/2 ⇒ localised with a slightly different decay
consequence: chiral condensate
semiclassical (saddle point) approximation to the path integral does not consist of justone instanton in the whole universe, but mainly of instanton-antiinstanton configura-tions = approximate saddles – as in the energy split derivation in Sec. 5 – with constantdensity
eigenvalues perturbed; if instanton gas dilute, then still near zero modes ⇒ nonzerodensity ρ(λ = 0)
Banks-Casher relation: ρ(0) ∝ 〈ψψ〉
instantons induce chiral symmetry breaking
mechanism for any object that supports zero modes
confinement, the other prominent nonperturbative feature of non-abelian gauge theories,remains a problem from instantons; it is not understood from first principles
12.4 Tunnelling picture, spectral flow and axial anomaly
solutions in Euclidean time represent tunnelling between energy minima in real time (see thevery beginning of the lecture)
Weyl gauge A0 = 0, Minkowski space L = kinetic - potential:
L =1
2
((∂0A
ai )
2 − (Bai )2), Πa
i =∂L
∂(∂0Aai )= ∂0A
ai = Eai , H=
∫d3x
1
2
((Πa
i )2 + (Ba
i )2)
(12.23)
H is a conserved quantity, thus all quantities are functions of ~x
vacua: Ai = 0 or better pure gauge Ai = − ig (∂iω)ω−1
(both E = Π and B are gauge invariants and thus still vanish)
ω’ that have the same limit for |~x| → ∞ can be see seen as mappings from R3 ∪ ∞ ∼=S3 → SU(N) and thus have a winding number
12-5
it is again given by the Chern-Simons current Kµ and called the Chern-Simons number:
⇒ in Yang-Mills theories there are infinitely many vacua of vanishing energy, just likein the Sine-Gordon model
they are labeled by the integer CS( ~A)
in between the vacua there are energy barriers
gauge transformations with a winding are also called ‘large gauge transformations’
tunneling between this vacua: Euclidean time as in kink, from t = −∞ to t =∞
note that CS is not concerved but changes by FF , see Eq. (12.9)
CS(t = +∞)− CS(t = −∞) =
∫ ∞∞dx0 ∂0CS =
∫ ∞∞d4x ∂0K0 (12.25)
=
∫ ∞∞d4x ∂µKµ −O(Ki at |~x| =∞) (12.26)
=
∫d4x
g2
16π2trFµνFµν = Q (12.27)
an instanton of charge Q changes the CS-number of the vacuum by Q
it is of nonzero energy in between (the analogue of visiting the false vacuum), by havinga bump of FF = action density at finite t
one big difference to QM though: decay is not exponential (no dim.full parameter likemass) and thus superposition is much less well-defined than for multiple kinks in theτ -chain in Sec. 5.2
space-time cylinder: distribute Q given by a boundary term over the three parts of theboundary, where the lateral surface [Mantelflache] does not contribute since A0 = 0
consequence for fermion spectra:
Dirac-Hamiltonian
H(x0)D = −γ0γiD
(x0)i ( ~A) (12.28)
where the time-dependence comes from the fact that ~A(x0, ~x)
at x0 = ±∞ we have vacua related by a large gauge transformation and thus HD hasthe same spectrum
in between that is not necessarily so, crossings of E = 0 can appear, after which theentire spectrum must rearrange (shifting rooms in an infinite hotel): ‘spectral flow’
such crossing can be shown to be related to the spectrum of the 4d Dirac operator andvia the index theorem to the topological charge Q
12-6
classical argument so far, in the quantum theory one has to use determinants of thesespectra and therefore has to cut them off in one way or another =regularisation
when viewed with cut-offs, an instanton creates a (left-handed) particle and removes a(right-handed) anti-particle and thus induces a change ∆Q5 = 1− (−1) = 2 – in general2Q – where Q5 is the charge of the axial current
j5µ = ψγµγ5ψ (12.29)
it is classically conserved, ∂µj5µ = 0, but anomalous in the quantum theory (from
regularisation):
∂µj5µ =
2
16π2trFαβFαβ axial anomaly (12.30)
the relation ∆Q5 = 2Q derived above is just the integrated version of the 0th component
in this setting simply a consequence of the number of crossings in the spectral flow(‘infrared-ultraviolet connection’) induced by the topological charge and thus indepen-dent of small deformations
“understandable why the anomaly is robust, namely fully determined by the lowestorder result in perturbation theory” [Pierre van Baal in hep-th/0309008]
12-7
13 Appendix
13.1 Uniqueness of solutions of (ordinary) differential equations
differential equation (no explicit time-dependence assumed) and boundary condition
x(t) = f(x(t)) , x(t0) = x0 (13.1)
def.: a function f is Lipschitz continuous (“cannot change arbitrarily fast”/“bound onsecant lines’ slope”) iff
|f(x1)− f(x2)| ≤ #|x1 − x2| ∀x1, x2 ∈ R (13.2)
(one constant # for all pairs x1,2)
Theorem [Picard/Lindelof/Cauchy/Lipschitz]:
if f in the problem (13.1) is Lipschitz continuous in x, then this problem has a uniquesolution x(t) in the interval [t0 − ε, t0 + ε] for some ε > 0
application to the QM soliton moving in V = −V and thus obeying:
x2 = 2(E − V ) = 2(E + V ) , V = a(1− cos 2π
x
v
)≥ 0 (13.3)
for the former see (2.7), but we kept a nonzero energy, so better (2.6)
classical solutions, so E ≥ 0 and E + V ≥ 0 X
specialize to
x = +√
2(E + V ) , i.e. f(x) =
√2(E + a(1− cos 2π
x
v)) (13.4)
which amounts to a particle moving forward only
f is indeed Lipschitz continuous in x
cases
– single (anti)soliton: x(t0) = ±v/2, where V is minimal ⇒ unique solution
lowest energy: stays, higher ones: oscillations in the valley, just the right energy:reaches the hill at infinite future (and came from the previous hill at infinite past)
– single (anti)soliton defined by x(t = −∞) = 0 and x(t =∞) = ±v
boundary condition at a nonfinite time ⇒ theorem does not apply ⇒ possibilityto have a family of solution with a free time parameter t0
– multisoliton: intermediately passing a hill in V , i.e. x(t0) = Zv for a finite time t0⇒ unique solution
energy such that no kinetic energy left: unique solution staying there, higher en-ergies: particle moves to infinity
– soliton in field theory with both t- and x-dependence ⇒ theorem doesn’t apply
13-1
13.2 A soliton in NLSE
take the normal form of the nonlinear Schrodinger equation for a complex ψ(t, x):
i∂tψ + ∂2xψ + σ|ψ|2ψ = 0 σ = ±1 (13.5)
other forms can be brought to this form by rescaling ψ, t and x
(with an ambiguity from rescaling solutions of the normal form itself, the next item)
rescaling a solution ψ0(t, x):
ψ(t, x) = Aψ0(A2t, Ax) , A ∈ R (13.6)
is another solution (dimensions are screwed as [∂t] = [∂2x])
boosting a solution:
ψ(t, x) = ψ0(t, x− vt) exp(iv
2(x− v
2t)), v ∈ R (13.7)
is another solution, whose absolute value does not feel the extra phase and thus moveswith a constant velocity v
plane wave solutions in time:
ψ(t) = ±√σω exp(iωt) , where sign(ω) = σ (13.8)
note that amplitude and frequency are coupled in this nonlinear system
(and that ω plays the role of the rescaling A from above)
towards soliton solutions make an ansatz:
ψ(t, x) = exp(iωt)η(x) (13.9)
with say a real η(x), then it must obey
η′′(x) = −ση3(x) + ωη(x) (13.10)
which is a mechanical problem (replace η(x) by x(t)) in a potential
V (η) =σ
4η4 − ω
2η2 (13.11)
perhaps the most interesting case is the mexican hat potential σ = +1, ω < 0, whichallows for a stable (bounded) motion; with energy conservation
(η′)2
2= E − V (η) (13.12)
perhaps the most interesting case is E = 0, for which η(x) comes from/arrives at η = 0in the infinite ‘past’/‘future’ x → ∓∞, and bounces off at ±
√2ω at some space point
x0 (the collective coordinate of this solution), η(x) is thus a hump function around x0
13-2
the exact solution is
η(x) =
√2ω
cosh(√ω(x− x0)
) (13.13)
(again, ω plays the role of the rescaling A from above)
it is called ‘bright soliton’ (since it approaches η = 0 asymptotically)
other solutions are possible: dark solutions in σ = −1, ω > 0 (for a specific energy)
and solutions periodic in x for other values of the energy
(also interpolating to η(x)=const. giving back the plane wave in t above)
