Instanton constituents in sigma modelsand Yang-Mills theory at finite temperature
Falk BruckmannUniv. of Regensburg
Extreme QCD, North Carolina State, July 2008
PRL 100 (2008) 051602 [0707.0775]EPJ Spec.Top.152 (2007) 61-88 [0706.2269]
partly with: D. Nogradi, P. van Baal, E.-M. Ilgenfritz,B. Martemyanov, M. Müller-Preussker
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Part I: The O(3) sigma model
a scalar field in 2D . . .
S =
∫d2x
12(∂µφ
a)2 a = 1,2,3 : global O(3) symmetry
. . . with a constraint
φaφa = 1 (circumvent Derrick’s theorem)
nontrivial properties:
asymptotic freedomdynamical mass gaptopology and instantons
condensed matter physics and toy model for gauge theories
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Topology
finite action:
r →∞ : φa → const.
as a mapping:φ : R2 ∪ ∞ ' S2
x −→ S2c
winding number/degree: all such φ’s are characterized by an integer Q= how often S2
c is wrapped by S2x through φ
here:Q =
18π
∫d2x εµνεabcφ
a∂µφb∂νφ
c ∈ Z
topological quantum number = invariant under small deformations of φ
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Classical solutions
Bogomolnyi trick. . .
(∂µφa ± εµνεabcφ
b∂νφc)2 = (∂µφ
a)2 ± 2εµνεabcφa∂µφ
b∂νφc + (∂µφ
a)2
. . . and bound (integrated):
S ≥ 4π|Q|
where the equality holds iff
∂µφa = ∓ εµνεabcφ
b∂νφc ‘selfduality equations’
first order (instead of second order in eqns. of motion)
classical solutions:instantons = localised in both directions
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Complex structure
introduce complex coordinates both in space and color space:
x1,2 → z = x1 + ix2
φa → u =φ1 + iφ2
1− φ3N : φa = (0,0,1) u = ∞S : φa = (0,0,−1) u = 0
⇒ self-duality equations become Cauchy-Riemann conditions on u
⇒ any meromorphic function u(z) is a solution
topological charge: Q = number of zeroes or poles
topological charge density:
q(x) =1π
1(1 + |u|2)2
∣∣∣∣∂u∂z
∣∣∣∣2
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Charge 1 instantons• simplest functions:
u(z) = λz−z0
u(z) = z−z0λ
q(x) = 1π
λ2
(|z−z0|2+λ2)2
Belavin-Polyakov monopoleare Q = 1 instantons: location z0, size λ
1 pole and 1 zero to cover S2c , one of them at infinity
• both, pole and zero, at finite z:
u(z) =z − zI
z − zII
constituents at z = zI , zII? ; ‘instanton quarks’?!
NO! same profile q(x) as above ⇒ one lump
conjecture: 2 complex moduli per Q ; locations of 2 constituents?!Falk Bruckmann Instanton constituents 6 / 19
Finite temperature= one compact direction, say: Im z = x2 ∼ x2 + β, β = 1/kBT• instantons:
use that higher charge solutions = products
u(z) =Q∏
k=1
λ
z − z0,kQ poles
and infinitely many copies: z0,k ≡ z0 + k · iβ, k ∈ Z
• a regularized u(z) is:
u(z) =λ
exp((z − z0)2πβ )− 1
has residues λ at z = z0 + k · iβand charge 1 over S1 · R1
small λ large λ
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Boundary conditionsq(x) and action density invariant under global SO(3) rotations
an SO(2) subgroup: φ→
rotationwith ω
1
φ, u → e 2πiωu
• let the fields φ and u be periodic up to that SO(2) subgroup: FB ’07
u(z + iβ) = e 2πiωu(z) ω ∈ [0,1]
q(x) strictly periodic
• novel solution:
u(z) =e ω(z−z0)
2πβ · λ
exp((z − z0)2πβ )− 1
has residues e 2πiωkλ at z = z0 + k · iβ
⇒ ‘different orientation’ of the instanton copies⇒ nontrivial overlaps ⇒ instanton constituents
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Topological profilesln q(x): (z0 = 0, cut off below e−5)
λ = β λ = 10β λ = 100β
periodic
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ω = 1/3
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antiper.
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⇒ for large size λ: 2 lumps with action ω and ω = 1− ωFalk Bruckmann Instanton constituents 9 / 19
‘Dissociation’• rewrite:
u(z) =1
exp(−ω(z − z1)2πβ )− exp(ω(z − z2)
2πβ )
locations: z1 = z0 − β ln λ2πω , z2 = z0 + β ln λ
2πω
instanton size → constituent distance: z2 − z1 ∼ lnλconstituent size: fixed by β and ω
• really locations of topological lumps?YES: corrections of the second term at z = z1 are exp. small
• individual constituent:
u(z) = exp(ωz2πβ
) ω analogous
top. charge:
q(x) =πω2
β2 cosh2(ω Re z 2πβ )
(static) Q = ω
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• possible values for Q: 0,1, . . . ω,1 + ω, . . . 1− ω,2− ω, . . .
asympt. φ−∞ → φ+∞: N→ N, S→ S N→ S S→ Nconstituents alternate
• why instanton quarks not visible for zero temperature, i.e. on R2?
β →∞: constituents large and overlap!no other scale competing with their distance
• generalisations: CP(N) models FB et al. in progress
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46 N constituents as expected
• realisation in condensed matter:cylinder of ..?.. with quasi-periodic bc.s
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Fermionic zero modesgauge field description ⇒ couple fermions ⇒ zero modesphase-bc.s ψ(x0 + iβ) = e 2πiζψ(x0), evolution with ζ: FB et al. in progress
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Part II: Gauge theories
pure Yang-Mills theory in (Euclidean) 4D:
S =
∫12
tr F 2µν ≥ |Q| = |
∫12
tr Fµν Fµν |
dual field strength F aµν =
12εµνρσF a
ρσ (~Ea ~Ba)
integer Q: instanton number/topological charge
topology:
Aµr→∞→ iΩ−1∂µΩ . . . pure gauge
Q = deg(Ω : S3r→∞ → SU(N)) . . . winding number
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Instantons
(anti)selfdual: F aρσ = ±F a
µν first order, nonlinear
charge 1: axially symmetric ansatz and solution BPST
Aaµ = ηa
µν
2xν
x2 + ρ2 trF 2 =ρ4
(x2 + ρ2)4 ηaµν ∈ −1,0,1
size ρlocalized in space and timealgebraic decay, similar to O(3) instantons on R2
instanton liquid model from semiclassical path integral Shuryak
chiral symmetry breakingaxial anomalytopological susceptibilityconfinement?
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Finite temperature: Calorons• use higher charge solutions of same color orientation CFTW
⇒ first calorons Harrington-Shepard ’78
•most general calorons: need ADHM formalism and Nahm transform⇒ calorons of nontrivial holonomy Kraan,van Baal; Lee, Lu ’98
space-space plot ofaction density for SU(2),intermediate holonomy
⇒ 2 lumps, almost staticNc for gauge group SU(Nc), like quarks in baryons
⇒magnetic monopoles of opposite magnetic chargein fact dyons with same electric as magn. charge (selfdual)
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Role of the holonomyrelative gauge orientation of instanton copies in the ADHM constr.
⇒ Aµ periodic up to a gauge transformation e 2πiωσ3/2 (cf. O(3))
gauge theory: compensated by time-dependent transf. e 2πiωσ3x0/2
⇒ introduces an asymptotic gauge field A0
⇒ asymptotic Polyakov loop = holonomy
P(~x) ≡ P exp(
i∫ β
0dx0 A0
)→ e 2πiωσ3/2 ≡ P∞
‘environment’
acts like a Higgs field, in the group: vev ω, direction σ3
•monopoles have masses ω/β and ω/β, ω = 1− ω
• Aa=3µ : power law decay (massless ‘photon’),
• Aa=1,2µ : exponential decay (massive ‘W -bosons’)
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• Polyakov loop in the bulk: P(~x) = ±12 at the monopolesHiggs field vanishes = ‘false vacuum’necessary for top. reasons Ford et al.; Reinhardt; Jahn et al.
• index theorem valid Nye, Singer
localisation depending on bc.s:
ψ(x0 + iβ) = e2πiζψ(x0) (Aµ still periodic)
ζ ∈ −ω2 ,
ω2 incl. periodic: localised at monopole Garcia Perez et al.
ζ ∈ rest incl. antiperiodic: localised at antimonopole
a zero in their profiles at the ‘other’ monopole, topological FB
• calorons can be studied on the lattice by cooling Ilgenfritz et al. ’02, FB et al.
• physical relevance of P∞:conjecture: holonomy trP∞ deconfinement order param. 〈trP〉x
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Calorons and the dynamics of YM theories
• eff. potential at 1-loop: triv. holonomy favored! Gross, Pisarski, Jaffe; Weiss
⇒ overruled by caloron gas contribution: Diakonov et al.
⇒minima at P = ±12 become unstable for low enough temperature⇒ onset of confinement
• gas of calorons and anticalorons put on the lattice: Gerhold et al.
⇒ linearly rising interquark potential just for nontrivial holonomy!
• confinement from a gas of purely selfdual dyons Diakonov, Petrov
⇒ unphysical (top. charge builds up)⇒ nevertheless interesting physical effects
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Summary
sigma models in 2D and YM in 4D admit instantons
instanton constituents for S1 × R1 and S1 × R3 = finite T
with fractional charges, say ω and 1− ω in the lowest models
when in compact direction periodic up to a subgroup, say e 2πiω..
= subgroup of global and local symmetry
Yang-Mills theory:can be made periodic → holonomy P∞⇒ caloron constituents as building blocks of semiclass. models
at finite temperature: (de)confinement?!
sigma models:quasi-periodic bc.s stay⇒ spin chains? skyrmion lattices? Quantum Hall effect?
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