Topology and Fermi liquids: the Kondo lattice and FL*
Subir Sachdev
Department of Physics, Harvard University,
Cambridge, Massachusetts, 02138, USA and
Perimeter Institute for Theoretical Physics,
Waterloo, Ontario N2L 2Y5, Canada
(Dated: April 8, 2016)
Abstract
We describe the Kondo model for magnetic impurities in metals. Then we discuss the Kondo lattice
model, appropriate for the physics of heavy fermion compounds, and the manner in which it realizes the
conventional Fermi liquids (FL) phase with heavy quasiparticles. An unconventional metal, FL*, is also
argued to be possible, where topological order (as described by a Chern Simons theory) allows a Fermi
surface enclosing a volume distinct from the Luttinger value. The FL* state can also appear in single
band models appropriate for the copper oxide superconductors.
1
18
I. MOMENTUM BALANCE IN Z2-FL*
We discussed in Lecture 1 the intimate connection between global U(1) symmetries and the
Luttinger relation between the size of the Fermi surface and the density of particles. The FL*
state clearly violates this relation, as the volume enclosed by the Fermi surface only counts the
conduction electrons, c, but not the d electrons. Here, we will revisit the ‘topological’ argument
for the Fermi surface volume, and show that the topological degeneracy of the spin liquid of the d
electrons on the torus will provide the missing ingredient needed to modify the Luttinger count.
This result reinforces the conclusion that the FL* saddle point of the large N theory is not merely
an artifact of the large N limit, but a state that is non-perturbatively stable.
The models we consider have two distinct U(1) symmetries that are important for our purposes:
the symmetries associated with conservation of total spin about the z axis, S
z
, and the total
electron number. Let us gauge these global symmetries by coupling them to external gauge fields
A
s
µ
and A
e
µ
, respectively. Then, proceeding as in Lecture 1, we place the system on a torus,
and adiabatically thread fluxes of these gauge fields through one of the cycles of the torus. We
characterize the system by low energy excitations which will respond to such a flux. Along with
the c
~
k�
quasiparticles around the Fermi surface, we also include the global topological excitations
of a Z2 spin liquid of the d electrons which are described by a Chern-Simons theory. We consider
the action
S =
Zd⌧
Zd
2k
4⇡2
X
�=",#
c
†~
k�
✓@
@⌧
� i
2�A
s
⌧
� iA
e
⌧
+ "(~k � �
~
A
s
/2 � ~
A
e)
◆c
~
k�
+
Zd
3x
i
⇡
✏
µ⌫�
a
µ
@
⌫
b
�
+i
2⇡✏
µ⌫�
a
µ
@
⌫
A
s
�
�. (1)
The Chern-Simons component is that introduced in Lecture 6 for the Z2 spin liquid with two
emergent U(1) gauge fields, aµ
and b
µ
. Also, as shown in Lecture 6, the a
µ
gauge fields couples to
the external As
µ
gauge field because the flux of aµ
is the spin current (the associated charge density
was labeled Q in Lecture 6).
Let us now thread a flux which couples only to the up-spin electrons. So we choose As
µ
= 2Ae
µ
⌘A
µ
. We work in real time, and thread a flux along the x-cycle of the torus. So we have
A
x
=�(t)
L
x
(2)
where �(t) is a function which increases slowly from 0 to 2⇡. The quasiparticles will respond to
this as discussed in Lecture 1. So we focus only on the response of the topological sector described
by the Chern-Simons theory. The A
x
gauge field couples only to a
y
, and so as in Lecture 9, we
parameterize
a
y
=✓
y
L
y
. (3)
19
Lx
Lyˆ
Wy(i)
ˆ
Wy(i + x)
FIG. 1. The torus geometry of size L
x
⇥ L
y
. Wilson loops encircling the torus along the y direction on
neighboring sites i and i + x are related by the flux through the shaded area.
Then the time evolution operator of the flux-threading operation can be written as
U = exp
✓i
2⇡
Zdt ✓
y
d�
dt
◆= e
i✓
y ⌘ W
y
(4)
So the time evolution operator is simply the Wilson loop operator Wy
. If the state of the system
before the flux-threading was |Gi, then the state after the flux threading will be W
y
|Gi.The key ingredient in the momentum balance argument for the Fermi surface size was a compu-
tation in the change in size of the Fermi surface before and after the flux threading. For that, we
need here a computation of the action of the lattice translational operator Tx
on |Gi and W
y
|Gi.In particular, to compute the change in momentum, we need the value of T�1
x
W
y
T
x
. From Fig. 1 we
see that T�1x
W
y
T
x
di↵ers from W
y
by a factor given by the aµ
flux through a strip of width equal to
a single lattice spacing, which encircles the torus along the y direction. The a
µ
flux through such
a narrow strip is not a continuum property that is described by the Chern-Simons theory. Rather,
we have to refer back to the lattice model from which the Chern-Simons topological description
was derived. Now recall in Lecture 6 we had obtained two classes of Z2 spin liquids, even or odd,
corresponding to whether the average boson density per site, Q, was integer or half-integer. In
Lecture 7 on antiferromagnets, these two classes correspond to systems in where the spin per unit
cell, S, is integer or half-integer. And for the half-integer cases of odd Z2 spin liquids, there was
a background ⇡ flux of aµ
per plaquette, while there was zero a
µ
flux for the integer cases. So we
have the important result that for spin liquids with half-integer spin per unit cell
W
y
T
x
= (�1)Ly
W
y
T
x
, (5)
while for integer spin W
y
and T
x
commute. We will only consider the half-integer spin case from
20
now on, as that corresponds to the Kondo lattice model.
The change in momentum from the flux threading is easily computed. The initial momentum
P
xi
is defined by
T
x
|Gi = e
iP
xi |Gi (6)
while for the final momentum P
xf
T
x
W
y
|Gi = e
iP
xf
W
y
|Gi . (7)
From (5) we therefore conclude
�P
x
= ⇡L
y
(mod 2⇡). (8)
Let us rewrite this result as
�P
x
=
✓2⇡
L
x
◆n
d
L
x
L
y
(mod 2⇡), (9)
where, comparing to the Lecture 1, we interpret nd
as the e↵ective density of up-spin particles per
site associated with the flux-threading operation in the Chern-Simons theory. And the value of nd
is
n
d
=1
2. (10)
This is precisely the density of up-spin electrons from the d band that would contribute to a
Fermi surface volume, had they not been in a spin-liquid sector described by the Chern-Simons
theory. So we have shown that the ‘small’ Fermi surface enclosing a volume given by the density
of conduction electrons alone is non-perturbatively compatible with the topological momentum
balance argument.
21
Arguments for the Fermi surface volume of the FL phase
Single ion Kondo effect implies at low energiesKJ →∞
( )( ) ( )
Fermi surface volume density of holes mod 2
1 1 mod 2c cn n
= −
= − − = +
Fermi liquid of S=1/2 holes with hard-core repulsion
( )† † † † 0i i i ic f c f↑ ↓ ↓ ↑
− † 0 , =1/2 holeif S↓
Arguments for the Fermi surface volume of the FL phase
Alternatively:
( )( )† † †
ij i j i i i i f fi fi fi fii j i
cT f
H t c c Vc f Vf c n n Un n
n n nσ σ σ σ σ σ ε
↑ ↓ ↑ ↓<
= + + + + + +
= +
∑ ∑ !
Formulate Kondo lattice as the large U limit of the Anderson model
( )For small , Fermi surface volume = mod 2.
This is adiabatically connected to the large limit where 1f c
f
U n n
U n
+
=
Topology and the Fermi surface size
Lx
Ly
Φ
We take N particles, each with charge Q, on a Lx ⇥ Ly lattice on a torus.
We pierce flux � = hc/Q through a hole of the torus.
An exact computation shows that the change in crystal momentum of the
many-body state due to flux piercing is
Pxf � Pxi =
2�N
Lx(mod 2�) = 2��Ly(mod 2�)
where � = N/(LxLy) is the density.
M. Oshikawa, PRL 84, 3370 (2000)
A. Paramekanti and A. Vishwanath,
PRB 70, 245118 (2004)
Proof of
Pxf � Pxi =
2�N
Lx(mod 2�) = 2��Ly(mod 2�).
The initial and final Hamiltonians are related by a gauge transformation
UGHfU�1G = Hi , UG = exp
�i
2�
Lx
X
i
xini
�.
while the wavefunction evolves from |�ii to UT |�ii, where UT is the time evolution
operator. We want to work in a fixed gauge in which the initial and final Hamilto-
nians are the same: in this gauge, the final state is |�f i = UG UT |�ii. Let
ˆ
Tx be
the lattice translation operator. Then we can establish the above result using the
definitions
ˆ
Tx |�ii = e
�iPxi |�ii ,
ˆ
Tx |�f i = e
�iPxf |�f i ,
and the easily established properties
ˆ
Tx UT = UTˆ
Tx ,
ˆ
Tx UG = exp
✓�i2�
N
Lx
◆UG
ˆ
Tx
Topology and the Fermi surface size
�Px = 2��Ly(mod 2�) , �Py = 2��Lx(mod 2�)
Now we compute the momentum balance assuming that the only low energy exci-
tations are quasiparticles near the Fermi surface, and these react like free particles
to a su�ciently slow flux insertion. So each quasiparticle picks up a momentum
�
�p ⌘ (2�/Lx, 0), and then we can write (with �np the quasiparticle density excited
by the flux insertion)
�Px =
X
p
�nppx.
Now �np = ±1 on a shell of thickness
�
�p · d�
Sp on the Fermi surface (where
�
Sp is an
area element on the Fermi surface). So we can write the above as a surface integral
�Px =
�
FSpx
✓LxLy
4�
2
◆�
�p · d
�
Sp
= (
�
�p · x)
Z
FV
✓LxLy
4�
2
◆dV
by the divergence theorem. So
�Px =
✓2�
Lx
◆LxLy
4�
2VFS , �Py =
✓2�
Ly
◆LxLy
4�
2VFS
where VFS is the volume of the Fermi surface. So, although the quasiparticles
are only defined near the Fermi surface, by using Gauss’s Law on the momentum
acquired by quasiparticles near the Fermi surface, we have converted the answer to
an integral over the volume enclosed by the Fermi surface.
Now we equate these values to those obtained above, and obtain
N � LxLyVFS
4�
2= Lxmx , N � LxLy
VFS
4�
2= Lymy
for some integers mx, my. By choosing Lx, Ly mutually prime integers we can now
show
� =
N
LxLy=
VFS
4�
2+ m
for some integer m: this is Luttinger’s theorem.
Topology and the Fermi surface size
�
�p
�Px = 2��Ly(mod 2�) , �Py = 2��Lx(mod 2�)
Now we compute the momentum balance assuming that the only low energy exci-
tations are quasiparticles near the Fermi surface, and these react like free particles
to a su�ciently slow flux insertion. So each quasiparticle picks up a momentum
�
�p ⌘ (2�/Lx, 0), and then we can write (with �np the quasiparticle density excited
by the flux insertion)
�Px =
X
p
�nppx.
Now �np = ±1 on a shell of thickness
�
�p · d�
Sp on the Fermi surface (where
�
Sp is an
area element on the Fermi surface). So we can write the above as a surface integral
�Px =
�
FSpx
✓LxLy
4�
2
◆�
�p · d
�
Sp
= (
�
�p · x)
Z
FV
✓LxLy
4�
2
◆dV
by the divergence theorem. So
�Px =
✓2�
Lx
◆LxLy
4�
2VFS , �Py =
✓2�
Ly
◆LxLy
4�
2VFS
where VFS is the volume of the Fermi surface. So, although the quasiparticles
are only defined near the Fermi surface, by using Gauss’s Law on the momentum
acquired by quasiparticles near the Fermi surface, we have converted the answer to
an integral over the volume enclosed by the Fermi surface.
Now we equate these values to those obtained above, and obtain
N � LxLyVFS
4�
2= Lxmx , N � LxLy
VFS
4�
2= Lymy
for some integers mx, my. By choosing Lx, Ly mutually prime integers we can now
show
� =
N
LxLy=
VFS
4�
2+ m
for some integer m: this is Luttinger’s theorem.
Topology and the Fermi surface size
�
�p
�Px = 2��Ly(mod 2�) , �Py = 2��Lx(mod 2�)
Now we compute the momentum balance assuming that the only low energy exci-
tations are quasiparticles near the Fermi surface, and these react like free particles
to a su�ciently slow flux insertion. So each quasiparticle picks up a momentum
�
�p ⌘ (2�/Lx, 0), and then we can write (with �np the quasiparticle density excited
by the flux insertion)
�Px =
X
p
�nppx.
Now �np = ±1 on a shell of thickness
�
�p · d�
Sp on the Fermi surface (where
�
Sp is an
area element on the Fermi surface). So we can write the above as a surface integral
�Px =
�
FSpx
✓LxLy
4�
2
◆�
�p · d
�
Sp
= (
�
�p · x)
Z
FV
✓LxLy
4�
2
◆dV
by the divergence theorem. So
�Px =
✓2�
Lx
◆LxLy
4�
2VFS , �Py =
✓2�
Ly
◆LxLy
4�
2VFS
where VFS is the volume of the Fermi surface. So, although the quasiparticles
are only defined near the Fermi surface, by using Gauss’s Law on the momentum
acquired by quasiparticles near the Fermi surface, we have converted the answer to
an integral over the volume enclosed by the Fermi surface.
Now we equate these values to those obtained above, and obtain
N � LxLyVFS
4�
2= Lxmx , N � LxLy
VFS
4�
2= Lymy
for some integers mx, my. Now choose Lx, Ly mutually prime integers; then
mxLx = myLy implies that mxLx = myLy = pLxLy for some integer p. Then
we obtain
� =
N
LxLy=
VFS
4�
2+ p.
This is Luttinger’s theorem.
Topology and the Fermi surface size
Lx
Ly
Φ
Topology and the Fermi surface size in FL*
T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004)
To obtain a di�erent Fermi surface size, we need low energy
excitations on a torus which are not composites of quasiparticles
around the Fermi surface. The degenerate ground states of a
Z2 spin liquid can provide the needed excitation, and lead to
a Z2-FL* state with a Fermi surface size of p, rather than the
Luttinger size of 1 + p.
Lx
Ly
Φ
Topology and the Fermi surface size in FL*
T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004)
The exact momentum transfers �Px = 2�(1 + p)Ly(mod2�)
and �Py = 2�(1+p)Lx(mod2�) due to flux piercing arise from
• A contribution 2�pLx,y from the small Fermi surface of
quasiparticles of size p.
• The remainder is made up by the topological sector: flux
insertion creates a “vison” in the hole of the torus.
Start with a spin liquid and then remove
electrons
= (|��i � |��i) /
�2
= (|��i � |��i) /
�2
Start with a spin liquid and then remove
electrons
= (|��i � |��i) /
�2
Start with a spin liquid and then remove
electrons
= (|��i � |��i) /
�2
Start with a spin liquid and then remove
electrons
= (|��i � |��i) /
�2
Start with a spin liquid and then remove
electrons
= (|��i � |��i) /
�2
Start with a spin liquid and then remove
electrons
= (|��i � |��i) /
�2
Start with a spin liquid and then remove
electrons
= (|��i � |��i) /
�2
A mobile charge +e, but
carrying no spin
= (|��i � |��i) /
�2
A mobile charge +e, but
carrying no spin
= (|��i � |��i) /
�2
S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987)
N. Read and B. Chakraborty, PRB 40, 7133 (1989)
Spin liquidwith density p of spinless, charge +e “holons”.
These can form a Fermi surface of size p, but
not of electrons
= (|��i � |��i) /
�2
S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987)
N. Read and B. Chakraborty, PRB 40, 7133 (1989)
Spin liquidwith density p of spinless, charge +e “holons”.
These can form a Fermi surface of size p, but
not of electrons
= (|��i � |��i) /
�2
S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987)
N. Read and B. Chakraborty, PRB 40, 7133 (1989)
Spin liquidwith density p of spinless, charge +e “holons”.
These can form a Fermi surface of size p, but
not of electrons
S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987)
N. Read and B. Chakraborty, PRB 40, 7133 (1989)
= (|��i � |��i) /
�2
Spin liquidwith density p of spinless, charge +e “holons”.
These can form a Fermi surface of size p, but
not of electrons
= (|��i � |��i) /
�2
= (|��i � |��i) /
�2
= (|��i � |��i) /
�2
= (|��i � |��i) /
�2
= (|��i � |��i) /
�2
= (|��i � |��i) /
�2
= (|��i � |��i) /
�2
= (|��i � |��i) /
�2
FL*
Mobile S=1/2, charge +e fermionic dimers: form
a Fermi surface of size p of electrons
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)
M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)
M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
FL*
Mobile S=1/2, charge +e fermionic dimers: form
a Fermi surface of size p of electrons
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)
M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
FL*
Mobile S=1/2, charge +e fermionic dimers: form
a Fermi surface of size p of electrons
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)
M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
FL*
Mobile S=1/2, charge +e fermionic dimers: form
a Fermi surface of size p of electrons
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)
M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
FL*
Mobile S=1/2, charge +e fermionic dimers: form
a Fermi surface of size p of electrons
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)
M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
FL*
Mobile S=1/2, charge +e fermionic dimers: form
a Fermi surface of size p of electrons
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)
M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
FL*
Mobile S=1/2, charge +e fermionic dimers: form
a Fermi surface of size p of electrons
R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, 235122 (2007)S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996)
M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
FL*
Mobile S=1/2, charge +e fermionic dimers: form
a Fermi surface of size p of electrons
Place FL* on a torus:
obtain “topological” states nearly
degenerate with quasiparticle
states: number of dimers
crossing red line is conserved
modulo 2
T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
2
FL*
Place FL* on a torus:
obtain “topological” states nearly
degenerate with quasiparticle
states: number of dimers
crossing red line is conserved
modulo 2
T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
0
FL*
Place FL* on a torus:
obtain “topological” states nearly
degenerate with quasiparticle
states: number of dimers
crossing red line is conserved
modulo 2
T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
0
FL*
Place FL* on a torus:
obtain “topological” states nearly
degenerate with quasiparticle
states: number of dimers
crossing red line is conserved
modulo 2
T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004)
= (|��i � |��i) /
�2
= (|� �i + |� �i) /
�2
2
FL*
We have described a metal with:
A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p Additional low energy quantum states on a torus not associated with quasiparticle excitations i.e. emergent gauge fields
FL*
We have described a metal with:
A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p Additional low energy quantum states on a torus not associated with quasiparticle excitations i.e. emergent gauge fields
There is a general and fundamental relationship between these two characteristics. Promising indications that such a metal describes the pseudogap of the cuprate supercondutors
FL*
66