Topology and Fermi liquids: the Kondo lattice and FL* - Subir...

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

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


�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.


�

�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.


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*


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

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

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

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

Place FL* on a torus:

obtain “topological” states nearly

degenerate with quasiparticle

states: number of dimers

crossing red line is conserved

modulo 2


= (|��i � |��i) /

�2

= (|� �i + |� �i) /

�2

2

FL*






modulo 2


= (|��i � |��i) /

�2

= (|� �i + |� �i) /

�2

0

FL*






modulo 2


= (|��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*

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