1. review of fermi liquid theory - harvard universityqpt.physics.harvard.edu/talks/natal15c.pdf ·...

1. Review of Fermi liquid theory Topological argument for the Luttinger theorem

2. Fractionalized Fermi liquid A Fermi liquid co-existing with topological order for the pseudogap metal

3. Quantum matter without quasiparticles(A) A mean-field model of a non-Fermi liquid, and

charged black holes(B) Field theory of a non-Fermi liquid (Ising-nematic

quantum critical point)(C) Theory of transport in strange metals: application to

the (less) strange metal in graphene

Quantum criticality of Ising-nematic ordering in a metal

x

yOccupied states

Empty states

A metal with a Fermi surfacewith full square lattice symmetry

x

y

Spontaneous elongation along y direction:Ising order parameter � < 0.


Spontaneous elongation along x direction:Ising order parameter � > 0.

x

yQuantum criticality of Ising-nematic ordering in a metal

Ising-nematic order parameter

� ⇠Z

d2k (cos kx

� cos ky

) c†k�

ck�

Measures spontaneous breaking of square latticepoint-group symmetry of underlying Hamiltonian

Spontaneous elongation along x direction:Ising order parameter � > 0.

x

yQuantum criticality of Ising-nematic ordering in a metal

x

y

Spontaneous elongation along y direction:Ising order parameter � < 0.


Pomeranchuk instability as a function of coupling �

��c

��⇥ = 0⇥�⇤ �= 0

or


T

Phase diagram as a function of T and �

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

TI-n


��c

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

TI-n Classicald=2 Isingcriticality



��c

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n


D=2+1 Ising

criticality ?


�

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n

D=2+1 Ising

criticality ?



�

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n

Strongly-coupled“non-Fermi liquid”

metal with no quasiparticles



�

T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n




Fermiliquid

Fermiliquid

Quantum critical


�

T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n



StrangeMetal


Fermiliquid

Fermiliquid


�

L = f†↵

✓@⌧ � r2

2m� µ

◆f↵

+ u f†↵f

†�f�f↵

The Fermi liquid

�� kF !

Occupied states

Empty states

The Fermi liquid: RG

FIG. 3: Alternative low energy formulation of Fermi liquid theory. We focus on an extended patch

of the Fermi surface, and expand in momenta about the point ~k0 on the Fermi surface. This yields

a theory of d-dimensional fermions in (7), with dispersion (14). The co-ordinate y represents the

d� 1 dimensions parallel to the Fermi surface.

(5) in one-dimensional. One benefit of (7) is now immediately evident: it has zero energy

excitations when

vFkx +

k

2y

2= 0, (8)

and so (8) defines the position of the Fermi surface, which is then part of the low energy

theory including its curvature. Note that (7) now includes an extended portion of the Fermi

surface; contrast that with (5), where the one-dimensional chiral fermion theory for each n̂

describes only a single point on the Fermi surface.

The gradient terms in (7) define a natural momentum space cuto↵, and associated scaling

limit. We will take such a limit at fixed ⇣, vF and . Notice that momenta in the x direction

scale as the square of the momenta in the y direction, and so we can choose v2Fk2x+

2k

4y < ⇤4.

Notice that as we reduce ⇤, we scale towards the single point ~k0 on the Fermi surface, as we

7

• Expand fermion kinetic energy at wavevectors about

~k0, bywriting f↵(~k0 + ~q) = ↵(~q)

L = f†↵

✓@⌧ � r2

2m� µ

◆f↵

+ u f†↵f

†�f�f↵


FIG. 3: Alternative low energy formulation of Fermi liquid theory. We focus on an extended patch

of the Fermi surface, and expand in momenta about the point ~k0 on the Fermi surface. This yields

a theory of d-dimensional fermions in (7), with dispersion (14). The co-ordinate y represents the

d� 1 dimensions parallel to the Fermi surface.

(5) in one-dimensional. One benefit of (7) is now immediately evident: it has zero energy

excitations when

vFkx +

k

2y

2= 0, (8)

and so (8) defines the position of the Fermi surface, which is then part of the low energy

theory including its curvature. Note that (7) now includes an extended portion of the Fermi

surface; contrast that with (5), where the one-dimensional chiral fermion theory for each n̂

describes only a single point on the Fermi surface.

The gradient terms in (7) define a natural momentum space cuto↵, and associated scaling

limit. We will take such a limit at fixed ⇣, vF and . Notice that momenta in the x direction

scale as the square of the momenta in the y direction, and so we can choose v2Fk2x+

2k

4y < ⇤4.

Notice that as we reduce ⇤, we scale towards the single point ~k0 on the Fermi surface, as we

7

L[ ↵

] = †↵

�@⌧

� i@x

� @2y

� ↵

+ u †↵

†�

�

↵

• Expand fermion kinetic energy at wavevectors about

~k0, bywriting f↵(~k0 + ~q) = ↵(~q)

L = f†↵

✓@⌧ � r2

2m� µ

◆f↵

+ u f†↵f

†�f�f↵


S[ ↵

] =

Zd

d�1y dx d⌧

h

†↵

�@

⌧

� i@

x

� @

2y

�

↵

+ u

†↵

†�

�

↵

i


S[ ↵

] =

Zd

d�1y dx d⌧

h

†↵

�@

⌧

� i@

x

� @

2y

�

↵

+ u

†↵

†�

�

↵

i

The kinetic energy is invariant under the rescaling x ! x/s,

y ! y/s

1/2, and ⌧ ! ⌧/s

z, provided z = 1 and

! s

(d+1)/4.

Then we find u ! us

(1�d)/2, and so we have the RG flow

du

d`

=

(1� d)

2

u

Interactions are irrelevant in d = 2 !

The fermion Green’s function to order u2has the form (upto logs)

G(~q,!) =A

! � qx

� q2y

+ ic!2

So the quasiparticle pole is sharp. And fermion momentum distribution

function n(~k) =Df†↵

(

~k)f↵

(

~k)Ehad the following form:


S[ ↵

] =

Zd

d�1y dx d⌧

h

†↵

�@

⌧

� i@

x

� @

2y

�

↵

+ u

†↵

†�

�

↵

i


S[ ↵

] =

Zd

d�1y dx d⌧

h

†↵

�@

⌧

� i@

x

� @

2y

�

↵

+ u

†↵

†�

�

↵

i

n(k)

kkF

A

The fermion Green’s function to order u2has the form (upto logs)

G(~q,!) =A

! � qx

� q2y

+ ic!2

So the quasiparticle pole is sharp. And fermion momentum distribution

function n(~k) =Df†↵

(

~k)f↵

(

~k)Ehad the following form:

• Fermi wavevector obeys the Luttinger relation kdF ⇠ Q, the

fermion density

• Sharp particle and hole of excitations near the Fermi surface

with energy ! ⇠ |q|z, with dynamic exponent z = 1.

• The phase space density of fermions is e↵ectively one-dimensional,

so the entropy density S ⇠ T . It is useful to write this is as S ⇠T (d�✓)/z

, with violation of hyperscaling exponent ✓ = d� 1.

The Fermi liquid

L = f†✓@⌧ � r2

2m� µ

◆f

+ 4 Fermi terms�� kF !

Occupied states

Empty states


fermion density






The Fermi liquid

L = f†✓@⌧ � r2

2m� µ

◆f

+ 4 Fermi terms

⇥| q|�

�� kF !

Occupied states

Empty states

L = f†✓@⌧ � r2

2m� µ

◆f

+ 4 Fermi terms


fermion density






The Fermi liquid

⇥| q|�

�� kF !

Occupied states

Empty states

Pomeranchuk instability as a function of coupling �

��c

��⇥ = 0⇥�⇤ �= 0

or


E�ective action for Ising order parameter

S⇥ =⇤

d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� c)⇤2 + u⇤4

⇥


E�ective action for Ising order parameter

S⇥ =⇤

d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� c)⇤2 + u⇤4

⇥

E�ective action for electrons:

Sc =⌃

d�

Nf⇧

�=1

�

⇤⇧

i

c†i�⇤⇥ ci� �⇧

i<j

tijc†i�ci�

⇥

⌅

⇥Nf⇧

�=1

⇧

k

⌃d�c†k� (⇤⇥ + ⇥k) ck�


��⇥ > 0 ��⇥ < 0

Coupling between Ising order and electrons

S�c

= � g

Zd⌧

NfX

↵=1

X

k,q

�q (cos kx� cos ky

)c†k+q/2,↵ck�q/2,↵

for spatially dependent �


S�c

= � g

Zd⌧

NfX

↵=1

X

k,q

�q (cos kx� cos ky

)c†k+q/2,↵ck�q/2,↵

S⇥ =⇤

d2rd⇥�(⌅�⇤)2 + c2(⇤⇤)2 + (�� c)⇤2 + u⇤4

⇥

Sc =Nf�

�=1

�

k

⇥d�c†k� (⇤⇥ + ⇥k) ck�


• � fluctuation at wavevector ~q couples most e�ciently to fermions

near ±~k0.

• Expand fermion kinetic energy at wavevectors about ±~k0 andboson (�) kinetic energy about ~q = 0.


L[ ±,�] =

†+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

��⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)


(a)

(b)Landau-damping

L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

One loop � self-energy with Nf

fermion flavors:

⌃�

(~q,!) = Nf

Zd2k

4⇡2

d⌦

2⇡

1

[�i(⌦+ !) + kx

+ qx

+ (ky

+ qy

)2]⇥�i⌦� k

x

+ k2y

⇤

=N

f

4⇡

|!||q

y

|


(a)

(b)Electron self-energy at order 1/N

f

:

⌃(

~k,⌦) = � 1

Nf

Zd2q

4⇡2

d!

2⇡

1

[�i(! + ⌦) + kx

+ qx

+ (ky

+ qy

)

2]

"q2y

g2+

|!||q

y

|

#

= �i2p3N

f

✓g2

4⇡

◆2/3

sgn(⌦)|⌦|2/3

L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2


(a)

(b)Electron self-energy at order 1/N

f

:

⌃(

~k,⌦) = � 1

Nf

Zd2q

4⇡2

d!

2⇡

1

[�i(! + ⌦) + kx

+ qx

+ (ky

+ qy

)

2]

"q2y

g2+

|!||q

y

|

#

= �i2p3N

f

✓g2

4⇡

◆2/3

sgn(⌦)|⌦|2/3

L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

⇠ |⌦|d/3 in dimension d.


L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

Schematic form of � and fermion Green’s functions in d dimensions

D(~q,!) =1/N

f

q2? +|!||q?|

, Gf

(~q,!) =1

qx

+ q2? � isgn(!)|!|d/3/Nf

In the boson case, q2? ⇠ !1/zb with zb

= 3/2.In the fermion case, q

x

⇠ q2? ⇠ !1/zf with zf

= 3/d.

Note zf

< zb

for d > 2 ) Fermions have higher energy thanbosons, and perturbation theory in g is OK.Strongly-coupled theory in d = 2.


L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

Schematic form of � and fermion Green’s functions in d = 2

D(~q,!) =1/N

f

q2y

+

|!||q

y

|

, Gf

(~q,!) =1

qx

+ q2y

� isgn(!)|!|2/3/Nf

In both cases qx

⇠ q2y

⇠ !1/z, with z = 3/2. Note that the

bare term ⇠ ! in G�1f

is irrelevant.

Strongly-coupled theory without quasiparticles.


Simple scaling argument for z = 3/2.

L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2



L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

X X



L = †+

�@⌧

� i@x

� @2y

� + + †

��@⌧

+ i@x

� @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@

y

�)2

X X

Under the rescaling x ! x/s, y ! y/s

1/2, and ⌧ ! ⌧/s

z, we

find invariance provided

� ! � s

! s

(2z+1)/4

g ! g s

(3�2z)/4

So the action is invariant provided z = 3/2.


⇥| q|�

�� kF !

FL Fermi liquid

• kdF ⇥ Q, the fermion density

• Sharp fermionic excitations

near Fermi surface with

⇥ ⇥ |q|z, and z = 1.

• Entropy density S ⇥ T (d��)/z

with violation of hyperscaling

exponent � = d� 1.

• Entanglement entropy

SE ⇥ kd�1F P lnP .

• Fermi surface

with kdF ⇠ Q.

• Di↵use fermionic

excitations with z = 3/2to three loops.

• S ⇠ T (d�✓)/z

with ✓ = d� 1.

• SE ⇠ kd�1F P lnP .

⇥| q|�

�� kF ! �� kF !

NFLNematic

QCP




⇥ ⇥ |q|z, and z = 1.






FL Fermi liquid

n(k)

kkF

• Fermi surface

with kdF ⇠ Q.



• S ⇠ T (d�✓)/z

with ✓ = d� 1.


M. A. Metlitski and S. Sachdev,Phys. Rev. B 82, 075127 (2010)

⇥| q|�

�� kF ! �� kF !⇥| q

|�




⇥ ⇥ |q|z, and z = 1.






FL Fermi liquid

NFLNematic

QCP

• Fermi surface

with kdF ⇠ Q.



• S ⇠ T (d�✓)/z

with ✓ = d� 1.


⇥| q|�

�� kF !⇥| q

|�

�� kF !




⇥ ⇥ |q|z, and z = 1.






FL Fermi liquid

NFLNematic

QCP

T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n



StrangeMetal


Fermiliquid

Fermiliquid


�

1. review of fermi liquid theory - harvard universityqpt.physics.harvard.edu/talks/natal15c.pdf ·...

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