Lec15 Quantum Criticality of Fermi surfacessachdev.physics.harvard.edu/phys268b/Lec15_Quantum... ·...

Quantum criticality of Fermi surfaces

Subir Sachdev

HARVARD

Physics 268br, Spring 2018

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.

Quantum criticality of Ising-nematic ordering in a metal

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 lattice

point-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.

Quantum criticality of Ising-nematic ordering in a metal

Pomeranchuk instability as a function of coupling �

��c

��⇥ = 0⇥�⇤ �= 0

or

Quantum criticality of Ising-nematic ordering in a metal

T

Phase diagram as a function of T and �

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

TI-n

Quantum criticality of Ising-nematic ordering in a metal

��c

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

TI-n Classicald=2 Isingcriticality

Quantum criticality of Ising-nematic ordering in a metal

Phase diagram as a function of T and �

��c

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n

Quantum criticality of Ising-nematic ordering in a metal

D=2+1 Ising

criticality ?

Phase diagram as a function of T and �

�

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n

D=2+1 Ising

criticality ?

Quantum criticality of Ising-nematic ordering in a metal

Phase diagram as a function of T and �

�

T

��⇥ = 0

Quantum critical

⇥�⇤ �= 0

rc

TI-n

Strongly-coupled“non-Fermi liquid”

metal with no quasiparticles

Quantum criticality of Ising-nematic ordering in a metal

Phase diagram as a function of T and �

�

T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n

Strongly-coupled“non-Fermi liquid”

metal with no quasiparticles

Quantum criticality of Ising-nematic ordering in a metal

Fermiliquid

Fermiliquid

Quantum critical

Phase diagram as a function of T and �

�

T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n

Strongly-coupled“non-Fermi liquid”

metal with no quasiparticles

StrangeMetal

Quantum criticality of Ising-nematic ordering in a metal

Fermiliquid

Fermiliquid

Phase diagram as a function of T and �

�

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 + k2y

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+2k4

y< ⇤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↵

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 + k2y

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+2k4

y< ⇤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↵

The Fermi liquid: RG

S[ ↵] =

Zdd�1y dx d⌧

h †↵

�@⌧ � i@x � @2y

� ↵ + u †

↵ †� � ↵

i

The Fermi liquid: RG

S[ ↵] =

Zdd�1y dx d⌧

h †↵

�@⌧ � i@x � @2y

� ↵ + u †

↵ †� � ↵

i

The kinetic energy is invariant under the rescaling x ! x/s,y ! y/s1/2, and ⌧ ! ⌧/sz, 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)

2u

Interactions are irrelevant in d = 2 !

The Fermi liquid: RG

S[ ↵] =

Zdd�1y dx d⌧

h †↵

�@⌧ � i@x � @2y

� ↵ + u †

↵ †� � ↵

i

The kinetic energy is invariant under the rescaling x ! x/s,y ! y/s1/2, and ⌧ ! ⌧/sz, 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)

2u

Interactions are irrelevant in d = 2 !

The fermion Green’s function to order u2 has 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:

The Fermi liquid: RG

S[ ↵] =

Zdd�1y dx d⌧

h †↵

�@⌧ � i@x � @2y

� ↵ + u †

↵ †� � ↵

i

The Fermi liquid: RG

S[ ↵] =

Zdd�1y dx d⌧

h †↵

�@⌧ � i@x � @2y

� ↵ + u †

↵ †� � ↵

i

n(k)

kkF

A

The fermion Green’s function to order u2 has 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, thefermion density

• Sharp particle and hole of excitations near the Fermi surfacewith 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

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

• Sharp particle and hole of excitations near the Fermi surfacewith 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

⇥| q|�

�� kF !

Occupied states

Empty states

L = f†✓@⌧ � r2

2m� µ

◆f

+ 4 Fermi terms

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

• Sharp particle and hole of excitations near the Fermi surfacewith 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

⇥| q|�

�� kF !

Occupied states

Empty states

Pomeranchuk instability as a function of coupling �

��c

��⇥ = 0⇥�⇤ �= 0

or

Quantum criticality of Ising-nematic ordering in a metal

E�ective action for Ising order parameter

S⇥ =⇤

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

⇥

Quantum criticality of Ising-nematic ordering in a metal

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�

Quantum criticality of Ising-nematic ordering in a metal

��⇥ > 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 �

Quantum criticality of Ising-nematic ordering in a metal

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�

Quantum criticality of Ising-nematic ordering in a metal

• � fluctuation at wavevector ~q couples most e�ciently to fermionsnear ±~k0.

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

Quantum criticality of Ising-nematic ordering in a metal

• � fluctuation at wavevector ~q couples most e�ciently to fermionsnear ±~k0.

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

Quantum criticality of Ising-nematic ordering in a metal

L[ ±,�] =

†+

�@⌧ � i@x � @2y

� + + †

��@⌧ + i@x � @2y

� �

��⇣ †+ + + †

� �

⌘+

1

2g2(@y�)

2

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

Quantum criticality of Ising-nematic ordering in a metal

(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⌦� kx + k2y

⇤

=Nf

4⇡

|!||qy|

Quantum criticality of Ising-nematic ordering in a metal

(a)

(b)Electron self-energy at order 1/Nf :

⌃(~k,⌦) = � 1

Nf

Zd2q

4⇡2

d!

2⇡

1

[�i(! + ⌦) + kx + qx + (ky + qy)2]

"q2yg2

+|!||qy|

#

= �i2p3Nf

✓g2

4⇡

◆2/3

sgn(⌦)|⌦|2/3

L = †+

�@⌧ � i@x � @2y

� + + †

��@⌧ + i@x � @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@y�)

2

Quantum criticality of Ising-nematic ordering in a metal

(a)

(b)Electron self-energy at order 1/Nf :

⌃(~k,⌦) = � 1

Nf

Zd2q

4⇡2

d!

2⇡

1

[�i(! + ⌦) + kx + qx + (ky + qy)2]

"q2yg2

+|!||qy|

#

= �i2p3Nf

✓g2

4⇡

◆2/3

sgn(⌦)|⌦|2/3

L = †+

�@⌧ � i@x � @2y

� + + †

��@⌧ + i@x � @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@y�)

2

⇠ |⌦|d/3 in dimension d.

Quantum criticality of Ising-nematic ordering in a metal

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/Nf

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, qx ⇠ q2? ⇠ !1/zf with zf = 3/d.

Note zf < zb for d > 2 ) Fermions have higher energy than

bosons, and perturbation theory in g is OK.

Strongly-coupled theory in d = 2.

Quantum criticality of Ising-nematic ordering in a metal

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/Nf

q2y +|!||qy|

, 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.

Quantum criticality of Ising-nematic ordering in a metal

Simple scaling argument for z = 3/2.

L = †+

�@⌧ � i@x � @2y

� + + †

��@⌧ + i@x � @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@y�)

2

Quantum criticality of Ising-nematic ordering in a metal

Simple scaling argument for z = 3/2.

L = †+

�@⌧ � i@x � @2y

� + + †

��@⌧ + i@x � @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@y�)

2

X X

Quantum criticality of Ising-nematic ordering in a metal

Simple scaling argument for z = 3/2.

L = †+

�@⌧ � i@x � @2y

� + + †

��@⌧ + i@x � @2y

� �

� �⇣ †+ + + †

� �

⌘+

1

2g2(@y�)

2

X X

Under the rescaling x ! x/s, y ! y/s1/2, and ⌧ ! ⌧/sz, wefind invariance provided

� ! � s

! s(2z+1)/4

g ! g s(3�2z)/4

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

Quantum criticality of Ising-nematic ordering in a metal

⇥| q|�

�� kF !

FL Fermi liquid

• kdF ⇥ Q, the fermion density

• Sharp fermionic excitationsnear Fermi surface with⇥ ⇥ |q|z, and z = 1.

• Entropy density S ⇥ T (d��)/z

with violation of hyperscalingexponent � = d� 1.

• Entanglement entropySE ⇥ kd�1

F P lnP .

• Fermi surfacewith kdF ⇠ Q.

• Di↵use fermionicexcitations with z = 3/2to three loops.

• S ⇠ T (d�✓)/z

with ✓ = d� 1.

• SE ⇠ kd�1F P lnP .

⇥| q|�

�� kF ! �� kF !

NFLNematic

QCP

• kdF ⇥ Q, the fermion density

• Sharp fermionic excitationsnear Fermi surface with⇥ ⇥ |q|z, and z = 1.

• Entropy density S ⇥ T (d��)/z

with violation of hyperscalingexponent � = d� 1.

• Entanglement entropySE ⇥ kd�1

F P lnP .

FL Fermi liquid

n(k)

kkF

• Fermi surfacewith kdF ⇠ Q.

• Di↵use fermionicexcitations with z = 3/2to three loops.

• S ⇠ T (d�✓)/z

with ✓ = d� 1.

• SE ⇠ kd�1F P lnP .

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

⇥| q|�

�� kF ! �� kF !⇥| q

|�

• kdF ⇥ Q, the fermion density

• Sharp fermionic excitationsnear Fermi surface with⇥ ⇥ |q|z, and z = 1.

• Entropy density S ⇥ T (d��)/z

with violation of hyperscalingexponent � = d� 1.

• Entanglement entropySE ⇥ kd�1

F P lnP .

FL Fermi liquid

NFLNematic

QCP

• Fermi surfacewith kdF ⇠ Q.

• Di↵use fermionicexcitations with z = 3/2to three loops.

• S ⇠ T (d�✓)/z

with ✓ = d� 1.

• SE ⇠ kd�1F P lnP .

⇥| q|�

�� kF !⇥| q

|�

�� kF !

• kdF ⇥ Q, the fermion density

• Sharp fermionic excitationsnear Fermi surface with⇥ ⇥ |q|z, and z = 1.

• Entropy density S ⇥ T (d��)/z

with violation of hyperscalingexponent � = d� 1.

• Entanglement entropySE ⇥ kd�1

F P lnP .

FL Fermi liquid

NFLNematic

QCP

T

��⇥ = 0⇥�⇤ �= 0

rc

TI-n

Strongly-coupled“non-Fermi liquid”

metal with no quasiparticles

StrangeMetal

Quantum criticality of Ising-nematic ordering in a metal

Fermiliquid

Fermiliquid

Phase diagram as a function of T and �

�

H = �X

i<j

tijc†i↵cj↵ + U

X

i

✓ni" �

1

2

◆✓ni# �

1

2

◆� µ

X

i

c†i↵ci↵

tij ! “hopping”. U ! local repulsion, µ ! chemical potential

Spin index ↵ =", #

ni↵ = c†i↵ci↵

c†i↵cj� + cj�c

†i↵ = �ij�↵�

ci↵cj� + cj�ci↵ = 0

The Hubbard Model

Will study on the square lattice

�

Fermi surfaces in electron- and hole-doped cuprates

Hole states

occupied

Electron states

occupied

�E↵ective Hamiltonian for quasiparticles:

H0 = �X

i<j

tijc†i↵cj↵ ⌘

X

k

"kc†k↵ck↵

with tij non-zero for first, second and third neighbor, leads to satisfactory agree-

ment with experiments. The area of the occupied electron states, Ae, from

Luttinger’s theory is

Ae =

⇢2⇡

2(1� x) for hole-doping x

2⇡2(1 + p) for electron-doping p

The area of the occupied hole states, Ah, which form a closed Fermi surface and

so appear in quantum oscillation experiments is Ah = 4⇡2 �Ae.

�

Hole states

occupied

Electron states

occupied

�

The electron spin polarization obeys�

⌃S(r, �)⇥

= ⌃⇥(r, �)eiK·r

where K is the ordering wavevector.

+

Fermi surface+antiferromagnetism

We use the operator equation (valid on each site i):

U

✓n" �

1

2

◆✓n# �

1

2

◆= �2U

3~S2 +

U

4(1)

Then we decouple the interaction via

exp

2U

3

X

i

Zd⌧ ~S2

i

!=

ZD ~Ji(⌧) exp

�X

i

Zd⌧

3

8U~J2i � ~Ji~Si

�!

(2)We now integrate out the fermions, and look for the saddle point of theresulting e↵ective action for ~Ji. At the saddle-point we find that the lowestenergy is achieved when the vector has opposite orientations on the A andB sublattices. Anticipating this, we look for a continuum limit in terms ofa field ~'i where

~Ji = ~'i eiK·ri (3)

Fermi surface+antiferromagnetism

In this manner, we obtain the “spin-fermion” model

Z =

ZDc↵D~' exp (�S)

S =

Zd⌧

X

k

c†k↵

✓@

@⌧� "k

◆ck↵

� �

Zd⌧

X

i

c†i↵~'i · ~�↵�ci�eiK·ri

+

Zd⌧d2r

1

2(rr ~')

2 +1

2(@⌧ ~')

2 +s

2~'2 +

u

4~'4

�

Fermi surface+antiferromagnetism

Fermi surface+antiferromagnetismIn the Hamiltonian form (ignoring, for now, the time depen-

dence of ~'), the coupling between ~' and the electrons takes the

form

Hsdw = �

X

k,q,↵,�

~'q · c†k+q,↵~�↵�ck+K,�

where ~� are the Pauli matrices, the boson momentum q is small,

while the fermion momenum k extends over the entire Brillouin

zone. In the antiferromagnetically ordered state, we may take

~' / (0, 0, 1) , and the electron dispersions obtained by diago-

nalizing H0 +Hsdw are

Ek± ="k + "k+K

2±

s✓"k � "k+K

2

◆2

+ �2|~'|2

This leads to the Fermi surfaces shown in the following slides

as a function of increasing |~'|.

Metal with “large” Fermi surface

Fermi surface+antiferromagnetism

Fermi surfaces translated by K = (�,�).

Fermi surface+antiferromagnetism

“Hot” spots

Fermi surface+antiferromagnetism

Electron and hole pockets inantiferromagnetic phase with h~'i 6= 0

Fermi surface+antiferromagnetism

Increasing SDW order

���

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

�

Hole pockets

Electron pockets

Square lattice Hubbard model with hole doping

Increasing SDW order

���

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

�

Hole pockets

Electron pockets

Square lattice Hubbard model with hole doping

Increasing SDW order

���

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

�

Hole pockets

Electron pockets

Hot spots

where "k = "k+K

Square lattice Hubbard model with hole doping

Increasing SDW order

���

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

�

Hole pockets

Electron pockets

Fermi surface breaks up at hot spotsinto electron and hole “pockets”

Hole pockets

Hot spots

where "k = "k+K

Square lattice Hubbard model with hole doping

Increasing SDW order

���

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

�

Hole pockets

Electron pockets

Fermi surface breaks up at hot spotsinto electron and hole “pockets”

Hot spots

where "k = "k+K

Square lattice Hubbard model with hole doping

Square lattice Hubbard model with hole doping

Metal with “large” Fermi

surface

s

Increasing SDW order

Metal with electron and hole pockets

Metal with hole pockets

Increasing SDW orderIncreasing SDW order

h~'i = 0h~'i 6= 0h~'i 6= 0and smalland large

Square lattice Hubbard model with electron doping

Metal with “large” Fermi

surface

s

Metal with electron and hole pockets

Metal with electron pockets

h~'i = 0h~'i 6= 0h~'i 6= 0and smalland large

Square lattice Hubbard model with no doping

Metal with “large” Fermi

surface

s

Increasing SDW order

Metal with electron and hole pockets

Insulator

Increasing SDW orderIncreasing SDW order

h~'i = 0h~'i 6= 0h~'i 6= 0and smalland large

Fermi surfaces translated by K = (�,�).

Fermi surface+antiferromagnetism

“Hot” spots

Low energy theory for critical point near hot spots

Theory has fermions 1,2 (with Fermi velocities v1,2)and boson order parameter ~',interacting with coupling �

kx

ky

v1 v2

�2 fermionsoccupied

�1 fermionsoccupied

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

v1 v2

�2 fermionsoccupied

�1 fermionsoccupied

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

Lf = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

kx

ky

v1 v2

“Hot spot”

“Cold” Fermi surfaces

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

Lf = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

kx

ky

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

Lf = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

Order parameter: L' =1

2(rr ~')

2+

1

2(@⌧ ~')

2+

s

2~'2

+u

4~'4

“Yukawa” coupling: Lc = ��~' ·⇣ †1↵~�↵� 2� + †

2↵~�↵� 1�

⌘

Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

Lf = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

Order parameter: L' =1

2(rr ~')

2+

1

2(@⌧ ~')

2+

s

2~'2

+u

4~'4

v1 v2

Fermion dispersions: "k1 = v1 · k and "k2 = v2 · k

"k1 < 0 "k2 < 0

Metal with “large” Fermi

surfaceh~'i = 0

Lf = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

Fermion dispersions: "k1 = v1 · k and "k2 = v2 · k

Metal with hole and electron pockets

Ek± ="k1 + "k2

2±

s✓"k1 � "k2

2

◆2

+ �2|~'|2

Ek± < 0

Ek± > 0

Ek+ > 0Ek� < 0

h~'i 6= 0

��~' ·⇣ †1↵~�↵� 2� + †

2↵~�↵� 1�

⌘Lf = †1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

Hertz action.

Upon integrating the fermions out, the leading term in the ~' e↵ectiveaction is �⇧(q,!n)|~'(q,!n)|2, where ⇧(q,!n) is the fermion polariz-ability. This is given by a simple fermion loop diagram

⇧(q,!n) =

Zddk

(2⇡)d

Zd✏n2⇡

1

[�i(✏n + !n) + v1 · (k+ q)][�i✏n + v2 · k].

We define oblique co-ordinates p1 = v1 · k and p2 = v2 · k. It isthen clear that the integrand is independent of the (d� 2) transversemomenta, whose integral yields an overall factor ⇤d�2 (in d = 2 thisfactor is precisely 1).

Also, by shifting the integral over k1 we note that the integral is

independent of q. So we have

⇧(q,!n) =⇤d�2

|v1 ⇥ v2|

Zdp1dp2d✏n

8⇡3

1

[�i(✏n + !n) + p1][�i✏n + p2].

Next, we evaluate the frequency integral to obtain

⇧(q,!n) =⇤d�2

⇣|v1 ⇥ v2|

Zdp1dp24⇡2

[sgn(p2)� sgn(p1)]

�i⇣!n + p1 � p2

= � |!n|⇤d�2

4⇡|v1 ⇥ v2|.

In the last step, we have dropped a frequency-independent, cuto↵-

dependent constant which can absorbed into a redefinition of r. In-

serting this fermion polarizability in the e↵ective action for ~', we

obtain the Hertz action for the SDW transition:

SH =

Zddk

(2⇡)dTX

!n

1

2

⇥k2 + �|!n|+ s

⇤|~'(k,!n)|2

+u

4

Zddxd⌧

�~'2

(x, ⌧)�2

.

Fate of the fermions.

Let us, for now, assume the validity of the Hertz Gaussian action, and compute

the leading correction to the electronic Green’s function. This is given by the

following Feynman graph for the electron self energy, ⌃. At zero momentum for

the 1 fermion we have

⌃1(0,!n) = �2Z

ddq

(2⇡)d

Zd✏n2⇡

1

[q2 + �|✏n| ] [�i(✏n + !n) + v2 · q ].

We first perform the integral over the q direction parallel to v2, while ignoring

the subdominant dependence on this momentum in the boson propagator. Then

we have

⌃1(0,!n) = i�2

|v2|

Zdd�1q

(2⇡)d�1

Zd✏n2⇡

sgn(✏n + !n)

|q|2 + �|✏n|

= i�2

⇡|v2|�sgn(!n)

Zdd�1q

(2⇡)d�1ln

✓|q|2 + �|!n|

|q|2

◆.

Evaluation of the q integral shows that

⌃1(0,!n) ⇠ |!n|(d�1)/2

The most important case is d = 2, where we have

⌃1(0,!n) = i�2

⇡|v2|p�sgn(!n)

p|!n| , d = 2.

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Fate of the fermions.

Let us, for now, assume the validity of the Hertz Gaussian action, and compute

the leading correction to the electronic Green’s function. This is given by the

following Feynman graph for the electron self energy, ⌃. At zero momentum for

the 1 fermion we have

⌃1(0,!n) = �2Z

ddq

(2⇡)d

Zd✏n2⇡

1

[q2 + �|✏n| ] [�i(✏n + !n) + v2 · q ].

We first perform the integral over the q direction parallel to v2, while ignoring

the subdominant dependence on this momentum in the boson propagator. Then

we have

⌃1(0,!n) = i�2

|v2|

Zdd�1q

(2⇡)d�1

Zd✏n2⇡

sgn(✏n + !n)

|q|2 + �|✏n|

= i�2

⇡|v2|�sgn(!n)

Zdd�1q

(2⇡)d�1ln

✓|q|2 + �|!n|

|q|2

◆.

Evaluation of the q integral shows that

⌃1(0,!n) ⇠ |!n|(d�1)/2

The most important case is d = 2, where we have

⌃1(0,!n) = i�2

⇡|v2|p�sgn(!n)

p|!n| , d = 2.

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v1 v2

kx

ky

Critical point theory is strongly coupled in d = 2Results are independent of coupling �

A. J. Millis, Phys. Rev. B 45, 13047 (1992)Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004)

Gfermion ⇠ 1pi! � v.k

kx

ky

M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010)

Critical point theory is strongly coupled in d = 2Results are independent of coupling �

k?

kk

Gfermion =Z(kk)

! � vF (kk)k?, Z(kk) ⇠ vF (kk) ⇠ kk

Strong coupling physics in d = 2

The theory so far has the boson propagator

⇠ 1

q2 + �|!|

which scales with dynamic exponent zb = 2, and now a fermion propagator

⇠ 1

�i! + c1|!|(d�1)/2 + v · q.

First note that for d < 3, the bare �i! term is less important than the contribu-

tion from the self energy at low frequencies. Ignoring the i! term, we see that

the fermion propagator scales with dynamic exponent zf = 2/(d�1). For d > 2,

zf < zb, and so at small momenta the boson fluctuations have lower energy than

the fermion fluctuations. Thus it seems reasonable to assume that the fermion

fluctuations are not as singular, and we can focus on an e↵ective theory of the

SDW order parameter ~' alone. In other words, the Hertz assumptions appear

valid for d > 2.

However, in d = 2, we have zf = zb = 2. Thus fermionic and bosonic fluctua-

tions are equally important, and it is not appropriate to integrate the fermions

out at an initial stage. We have to return to the original theory of coupled

bosons and fermions. This turns out to be strongly coupled, and exhibits com-

plex critical behavior. For more details, see

M. A. Metlitski and S. Sachdev, arXiv:1005.1288 (Physical Review B 82, 075127(2010)).

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Strong coupling physics in d = 2

The theory so far has the boson propagator

⇠ 1

q2 + �|!|

which scales with dynamic exponent zb = 2, and now a fermion propagator

⇠ 1

�i! + c1|!|(d�1)/2 + v · q.

First note that for d < 3, the bare �i! term is less important than the contribu-

tion from the self energy at low frequencies. Ignoring the i! term, we see that

the fermion propagator scales with dynamic exponent zf = 2/(d�1). For d > 2,

zf < zb, and so at small momenta the boson fluctuations have lower energy than

the fermion fluctuations. Thus it seems reasonable to assume that the fermion

fluctuations are not as singular, and we can focus on an e↵ective theory of the

SDW order parameter ~' alone. In other words, the Hertz assumptions appear

valid for d > 2.

However, in d = 2, we have zf = zb = 2. Thus fermionic and bosonic fluctua-

tions are equally important, and it is not appropriate to integrate the fermions

out at an initial stage. We have to return to the original theory of coupled

bosons and fermions. This turns out to be strongly coupled, and exhibits com-

plex critical behavior. For more details, see

M. A. Metlitski and S. Sachdev, arXiv:1005.1288 (Physical Review B 82, 075127(2010)).

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