these phase diagrams? captures the “non-Fermi liquid ...conferences.physics.illinois.edu/JointPerimeter2014/talks/raghu... · rule (together with Maxwell’s equations) ... NFL

Srinivas Raghu (Stanford)

Quantum critical metals and their instabilities

Can we understand a modification of Landau’s theory that captures the “non-Fermi liquid” states that may occur in

these phase diagrams?

We will try to find controlled examples in well motivated (but toy) field theories.

The new ingredient which causes breakdown of LFLTis hypothesized by many to be a gapless scalar:

L = (@�)2 �m2�2 + ...

m2 ⇠ (g � gc)

Saturday, September 13, 14

With Liam Fitzpatrick, Jared Kaplan, Shamit Kachru Gonzalo Torroba, Huajia Wang

R. Mahajan, D. Ramirez, S. Kachru, and SR, PRB 88, 115116 (2013).

A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, PRB (2014).

A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, PRB 88, 125116 (2013).

Collaborators and References

A. Liam Fitzpatrick, S. Kachru, J. Kaplan, and SR, G. Torroba, H. Wang, to appear.

Basic themes to be explored

Basic issues related to metallic quantum criticality.

Hertz: metallic quasiparticles + order parameter fluctuations.

Current frontier: treatment of the interplay between incoherent fermions and order parameter fluctuations.

Goal: treat fermions and order parameter fields on equal footing, find controllable limits and associated scaling laws.

Instabilities towards conventional ground states?

Ordinary metals

Free electrons

k(t) = ei✏(k)t

~ k(0)

quasiparticles

q.p. scattering rate

k(t) ⇡ ei✏(k)t

~ e��(k)t

~ k(0)switch on short-range interactions

1. The basic problem and setup

Since this is a mixed audience, let me review the basic physics we’re trying to understand. Many metals are

well described by adiabatic modifications of a free electronmodel:

Non-Fermi liquids A. J. Schofield 2

can be obtained relatively simply using Fermi’s goldenrule (together with Maxwell’s equations) and I have in-cluded these for readers who would like to see wheresome of the properties are coming from.

The outline of this review is as follows. I begin witha description of Fermi-liquid theory itself. This the-ory tells us why one gets a very good description of ametal by treating it as a gas of Fermi particles (i.e. thatobey Pauli’s exclusion principle) where the interactionsare weak and relatively unimportant. The reason isthat the particles one is really describing are not theoriginal electrons but electron-like quasiparticles thatemerge from the interacting gas of electrons. Despite itsrecent failures which motivate the subject of non-Fermiliquids, it is a remarkably successful theory at describ-ing many metals including some, like UPt3, where theinteractions between the original electrons are very im-portant. However, it is seen to fail in other materialsand these are not just exceptions to a general rule butare some of the most interesting materials known. Asan example I discuss its failure in the metallic state ofthe high temperature superconductors.

I then present four examples which, from a theo-retical perspective, generate non-Fermi liquid metals.These all show physical properties which can not beunderstood in terms of weakly interacting electron-likeobjects:

• Metals close to a quantum critical point. When aphase transition happens at temperatures close toabsolute zero, the quasiparticles scatter so stronglythat they cease to behave in the way that Fermi-liquid theory would predict.

• Metals in one dimension–the Luttinger liquid. Inone dimensional metals, electrons are unstable anddecay into two separate particles (spinons andholons) that carry the electron’s spin and chargerespectively.

• Two-channel Kondo models. When two indepen-dent electrons can scatter from a magnetic impu-rity it leaves behind “half an electron”.

• Disordered Kondo models. Here the scatteringfrom disordered magnetic impurities is too strongto allow the Fermi quasiparticles to form.

While some of these ideas have been used to try and un-derstand the high temperature superconductors, I willshow that in many cases one can see the physics illus-trated by these examples in other materials. I believethat we are just seeing the tip of an iceberg of new typesof metal which will require a rather different startingpoint from the simple electron picture to understandtheir physical properties.

Figure 1: The ground state of the free Fermi gas in mo-mentum space. All the states below the Fermi surfaceare filled with both a spin-up and a spin-down elec-tron. A particle-hole excitation is made by promotingan electron from a state below the Fermi surface to anempty one above it.

2. Fermi-Liquid Theory: the electron quasi-particle

The need for a Fermi-liquid theory dates from thefirst applications of quantum mechanics to the metallicstate. There were two key problems. Classically eachelectron should contribute 3kB/2 to the specific heatcapacity of a metal—far more than is actually seen ex-perimentally. In addition, as soon as it was realizedthat the electron had a magnetic moment, there wasthe puzzle of the magnetic susceptibility which did notshow the expected Curie temperature dependence forfree moments: χ ∼ 1/T .

These puzzles were unraveled at a stroke whenPauli (Pauli 1927, Sommerfeld 1928) (apparentlyreluctantly—see Hermann et al. 1979) adopted Fermistatistics for the electron and in particular enforced theexclusion principle which now carries his name: No twoelectrons can occupy the same quantum state. In theabsence of interactions one finds the lowest energy stateof a gas of free electrons by minimizing the kinetic en-ergy subject to Pauli’s constraint. The resulting groundstate consists of a filled Fermi sea of occupied statesin momentum space with a sharp demarcation at theFermi energy ϵF and momentum pF = hkF (the Fermisurface) between these states and the higher energy un-occupied states above. The low energy excited statesare obtained simply by promoting electrons from justbelow the Fermi surface to just above it (see Fig. 1).They are uniquely labelled by the momentum and spinquantum numbers of the now empty state below theFermi energy (a hole) and the newly filled state aboveit. These are known as particle-hole excitations.

This resolves these early puzzles since only a smallfraction of the total number of electrons can take part

CV ⇠ T

⇢ ⇠ T 2

Fairly robustly in the simplestsetting, fermions contribute:

Sunday, February 23, 14

plane-wave eigenstates

Quasiparticle description is viable when

C ⇠ T

� ⌧ ✏

Near Fermi surface, this is increasingly well-satisfied:

✏ ! 0, � ⇠ ✏2 ⌧ ✏

Consequences: Universal thermodynamics

e.g.

Non-interacting Fixed point action, S: Landau Fermi liquid theory

Effective field theory of ordinary metals

Fermi surface

(Shankar 1991, Polchinski 1992)

qx

qy

qx

qy

(a) (b)

(c)

~k

~k + ~q

~q

empty states

filled states

empty states

filled states

S =

Z

k,! (i! � v`)

Z

!,k⌘

Z

⇤

d!

2⇡

Z

⇤k

d`dd�1kk(2⇡)d

Most interactions are irrelevant.

Clean metal - unstable only to BCS (at exponentially small energy scales).

BCS -> marginal.

` : perpendicular to F. S.

Breakdown of fermion quasiparticles

A recurring theme: Fermi liquid theory breaks down at a quantum phase transition.

NFL emanates from a critical point at T=0.

NFL can give way to higher Tc superconductivity.

QCPs in metals: wide-open problem especially in d=2+1.

This theory can be promoted to a full “Landau Fermi liquid theory” which has enjoyed great success.

However, in some of the most interesting modern materials, LFLT apparently breaks down:

Saturday, September 13, 14

NFL

How do Fermi liquids break down?

New IR effects can occur if additional low energy modes couple to electrons: e.g. order parameter fluctuations near criticality.

This occurs at quantum critical points in metals.

Two types of quantum critical points: 1) transition preserves translation symmetry (e.g. Ferromagnetism, nematic ordering) ! 2) transition breaks translation symmetry (e.g. CDW, SDW ordering)

I will focus on the class 1) of critical points above.

Shankar/Polchinski’s theory cannot capture such behavior.

thermal melting of a stripe state to form a nematic fluid is readily understood theoretically

(9, 18, 29–31) and, indeed, the resulting description is similar to the theory of the nearly

smectic nematic fluid that has been developed in the context of complex classical fluids

(2, 32). Within this perspective, the nematic state arises from the proliferation of disloca-

tions, the topological defects of the stripe state. This can take place either via a thermal

phase transition (as in the standard classical case) or as a quantum phase transition.

Whereas the thermal phase transition is well understood (2, 32), the theory of the quantum

smectic-nematic phase transition by a dislocation proliferation mechanism is largely an

open problem. Two notable exceptions are the work of Zaanen et al. (31) who studied this

phase transition in an effectively insulating system, and the work of Wexler & Dorsey (33)

who estimated the core energy of the dislocations of a stripe quantum Hall phase.

Nevertheless, the microscopic (or position-space) picture of the quantum nematic phase

that results consists of a system of stripe segments, the analog of nematogens, whose

typical size is the mean separation between dislocations. The system is in a nematic state

if the nematogens exhibit long-range orientational order on a macroscopic scale (8, 9).

However, since the underlying degrees of freedom are the electrons from which these nano-

structures form, the electron nematic is typically an anisotropic metal. Similarly, nematic

order can also arise from thermal or quantum melting a frustrated quantum antiferromag-

net (34–36).

An alternative picture of the nematic state (and of the mechanisms that may give rise to

it) can be gleaned from a Fermi-liquid-like perspective. In this momentum space picture

one begins with a metallic state consisting of a system of fermions with a Fermi surface (FS)

and well-defined quasiparticles (QP). In the absence of any sort of symmetry breaking, the

shape of the FS reflects the underlying symmetries of the system. There is a classic result

Nematic phases

Via Pomeranchukinstability

Via meltingstripes

Fermiliquid

Smecticor stripe

phase

Figure 1

Two different mechanisms for producing a nematic phase with point particles: The gentle melting of astripe phase can restore long-range translational symmetry while preserving orientational order (8).Alternatively, a nematic Fermi fluid can arise through the distortion of the Fermi surface of a metal viaa Pomeranchuk instability (27, 28). (After, in part, Reference 8.)

158 Fradkin et al.

Ann

u. R

ev. C

onde

ns. M

atte

r Phy

s. 20

10.1

:153

-178

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rgby

Sta

nfor

d U

nive

rsity

- M

ain

Cam

pus -

Lan

e M

edic

al L

ibra

ry o

n 04

/02/

13. F

or p

erso

nal u

se o

nly.

Isotropic Nematic

Pomeranchuk instability

thermal melting of a stripe state to form a nematic fluid is readily understood theoretically

(9, 18, 29–31) and, indeed, the resulting description is similar to the theory of the nearly

smectic nematic fluid that has been developed in the context of complex classical fluids

(2, 32). Within this perspective, the nematic state arises from the proliferation of disloca-

tions, the topological defects of the stripe state. This can take place either via a thermal

phase transition (as in the standard classical case) or as a quantum phase transition.

Whereas the thermal phase transition is well understood (2, 32), the theory of the quantum

smectic-nematic phase transition by a dislocation proliferation mechanism is largely an

open problem. Two notable exceptions are the work of Zaanen et al. (31) who studied this

phase transition in an effectively insulating system, and the work of Wexler & Dorsey (33)

who estimated the core energy of the dislocations of a stripe quantum Hall phase.

Nevertheless, the microscopic (or position-space) picture of the quantum nematic phase

that results consists of a system of stripe segments, the analog of nematogens, whose

typical size is the mean separation between dislocations. The system is in a nematic state

if the nematogens exhibit long-range orientational order on a macroscopic scale (8, 9).

However, since the underlying degrees of freedom are the electrons from which these nano-

structures form, the electron nematic is typically an anisotropic metal. Similarly, nematic

order can also arise from thermal or quantum melting a frustrated quantum antiferromag-

net (34–36).

An alternative picture of the nematic state (and of the mechanisms that may give rise to

it) can be gleaned from a Fermi-liquid-like perspective. In this momentum space picture

one begins with a metallic state consisting of a system of fermions with a Fermi surface (FS)

and well-defined quasiparticles (QP). In the absence of any sort of symmetry breaking, the

shape of the FS reflects the underlying symmetries of the system. There is a classic result

Nematic phases

Via Pomeranchukinstability

Via meltingstripes

Fermiliquid

Smecticor stripe

phase

Figure 1

Two different mechanisms for producing a nematic phase with point particles: The gentle melting of astripe phase can restore long-range translational symmetry while preserving orientational order (8).Alternatively, a nematic Fermi fluid can arise through the distortion of the Fermi surface of a metal viaa Pomeranchuk instability (27, 28). (After, in part, Reference 8.)

158 Fradkin et al.

Ann

u. R

ev. C

onde

ns. M

atte

r Phy

s. 20

10.1

:153

-178

. Dow

nloa

ded

from

ww

w.a

nnua

lrevi

ews.o

rgby

Sta

nfor

d U

nive

rsity

- M

ain

Cam

pus -

Lan

e M

edic

al L

ibra

ry o

n 04

/02/

13. F

or p

erso

nal u

se o

nly.

Fermi liquids:

Analogy with classical liquid crystals

At the critical point is a massless field.

Ising nematic transition: breaking of point group symmetry.

Concrete model system

thermal

meltingofastripestateto

form

anem

aticfluid

isread

ilyunderstoodtheoretically

(9,18,29–3

1)an

d,indeed,theresultingdescriptionissimilar

tothetheory

ofthenearly

smecticnem

atic

fluid

that

has

beendeveloped

inthecontextofcomplexclassicalfluids

(2,32).Within

thisperspective,thenem

atic

statearises

from

theproliferationofdisloca-

tions,

thetopologicaldefects

ofthestripestate.

This

cantakeplace

either

viaathermal

phasetran

sition

(asin

thestan

dard

classicalcase)oras

aquan

tum

phasetran

sition.

Whereasthethermal

phasetran

sitioniswellu

nderstood(2,3

2),thetheory

ofthequan

tum

smectic-nem

atic

phasetran

sitionbyadislocationproliferationmechan

ism

islargelyan

open

problem.T

wonotableexceptionsarethework

ofZaanen

etal.(31)whostudiedthis

phasetran

sitionin

aneffectivelyinsulatingsystem

,an

dthework

ofWexler&

Dorsey

(33)

whoestimated

thecore

energy

ofthedislocationsofastripequan

tum

Hallphase.

Nevertheless,themicroscopic(orposition-space)

picture

ofthequan

tum

nem

aticphase

that

resultsconsistsofasystem

ofstripesegm

ents,thean

alogofnem

atogens,

whose

typical

size

isthemeanseparationbetweendislocations.Thesystem

isin

anem

atic

state

ifthenem

atogensexhibit

long-range

orientational

order

onamacroscopic

scale(8,9).

However,since

theunderlyingdegrees

offreedom

aretheelectronsfrom

whichthesenan

o-

structuresform

,theelectronnem

atic

istypically

anan

isotropic

metal.Similarly,nem

atic

order

canalso

arisefrom

thermal

orquan

tum

meltingafrustratedquan

tum

antiferromag-

net

(34–3

6).

Analternativepicture

ofthenem

aticstate(andofthemechan

ismsthat

may

give

rise

to

it)canbegleaned

from

aFermi-liquid-likeperspective.In

this

momentum

spacepicture

onebeginswithametallicstateconsistingofasystem

offerm

ionswithaFermisurface(FS)

andwell-defined

quasiparticles

(QP).In

theab

sence

ofan

ysortofsymmetry

break

ing,

the

shap

eoftheFSreflects

theunderlyingsymmetries

ofthesystem

.Thereisaclassicresult

Nem

atic

pha

ses

Via

Pom

eran

chuk

inst

abili

tyVi

a m

eltin

gst

ripes

Ferm

iliq

uid

Smec

ticor

strip

eph

ase

Figure

1

Twodifferentmechan

ismsforproducinganem

aticphasewithpointparticles:Thegentlemeltingofa

stripephasecanrestore

long-range

tran

slational

symmetry

whilepreservingorientational

order

(8).

Alternatively,anem

aticFermifluid

canarisethrough

thedistortionoftheFermisurfaceofametal

via

aPomeran

chukinstab

ility(27,2

8).(A

fter,in

part,Reference

8.)

158

Fradkin

etal.

Annu. Rev. Condens. Matter Phys. 2010.1:153-178. Downloaded from www.annualreviews.orgby Stanford University - Main Campus - Lane Medical Library on 04/02/13. For personal use only.

“spin” up “spin” down

2 possible ground states

t+ �

t� �order parameter:�

�

Effective theory: Fermion-boson problem

Landau Fermi liquidLandau-Ginzburg-Wilson theory for order parameter.

Fermion-boson “Yukawa” coupling

Starting UV action: S = S + S� + S ��

S

S�

S ��

Obtaining such an action: Start with electrons strongly interacting (“Hubbard model”). “Integrate out” high energy modes from lattice scale down to a new UV cutoff ⇤ << EF .

= Scale below which we can linearize the fermion Kinetic energy. ⇤

Effective theory: Fermion-boson problem

Fermions

bosons

“Yukawa” coupling

The lesson I will take from this is the following: it will be useful to find controlled approaches to non-Fermi liquid

fixed points using toy models, even unrealistic toy models.

II. Our toy model & RG philosophy

We will study the theory with UV action:

Fermi interactions. We describe the correlation functionsof both the boson and fermion degrees of freedom at thenon-Fermi liquid fixed point; they di⇥er from the resultsobtained in alternative treatments. In §4, we re-introducethe four-Fermi interactions and describe subtleties associ-ated with log2 divergences that arise in their presence. In§5, we discuss controlled large N theories where the sub-tleties of §4 do not arise, and we find fixed points whichgeneralize those of §3 to include four-Fermi interactions.We show that these fixed points have no superconductinginstabilities. We close with a discussion of open issues in§6. Explicit calculations which we refer to in the mainbody are presented in several appendices.

II. EFFECTIVE ACTION AND SCALINGANALYSIS

Let ⌥� denote a fermion field with spin ⌅ =⇤, ⌅, anddispersion �(k), defined relative to the Fermi level. Let ⌃be the scalar boson field corresponding to the order pa-rameter for the quantum phase transition. The e⇥ectivelow energy Euclidean action consists of a purely fermionicterm, a purely bosonic term and a Yukawa coupling be-tween bosons and fermions:

S =

⇤d⇧

⇤ddx L = S⌅ + S⇤ + S⌅�⇤

L⌅ = ⌥� [�⇥ + µ� �(i⇧)]⌥� + ⇥⌅⌥�⌥��⌥��⌥�

L⇤ = m2⇤⌃

2 + (�⇥⌃)2 + c2

� ⇧⌃

⇥2+

⇥⇤

4!⌃4

S⌅,⇤ =

⇤dd+1kdd+1q

(2⇤)2(d+1)g(k, q)⌥(k)⌥(k + q)⌃(q), (1)

where repeated spin indices are summed. The first term,L⌅, represents a Landau Fermi liquid, with weak residualself-interactions incorporated in forward and BCS scat-tering amplitudes. The second term represents an in-teracting scalar boson field with speed c and mass m⇤

(which corresponds to the inverse correlation length thatvanishes as the system is tuned to the quantum criti-cal point). The third term is the Yukawa coupling be-tween the fermion and boson fields and is more naturallydescribed in momentum space. The quantity g(k, q) isa generic coupling function that depends both on thefermion momentum k, as well as the momentum trans-fer q (we have suppressed spin indices for clarity). For aspherically symmetric Fermi system, the angular depen-dence of g(k, q) for |k| = kF can be decomposed into dis-tinct angular momentum channels, each of which marksa di⇥erent broken symmetry. Familiar examples includeferromagnetism (angular momentum zero) and nematicorder (angular momentum 2). More generally, the cou-pling can be labelled by the irreducible representation ofthe crystal point group and it respects symmetry trans-formations under which ⌃ and ⌥⌥ both change sign.

Before proceeding, we make a few comments on the ori-gins of the e⇥ective action above. One starts with a the-ory involving fermions interacting at short distances with

qx

qy

qx

qy

(a) (b)

(c)

�k

�k + �q

�q

empty states

filled states

empty states

filled states

FIG. 1. Summary of tree-level scaling. High energy modes(blue) are integrated out at tree level and remaining low en-ergy modes (red) are rescaled so as to preserve the boson andfermion kinetic terms. The boson modes (a) have the lowenergy locus at a point whereas the fermion modes (b) havetheir low energy locus on the Fermi surface. The most rele-vant Yukawa coupling (c) connects particle-hole states nearlyperpendicular to the Fermi surface; all other couplings areirrelevant under the scaling.

strong repulsive forces. These interactions are decoupledby an auxiliary boson field ⌃ representing a fermion bilin-ear, and the partition function is obtained by averagingover all possible values of both the fermion and bosonfields. Initially, the auxiliary field has no dynamics andis massive. However, as high energy modes of the ma-terial of interest are integrated out, radiative correctionsinduce dynamics for the bosons. In a Wilsonian theory,the dynamics are encapsulated only in local, analytic cor-rections to the bare action. This mode elimination is con-tinued until eventually, the UV cuto⇥ � ⇥ EF representsthe scale up to which the quasiparticle kinetic energy canbe linearized about the Fermi level. At these low ener-gies, and in the vicinity of the quantum critical pointwhere the field ⌃ condenses, it is legitimate to view ⌃as an independent, emergent fluctuating field that cou-ples to the low energy fermions via a Yukawa coupling aswritten above23. This will be the point of departure ofour analysis below.We first describe a consistent scaling procedure for the

action in Eq. 1. The key challenge stems from thefact that the boson and fermion fields have vastly dif-ferent kinematics. Our bosons have dispersion relationk20 = c2k2+m2

⇤, so that low energies correspond as usualto low momentum, and their scaling is that of a standardrelativistic field theory where all components of momen-tum scale the same way as k0. By contrast, the fermiondispersion relation is k0 = �(k) � µ, so their low en-ergy states occur close to the Fermi surface (Fig. 1).Moreover, the Yukawa coupling between the two sets offields must conserve energy and momentum in a coarse-

2

Wednesday, June 26, 13

Starting UV action (in imaginary time):

g=0: decoupled limit (Fermi liquid + ordinary critical point).

non-zero g: complex tug-of-war between bosons and fermions.

Ising nematic theory: g(k, q) = g (cos kx

� cos ky

) .

Tug-of-war between bosons and fermions

Non-zero g: Bosons can decay into particle-hole continuum -> overdamped bosons.

(a)ab

a

ab

b

(b)a b

ab

a

(c)ab

a

a

b

ab

b

Non-zero g: Quasiparticle scattering enhanced due to bosons.

(a)ab

a

ab

b

(b)a b

ab

a

(c)ab

a

a

b

ab

b

+ · · ·

+ · · ·q.p. Scattering rate can exceed its energy.

Fermion propagators: poles become branch cuts.

Result: breakdown of Landau quasiparticle.

How to proceed???

Mainstream view: Hertz (1976)

This approach takes the viewpoint that damping of bosons due to fermions is the most significant effect.

Idea: integrating out all fermions results in a non-local theory of nearly free, overdamped bosons:

Interactions among bosons are irrelevant but singular: ignoring them could be dangerous!

Seff =

Z

k,!

⇥!2 � c2k2 + i�(!, k)

⇤�2 �(!, k) = g2

kd�1F

v

!

k

Landau damping -> bosons governed by z=3 dynamic scaling.

A long line of works building along this direction exists:

Hertz 1976 Millis 1993 Polchinski 1994 Altshuler, Ioffe, Millis, 1994 Nayak, Wilczek, 1994 Oganesyan, Fradkin, Kivelson, 2001 Chubukov et al, 2006 Sung-Sik Lee, 2009 Metlitski, Sachdev 2010 Mross, Mcgreevy, Liu, Senthil (2010) Davidovic, Sung-Sik Lee (2014) ……….

We go in a different direction

Outline of the rest of the talk

I. Large N theory

II. Renormalization group analysis

III. Superconducting “domes”

{Ignore SC here and address “normal” state

I. Large N limits

Large N limits

Essence of the problem: dissipative coupling between bosons and fermions.

Large N limits: particles with many (N) flavors act as a dissipative “bath” while remaining degrees of freedom become overdamped.

e.g. Large number of fermion flavors (Nf). Boson can decay in many channels -> Overdamped bosons (NFL is subdominant). Mainstream (Hertz) theory captures the IR behavior in this regime.

e.g. Large number of boson flavors (Nb). Fermion can decay in many channels -> NFL is strongest effect (boson damping is subdominant).

Large N limits present us with sharp separation of energy scales.

Large N limits

Large NB:

O(1/NB) :

Large NF:

O(1/NF ) :

Implementation of large N limits

I will consider the case: NF = 1, NB ! 1.

(repeated indices summed).

↵ = 1 · · ·NF

i, j = 1 · · ·NB

g �! g i↵

↵j �

ji

! i↵

! ↵i

� ! �ji

Focus today on SO(N) global symmetry

! !⇤

e�s⇤ ⇤

e�s⇤

! ⇤

e�s⇤

a) b)

c)

kk

k

FIG. 4. Examples of possible schemes for decimating high energy modes. Scheme a), which we adopted in ? , integrates outshells in ! but integrates out all momenta in a given frequency shell. Scheme b), which integrates out both frequencies andmomenta, is better for our purposes (as explained below and in §6), and we adopt it in this paper. Scheme c) is recommendedas an assignment for graduate students one wishes to avoid.

L = i [@⌧ + µ � ✏(ir)] i +� NB

i i j j

L� = tr

✓m2

��2 + (@⌧�)2 + c2

⇣~r�

⌘2

◆

+�

(1)

�

8NBtr(�4) +

�(2)

�

8N2

B

(tr(�2))2

L ,� =gpNB

i j�ji (II.1)

The (spinless) fermions are in an NB-vector i, while the scalar �ji is an NB ⇥ NB complex matrix. We take the

global symmetry group to be SU(NB), and as in ? , we will set �(1)

� = 0. This choice is technically natural (as there

is an enhanced SO(N2

B) symmetry broken softly by the Yukawa coupling), and makes the analysis far more tractable.In this section, we describe the perturbative RG approach to studying this system, following ? . We start with

the same RG scaling as in that paper, scaling boson and fermion momenta di↵erently as in Figure 3 (in a way thatis completely determined by the scaling appropriate to the relevant decoupled fixed points at g = 0). However, wedepart in one important way from the philosophy of ? - instead of decimating in ! between ⇤ and ⇤ � d⇤ at eachRG step, but integrating out all momenta (as in Figure 4 a)), we instead do a more ‘radial’ decimation, integratingout shells in both ! and k (as in Figure 4 b)). This introduces two UV cuto↵s in the problem, ⇤ and ⇤k. We findthis procedure superior because it avoids the danger of retaining very high energy modes (at large k) at late stepsof the RG. While of course observables will agree in the di↵erent schemes, aspects of the physics which are obscurein the scheme of Figure 4 a) become manifest in the scheme we have chosen here. This elementary (but sometimesconfusing) point is discussed in more detail in §6, which can be read more or less independently of the rest of thepaper.

The large NB RG equations are quite simple. The Yukawa vertex renormalization and the boson wave-functionrenormalization due to fermion loops are both O(1/NB) e↵ects. Therefore, at leading order, the boson is governedby an O(N2

B) Wilson-Fisher fixed point, while the fermion wave-function renormalization governs the non-trivial betafunctions. Here and throughout the paper we will use the notation ‘`’ to represent the component of the fermionmomentum perpendicular to the fermi surface and ‘!’ to represent fermion energies. Writing

L� = �2(!2 + c2k2),

L = (1 + �Z) †i! � (v + �v) †` ,

L � = (g + �g)� † ,

�Z ⌘ Z � 1, �v ⌘ v0

Z � v, �g ⌘ g0

Z � g , (II.2)

we simply need to compute the logarithmic divergences in �Z and �v to find the one-loop running. �Z and �v are

4

Large NB action

i, j = 1 · · ·NB

Emergent SO(NB2) symmetry when �(1)� = 0

This symmetry is softly broken: i.e., only at O(1/N2B).

Large NB solution

NB ! 1 : ⌃ =

1) Fermi velocity vanishes at infinite NB. 2) Green function has branch cut spectrum. 3) Damping of order parameter is a 1/NB

effect.

Properties of the solution: G(k,!) =1

!1�✏/2f⇣!k;NB

⌘✏ = 3� d

f⇣!k;NB ! 1

⌘= 1

The theory can smoothly be extended to d=2. The theory describes infinitely heavy, incoherent fermionic quasiparticles.

The solution matches on to perturbation theory in the UV.

Large N limits

First 1/NB correction: Landau damping of the boson due to incoherent particle-hole fluctuations.

The form of Landau damping here very different than in the standard approach:

Landau damping due to ill-defined quasiparticles is weaker.

�(k,!) = ! log

�!2 � v2F k

2�

The theory can be solved at large N even in d=2+1.

This leads to a broad energy regime governed by a z=2 boson.

NF

NB

Hertz

Our theory

??

Real materials

Scaling landscape

Moral of the story: there may be several distinct asymptotic limits with different scaling behaviors, dynamic crossovers in this problem.

II. RG analysis

Scaling near the upper-critical dimensionThe unconventional large N we use here is in part inspired

by AdS/CFT examples.

What scaling do we use?

Fermi interactions. We describe the correlation functionsof both the boson and fermion degrees of freedom at thenon-Fermi liquid fixed point; they di⇥er from the resultsobtained in alternative treatments. In §4, we re-introducethe four-Fermi interactions and describe subtleties associ-ated with log2 divergences that arise in their presence. In§5, we discuss controlled large N theories where the sub-tleties of §4 do not arise, and we find fixed points whichgeneralize those of §3 to include four-Fermi interactions.We show that these fixed points have no superconductinginstabilities. We close with a discussion of open issues in§6. Explicit calculations which we refer to in the mainbody are presented in several appendices.

II. EFFECTIVE ACTION AND SCALINGANALYSIS

Let ⌥� denote a fermion field with spin ⌅ =⇤, ⌅, anddispersion �(k), defined relative to the Fermi level. Let ⌃be the scalar boson field corresponding to the order pa-rameter for the quantum phase transition. The e⇥ectivelow energy Euclidean action consists of a purely fermionicterm, a purely bosonic term and a Yukawa coupling be-tween bosons and fermions:

S =

⇤d⇧

⇤ddx L = S⌅ + S⇤ + S⌅�⇤

L⌅ = ⌥� [�⇥ + µ� �(i⇧)]⌥� + ⇥⌅⌥�⌥��⌥��⌥�

L⇤ = m2⇤⌃

2 + (�⇥⌃)2 + c2

� ⇧⌃

⇥2+

⇥⇤

4!⌃4

S⌅,⇤ =

⇤dd+1kdd+1q

(2⇤)2(d+1)g(k, q)⌥(k)⌥(k + q)⌃(q), (1)

where repeated spin indices are summed. The first term,L⌅, represents a Landau Fermi liquid, with weak residualself-interactions incorporated in forward and BCS scat-tering amplitudes. The second term represents an in-teracting scalar boson field with speed c and mass m⇤

(which corresponds to the inverse correlation length thatvanishes as the system is tuned to the quantum criti-cal point). The third term is the Yukawa coupling be-tween the fermion and boson fields and is more naturallydescribed in momentum space. The quantity g(k, q) isa generic coupling function that depends both on thefermion momentum k, as well as the momentum trans-fer q (we have suppressed spin indices for clarity). For aspherically symmetric Fermi system, the angular depen-dence of g(k, q) for |k| = kF can be decomposed into dis-tinct angular momentum channels, each of which marksa di⇥erent broken symmetry. Familiar examples includeferromagnetism (angular momentum zero) and nematicorder (angular momentum 2). More generally, the cou-pling can be labelled by the irreducible representation ofthe crystal point group and it respects symmetry trans-formations under which ⌃ and ⌥⌥ both change sign.

Before proceeding, we make a few comments on the ori-gins of the e⇥ective action above. One starts with a the-ory involving fermions interacting at short distances with

qx

qy

qx

qy

(a) (b)

(c)

�k

�k + �q

�q

empty states

filled states

empty states

filled states

FIG. 1. Summary of tree-level scaling. High energy modes(blue) are integrated out at tree level and remaining low en-ergy modes (red) are rescaled so as to preserve the boson andfermion kinetic terms. The boson modes (a) have the lowenergy locus at a point whereas the fermion modes (b) havetheir low energy locus on the Fermi surface. The most rele-vant Yukawa coupling (c) connects particle-hole states nearlyperpendicular to the Fermi surface; all other couplings areirrelevant under the scaling.

strong repulsive forces. These interactions are decoupledby an auxiliary boson field ⌃ representing a fermion bilin-ear, and the partition function is obtained by averagingover all possible values of both the fermion and bosonfields. Initially, the auxiliary field has no dynamics andis massive. However, as high energy modes of the ma-terial of interest are integrated out, radiative correctionsinduce dynamics for the bosons. In a Wilsonian theory,the dynamics are encapsulated only in local, analytic cor-rections to the bare action. This mode elimination is con-tinued until eventually, the UV cuto⇥ � ⇥ EF representsthe scale up to which the quasiparticle kinetic energy canbe linearized about the Fermi level. At these low ener-gies, and in the vicinity of the quantum critical pointwhere the field ⌃ condenses, it is legitimate to view ⌃as an independent, emergent fluctuating field that cou-ples to the low energy fermions via a Yukawa coupling aswritten above23. This will be the point of departure ofour analysis below.We first describe a consistent scaling procedure for the

action in Eq. 1. The key challenge stems from thefact that the boson and fermion fields have vastly dif-ferent kinematics. Our bosons have dispersion relationk20 = c2k2+m2

⇤, so that low energies correspond as usualto low momentum, and their scaling is that of a standardrelativistic field theory where all components of momen-tum scale the same way as k0. By contrast, the fermiondispersion relation is k0 = �(k) � µ, so their low en-ergy states occur close to the Fermi surface (Fig. 1).Moreover, the Yukawa coupling between the two sets offields must conserve energy and momentum in a coarse-

2

Wednesday, June 26, 13

UV theory: decoupled Fermi liquid + nearly free bosons (g=0).

Scaling must contend with vastly different kinematics of bosons and fermions.

Fermions: low energy = Fermi surface.

Bosons: low energy = point in k-space.

-> anisotropic scaling.

-> isotropic scaling. d=3 is the upper critical dimension.

} [g] =1

2(3� d)

Note for experts: this can also be seen readily in z=1 “patch” scaling.

Renormalization group analysis

Integrate out modes with energy ⇤e�t < E < ⇤

Following Wilson, we will integrate out only high-energy modes to obtain RG flows.

K. G. Wilson

This is a radical departure from the standard approach to this problem.

UV cutoff: scale below which fermion dispersion can be linearized (with a well-defined Fermi velocity).

⇤ =

Integrate out modes with momenta⇤k / ⇤

⇤ke�t < k < ⇤k

I will present RG results at large Nb.

Renormalization group analysis

��4 term :

g � term :

a > 0

b > 0

�⇤ = O(✏)

g⇤ = O(p✏)

d��

dt= ✏�� a�2

�

dg

dt=

✏

2g � bg3

Naive fixed point:

RG flows at one-loop:

Fermi velocity:

dv

dt= �cg2S(v) S(v) ⇠ sgn(v)

v⇤ = 0

✏ = 3� d NB � 1

A few heretical remarks

Landau damping, which plays a central role in Hertz’s theory never contributes to RG running!

(a)ab

a

ab

b

(b)a b

ab

a

(c)ab

a

a

b

ab

b

The same is not true for the fermion. It obtains non-trivial wave-function renormalization:

(a)ab

a

ab

b

(b)a b

ab

a

(c)ab

a

a

b

ab

b

⌃t+dt(!, k)� ⌃t(k) = i!g2agdt

This effect (and vertex corrections) produce the NFL.

a positive constant.

This effect gives rise to anomalous dimensions for the fermions (i.e. Green functions have branch cuts, no poles) and to velocity running.

Properties of the naive fixed point

Fermion 2-pt function takes the form:

f(x) = scaling functionG(!, k) =

1

!1�2� f

✓!

k?

◆

v/c: vanishes before the system reaches the fixed point!

This feature shuts down Landau damping.

Is this too much of a good thing?? Infinitely heavy, incoherent fermions + non-mean-field critical exponents!

Consistent with the large NB solution.

Introducing leading irrelevant couplings

w ~ band curvature✏(k)� µ = v`+ w`2 + · · ·

dv

dt= �cg2S(v) S(v) ⇠ sgn(v)

RG flow equations

dw

dt= �w

w cannot be neglected below an emergent energy scale:

µ⇤ ⇠ ⇤e�↵v0/g20 , ↵ ⇠ O(1)

We don’t know what happens below this scale (Lifshitz transition?)

w is dangerously irrelevant

Summary so far

We studied a metal near a nematic quantum critical point and found non-Fermi liquid phenomena via 1) large N and 2)RG methods.

The fixed point corresponds to an infinitely heavy incoherent soup of fermions + order parameter fluctuations.

This fixed point is unstable, but it governs scaling laws over a broad range of energy/temperature scales.

Both methods produce consistent results.

Summary so far

EF

!LD ⇠ gEF1pN

Wilson-Fisher+ dressed non-Fermi liquid

Scale where Landau damping sets in

???

III. Superconducting “domes”

Effect of QCP on pairing

The order parameter has two effects on the fermions:

1) It destroys fermion quasiparticle. Bad for pairing.

2) It enhances the pairing interaction: (like a critical optical phonon). Good for pairing.

Which of these effects dominates??

1 RG approach to BCS instabilities

In the renormalization of finite density fermions coupled to a boson, we found that it is crucial toinclude a tree-level counterterm for forward scattering from ( † )2. Such a counterterm cancels alog divergence in the 4-Fermi interaction mediated by boson exchange. With this counterterm inplace, all nonlocal renormalization e↵ects are cancelled, and renormalization for NFLs proceedsin the usual fashion.

This procedure should also be applied to the BCS channel, and here we describe the results.We will find that this will cancel the multilogs in BCS loop diagrams, rendering the RG linearin logs. In this approach, the multilogs are equivalent to insertions of the 4 counterterm assubdivergences inside a loop diagram. This is also familiar from relativistic theories: multilogs inhigher loop diagrams are simply cancelled against counterterms associated to lower loop diagrams,and they do not contribute to the beta functions. The only new element here is that the requiredcounterterm appears already at tree level, something that does not occur in the relativistic case.

1.1 Tree level running of the BCS coupling

Our starting tree level interaction potential is

Lint = g0

� † +1

2�0

( † )2 , (1.1)

where the subindex ‘0’ denotes a bare quantity. Although some of the physics is more transparentin the two-cuto↵ approach, let us for simplicity consider a single cuto↵ on momenta, |~q| < ⇤, or,equivalently, dimensional regularization.

We now integrate a shell of high momentum bosons. This generates a contribution to ( † )2,as shown in Figure 1.

Figure 1: Boson-mediated 4 interaction at tree level.

1

vv v

v






Lint = g0

� † +1

2�0

( † )2 , (1.1)




1

vv v

v

Figu

re2:

One

loop

diagramswithlog2

⇤depe

ndence.

The

doub

lelogs

cancel

whenad

ding

thetw

odiagramson

each

column.

theon

eloop

reno

rmalizationof

thecubicvertex

g�

† ,w

hile

thesecond

oneissim

plyaferm

ion

wavefun

ctioninsertion.

Inpa

rticular,from

thecoun

terterm

wewill

geta

(g2

log⇤)��⇠

g4log2

⇤.

Now

consider

thetw

odiagramson

thebo

ttom

;theyarebo

thprop

ortio

naltog4

log2

⇤,w

here

one

factor

oflog⇤comes

asbe

fore

in(1.3),while

theotheron

ecomes

from

theferm

ionloop

.Clearly,

thefirst

diagram

onthebo

ttom

cancelsthe(g

2

log⇤)��contrib

utionfrom

thefirst

diagram

onthetop,

andthesameregardingthesecond

diagramson

thetopan

dbo

ttom

ofthefig

ure.

As

aresult,

allof

thesediagramsareagainlin

earin

logs

whenexpressedin

term

sof

theph

ysical

coup

ling�,

⇠�g

2

log⇤.

(1.7)

Asweexpe

ct,thesecancellatio

nscontinue

toha

ppen

inotherloop

diagrams,

themultilogs

beingjust

aconsequenceof

thetree

levelcoun

terterm

foun

din

theprevious

section.

Letus

illustratethiswith

thetriple

logs

aton

eloop

,sho

wnin

Figu

re3.

Forexam

ple,

thefirst

diagram

inthefig

uregivesafactor

⇠g4

log3

⇤;a

log2

⇤comes

from

theplan

ewavedecompo

sition,

while

theextralog⇤comes

from

theremaining

loop

integral

over

thetw

ointernal

ferm

ionlin

es.In

the

second

diagram

wewill

have

ag2

log⇤from

theloop

times�+��

from

the4-ferm

ionvertex.The

3

v

vv

v

Figure 2: One loop diagrams with log2 ⇤ dependence. The double logs cancel when adding the twodiagrams on each column.

the one loop renormalization of the cubic vertex g� † , while the second one is simply a fermionwavefunction insertion. In particular, from the counterterm we will get a

(g2 log⇤)�� ⇠ g4 log2 ⇤ .

Now consider the two diagrams on the bottom; they are both proportional to g4 log2 ⇤, where onefactor of log⇤ comes as before in (1.3), while the other one comes from the fermion loop. Clearly,the first diagram on the bottom cancels the (g2 log⇤)�� contribution from the first diagram onthe top, and the same regarding the second diagrams on the top and bottom of the figure. Asa result, all of these diagrams are again linear in logs when expressed in terms of the physicalcoupling �,

⇠ �g2 log⇤ . (1.7)

As we expect, these cancellations continue to happen in other loop diagrams, the multilogsbeing just a consequence of the tree level counterterm found in the previous section. Let usillustrate this with the triple logs at one loop, shown in Figure 3. For example, the first diagramin the figure gives a factor ⇠ g4 log3 ⇤; a log2 ⇤ comes from the plane wave decomposition, whilethe extra log⇤ comes from the remaining loop integral over the two internal fermion lines. In thesecond diagram we will have a g2 log⇤ from the loop times �+ �� from the 4-fermion vertex. The

3

v

v

v

vFigure 2: One loop diagrams with log2 ⇤ dependence. The double logs cancel when adding the twodiagrams on each column.


(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

vv

vv

Figure 3: One loop diagrams with log3 ⇤ dependence. The triple logs cancel in the sum of the diagrams

last diagram gives the usual log⇤ from the BCS calculation times (� + ��)2. Adding all thesecontributions gives a term

⇠ �2 log⇤ , (1.8)

where again the multilogs cancel.Combining (1.6), (1.7) and (1.8) gives a one loop beta function for the BCS coupling

�� = g2 + �2 + c1

g2� (1.9)

where c1

is some constant which also depends on group theory factors (ignored so far). The lastterm is typically small, and the solution to the beta function with the first two terms correctlyreproduces the enhanced BCS Landau pole,

⇤BCS ⇠ e�const/g⇤ . (1.10)

We note that (1.9) is very similar to the result obtained by Son, although the approaches areslightly di↵erent.

1.3 Comparison to the approach with multilogs

Let us now compare our results to the approach where the BCS beta function contains multilogs(Fitzpatrick et al).

Here one neglects the tree level log divergence from the boson exchange (1.3) and one loopcontributions such as the last diagrams in Figure 2. Without such diagrams, the multilogs are notcancelled and one obtains beta functions of the form

�˜� = (�+ g2t)2 + c

2

�+ . . . (1.11)

Clearly part of the di↵erence between the two approaches is an issue of coupling redefinition:setting � = �+ g2t gives

�� = g2 + �2 + c2

(�� g2t) + . . . . (1.12)

The first two terms here are the same as in (1.9) and hence this also leads to an enhanced BCSgap. However, the two approaches are physically di↵erent, since here one is neglecting some of the

4

vv

vv

vv



⇠ �2 log⇤ , (1.8)


�� = g2 + �2 + c1

g2� (1.9)

where c1


⇤BCS ⇠ e�const/g⇤ . (1.10)





�˜� = (�+ g2t)2 + c

2

�+ . . . (1.11)


�� = g2 + �2 + c2

(�� g2t) + . . . . (1.12)


4

vv v v

v v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

v

v

v

v vv v

v

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

k

�k �k0

k0

(a)ab

a

ab

b

(b)a b

ab

a

(c)ab

a

a

b

ab

b

Perturbation theory near d=3

There are log-squared divergences in the Cooper channel in the vicinity of the quantum critical point.






Lint = g0

� † +1

2�0

( † )2 , (1.1)




1

vv v

v






Lint = g0

� † +1

2�0

( † )2 , (1.1)




1

vv v

v

Figu

re2:

One

loop

diagramswithlog2

⇤depe

ndence.

The

doub

lelogs

cancel

whenad

ding

thetw

odiagramson

each

column.

theon

eloop

reno

rmalizationof

thecubicvertex

g�

† ,w

hile

thesecond

oneissim

plyaferm

ion

wavefun

ctioninsertion.

Inpa

rticular,from

thecoun

terterm

wewill

geta

(g2

log⇤)��⇠

g4log2

⇤.

Now

consider

thetw

odiagramson

thebo

ttom

;theyarebo

thprop

ortio

naltog4

log2

⇤,w

here

one

factor

oflog⇤comes

asbe

fore

in(1.3),while

theotheron

ecomes

from

theferm

ionloop

.Clearly,

thefirst

diagram

onthebo

ttom

cancelsthe(g

2


utionfrom

thefirst

diagram

onthetop,


diagramson

thetopan

dbo

ttom

ofthefig

ure.

As

aresult,

allof


earin

logs

whenexpressedin

term

sof

theph

ysical

coup

ling�,

⇠�g

2

log⇤.

(1.7)

Asweexpe

ct,thesecancellatio

nscontinue

toha

ppen

inotherloop

diagrams,

themultilogs

beingjust

aconsequenceof

thetree

levelcoun

terterm

foun

din

theprevious

section.

Letus

illustratethiswith

thetriple

logs

aton

eloop

,sho

wnin

Figu

re3.

Forexam

ple,

thefirst

diagram

inthefig

uregivesafactor

⇠g4

log3

⇤;a

log2

⇤comes

from

theplan

ewavedecompo

sition,

while

theextralog⇤comes

from

theremaining

loop

integral

over

thetw

ointernal

ferm

ionlin

es.In

the

second

diagram

wewill

have

ag2

log⇤from

theloop

times�+��

from

the4-ferm

ionvertex.The

3

v

vv



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

v

v

v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

vv

vv



⇠ �2 log⇤ , (1.8)


�� = g2 + �2 + c1

g2� (1.9)

where c1


⇤BCS ⇠ e�const/g⇤ . (1.10)





�˜� = (�+ g2t)2 + c

2

�+ . . . (1.11)


�� = g2 + �2 + c2

(�� g2t) + . . . . (1.12)


4

vv

vv

vv



⇠ �2 log⇤ , (1.8)


�� = g2 + �2 + c1

g2� (1.9)

where c1


⇤BCS ⇠ e�const/g⇤ . (1.10)





�˜� = (�+ g2t)2 + c

2

�+ . . . (1.11)


�� = g2 + �2 + c2

(�� g2t) + . . . . (1.12)


4

vv v v

v v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

v

v

v

v vv v

v

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

k

�k �k0

k0






Lint = g0

� † +1

2�0

( † )2 , (1.1)




1

vv v

v






Lint = g0

� † +1

2�0

( † )2 , (1.1)




1

vv v

v

Figu

re2:

One

loop

diagramswithlog2

⇤depe

ndence.

The

doub

lelogs

cancel

whenad

ding

thetw

odiagramson

each

column.

theon

eloop

reno

rmalizationof

thecubicvertex

g�

† ,w

hile

thesecond

oneissim

plyaferm

ion

wavefun

ctioninsertion.

Inpa

rticular,from

thecoun

terterm

wewill

geta

(g2

log⇤)��⇠

g4log2

⇤.

Now

consider

thetw

odiagramson

thebo

ttom

;theyarebo

thprop

ortio

naltog4

log2

⇤,w

here

one

factor

oflog⇤comes

asbe

fore

in(1.3),while

theotheron

ecomes

from

theferm

ionloop

.Clearly,

thefirst

diagram

onthebo

ttom

cancelsthe(g

2


utionfrom

thefirst

diagram

onthetop,


diagramson

thetopan

dbo

ttom

ofthefig

ure.

As

aresult,

allof


earin

logs

whenexpressedin

term

sof

theph

ysical

coup

ling�,

⇠�g

2

log⇤.

(1.7)

Asweexpe

ct,thesecancellatio

nscontinue

toha

ppen

inotherloop

diagrams,

themultilogs

beingjust

aconsequenceof

thetree

levelcoun

terterm

foun

din

theprevious

section.

Letus

illustratethiswith

thetriple

logs

aton

eloop

,sho

wnin

Figu

re3.

Forexam

ple,

thefirst

diagram

inthefig

uregivesafactor

⇠g4

log3

⇤;a

log2

⇤comes

from

theplan

ewavedecompo

sition,

while

theextralog⇤comes

from

theremaining

loop

integral

over

thetw

ointernal

ferm

ionlin

es.In

the

second

diagram

wewill

have

ag2

log⇤from

theloop

times�+��

from

the4-ferm

ionvertex.The

3

v

vv

v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

v

v

v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

vv

vv



⇠ �2 log⇤ , (1.8)


�� = g2 + �2 + c1

g2� (1.9)

where c1


⇤BCS ⇠ e�const/g⇤ . (1.10)





�˜� = (�+ g2t)2 + c

2

�+ . . . (1.11)


�� = g2 + �2 + c2

(�� g2t) + . . . . (1.12)


4

vv

vv

vv



⇠ �2 log⇤ , (1.8)


�� = g2 + �2 + c1

g2� (1.9)

where c1


⇤BCS ⇠ e�const/g⇤ . (1.10)





�˜� = (�+ g2t)2 + c

2

�+ . . . (1.11)


�� = g2 + �2 + c2

(�� g2t) + . . . . (1.12)


4

vv v v

v v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

v

v

v

v vv v

v

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

k

�k �k0

k0

+ + · · ·

�eff (!) ⇠ �+ �g2 log2⇤

!

�+ · · ·

Naively summing these up in the spirit of BCS, we find a parametrically higher instability scale:

!⇤ ⇠ exp (�1/|g|)

Exponentially enhanced pairing at QCP is encouraging.

However, log2 divergences pose a significant challenge for the RG.

Notions of universality and fixed points could be destroyed.

RG time: t = log [⇤/!]

When log2 divergences are present, RG flows are explicitly t-dependent:

d

dt(coupling) / t

How to proceed?

Log2 divergences and universality

Key point: ordinary log-divergence is hidden in a tree-level interaction:

RG Analysis






Lint = g0

� † +1

2�0

( † )2 , (1.1)




1

vv v

v1 RG approach to BCS instabilities





Lint = g0

� † +1

2�0

( † )2 , (1.1)




1

vv v

v

Figu

re2:

One

loop

diagramswithlog2

⇤depe

ndence.

The

doub

lelogs

cancel

whenad

ding

thetw

odiagramson

each

column.

theon

eloop

reno

rmalizationof

thecubicvertex

g�

† ,w

hile

thesecond

oneissim

plyaferm

ion

wavefun

ctioninsertion.

Inpa

rticular,from

thecoun

terterm

wewill

geta

(g2

log⇤)��⇠

g4log2

⇤.

Now

consider

thetw

odiagramson

thebo

ttom

;theyarebo

thprop

ortio

naltog4

log2

⇤,w

here

one

factor

oflog⇤comes

asbe

fore

in(1.3),while

theotheron

ecomes

from

theferm

ionloop

.Clearly,

thefirst

diagram

onthebo

ttom

cancelsthe(g

2


utionfrom

thefirst

diagram

onthetop,


diagramson

thetopan

dbo

ttom

ofthefig

ure.

As

aresult,

allof


earin

logs

whenexpressedin

term

sof

theph

ysical

coup

ling�,

⇠�g

2

log⇤.

(1.7)

Asweexpe

ct,thesecancellatio

nscontinue

toha

ppen

inotherloop

diagrams,

themultilogs

beingjust

aconsequenceof

thetree

levelcoun

terterm

foun

din

theprevious

section.

Letus

illustratethiswith

thetriple

logs

aton

eloop

,sho

wnin

Figu

re3.

Forexam

ple,

thefirst

diagram

inthefig

uregivesafactor

⇠g4

log3

⇤;a

log2

⇤comes

from

theplan

ewavedecompo

sition,

while

theextralog⇤comes

from

theremaining

loop

integral

over

thetw

ointernal

ferm

ionlin

es.In

the

second

diagram

wewill

have

ag2

log⇤from

theloop

times�+��

from

the4-ferm

ionvertex.The

3

v

vv

v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

v

v

v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

vv

vv



⇠ �2 log⇤ , (1.8)


�� = g2 + �2 + c1

g2� (1.9)

where c1


⇤BCS ⇠ e�const/g⇤ . (1.10)





�˜� = (�+ g2t)2 + c

2

�+ . . . (1.11)


�� = g2 + �2 + c2

(�� g2t) + . . . . (1.12)


4

vv

vv

vv



⇠ �2 log⇤ , (1.8)


�� = g2 + �2 + c1

g2� (1.9)

where c1


⇤BCS ⇠ e�const/g⇤ . (1.10)





�˜� = (�+ g2t)2 + c

2

�+ . . . (1.11)


�� = g2 + �2 + c2

(�� g2t) + . . . . (1.12)


4

vv v v

v v



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3



(g2 log⇤)�� ⇠ g4 log2 ⇤ .


⇠ �g2 log⇤ . (1.7)


3

v

v

v

v vv v

v(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

k

�k �k0

k0

To see this: consider the angular momentum basis:

V (`) =2`+ 1

2

Z 1

�1d(cos ✓)�(✓)P`(cos ✓)

=g2

(k0 � k00)2 + c2|k � k0|2

k

�k �k0

k0

RG Analysis

S-wave case for simplicity:

V (0) =

g2

2

Z 1

�1

d(cos ✓)

(k0 � k00)2+ |k � k0|2

(` = 0)

Momentum transfer: q2 ⌘ |k � k0|2 ' 2k2F (1� cos ✓)

d(cos ✓) = �d(q2)/2k2F

Note that kF as a scale is crucial here: it converts an angle to a momentum transfer q imparted by the boson:

V (0) = � g2

4k2F

Z 2k2F

�2k2F

d(q2)

(k0 � k00)2 + q2

RG Analysis

Idea: treat V(0) in a Wilsonian spirit, decimating only “fast” q modes.

The contribution from decimating a thin shell

�V (0) = �g2d⇤

⇤

⇤� d⇤ < q < ⇤ :

Adding the Fermi liquid contribution, which is present even when g=0, we obtain the RG flow for the BCS pairing amplitude:

dV (`)

dt= �

⇥g2 + V (`)2

⇤

Note that flows are not explicitly t-dependent. It’s solution contains the exponential enhancement of pairing that we “guessed” using perturbation theory.

[dt = d log⇤]

Accounting for anomalous dimensions

Last step in the RG: after integrating out a thin shell of modes, we must rescale fields.

Fermion fields acquire anomalous dimension in the presence of the boson. This will affect the flow of BCS couplings after field rescaling.

Final result: competition between 2 effects.

(a)ab

a

ab

b

(b)a b

ab

a

(c)ab

a

a

b

ab

b

dV (`)

dt= �

⇥g2 + V (`)2

⇤� agg

2V (`)

ag =1

2⇡2(1 + |v|/c)

{

enhancement of pairing by scalar

{suppression due to qp destruction.

Previous work on dense QCD

D.T. Son considered superconductivity of quarks due to gluon exchange in finite density QCD.

Similar log2 divergences occur here.

In a remarkable paper, Son concluded (based on intuitive reasoning):

d

dt� = �(g2 + �2)

Our result agrees (upto anomalous dimensions) with Son’s.

Similar recent work: Metlitski et al., 1403.3694.

“Phase diagrams” at large Nb

Consider the large Fermion velocity limit: c/v ->0.

Taking g, v large but with fixed (and small)

we can still control the theory.

⇤BCS ⇠ kF e�1/

p↵

⇤NFL ⇠ kF e�1/↵

In the large v limit, with “moderately” large N, we can arrange for BCS instability to occur above the scale where our theory breaks down (due to Landau damping).

↵ = g2/v

Note: there is a subdominant CDW instability also present in this theory.

⇤LD ⇠p↵kF

vpN

Conclusions and outlook

Outstanding goal: to demonstrate enhanced superconductivity out of a non-Fermi liquid.

Near d=3+1, superconductivity forms at higher energy scales than the formation of a non-Fermi liquid in the large Nb limit. Anomalous dimension contribution is small. Also Landau damping is unimportant.

Figure 5: Schematic representation of the phase diagram as a function of the parameter u that

tunes to criticality and temperature, and di↵erent values of Nc

. For small Nc

the superconducting

region (SC) completely coveres the non-Fermi liquid (NFL) region. As Nc

increases, the NFL region

extends beyond the SC phase

Ignoring for the moment other possible instabilities, the phase diagram of the theory is

illustrated schematically in Figure 5. Here u is a control parameter that tunes the boson to

criticality, in the way described in §3.3.It is very encouraging that already the simple theory considered in this work, in its weak

coupling expansion, displays such a rich phenomenology. This model can be potentially

relevant for various strongly correlated systems, both in d = 3 space dimensions, or by

developing higher orders of the ✏–expansion to try to model quasiplanar systems. Changing

the parameterNc

, the generalization of SU(2) spin, leads to a very di↵erent interplay between

superconductivity and quantum criticality. It is intriguing that compounds dominated by

single-band or multi-band interactions also display di↵erent phase diagrams [6].

The prediction of a NFL regime above the SC dome for moderately large N

c

should be

highlighted.5 Over a broad range of initial parameters the theory flows to this regime, which

incorporates Landau damping backreaction on the Fermi surface, and where the velocity and

quasiparticle residue flow to zero as described in (6.6). This robust prediction makes this

model an ideal candidate to study how quantum criticality can drive superconductivity and

strange metallic behavior, a subject of importance for understanding high T

c

materials.

Finally, it will clearly be important to study other possible IR phases of the theory. We

have focused on the SC instability, but there can be other instabilities such as stripe order,

5Estimating the critical value of Nc for which the NFL extends above the SC dome requires taking into

account order one factors which we haven’t done here, especially in the calculation of the gap. For ✏ ⇠ 0.1,already for Nc ⇠ 5 NFL e↵ects start to become important near �.

29

Near d=2+1, other possibilities may occur. !With both overdamped fermions and bosons, pairing can occur out of a non-Fermi liquid with a large anomalous dimension (work in progress).

Documents

these phase diagrams? captures the “non-Fermi liquid ...conferences.physics.illinois.edu/JointPerimeter2014/talks/raghu... · rule (together with Maxwell’s equations) ... NFL