these phase diagrams? captures the “non-fermi liquid...
TRANSCRIPT
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
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
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.
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
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
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 v v
v 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
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
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.
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
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
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
vFigure 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.
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
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
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 v v
v 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
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
v vv v
v
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
k
�k �k0
k0
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
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
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.
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
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
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 v v
v 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
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
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
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
v1 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
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.
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
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
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 v v
v 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
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
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).