Are non-Fermi-liquids stable to Cooper pairing?

Max A. Metlitski,1 David F. Mross,2 Subir Sachdev,3, 4 and T. Senthil5

1Kavli Institute for Theoretical Physics, UC Santa Barbara, CA 931062Physics Department, California Institute of Technology, Pasadena, CA 91125

3Department of Physics, Harvard University, Cambridge, MA 021384Perimeter Institute of Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada.

5Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139(Dated: March 11, 2015)

States of matter with a sharp Fermi-surface but no well-defined Landau quasiparticles arise ina number of physical systems. Examples include: (i) quantum critical points associated with theonset of order in metals; (ii) spinon Fermi-surface (U(1) spin-liquid) state of a Mott insulator;(iii) Halperin-Lee-Read composite fermion charge liquid state of a half-filled Landau level. In thiswork, we use renormalization group techniques to investigate possible instabilities of such non-Fermi-liquids in two spatial dimensions to Cooper pairing. We consider the Ising-nematic quantum criticalpoint as an example of an ordering phase transition in a metal, and demonstrate that the attrac-tive interaction mediated by the order parameter fluctuations always leads to a superconductinginstability. Moreover, in the regime where our calculation is controlled, superconductivity preemptsthe destruction of electronic quasiparticles. On the other hand, the spinon Fermi-surface and theHalperin-Lee-Read states are stable against Cooper pairing for a su�ciently weak attractive short-range interaction; however, once the strength of attraction exceeds a critical value, pairing sets in.We describe the ensuing quantum phase transition between (i) U(1) and Z2 spin-liquid states; (ii)Halperin-Lee-Read and Moore-Read states.

I. INTRODUCTION

It is well-known that ordinary metals described byFermi-liquid (FL) theory are unstable to an arbitrarilyweak attractive interaction in the BCS channel, whichleads to Cooper pairing of electrons and drives the sys-tem into a superconducting phase. The purpose of thepresent paper is to examine the stability of certain non-Fermi-liquid (nFL) states in two dimensions to Cooperpairing. We study systems where the non-Fermi-liquidbehavior arises as a result of the interaction of a gaplessbosonic mode with fermions in the vicinity of the Fermi-surface (FS). Specific examples we analyze are describedin the following subsections.

A. Quantum critical points in metals

Many correlated metals appear to possess quantumcritical points (QCPs) with fascinating properties.1–3

Frequently, there is a striking breakdown of Fermi liquidtheory in the vicinity of the QCP. Equally strikingly su-perconductivity is often but not always strengthened nearthe QCP. Indeed, a fairly common phase diagram (seeFig. 1, top), shared for instance by cuprate, pnictide andcertain heavy-fermion materials, has a superconductingdome around the putative ‘metallic’ QCP with ‘optimal’transition temperature T

c

right at the QCP. On the otherhand, there are prominent quantum critical heavy elec-tron metals such as CeCu

6�x

Aux

and YbRh2

Si2

wheresuperconductivity does not appear down to very lowtemperatures.1 It thus appears that superconductivity isenhanced at some but not all quantum critical pointsin metals. Despite this there is currently limited under-

standing of the interplay between the quantum criticality,the non-Fermi liquid ‘normal’ state, and the possible su-perconductivity. Clearly, a theory of the relationship ofsuperconductivity and quantum criticality has to accom-modate the absence of superconductivity at some andenhancement at other quantum critical points.

It is important right away to recognize that there aretwo fundamentally distinct classes of quantum critical-ity in metallic systems. They are distinguished by thefate of the electron Fermi surface as the metal undergoesthe quantum phase transition. In one class, the electronFermi surface evolves continuously through the criticalpoint but is distorted in some way. These QCPs aretypically associated primarily with the onset of a brokensymmetry characterized by a Landau order parameter ina metal. Examples include the onset of ferromagnetismor antiferromagnetism in a paramagnetic metal. Theproper theoretical framework to describe such a phasetransition is through coupling the low energy electronicdegrees of freedom at the Fermi surface to fluctuationsof the Landau order parameter.4 An alternate class ofquantum phase transitions involves a more violent trans-formation of the electronic structure where the electronFermi surface (or a sheet of it) disappears completelyon crossing the critical point.5 Surprisingly, such a dis-continuous evolution of the electron Fermi surface canhappen through a continuous phase transition. Exam-ples include the so-called Kondo breakdown transition inKondo lattices6,7 and continuous Mott metal-insulatortransitions in two8 or three dimensions.9 There is cur-rently only one known theoretical framework that yieldssuch a phase transition: this is based on slave-particlemethods and inevitably leads to a description in terms offractionalized slave-particles coupled to fluctuating emer-

arX

iv:1

403.

3694

v2 [

cond

-mat

.str-e

l] 1

0 M

ar 2

015

2

Non-Fermi-liquid

Fermi liquidSC

h�i 6= 0 h�i = 0x

xc

T

Fermi liquid

SC

Coherent electrons Quantum critical �

Fermi liquid

h�i 6= 0x

xc

T

Fermi liquid

h�i = 0

FIG. 1: Top: Conventional phase diagram of a quantum crit-ical point (QCP) associated with an order parameter �, witha superconducting dome (SC) partially overlapping the quan-tum critical region of the ‘bare’ QCP of a metal. Bottom:the phase diagram obtained in the present paper, with theSC dome fully overlapping the incipient regime of incoherentfermionic quasiparticles, while the quantum critical � fluctu-ations survive into higher temperatures in the normal state.

gent gauge fields.

In this paper we will consider examples of both kindsof quantum critical points as case studies for the relation-ship between quantum criticality, superconductivity, andnon-Fermi liquid physics. In the example studied of thefirst class, where the Fermi surface is distorted throughthe development of a broken symmetry, we show thatsuperconductivity is strongly enhanced near the criticalpoint. We suggest that this may be more generally true:order parameter fluctuations enhance superconductivity.In the example studied of the second class where the en-tire electron Fermi surface is annihilated, we argue thatsuperconductivity is suppressed. This dichotomy mayexplain the phenomenology described above where somebut not all QCPs show an enhancement of superconduc-tivity.

We begin with QCPs associated with the onset of asymmetry breaking order. Strong fluctuations of the or-der parameter present at the QCP tend to decohere theelectronic quasiparticles: as the system is tuned to thecritical point, the residue Z and the Fermi-velocity v

F

ofquasiparticles approach zero. A common feature of suchQCPs is that there exists some pairing channel in which

the order parameter fluctuations mediate attraction. Thestrength of the attraction increases as one approaches theQCP, yet the same order parameter fluctuations, whichprovide the pairing glue, also destroy the very quasiparti-cles that are trying to pair. The central question is whichof these two competing e↵ects wins. In particular, is sucha QCP in a metal inherently unstable to superconductiv-ity, as empirical observations suggest?10

In the present paper we address the above question forthe class of metallic QCPs, where the order parametercarries a wave-vector ~Q = 0 (for recent progress on the~Q 6= 0 case, see Refs. 11,12). The most familiar exam-ple of such a phase transition is the Stoner instabilityassociated with the development of ferromagnetic order.Modern developments show that due to fluctuation ef-fects the Stoner transition is likely modified at low tem-perature and becomes first order (or develops an inter-mediate spiral ordered phase).13–17 A di↵erent examplewhich does not su↵er from these complications16,18 (see,however, footnote 19) is the transition associated withthe onset of Ising-nematic order, characterized by spon-taneous breaking of a four-fold rotational symmetry ofthe lattice to a two-fold subgroup.18,20–29,31–36 The orderparameter in this case is just a real Ising field �(x). Grow-ing evidence for such order has been found in a numberof physical systems including cuprate,37–43 pnictide44–51

and ruthenate52 materials. From a theoretical viewpoint,the Ising-nematic QCP is perhaps one of the simplestphase transitions in metals. It, thus, provides a conve-nient setting for studying the interplay between quantumcriticality, nFL and pairing physics.53

We perform a systematic renormalization group (RG)analysis of the Ising-nematic QCP. Our approach uti-lizes an idea introduced by D. T. Son in his study ofquark pairing by the color gauge field in dense baryonicmatter.54 We combine the conventional Fermi-liquid RGtreatment of Refs. 55,56 with the so-called “two-patch”scaling approach of Refs. 18,57–59. Analytical control isgained through the ✏-expansion introduced in Ref. 60 andits subsequent large-N improvement.32 We find that theIsing-nematic QCP is always unstable to superconduc-tivity. In particular, attractive pairing interaction medi-ated by the order parameter fluctuations dominates overother residual short range interactions (even if they arerepulsive) and drives a pairing instability as the QCP isapproached. However, the residual short range interac-tions determine the angular momentum/spin channel inwhich the pairing instability occurs; as a result, the pair-ing symmetry is non-universal. The usual weak couplingBCS formula, T

c

⇠ exp(�1/|V |), relating the supercon-ducting T

c

to the strength of the short-range interac-tion V clearly does not hold in the vicinity of the QCP.Rather the superconductivity is strongly enhanced, andT

c

at the QCP scales in a power-law manner with thecoupling between order-parameter fluctuations and theelectrons. Thus, in this example we clearly demonstratethe importance of quantum criticality in optimizing thesuperconducting T

c

. Moreover, in the regime where our

3

calculation is controlled (small ✏), the energy scale atwhich superconductivity sets in is parametrically largerthan the energy scale at which electronic quasiparticlesare destroyed. Thus, the superconducting instability is sostrong that it preempts the nFL physics (see the bottomfigure in Fig. 1). The above results of our RG analysis arein exact agreement with a direct solution of Eliashberg-like integral equations, as is shown elsewhere by one ofus.61,62

Next, we proceed to the class of QCPs associated withannihilation of the Fermi-surface. We take as an examplethe Mott transition from a Fermi-liquid to an insulatingspin-liquid with a spinon Fermi-surface.8 Applying theRG procedure described above, we show that spinon pair-ing is suppressed both in the spinon Fermi-surface phaseand at the Mott transition itself. As a result, the Motttransition and the FL phase in its vicinity will be stableto superconductivity. We expect similar conclusions tohold for the Kondo breakdown transition in the Kondolattice. First, however, we review the construction andproperties of the spinon Fermi-surface phase.

B. Spinon Fermi-surface phase

The spinon Fermi-surface phase is an exotic Mott-insulating spin-liquid with emergent spin-1/2 fermionicspinon excitations, f

↵

(x), ↵ =", #.63 The spinon disper-sion is such that they form a Fermi-surface. This phasemay be accessed in the slave-particle (parton) treatmentof a lattice spin model, where electron spin operators ~S

i

are represented as ~Si

= 1

2

f†i↵

~�↵�

fi�

, subject to the local

constraint, f†i↵

fi↵

= 1. While the spinons are neutral un-der the physical electromagnetic field, they carry a chargeunder an emergent U(1) gauge field a

µ

, hence this phaseis also often referred to as a U(1) spin-liquid. An e↵ec-tive Lagrangian of the spinon FS phase may be writtenas,

Lf

= f†↵

[@⌧

� ia⌧

+ ✏(�ir�~a)]f↵

+1

2g2

(✏µ⌫�

@⌫

a�

)2 + . . .

(1.1)

where ✏(~k) is the spinon dispersion and the ellipses de-note additional perturbations, such as four-spinon inter-actions.

The spinon FS phase is expected to naturally arise inso-called “weak” Mott insulators - ones proximate to ametal-insulator transition. In this situation the spinonFS state may be conveniently described within a slaveparticle description of an electronic Hubbard model. Wewrite the electron operator as c

i↵

= bi

fi↵

, where b is acharge-e boson with zero spin, and f

↵

is the spinon asdescribed above. This representation introduces a U(1)gauge redundancy under which b carries gauge charge�1 and f

↵

carries gauge charge 1. We consider a state inwhich f

↵

form a Fermi surface. If in addition the boson bis condensed, we obtain the usual metallic Fermi-liquid.If, however, b is gapped, we obtain an electrical insulator

but with a spinon Fermi-surface coupled to a fluctuat-ing U(1) gauge field. This is the spinon Fermi-surfacestate introduced above. We note that right at the Mottmetal-insulator transition, the boson b is critical whilethe spinon continues to form a Fermi-surface. For now,we focus on the insulating spinon Fermi-surface phase;properties of the Mott transition will be reviewed in thenext section.

There is numerical evidence for the presence of thespinon FS spin-liquid phase in the triangular latticeHubbard model in the intermediate range of U/t.66–69

Moreover, it has been proposed as a candidate forthe quasi-2d triangular lattice organic insulators �(BEDT � TTF)

2

Cu2

(CN)3

and EtMe3

Sb[Pd(dmit)2

]2

(abbreviated ET and DMIT below).66,67 These mate-rials have an estimated spin-exchange coupling J ⇠ 250K, yet display no magnetic order down to 20 � 30 mKtemperature. Moreover, these electrical insulators, sur-prisingly, show metallic behavior in their low tempera-ture spin-susceptibility (�

s

! const)70–72 and specific-heat (C/T ! const).73,74 DMIT also exhibits metallicthermal transport at low temperature, /T ! const,75

while ET shows activated thermal transport, albeit witha rather small gap � ⇡ 0.46 K.76 Both materials can bedriven metallic by an application of a moderate pressureof ⇠ 0.4 GPa, with ET developing superconductivitybelow ⇠ 3 K on the high-pressure side.77 At ambientpressure, ET displays a phase transition (or a very rapidcrossover) at 6K,78 resulting in partial loss of low-energyexcitations as evidenced by specific heat.73 It has beensuggested that this low-temperature anomaly may be dueto a pairing instability of the spinon FS.80,81

Current theoretical understanding of the spinon FSphase is based on the following observations. The pres-ence of gapless spinon excitations in the vicinity of theFS strongly a↵ects the gauge field dynamics. The longi-tudinal fluctuations of the emergent electric field are De-bye screened by the spinon FS and become gapped. Thefluctuations of the emergent magnetic field are Landau-damped by the FS, but remain gapless. The couplingof these Landau-damped magnetic field fluctuations tospinons is expected to lead to “non-Fermi-liquid” behav-ior of the spinon FS,57,58,64,65 e.g. the anomalous scalingof specific heat C ⇠ T 2/3.

In this paper we analyze whether the spinon FS phaseis stable to BCS pairing of spinons. We first observethat the gapless fluctuations of the magnetic field medi-ate a long-range repulsive interaction in the BCS channeland hence are not expected to cause spinon pairing. In-deed, fluctuations of the magnetic field mediate a current-current interaction. The spinons in a BCS pair haveopposite momenta and opposite currents and hence, byAmpere’s law, repel. Therefore, gapless gauge field fluc-tuations suppress spinon pairing.82 However, in additionto gauge field mediated long-range interactions, short-range interactions between the spinons will generally bepresent. Depending on the microscopic details of the sys-tem, such short range interactions may be attractive in

4

the BCS channel with some angular momentum and spin.If the short range attraction is su�ciently strong, we ex-pect the spinons to pair, developing a condensate hffi(we leave the angular/spin structure of the pair wave-function implicit for now). As in an ordinary supercon-ductor, the spinon excitations acquire a gap, except pos-sibly at symmetry dictated (or accidental) point nodeson the FS. The pair condensate spontaneously breaks theemergent U(1) gauge symmetry down to a Z

2

subgroup.As a result, the gauge field becomes gapped through theHiggs mechanism. Gauge excitations now take the formof gapped vortices carrying a magnetic flux ⇡. Suchexcitations are often referred to as visons. Visons andspinons possess mutual semionic statistics. Thus, thepaired phase of spinons is just a Z

2

spin-liquid.

We confirm the above intuitive picture with a system-atic RG calculation. We show that the spinon FS phase,is, indeed, stable as long as the strength of the short-range attractive BCS interactions |V

m

| is smaller than acritical value |V

c

| for all angular momentum channels m(we employ a sign convention where V < 0 representsan attractive interaction). However, once |V

m

| > |Vc

| forsome m, the spinon FS develops an instability to pair-ing in angular momentum channel m. V

m

= Vc

, thus,marks the quantum phase transition between the U(1)spin-liquid and the Z

2

spin-liquid. We find the phasetransition to be continuous and calculate the critical ex-ponents using the ✏-expansion of Refs. 32,60. Our find-ings are contrary to previous claims83 that this phasetransition is driven first order by gauge field fluctuations.We discuss the properties of the paired phase in the vicin-ity of the transition. Right at the critical point we find(at least to the order of the ✏-expansion that we study)that most experimentally accessible properties (specificheat, uniform and finite wave-vector spin-susceptibility,spin-chirality correlations) are not modified from thosein the spinon Fermi-surface phase itself. Our findings arein exact agreement with an Eliashberg-like treatment ofthe problem.61

Previously, the pairing quantum phase transition fromthe spinon Fermi-surface state was considered in 3 di-mensions by Chung et al.

84 within an Eliashberg-like ap-proximation. Our paper presents an RG analysis directlyin 2 dimensions, although there are some qualitative sim-ilarities with the results of Chung et al.

84 In particular,Chung et al. have also concluded that a continuous pair-ing transition is possible. However, we believe that someof the results of Chung et al. are not generic. In par-ticular, Chung et al. find that pairing can only occurin angular momentum channels m � 2. In contrast, webelieve that both in 2d and 3d pairing with arbitraryangular momentum can be induced by tuning the appro-priate V

m

. Furthermore, we expect the power-law onsetof the pairing gap found by Chung et al. in 3d to bemodified by the renormalization of spinon quasiparticleresidue and Fermi-velocity. In fact, we anticipate thatthe precise critical properties of the pairing transitionin 3d will be very similar to those of the 2d Halperin-

Lee-Read phase in the presence of long-range Coulombinteractions, discussed in section ID.

C. Mott transition from a Fermi-liquid to a spinonFermi-surface phase

The Mott transition from a Fermi-liquid to a spinon FSphase is an example of a QCP where the entire electronFS disappears. As noted in the previous section, thistransition is driven by condensation of the slave boson b.The transition may be described by the e↵ective theory,

L = Lb

+ Lf

(1.2)

where the Lagrangian Lb

for the complex scalar field b is

Lb

= |(@⌧

� ia⌧

)b|2 + v2

b

|(r � i~a)b|2 + t|b|2 + u|b|4 (1.3)

and Lf

is still given by Eq. (1.1). Note that here we areconsidering a Mott transition occurring at fixed electrondensity. When t is large and positive, the boson b isgapped and can be integrated out, so the system is inthe spinon FS phase. On the other hand, when t is largeand negative, b is condensed, hbi 6= 0. As a result, thegauge field a

µ

becomes gapped via the Higgs mechanism;furthermore, the electron c

↵

= bf↵

and the spinon f↵

are identified, c↵

! hbif↵

. Thus, the system is in theordinary FL phase.

We now discuss the fate of the system when t is tunedto a critical value t

c

where b is gapless (for more de-tails, see Ref. 8, whose findings we summarize here). Ifthe fluctuations of the gauge field a

µ

are ignored thenthe spinon and boson sectors in Eq. (1.2) decouple, andthe boson sector undergoes a transition in the XY uni-versality class, while the spinon sector remains a “spec-tator” Fermi-liquid across the transition. Proceeding toinclude gauge field fluctuations, we note that the longitu-dinal electric field is again Debye screened by the spinonFermi-surface and so can be ignored. The fluctuationsof the magnetic field are again Landau-damped by thespinon Fermi-surface, but remain gapless. It turns outthat such Landau-damped gauge fluctuations do not af-fect the b-sector of the theory, which remains decoupledfrom the spinon sector and continues to be described bythe XY critical theory. On the other hand, the b-sectordoes a↵ect the low energy gauge fluctuations. Integratingthe gapless b boson at the XY critical point out, one ob-tains the following e↵ective action for the magnetic fieldfluctuations,

Sa

=1

2

Zd2~xd2~x0d⌧(r ⇥~a)(~x, ⌧)⇧(~x � ~x0)(r ⇥~a)(~x0, ⌧)

(1.4)where ⇧(~x) = v

b

�/(4⇡2|~x|), and � ⇡ 0.36 is the universalconductivity of the XY model.85 Thus, the gauge-spinonsector of the theory is described by the action

S =

Zd2xd⌧L

f

+ Sa

(1.5)

5

which will be the starting point for our theoretical anal-ysis in this paper. We note that this action coincideswith that of the Halperin-Lee-Read state with Coulombinteractions, discussed in section I D. Studying the the-ory (1.5), one finds that gauge field fluctuations turn thespinon FS at the Mott transition into a marginal Fermi-liquid with a specific heat C ⇠ �T log T , which domi-nates the overall specific heat of the system. We remindthe reader that since the physical electron c

↵

is a prod-uct of the boson b and the spinon f

↵

, the actual physi-cal electron Green’s function displays a strong deviationfrom Fermi-liquid theory at the Mott transition.

One may now ask whether the spinon FS at the Motttransition is stable to BCS pairing of spinons. Beforewe address this question, we would like to stress thatindependent of whether the spinons pair, we expect nolong-range superconductivity exactly at the Mott transi-tion. After all, at the Mott transition charge degreesof freedom are on the verge of becoming localized solong-range phase coherence will be suppressed. Instead,spinon pairing should be interpreted as a local tendencyof electrons to pair. Let us discuss the scenario wherespinon pairing does occur at the transition. In this case,as we tune the system away from the Mott transitionto t < t

c

, a condensate hcci ⇠ hbi2hffi appears, i.e.

the compressible phase adjacent to the Mott transitionis a superconductor rather than a Fermi-liquid. On theother hand, the phase with t > t

c

, where the boson bis gapped, is a Z

2

spin-liquid insulator as discussed inthe previous section. Thus, if the spinons are paired theMott transition occurs between a superconductor and aZ

2

spin-liquid insulator.86 As we approach the transitionfrom the superconducting side, both the superconductingT

c

and the superconducting condensate hcci will vanish,however, the gap to a single electron c

↵

will remain fi-nite across the transition. In contrast, if the spinon FSis stable against pairing then the single electron gap atthe transition will vanish.

With the above remarks in mind, we now summarizethe conclusions of our RG analysis. As with the spinonFS phase, we show that repulsive current-current interac-tions mediated by the gauge field suppress spinon pairingat the Mott transiton. As a result, as long as the strengthof short-range attraction between spinons |V

m

| is belowa critical value |V

c

|, the spinon FS at the Mott transitionis stable. We believe that in this regime a stronger state-ment actually holds: no spinon pairing occurs on eitherside of the Mott transition, in particular, no supercon-ductivity develops in the FL phase adjacent to the Motttransition. Thus, the Mott transition is an example of aQCP in a metal, which is stable to superconductivity.

On the other hand, once |Vm

| > |Vc

| for some m, thespinons at the Mott transition pair, developing a con-densate hffi 6= 0. Thus, in this parameter regime theMott transition occurs between a superconductor and aZ

2

spin-liquid insulator, and the single electron gap re-mains finite across the transition.

D. Halperin-Lee-Read phase

The RG formalism developed in this paper can be ap-plied to analyze the stability of yet another exotic phase:the Halperin-Lee-Read (HLR) phase. The HLR phase isa compressible phase of the quantum Hall (QH) fluid ata filling fraction ⌫ = 1/2.87 It is believed to be experi-mentally realized by the conventional 2DEG in the low-est Landau level.88 When the Landau level is half-filled,there are two magnetic flux quanta per each electron.If one performs a transformation to composite fermions(CF) by attaching two flux quanta to each electron, thecomposite fermions will, on average, see no magnetic fieldand form a Fermi-surface. Technically, flux attachmentis performed with an aid of a Chern-Simons (CS) U(1)gauge field a

µ

, leading to the action

S =

Zd2xd⌧(L

f

+ LCS

) + SU

,

Lf

= f†[@⌧

� ia⌧

� 1

2m(@

i

� iai

+ iAi

)2]f (1.6)

LCS

=i

2(4⇡)✏µ⌫�

aµ

@⌫

a�

(1.7)

SU

=1

2

Zd2~xd2~x0d⌧f†f(~x, ⌧)U(~x � ~x0)f†f(~x0, ⌧)

(1.8)

Here, f(x) is the composite fermion operator, ~A is thevector potential for the external magnetic field and U(~x)is the microscopic electron-electron interaction potential.Integration over a

⌧

produces the constraint,

r ⇥ ~a = 2(2⇡)f†f (1.9)

linking the magnetic flux density of the CS field aµ

tothe electron density f†f . This constraint can be used torewrite S

U

in terms of r ⇥ ~a.At ⌫ = 1/2, the flux of the CS gauge field a

µ

on aver-age cancels the external magnetic field, however, fluctua-tions of a

µ

about the average flux persist. The dynamicsof a

µ

are nearly the same as in the spinon FS phase withthe longitudinal electric field Debye screened and gapped,and the magnetic field Landau damped and gapless. Asthe electric field is gapped, the CS term in Eq. (1.7) isirrelevant in the RG sense (more precisely, it generates acharge-current interaction of composite fermions which issuppressed in the small momentum limit compared to thecurrent-current interaction). Therefore, the low-energye↵ective theory of the HLR phase is nearly identical tothat of the spinon FS phase when the microscopic elec-tron interaction U(~x) in the QH fluid is short ranged.For a power law interaction,

U(~x) ⇠ 1

|~x|1+✏

, (1.10)

with ✏ < 1, the electron density fluctuations and hencethe gauge field fluctuations are suppressed.87 In fact,

6

for ✏ < 0, the composite fermion quasiparticles remainsharply defined, while for Coulomb interaction, ✏ = 0,the HLR phase is believed to be a marginal Fermi-liquidwith a specific heat C ⇠ �T log T .60,87 For ✏ > 0, theHLR phase is a true nFL,65,87,89,90 with a power law spe-cific heat C ⇠ T 2/(2+✏), however, the theory is underanalytical control in the limit ✏ ⌧ 1.32,60.

In passing, we note that the HLR phase may alter-nately be described within a slave particle approach thatexposes the conceptual similarity to a spin-liquid Mottinsulator with a spinon Fermi-surface discussed above.We represent the electron operator c as a product of acharge-e boson b and a charge neutral fermion f : c = bf .Then the bosons are at filling factor ⌫ = 1/2 and we takethem to be in the bosonic Laughlin state at that filling.Being neutral, the fermions f see no magnetic field, andform a Fermi-surface. This slave particle description in-troduces a U(1) gauge redundancy, with b and f carryingopposite charges under an emergent gauge field a

µ

. Thecorresponding gauge constraint fixes the number densityof the bosons to equal that of the f fermions. Being elec-trically charged, the boson density is simply equal to thephysical electron density. Thus, the density of f fermionsalso equals the physical electron density. Consequently,the size of the f Fermi-surface is set by the physical elec-tron density. Since the bosonic ⌫ = 1/2 Laughlin stateis gapped, we can integrate the boson degrees of freedomout, generating a Chern-Simons term (1.7) for the emer-gent gauge field a

µ

. Thus, the slave particle descriptionis completely equivalent to the familiar flux-attachmentpicture described above.

In this paper, we address the stability of the HLRphase to BCS pairing of composite fermions. As withthe spinon FS phase, the long-range current-current in-teraction mediated by gapless gauge field fluctuationssuppresses pairing in the BCS channel. Thus, we findthat the HLR phase is stable as long as the strength ofthe short-range attractive BCS interaction |V | is smallerthan a critical value |V

c

|. However, once |V | > |Vc

|,pairing of composite fermions will occur, giving rise toan incompressible QH phase with a Hall conductivity�

xy

= 1/2. A possible “microscopic” source of an at-tractive BCS interaction is the short-distance part ofthe charge-current interaction mediated by the CS gaugefield, which produces attraction in the p+ip channel.92 Infact, if pairing occurs in the p + ip channel, the resultingphase is just the familiar Moore-Read (MR) “Pfa�an”state.92 After the pairing transition, composite fermionsbecome gapped neutral fermion excitations of the MRphase. Gauge excitations are also gapped through theHiggs mechanism and appear in the form of vortices car-rying magnetic flux ⇡ of a

µ

, which via Eq. (1.9), trans-lates into physical electric charge q = e/4. Furthermore,these vortices support Majorana zero modes of compos-ite fermions in their core and, therefore, can be identi-fied with q = e/4 non-Abelian quasiparticles of the MRstate. We find the phase transition between the HLR andthe MR phases to be continuous, consistent with numeri-

cal simulations94–96, but contrary to previous theoreticalclaims.97 We describe how the neutral fermion gap andthe charge gap vanish as one approaches the QCP fromthe MR side, and discuss the phenomenology of the MRphase in the vicinity of the transition.

II. RENORMALIZATION GROUP ANALYSIS

Although various nFL states described above arise invery di↵erent physical systems, they admit a unified the-oretical treatment involving a gapless ~Q = 0 boson in-teracting with the FS. We denote the boson as �(x): itrepresents the order parameter in the case of the Ising-nematic QCP and the transverse component of the vec-tor potential ~a in the case of the spinon FS phase, theMott critical point, and the HLR phase. We denote thefermions (physical electrons in the Ising-nematic case,spinons in the spinon FS phase/Mott transition case andcomposite fermions in the HLR case) as f

↵

. We take theflavor index ↵ to run from 1 to N . Physically, N = 2 (twospin flavors) for the Ising-nematic QCP and spinon FSphase/Mott transition, and N = 1 for the spin-polarizedHLR phase.

Due to Landau-damping, boson fluctuations withwave-vector ~q ! 0 interact most strongly with fermionsin the regions of the FS to which ~q is nearly tangent.57–59

We divide the FS into pairs of antipodal patches, labelledby an index j, with

width ⇤y

⌧ kF

and thickness ⇤x

⇠⇤2

y

kF

⌧ ⇤y

⌧ kF

, (2.1)

where kF

is the Fermi momentum; see Fig. 2. For simplic-ity, we assume that the Fermi-surface is connected andconvex, and furthermore, that the local Fermi-surfacecurvature K and Fermi momentum k

F

are comparable.

⇤y

⇤x

f+f�

FIG. 2: A pair of antipodal patches, labeled by a fixed j, onthe Fermi surface. The values of ⇤

x

and ⇤y

are constrainedas in Eq. (2.1).

7

Antipodal points ±~kj

on the FS are chosen in patchpair j and directions perpendicular (x

j

) and tangent (yj

)

to the FS at ~kj

defined. The fermion operator f↵

is then

expanded in terms of patch fields f j

±,↵

(x) as,

f↵

(x) =X

j

(f j

+↵

(x)ei

~

kj ·~x + f j

�↵

(x)e�i

~

kj ·~x). (2.2)

We also define boson patch fields �j(x) to include onlymomenta nearly tangent to the FS in patch j:

|qx

| < |qy

|⇤y

kF

, |qy

| < ⇤y

. (2.3)

The e↵ective action S describing the fermion-boson in-teraction then breaks up into decoupled actions for eachpatch pair,

S =X

j

Sj , (2.4)

with18,32,59

Sj =

Zd2xd⌧(L

f

[f j ] + Lint

[f j , �j ]) + S�

[�j ], (2.5)

The Lagrangian densities are

Lf

= f†+↵

@

⌧

+ vF

(�i@x

�@2

y

2K)

!f+↵

+ f†�↵

@

⌧

+ vF

(i@x

�@2

y

2K)

!f�↵

(2.6)

Lint

= vF

�(f†+↵

f+↵

+ ⇣f†�↵

f�↵

) (2.7)

S�

=N

2g2

Zd2~qd!

(2⇡)3|q

y

|1+✏|�(~q, !)|2 (2.8)

Here, we have suppressed the patch index j. The Fermi-surface curvature K, the Fermi-velocity v

F

and, in thecase of the Ising nematic transition, the coupling con-stant g2, will generally vary along the Fermi-surface (i.e.will be patch-dependent). The constant ⇣ = 1 for thenematic QCP and ⇣ = �1 for the spinon FS phase/Motttransition and the HLR phase.

For general ✏, the action S�

is non-local. For theHLR state, this term encodes the long-range microscopicelectron-electron interation, U(~x) ⇠ 1/|~x|1+✏. The im-portant case of a Coulomb interaction corresponds to✏ = 0, while for a short-range interaction, ✏ = 1, and theterm (2.8) is local. In case of the nematic QCP or spinonFS phase, the physical value of ✏ is ✏ = 1. However, onemay be able to access ✏ = 1 via an expansion around✏ = 0.32,60 We, thus, work in the regime 0 ✏ ⌧ 1below. Proceeding finally to the case of the Mott transi-tion, the screening of the gauge field by the gapless bosonb also generates a non-local term (1.4) in the gauge ac-tion, i.e. the e↵ective action for the gauge-spinon sectoris described by Eqs. (2.6)-(2.8) with ✏ = 0.

In the case of the Ising nematic transition, the La-grangian also allows for a perturbation r�2, which tunesthe system across the QCP. Below, we will work directlyat the QCP, setting r = 0. We also perform all ourRG calculations at temperature T = 0. As usual, wetreat finite T as an infra-red cut-o↵ when running theRG equations.

As already noted, distinct pairs of patches j 6= j0 aredecoupled in the above description and can be treatedindependently. We will shortly discuss the crucial roleplayed by the inter-patch interactions in the pairingphysics, however, for now, let us ignore such couplingsand review the RG analysis of the two-patch theory (2.6)- (2.8).32,60 The two-patch theory is described by a singledimensionless coupling constant,

↵ ⌘g2v

F

⇤�✏

y

(2⇡)2. (2.9)

The fermion part of the action (2.6) dictates the scalingof frequency and momenta:

! ! e�zf `!, qx

! e�`qx

, qy

! e�`/2qy

, (2.10)

with the bare dynamical exponent, zf

= 1. As we willsee below, z

f

will generally be renormalized by inter-actions, however, the “anisotropic” momentum scaling,qx

⇠ q2

y

, is exact due to the non-renormalization of theFS curvature K.18 The full interacting fermion Green’sfunction G(!, ~q) depends only on the distance to the FS,qx

+ q2

y

/(2K), so we may identify zf

with the fermiondynamical exponent. On the other hand, the full bosonpropagator D(!, ~q) of the two-patch theory depends onlyon the momentum tangent to the FS, q

y

, so the abovescaling fixes the relationship18 between the boson dynam-ical exponent z

b

and the fermion dynamical exponent zf

,

zb

= 2zf

. (2.11)

Under the above scaling with bare zf

= 1, ↵ flows asd↵/d` = (✏/2)↵. Hence, the fermion-boson interaction isirrelevant for ✏ < 0, relevant for ✏ > 0 and marginal attree-level for ✏ = 0. To compute quantum corrections tothe RG flow one can utilize either a perturbative expan-sion in ↵ (Ref. 60; however, see footnote 98) or a 1/Nexpansion (Ref. 32). At leading order both expansionsgive the same result. To one loop order (first order in1/N), ↵ and v

F

run as,

d↵

d`=

✏

2↵ � ↵2

N(2.12)

dvF

d`= � ↵

Nv

F

(2.13)

and the fermion field acquires an anomalous dimension,

f(!, qx

, qy

) !1 +

✓7

4� ⌘

f

2

◆d`

�f(ed`!, ed`q

x

, ed`/2qy

)

(2.14)

8

with ⌘f

= ↵/N . For ✏ = 0, ↵ flows logarithmically tozero, and the system is a marginal Fermi-liquid with thefermion self-energy,

⌃(!) ⇠ �i↵

N! log

⇤!

|!| (2.15)

with ⇤!

⇠ vF

⇤x

- the energy cut-o↵. For ✏ > 0, the flow(2.12) has an infra-red stable fixed point at ↵⇤ = N✏/2.If N is of O(1) then ✏ ⌧ 1 ensures that the fixed-pointoccurs at weak coupling. On the other hand, if N � 1,we take ✏ ⇠ O(1/N) to make ↵⇤ ⇠ O(1) and obtain awell-defined large-N limit. In either case, at the fixedpoint,

dvF

d`= � ✏

2v

F

, ⌘f

=✏

2(2.16)

implying a fermionic dynamical exponent

zf

= 1 +✏

2, (2.17)

and a fermion self-energy

⌃f

(!) ⇠ !1�⌘f . (2.18)

The exponent zf

directly manifests itself in the nFL spe-cific heat,

C ⇠ T 1/zf . (2.19)

We note that the expression for zf

in Eq. (2.17) holdsto all orders in ✏: this is tied to the non-analytic nature ofthe q

y

dependence in S�

, which undergoes no renormal-ization. On the other hand, for ✏ = 1, S

�

is analytic in qy

and, in principle, can undergo renormalization. Our abil-ity to access the physically important ✏ = 1 point throughan expansion around ✏ = 0 is, thus, tied to such renormal-izations being absent. No renormalization of S

�

in the✏ = 1 theory has been found up to three loop order,18

however, a general proof of this statement is currentlylacking.

We next return to consider the e↵ect of inter-patch in-teractions, which have been mostly ignored in previousstudies. However, as we demonstrate below, such cou-plings must be included in the theory, as they are auto-matically generated in the RG process. This fact was firstnoted in Ref. 54 in the context of 3d QCD at finite quarkdensity, and here we closely follow the RG treatment pro-posed by Ref. 54. So far, we have left the precise RG pro-cedure somewhat implicit. Recall that under the scalingwe advocated for the two-patch theory, q

x

! e�`qx

andqy

! e�`/2qy

, so in the RG process we reduce both thefermion momentum cut-o↵ perpendicular to the FS, ⇤

x

,and the cut-o↵ tangent to the FS, ⇤

y

. While ⇤x

canbe, as usual, shrunk by integrating out gapped fermionexcitations away from the FS, reducing ⇤

y

in the samemanner would require integration over gapless fermionson the FS, which is illegal. Instead, during each RG stepwe re-partition the FS into smaller patches with width⇤0

y

= e�`/2⇤y

, while the reduction in the patch thickness

⇤0x

= e�`⇤x

is still performed by integrating out gappedfermions away from the FS. Simultaneously, in each RGstep we integrate out boson fluctuations with momentae�`/2⇤

y

< |~q| < ⇤y

; see Fig. 3. Before the RG step, suchboson fluctuations mediate non-local intra-patch interac-tions between the fermions. However, after the RG step,these generate a local four-fermion inter-patch interac-tion, as shown in Fig. 3 (bottom).

As is well known from ordinary FL theory, a very re-stricted set of four-fermion inter-patch couplings on theFS is kinematically allowed.55,56 Only forward-scatteringand BCS scattering interactions survive as the shell offermion states around the FS is shrunk in the RG pro-cess. As we are interested in the physics of pairing, inthe present paper we concentrate only on four-fermioninteractions in the BCS channel, which can be describedby the action,

SBCS = �1

4

Z4Y

i=1

d2~ki

d!i

(2⇡)3f†

↵

(k1

)f†�

(k2

)f�

(k3

)f�

(k4

)(2⇡)3�3(k1

+ k2

� k3

� k4

)

⇥⇣(�

↵�

���

+ �↵�

���

)V a(~k1

,~k2

;~k3

,~k4

) + (�↵�

���

� �↵�

���

)V s(~k1

,~k2

;~k3

,~k4

)⌘

(2.20)

Here, V s/V a are, respectively, symmetric/antisymmetric

under exchanging ~k1

$ ~k2

, and ~k3

$ ~k4

. Only thevalues of the interaction for BCS-matched momenta,V s,a(~k

1

, �~k1

;~k2

, �~k2

), play a role; furthermore, ~k1,2

canbe taken to lie on the FS. From now on, we assume thatthe system is rotationally invariant,99 so we may writeV s,a(~k

1

, �~k1

;~k2

, �~k2

) = V s,a(✓1

� ✓2

), with ✓1,2

- angleson the FS. Performing an expansion in angular harmon-

ics,

V s,a(✓1

� ✓2

) =1X

m=�1V s,a

m

eim(✓1�✓2) (2.21)

V s involves only even angular momentum componentsand V a - odd. It is convenient to define dimensionless

9

⇤y

k!2

!k!2 !k

!1

k!1

k!1!k!2

∆V

k!2

!k!2 !k

!1

k!1

#RG

⇤y

/2

RG

RG

⇤y

/2

!1FIG. 3: Top: Our RG procedure. During each RG step, each patch of the Fermi-surface is divided into two smaller patches.The relationship between the widths and heights of the patches remains as in Eq. (2.1). Bottom: Single boson exchangemediates a non-local intra-patch interaction (left). Here and below, solid/dashed lines are fermion/boson propagators. As highmomentum boson modes are integrated out in the RG process, a local inter-patch four-Fermi interaction in the BCS channel�V (~k1,�~k1;~k2,�~k2) is generated (right).

BCS interaction constants,

eV s,a

m

⌘ kF

2⇡vF

V s,a

m

. (2.22)

In the absence of the coupling to the gapless boson (i.e.in a Fermi-liquid), the RG flow of the BCS interaction(2.20) can be determined as in Refs. 55,56. The RG intheir work involves only the rescaling of ⇤

x

, which is thesame as that in Fig. 3 (top). Our rescaling of ⇤

y

plays norole in the renormalization of the BCS interaction, andso we can read o↵ the renormalization of eV s,a

m

from theirresults: this interaction is marginal at tree level, andacquires the following flow at one-loop level (see Fig. 4),

deV s,a

m

d`= �(eV s,a

m

)2 (2.23)

Thus, in a Fermi-liquid, if the initial value of the BCSinteraction is repulsive, V s,a

m

> 0, then V s,a

m

flows loga-rithmically to zero, while if the initial value of the inter-action is attractive, V s,a

m

< 0, V s,a

m

runs away to �1 atan energy scale, �

BCS

⇠ ⇤!

exp(�1/|eV s,a

m

|), signaling aninstability to fermion pairing.

Next, we study how the flow of the four-fermion BCSinteractions (2.23) is modified by the presence of thegapless boson �. In the limit ↵ ⌧ 1 (or N � 1),eV ⌧ 1, the leading modification comes from the diagramin Fig. 3 (bottom, left), which represents the one-boson

exchange contribution to the four-fermion BCS ampli-tude. As already noted, integration over intermediatelarge-momentum � modes in Fig. 3 generates an inter-patch four-fermion interaction,

�V s,a(~k1

, �~k1

;~k2

, �~k2

) = �⇣

2v2

F

D>

(0,~k1

� ~k2

) (2.24)

where D(!, ~q) is the boson propagator and the subscript“>” indicates that only modes in the momentum shelle�`/2⇤

y

< |~q| < ⇤y

should be kept. We remind thereader that the constant ⇣ distinguishes between the dif-ferent nFLs: we have ⇣ = 1 for the Ising-nematic case,and ⇣ = �1 for the spinon Fermi-surface phase/Motttransition and HLR cases. Note that the frequencies ofthe external fermions and, hence, of the boson in Fig. 3(bottom) can be set to 0. Eq. (2.24) gives �V for the

case of small angle scattering, ~k1

! ~k2

; the result for~k

1

! �~k2

is determined by symmetry. The static bosonpropagator is given by D(0, ~q) = g2/(N |~q|1+✏). Comput-ing the angular harmonics corresponding to (2.24),

�eV s,a

m

= �2

✓k

F

2⇡vF

◆⇣

2v2

F

Zd✓

2⇡D

>

(0, kF

✓)e�im✓

= �⇣g2vF

2⇡2N

Z⇤y

e

�`/2⇤y

dq

q1+✏

cos(mq/kF

) = �⇣↵

N`

(2.25)

10

k!

2

!k!

2 !k!

1

k!

1

"lΤ,l!#

"!lΤ,!l!#

FIG. 4: Renormalization of the BCS interactionV (~k1,�~k1;~k2,�~k2) in a FL.

In the last step, we have dropped the factor cos(mq/kF

)as ⇤

y

⌧ kF

. Thus, the process in Fig. 3 (bottom) con-

tributes a term deV s,a

m

/d` = �⇣↵/N to the RG flow of V ,which combines with Eq. (2.23) to give,

deV s,a

m

d`= �⇣

↵

N� (eV s,a

m

)2 (2.26)

There are also terms of order ↵eV s,a

m

which arise from ver-tex corrections and the flow of v

F

in the definition (2.22),but we have dropped them because they are are higherorder in ✏. Note that the flow (2.26) is independent ofthe angular momentum and spin channel; hence we dropthe angular momentum/spin indices on V below. Theflow equation (2.26) for the inter-patch BCS interactionin conjunction with the flow of the intra-patch couplingconstant ↵ in Eq. (2.12) determines the physics of thenFL states considered. We next analyze these RG equa-tions and discuss their consequences. However, we firstpoint out that in the regime of analytical control ✏ ⌧ 1,all the conclusions of our RG treatment can be repro-duced by solving the Eliashberg equation for the pairingvertex, as has been shown elsewhere.61 This lends furthersupport to our results.

III. RESULTS: ISING-NEMATIC QCP

We first discuss the solution to RG Eqs. (2.12), (2.26)for the nematic QCP. In this case, the constant ⇣ = 1 inEq. (2.26), so the fluctuations of the order parameter cap-tured by the first term in Eq. (2.26) drive the short-rangeinteraction V negative (attractive), as expected. In fact,as discussed in appendix A, we find that independent ofthe initial values of ↵ and V , V flows to �1 at a fi-nite ` = `

p

, indicating an instability of the Ising-nematicQCP to superconductivity. Thus, we expect both thezero temperature electron pairing gap � and the super-conducting critical temperature T

c

to be proportional to� ⇠ T

c

⇠ ⇤!

e�`p . Unlike in the ordinary Fermi-liquid,the run-away flow V ! �1 occurs even if the initialvalue of V is repulsive: gapless order parameter fluctua-tions eventually drive V attractive. However, the magni-tude of `

p

and hence the pairing gap does depend on the

initial value of V : the smaller the initial V - the largerthe gap. As already noted, the flow equations for V s,a

m

in di↵erent angular momentum/spin channels decoupleand are identical. We, thus, expect pairing to occur inthe channel where V s,a

m

diverges first, i.e. one which hasthe smallest initial V s,a

m

. Hence, the pairing symmetry isnon-universal.

It is interesting to compare the pairing scale � withthe energy scale E

nFL

= ⇤!

e�`nFL at which electronicquasiparticles get destroyed. Here, we identify E

nFL

asthe energy at which the Fermi-velocity v

F

, whose flow isdetermined by Eq. (2.13), starts to deviate significantlyfrom its bare value. We find that as long as our cal-culation is controlled, (i.e. ✏ ⌧ 1), E

nFL

⌧ �, so thesuperconducting instability preempts the destruction ofquasiparticles and associated nFL behavior. This is quitedistinct from the physics of many materials where nFLbehavior is observed at energies/temperatures well abovethe superconducting T

c

. As we take the artificial controlparameter ✏ to its physical value ✏ = 1, the two scalesE

nFL

and � approach each other, however, at this pointwe lose analytical control.

We now briefly illustrate the above conclusions for sev-eral regimes of ✏, ↵, V (see appendix A for more details).First, consider the case ✏ = 0. Here, we find

� = ⇤!

exp

"� 1p

e↵

⇡

2+ tan�1

eVpe↵

!#(3.1)

with

e↵ ⌘ ↵

N. (3.2)

If the bare short-range interaction eV is small compared tothe long-range interaction, |eV | ⌧

pe↵, then Eq. (3.1) re-

duces to � = ⇤!

exp(�⇡/(2pe↵)). On the other hand, if

the bare short-range interaction eV is large and repulsive,eV �

pe↵, � = ⇤

!

exp(�⇡/pe↵), i.e. the gap is reduced

by a factor of two on the logarithmic scale compared tothe case of small eV . Finally, if the bare short-range in-teraction is large and attractive, eV < 0, |eV | �

pe↵, the

gap takes the standard BCS form, � = ⇤!

exp(�1/|eV |).The scale at which nFL e↵ects become appreciable isE

nFL

⇠ ⇤!

exp(�1/e↵). Thus, as long as e↵ ⌧ 1, thepairing gap � is parametrically larger than the nFL scaleE

nFL

. We note in passing that the result (3.1) is identi-cal to one obtained for the problem of quark pairing bycolor gauge fields in 3d dense baryonic matter,54 and forelectron pairing near a ferromagnetic QCP in 3d.100

Proceeding to the case ✏ > 0 (which may be continu-ously connected to the physical case ✏ = 1), we find thatthe nFL scale is still given by E

nFL

⇠ ⇤!

exp(�1/e↵)for e↵ � ✏, as well as for e↵ ⇠ O(✏), while for e↵ ⌧ ✏,E

nFL

⇠ ⇤!

(↵/✏)2/✏. The pairing scale � is still given bythe expression in Eq. (3.1) for e↵ � ✏2, so the relationE

nFL

⌧ � holds. For e↵ ⌧ ✏2 and eV > 0 (or eV < 0, but

11

V!

V!

"

#

$ Ε !2

Gapless FSPaired FSPairing

transition

V!

"

&

$& Ε !2&'

FIG. 5: RG flow of the inter-patch BCS interaction eV s,a

m

in thespinon FS/HLR phase with 0 < ✏ ⌧ 1. The IR stable fixed-

point eV +⇤ ⇡

p✏/2 controls the gapless FS phase, while the

IR unstable fixed point eV �⇤ ⇡ �

p✏/2 controls the continuous

transition to the paired phase.

|eV | ⌧ ✏/ log ✏

2

e↵ ), we obtain

� ⇠ ⇤!

✓e↵✏2

◆2/✏

(3.3)

so � depends on the coupling constant ↵ in a power-lawmanner and E

nFL

/� ⇠ ✏2/✏ ⌧ 1. Eq. (3.3) has beenpreviously obtained within an Eliashberg-like treatmentin Ref. 62. Naive extrapolation of the above result to thephysically relevant value ✏ = 1 gives, � ⇠ E

nFL

⇠ ↵2⇤!

,i.e. the pairing and nFL scales become parametricallyequal. This conclusion is again supported by the directsolution of Eliashberg-like equations.101,102

Note that in the above analysis, we have ignoredthe d-wave dependence of the coupling between theIsing-nematic order parameter and electrons on the an-gle around the FS. We don’t expect the d-wave form-factor to a↵ect the maximum magnitude of the pairinggap strongly, however, it will certainly a↵ect its angu-lar dependence. In fact, recent results of Maier andScalapino103 and Lederer et al.104 suggest that the angu-lar dependence of the gap at the QCP may be quite singu-lar. These authors study the regime where the system istuned su�ciently away from the Ising-nematic QCP thatthe standard weak coupling BCS machinery can be ap-plied. They find that as the QCP is approached, the su-perconducting gap becomes strongly peaked around theangle where the coupling between the order parameterand the electrons is maximal (i.e. around the anti-node).It is interesting whether this result survives all the wayto the QCP. In the future, we hope to settle this questionby extending our RG analysis to the physical case withno rotational symmetry.

While our RG analysis is performed exactly at themetallic critical point, superconductivity will survivewhen one tunes the system slightly away from the QCPwith the perturbation r�2. Recall that r induces a finitecorrelation length for the order parameter, ⇠

�

/ r�⌫ ,⌫ = (1 + ✏)�1, with the corresponding energy scaleE

�

⇠ ⇠�zb�

⇠ r(2+✏)/(1+✏). Away from the QCP, E�

servesas an IR cut-o↵ on the RG equation (2.26). Therefore,the pairing gap �(r) will be essentially unmodified fromits QCP value �

0

= �(r = 0), as long as E�

(r) ⌧ �0

.We may then estimate the characteristic width of thesuperconducting dome as �r ⇠ �

0

(1+✏)/(2+✏) (here, weconsider the most interesting regime when pairing at the

QCP is dominated by order parameter fluctuations ratherthan the bare short-range BCS attraction V ). As always,the precise shape of the dome for |r| ⇠ �r cannot be de-termined from RG considerations alone. The dome willgenerally have tails extending to r � �r, where the gap(T

c

) will be strongly suppressed compared to �0

. Theprecise form of �(r) in these tails can be obtained byrunning the RG equation (2.26) for the BCS coupling upto the energy scale E

�

. It is easy to see that for r � �r,|V (E

�

)| ⌧ 1, i.e. the system at energy E�

is in the weak-coupling regime. Below the energy E

�

, � is not critical,and the system is described by Fermi-liquid theory, so theBCS coupling continues to flow according to Eq. (2.23).Therefore, if V (E

�

) < 0, then the system will develop su-perconductivity, with �(r) = E

�

exp(�1/|V (E�

)|). Onthe other hand, if V (E

�

) > 0, no superconducting in-stability will occur. Thus, if the bare V is attractive,the tails of the superconducting dome will extend to allr (as long as one remains in the regime of applicabilityof the critical theory). On the other hand, if the bare Vis repulsive, then the dome will terminate at a finite r,corresponding to V (E

�

) = 0.We would like to note that in the above discussion, r

denoted the deviation from the “metallic” QCP. As wesaw, this QCP is unstable to superconductivity, so thetrue Ising-nematic QCP will occur inside the supercon-ducting dome. In addition, its location will generallyshift away from that of the putative metallic QCP atr = 0 to r = r

c

. On general scaling grounds, we ex-pect |r

c

| ⇠ �r. The universality class of the true QCP atr = r

c

depends on whether pairing gives rise to a fullygapped or a nodal superconductor. If the superconductoris fully gapped, this transition will be in the classical 3DIsing univerality class. The character of this transitionin a nodal d-wave superconductor has been discussed inRef. 105. The critical behavior associated with the trueQCP will only be observable for T ⌧ �

0

, and the systemwill cross over to the metallic critical behavior discussedin this paper for T � �

0

.

IV. RESULTS: SPINON FS PHASE AND HLRPHASE

We now turn to the solution of the RG equationsEqs. (2.12), (2.26) for the spinon FS phase and HLRphase. The constant ⇣ in Eq. (2.26) now takes the value⇣ = �1, hence gauge field fluctuations drive eV repulsive,in accordance with intuition. We first solve Eqs. (2.12),(2.26) when ✏ > 0 (with an eye to describing the phys-ical spinon FS phase and the HLR phase with short-range interactions, where ✏ = 1). Here, the coupling↵ flows to the fixed point ↵⇤ = N✏/2, and we may sub-stitute this fixed point value into the RG equation forV , (2.26). We then find two perturbatively accessiblefixed points for eV : eV ±

⇤ = ±p

✏/2, see Fig. 5. The fixed

point eV +

⇤ is infra-red stable; as long as the initial value

12

of eV is greater than eV �⇤ , eV flows to eV +

⇤ . Thus, thespinon FS and HLR phases are controlled by the fixedpoint (↵⇤, eV +

⇤ ) and are stable to fermion pairing. How-ever, if the initial value of eV s,a

m

in some angular momen-tum/spin channel is smaller than eV �

⇤ , eV s,a

m

runs away to�1, and fermion pairing occurs. eV = eV �

⇤ , thus, marksthe phase transition between the U(1) and Z

2

spin-liquidphases (HLR and incompressible QH phases). Note thatunlike in a Fermi-liquid, a finite strength of the attrac-tive short-range interaction |eV | > |eV �

⇤ | > 0 is neededto overcome the long-range repulsion mediated by thegauge field and cause fermion pairing. Pairing in a givenangular-momentum/spin channel can be driven by tun-ing the corresponding eV s,a

m

. The pairing transition iscontinuous and the spinon (neutral fermion) gap onsetsin a power law fashion, � ⇠ (eV �

⇤ � eV )z⌫ , where

1

z⌫=

d

deV

deVd`

!����eV =

eV

�⇤

=p

2✏ (4.1)

This is, again, distinct from an ordinary FL where theelectron gap has the familiar exponential form � ⇠exp(�1/|eV |).

We note that to the leading order in ✏ discussed above,the presence of inter-patch interactions V does not af-fect the flow of the intra-patch coupling constant ↵,Eq. (2.12), and the Fermi-velocity v

F

, Eq. (2.13). As aresult, most physical properties (fermion and boson dy-namical exponents z

f

, zb

; specific heat; 2kF

exponents32

etc.), at the two fixed points V = V ±⇤ are identical. This

conclusion may be true to all orders in ✏, since, perturba-tively, BCS interactions do not influence the single par-ticle properties (v

F

, Z) in a FL.We next discuss the marginal case ✏ = 0, which de-

scribes the QH fluid with Coulomb interactions. Here,the coupling constant ↵ logarithmically flows to 0. Thecombined flow of e↵, eV is shown in Fig. 6 (see appendixB for details). The flow is characterized by a singlefixed-point e↵ = 0, eV = 0 and features an attractor lineeV =

pe↵ and a separatrix eV = �

pe↵. As long as the

initial values of eV , e↵ satisfy eV > �pe↵, the couplings

flow to the attractor line eV =pe↵ and then into the fixed

point e↵ = 0, eV = 0. So, the HLR phase with Coulombinteractions is stable in a finite region of parameter space.On the other hand, if the initial eV < �

pe↵, eV runs away

to �1 and fermion pairing occurs. Thus, the separa-trix eV = �

pe↵ describes the transition between the HLR

phase and the paired QH phase. Note that this separa-trix also logarithmically flows into the fixed point e↵ = 0,eV = 0, so the stable and the unstable fixed points eV ±

⇤ ,found for ✏ > 0, merge into a single fixed-point here. Thepairing transition is continuous and the fermion gap turnson as the separatrix is crossed in an unusual super-powerlaw fashion,

� ⇠ exp

� 1

16log2(V

c

� V )

�(4.2)

with Vc

⇡ �pe↵.

V. RESULTS: MOTT TRANSITION

As already noted, the ✏ = 0 theory also describes theQCP between the spinon FS phase and a Fermi-liquidphase. Thus, the results in section IV imply that thespinon Fermi-surface at the Mott transition is stable aslong as the initial values of (eV , e↵) lie to the right of theseparatrix in Fig. 6. On the other hand, if the initialvalues of (eV , e↵) lie to the left of the separatrix, the spinonacquires a gap, and the Mott transition occurs betweena Z

2

spin-liquid and a superconductor.In the former regime eV > �e↵, where the spinon FS

at the Mott transition is stable, we expect that an evenstronger statement holds: the spinon Fermi-surface re-mains stable as one tunes the system slightly away fromthe Mott transition. Indeed, if one tunes the system intothe compressible phase, t < t

c

in Eq. (1.3), the gaugefield becomes Higgsed by the condensate hbi 6= 0 belowa momentum scale q

a

⇠ (tc

� t)⌫ , where ⌫ is the cor-relation length exponent of the XY universality class.The corresponding energy scale E

a

⇠ q2

a

will serve asan IR cut-o↵ on the RG equations for the flow of (V ,↵), (2.12), (2.26). Below this energy scale, gauge fluc-tuations become non-critical and the spinon FS will bedescribed by FL theory. Now, as we discussed in sectionIV, for energies above E

a

, the flow of (eV , e↵) tends to theattractor line V (`) =

p↵(`) ⇡ `�1/2 > 0. Thus, at the

crossover scale V (Ea

) > 0, so no spinon pairing will oc-cur as one further lowers the energy into the Fermi-liquidregime. Consequently, the Fermi-liquid phase adjacent tothe Mott transition will not develop superconductivity.

Likewise, if one tunes the system into the insulatingphase, t > t

c

in Eq. (1.3), the screening (1.4) of the gaugefield by b will cease at a momentum scale q

a

⇠ (t � tc

)⌫ ,with the corresponding energy scale E

a

⇠ q2

a

. Below thisenergy scale the system is e↵ectively in the spinon FSphase. Again, by the time the scale E

a

is reached, (eV , e↵)will approach the attractor line V (`) =

p↵(`) ⇡ `�1/2 >

0. Since, as we discussed in section IV, a finite strengthof attraction V < V �

⇤ is needed to destabilize the spinonFermi-surface phase, we conclude that no spinon pairingwill occur on the insulating side of the transition, as well.

VI. Z2 SPIN-LIQUID AND QH STATES NEARTHE PAIRING TRANSITION

As we showed in section IV, spinon FS and HLR phasescan be driven through a continuous pairing transition.We now comment on some properties of the paired phasein the vicinity of the transition. In many ways, thesepaired states are analogous to ordinary superconduc-tors. As we already noted, the paired phase supports

13

0

0

V!

Α!

FIG. 6: RG flow of the intra-patch coupling constant e↵ andthe inter-patch BCS interaction eV s,a

m

in the HLR phase withCoulomb interactions (✏ = 0). Note the attractor line eV ⇡

pe↵

(dashed red curve) and the separatrix eV ⇡ �pe↵ (solid red

curve). The HLR phase is controlled by the logarithmic flow

of the attractor line into the fixed point (eV , e↵) = (0, 0). Thephase transition to the paired CF phase is controlled by thelogarithmic flow of the separatrix into the same fixed point(eV , e↵) = (0, 0).

two kinds of fundamental excitations: spinons/neutralfermions and vortices of the gauge field. The latter arevisons of the Z

2

spin-liquid/charge e/4 excitations of thepaired CF phase. The vortex excitations are, thus, par-ticularly important in the QH context as their energydetermines the charge gap. So far, we have only deter-mined the scaling of the fermion gap � near the pair-ing transition. We now crudely estimate the magnitudeof the vortex gap. The fermion pair-condensate is sup-pressed in the vortex core, whose radius we take to bethe fermion correlation length ⇠ ⇠ ��1/zf . Thus, thevortex gap E

v

⇠ (✏n

� ✏p

)⇠2, where ✏n

� ✏p

is the energydensity di↵erence between the “normal” phase and thepaired phase. The scaling of the energy density at thepairing transition is ✏ ⇠ !1+1/zf (e.g. recall the specificheat C ⇠ T 1/zf both in the gapless FS phase and atthe pairing transition), so setting the characteristic en-ergy scale ! in the paired phase to the fermion gap �,✏n

� ✏p

⇠ �1+1/zf and Ev

⇠ �1�1/zf . Therefore, the vi-son/charge gap vanishes as one approaches the de-pairingtransition, although more slowly than the spinon/neutralfermion gap. For the physically interesting case of thespinon FS or the QH system with short-range electron-electron interactions, ✏ = 1, z

f

= 3/2 and Ev

⇠ �1/3.Note that our estimate of the vortex gap strictly only ap-plies to the case ✏ > 0, for ✏ = 0, z

f

= 1+ and we expect

Ev

to vanish logarithmically as � ! 0.As is well-known, superconductors can be classified as

type-I or type-II depending on their response to an ex-ternal (orbital) magnetic field H. Both types of super-conductors are characterized by a Meissner e↵ect (fullexpulsion of magnetic flux) for small H. As the mag-netic field is increased, a (3d bulk) type-I superconductorundergoes a 1st order transition to a fully normal stateat a critical value H = H

c

. On the other hand, in atype-II superconductor, an array of Abrikosov vorticesis induced for magnetic fields H > H

c1

and the normalstate is recovered only for H > H

c2

> Hc1

. The type ofa conventional superconductor is determined by the ra-tio of the electron correlation length ⇠ and the magneticpenetration depth �. For � ⌧ ⇠, the superconductor istype-I, while for � � ⇠ - it is type-II.

Related “typology” also exists in pairedspinon/composite fermion phases.106,107. Howeverwe first need to understand what plays the role of theexternal magnetic field H in these systems. In thequantum Hall case, the flux of the emergent magneticfield is simply the electron density. Thus, the analogof the external magnetic field is the electron chemicalpotential µ. For the spinon FS phase on the trian-gular lattice, based on symmetry considerations, weexpect an external orbital magnetic field H to couplelinearly to the flux of the emergent gauge field, r ⇥ ~a:�L = ��H(r ⇥ ~a), with � - a coupling constant.108.Recall that the emergent gauge flux is physicallyidentified with the spin-chirality ~S

1

· (~S2

⇥ ~S3

) of theelementary triangles.108 Moreover, starting with theelectron Hubbard model on the triangular lattice, in theinsulating limit t ⌧ U , one finds that a coupling of theexternal orbital magnetic field to the spin-chirality is,indeed, induced at order t3/U2, so � ⇠ (t3/U2)(a2/�

0

),where a is the lattice spacing and �

0

- the elementaryflux quantum.108 Thus, in this case a physical orbitalmagnetic field directly plays the analog of an externalmagnetic field, coupling to the emergent magnetic fieldr⇥~a and, thereby, to the spinons, albeit with a reducedstrength.

Like ordinary superconductors, the paired spin liq-uid/paired quantum Hall phases exhibit two lengthscales: ⇠ and �, characteristic of fermion (spinon/neutralfermion) excitations and gauge field fluctuations, respec-tively. In the vicinity of the de-pairing transition, the be-havior of these length scales is controlled by the RG fixedpoint describing the transition. Our scaling theory indi-cates that fermions disperse as ! ⇠ (|~k|�k

F

)zf and gaugefluctuations disperse as ! ⇠ qzb with z

b

= 2zf

= 2 + ✏.As noted above, this relation holds both in the gap-less spinon FS/HLR phase and at the pairing transition.Upon entering the paired phase, a characteristic energyscale ! = � - the fermion gap is generated, which gives⇠ ⇠ �2 ⇠ ��1/zf . So, as one approaches the transi-tion, � ! 0, and both the correlation length ⇠ and the“penetration depth” � diverge, albeit with di↵erent ex-ponents. In particular, ⇠ � �, so the paired phase in the

14

vicinity of the transition is in the type-I regime, as wasargued on general grounds in Ref. 106. Further, the rela-tion E

v

� � obtained above is again typical of a type-Isuperconductor. Note that for ✏ = 1, � ⇠ ��1/3, whichis the standard expression for the scaling of the physical(non-local Pippard) penetration depth in a conventionaltype-I superconductor.109

As discussed above, the most dramatic manifestationof the type-I/type-II distinction in an ordinary super-conductor is the response to an external magnetic fieldH. There is also an analog of this phenomenon forpaired spinon/composite fermion phases.106,107 Let usbegin with the QH case and first consider short-rangeelectron-electron interactions. In this case the analogof the external magnetic field H is the electron chemi-cal potential µ. The paired QH phase is incompressible,so for deviations of chemical potential |µ| smaller thansome critical value, the system does not respond. (Thisis the analog of the Meissner e↵ect in the superconduc-tor). However, above a critical µ, the electron densitystarts to change. This can occur in two ways: i) onceµ > µ

c1

= 4Ev

, charge e/4 quasiparticles (gauge fieldvortices) start to be nucleated. If the interactions be-tween these quasiparticles are repulsive, we expect a sta-ble dilute quasiparticle lattice to form. The Hall plateauxthen persists for µ > µ

c1

, as well as when one sweepsthe physical magnetic field away from half-filling (hold-ing the electron density constant). This QH counter-part of type-II superconducting behavior is thought tobe realized in most conventional QH experiments. ii)It is possible that the charge e/4 quasiparticles attractrather than repel, making the vortex lattice unstable.We then expect a first order phase transition betweenthe paired QH phase and the HLR phase to occur atµ = µ

c

< 4Ev

, accompanied by a jump in the electrondensity. This is the QH analogue of type-I superconduct-ing behavior. We now show that this type-I scenario is,indeed, realized by paired QH states in the vicinity ofthe de-pairing transition. We can estimate the “thermo-dynamic” critical chemical potential µ

c

, by equating theenergy-densities of the paired state and the HLR state:✏p

= ✏n

� 1

2

µ2

c

, where is the compressibility of the HLRphase. (We are measuring the chemical potential relativeto the chemical potential of the HLR state at half-filling).Recalling our estimate, ✏

n

� ✏p

⇠ �1+1/zf , we concludeµ

c

⇠ �1/2+1/(2zf ) = �5/6 ⌧ Ev

⇠ �1/3. Thus, the firstorder transition to the HLR phase occurs before individ-ual e/4 quasiparticles can be excited, so the system is inthe type-I regime. The magnitude of the density jumpacross the first-order transition is �n

c

= µc

⇠ �5/6 (seefootnote 110 for some caveats). For short-range electron-electron interactions, if one sweeps the magnetic field(holding the electron density fixed) away from ⌫ = 1/2,the system phase separates into macroscopic domains ofthe paired CF phase and the HLR phase. In practice,however, the first order transition between paired quan-tum Hall and the HLR phases will be rendered secondorder by the e↵ect of impurities. Nevertheless, it is con-

ceptually important to understand the nature of the tran-sition in the clean limit. Long-range electron-electron in-teractions U(~x) ⇠ 1/|~x|1+✏ with 0 ✏ 2 frustrate themacroscopic phase separation, so one expect the forma-tion of “micro-emulsion”-like bubble/stripe phases in thevicinity of ⌫ = 1/2.106

A similar phenomenon can also occur in the Z2

spin-liquid phase in the vicinity of the de-pairing transition tothe spinon FS phase. Now, repeating the arguments pre-sented above for the QH case, we expect an application ofan external orbital magnetic field to induce a first ordertransition from the Z

2

spin-liquid phase to the spinonFS phase at �H

c

⇠ �5/6, accompanied by a jump ofmagntidue ⇠ �5/6 in the spin-chirality. For spin-singletpairing of spinons, the critical orbital field H

c

should becompared to the critical Zeeman field H

Z

= �/(ge

µB

)needed to break up the Cooper pairs. In the strict � ! 0,H

Z

is parametrically smaller than Hc

, so the orbital ef-fects may be neglected. This trend is further enhancedby the suppression of the coupling constant � in the in-sulating regime t ⌧ U .

VII. DISCUSSION

We briefly discuss in sections VII A - VII C a number ofexperimental phenomena to which our work is pertinent.In section VII D, we briefly note another system for whichour RG results are relevant: the nematic phase in thecontinuum.

A. Superconductivity near quantum critical points

One of the main results of this paper is a controlledtheory of the superconducting instability of a quantumcritical metal. As an illustration we studied the Ising-nematic quantum critical point. Many of our resultsare expected to carry over to metals near other Pomer-anchuk transitions. One of our main conclusions is thatsuperconductivity is strongly enhanced near such quan-tum critical points. This gives a firm theoretical basisto the empirical observation of superconducting domeswith T

c

optimized near some putative quantum criti-cal points. The results on the Ising-nematic transitionshould be contrasted with those on the Mott transitionfrom the spinon Fermi-surface spin-liquid insulator to aFermi-liquid. We find that at such a Mott transition thepairing instability is suppressed. The Mott transition be-longs to a qualitatively di↵erent class of QCPs in metals,one where an entire sheet of the electronic Fermi sur-face disappears through a continuous transition,6–9 andso displays a very di↵erent behavior compared to “con-ventional” symmetry-breaking transitions.

Returning to symmetry-breaking transitions, we recallthat the problem of superconductivity near the spin-density-wave quantum critical point was addressed inRefs. 11,12. It was found there that non-Fermi liquid

15

e↵ects in the electron spectrum and pairing correctionsarose at similar energy scales, which preempted identifi-cation of a clear-cut non-Fermi liquid regime in the nor-mal state.

For the specific case of the Ising-nematic transition,experiments48 on electron-doped iron superconductorBa(Fe

1�x

Cox

)2

As2

show that a quantum critical pointassociated with such order likely lies directly underneaththe superconducting dome. This quantum critical pointappears to be separated from a di↵erent one associatedwith onset of spin density wave order that occurs at lowerx. It is tempting to attribute the optimality of the su-perconducting T

c

to the enhanced fluctuations of the un-derlying Ising-nematic quantum critical point. For thisscenario to be legitimate it is necessary that other fluc-tuations (for instance in the spin density) have muchweaker e↵ects in the normal state at optimal doping.A contrasting scenario likely applies to a di↵erent ironsuperconductor BaFe

2

(As1�x

Px

)2

obtained by isovalentsubstitution.3 In this case, optimal T

c

occurs aroundx = 0.30, which is roughly where the Neel temperatureassociated with spin density wave (SDW) order (presentat low x) extrapolates to zero. The strong enhancementof the NMR relaxation rate near the optimal doping fur-ther suggests the presence of an SDW critical point.111

The low-x material also displays Ising-nematic order but,according to some reports,49 it disappears at a larger xthat is near the overdoped edge of the superconductingdome. So the SDW fluctuations seem to dominate overany nematic fluctuations near optimal doping in this ma-terial.

Quantum critical nematic fluctuations may also playa role in the physics of nearly optimally doped cuprates,and our results may provide a foundation to assessingtheir e↵ects.

A di↵erent aspect of our theoretical results is the re-lationship between non-Fermi liquid physics and super-conductivity near the Ising-nematic QCP. In the small-✏ regime where our calculations are controlled we foundthat the superconductivity is so strong that it sets in at atemperature scale parametrically larger than the scale atwhich non-Fermi liquid e↵ects set in. For ✏ = 1 we expectthat there is no such separation and the two phenomenahappen at parametrically the same scale. In this case, itis possible that the superconductivity will rear its headbefore the non-Fermi liquid physics has fully developed.It is then interesting to ask what happens when the su-perconductivity is suppressed with an external magneticfield. Presumably, this will expose the non-Fermi liquidphysics of the Ising-nematic quantum critical point downto low temperatures. In particular, the specific heat willfollow the predicted power law C ⇠ T

23 . Some aspects

of the non-Fermi liquid physics predicted for the criti-cal point will likely be suppressed by the magnetic field,along with the superconductivity. A good example is thelow-energy tunneling density of states, which in the ab-sence of the magnetic field was found to be power-lawsuppressed at the QCP18,32. This e↵ect arises primar-

ily from enhanced Cooper pairing fluctuations32. Sincethese will be suppressed in a magnetic field, so will thesingularity in the tunneling density of states.

B. When is superconductivity enhanced near aquantum critical point?

Based on our results we now suggest an answer to theempirical puzzle that superconductivity is enhanced atsome, but not all, metallic quantum critical points. Aswe emphasized in the introduction, there are actuallytwo qualitatively distinct classes of such quantum criticalpoints distinguished primarily by the fate of the electronFermi surface. For criticality associated primarily withthe onset of broken symmetry order, the electron Fermisurface evolves continuously but is distorted by the bro-ken symmetry. On the other hand, there are continuousquantum phase transitions where the electron Fermi sur-face evolves discontinuously. The associated quantumcritical points are dominated by fluctuations of the elec-tronic structure itself.

The specific examples studied in this paper lead us tosuggest more generally that quantum critical points asso-ciated primarily with the onset of broken symmetry willshow enhanced superconductivity, while those associatedprimarily with a discontinuous jump of the Fermi surfacemay not. Apart from the specific results in this paper,this suggestion finds theoretical support in many pre-vious approximate treatments of superconductivity dueto order parameter fluctuations, e.g. at the SDW onsetQCP. There are currently very few theoretical examplesof transitions in the second class (with a disappearingFermi surface). One known example is given by the Motttransition from a spinon FS spin-liquid insulator to aFermi-liquid; here we showed that pairing is suppressedat the QCP. Whether this is a general property of thesecond class of transitions is a good target for future re-search. As we noted in section I C, the lack of long-rangesuperconducivity at a Mott transition is not surprising;a far more non-trivial conclusion of our analysis is thesuppression of local pairing correlations near the transi-tion, reflected in the vanishing of the single electron gapat the QCP, and the absence of superconductivity in theFermi-liquid phase adjacent to the QCP.

Our suggestion finds empirical support in the ab-sence of superconductivity in the heavy fermion materi-als CeCu

6�x

Aux

and YbRh2

Si2

near their quantum crit-ical points. These have long been thought1 to be sys-tems where the Fermi surface evolves discontinuously andthe electronic structure changes due to the breakdown ofKondo screening. A futher complication in these materi-als is that the critical point seems to involve fundamentalchanges of the electronic structure accompanied also bythe onset of antiferromagnetic order. There is currentlyno theoretical understanding of why the discontinuousFermi surface change coincides with the onset of bro-ken symmetry. Nevertheless, it is natural to expect that

16

the fate of superconductivity will then be determined bya delicate interplay between two competing e↵ects: thepairing tendencies of order parameter fluctuations andthe strong non-Fermi liquid e↵ects due to the electronicfluctuations. We hope that the present paper sets thestage for future progress on this issue.

C. Quantum Hall states at ⌫ = 1/2.

In this paper, we have developed a systematic theoryof the transition between the compressible HLR phaseand the incompressible Moore-Read phase of the quan-tum Hall fluid. We have demonstrated that contrary toprevious theoretical claims97 a direct continuous transi-tion between these phases is allowed. In a conventionalGaAs system, the HLR phase is believed to be realizedat filling factors ⌫ = 1/2, ⌫ = 3/2, while the Moore-Read phase is a candidate for the plateaux at ⌫ = 5/2.In a large magnetic field, when the mixing between Lan-dau levels can be neglected, the physics of a partiallyfilled Landau level is determined by the projection of theelectron-electron interaction onto the states in the Lan-dau level. The projection is di↵erent for di↵erent Landaulevels, which is believed to explain the above contrastingbehaviors observed in the lowest (n = 0) and first (n = 1)Landau levels.93,94 Since both the electron-electron in-teractions and the form of the single-particle states inGaAs are di�cult to tune, the realization of a directtransition between the Moore-Read phase and the HLRphase in GaAs is challenging. However, it was recentlysuggested that this transition may be realized in bilayergraphene, by tuning the strength of the perpendicularelectric field.112,113 The introduction of the perpendicu-lar electric field modifies the form of the single-particlestates forming the Landau level and hence the e↵ectiveinteractions within the Landau level. We, thus, hope thatbilayer graphene will provide an experimental avenue totest our predictions.

D. Nematic phase in the continuum

In this paper, we have concentrated on the Pomer-anchuk instability in the presence of a lattice, where adiscrete rotational symmetry is spontaneously broken. Itis also interesting to consider a system in the continuum,which possesses a full SO(2) rotational symmetry. In thiscase, the nematic phase where SO(2) is spontaneouslybroken to a discrete two-fold subgroup will possess a gap-less Goldstone mode. A curious feature of this phase isthat unlike in systems involving spontaneous breakingof internal (non-spatial) symmetries, here the Goldstonemode couples to electrons near the FS in a non-derivativemanner.23 As a result, the entire nematic phase was pre-dicted to exhibit non-Fermi-liquid behavior.23 In fact, itis easy to see, that the coupling of the Goldstone modeto the FS is essentially the same as for the gapless criti-

cal mode � at the Ising-nematic QCP. Therefore, the ef-fective theory of the “continuum”-nematic phase will beidentical to that of the Ising-nematic QCP studied above.In particular, according to the results of our ✏-expansion,the entire continuum-nematic phase will become super-conducting and the non-Fermi-liquid behavior of Ref. 23will be preempted.Acknowledgements. We would like to thank

S. Parameswaran, Z. Papic, A. Chubukov, S.-S. Lee,P. A. Lee, D. Scalapino, S. Kivelson, C. Nayak, L. Balentsand M. P. A. Fisher for useful discussions. This researchwas supported in part by the National Science Founda-tion under Grant Nos. NSF PHY11-25915 and DMR-1360789, and by the Templeton Foundation. The workof TS was supported by Department of Energy DESC-8739- ER46872, and partially by a Simons Investigatorgrant from the Simons Foundation. Research at Perime-ter Institute is supported by the Government of Canadathrough Industry Canada and by the Province of Ontariothrough the Ministry of Research and Innovation.

Appendix A: Solution of RG equations:Ising-nematic QCP

In this appendix we provide a detailed solution of theRG equations, (2.12), (2.26) for the Ising-nematic QCP(⇣ = 1 in Eq. (2.26)). We use the notation e↵ = ↵/N .

We begin by considering the case ✏ = 0. Then e↵ runslogarithmically to zero,

e↵(`) =e↵(0)

1 + e↵(0)`, v

F

(`) =v

F

(0)

1 + e↵(0)`(A1)

and the non-Fermi liquid corrections become appreciablebelow an energy scale of order e�`nFL with `

nFL

⇠ 1/e↵.Since the flow of e↵ is slow, in order to analyze the flow

of eV , let us first assume that e↵ is constant. We see thatthe flow is then towards eV = �1 signaling a pairinginstability. Solving Eq. (2.26),

eV (`) =pe↵ tan

�

pe↵` + tan�1

eV (0)pe↵

!(A2)

We see that eV diverges at

`p

=1pe↵

⇡

2+ tan�1

eV (0)pe↵

!(A3)

and we expect a pairing gap � ⇠ ⇤!

e�`p . Let us dis-cuss various limits of Eq. (A3). If the “bare” short rangeinteraction is small compared to the long-range interac-tion,

`p

⇡ ⇡

2pe↵

, |eV | ⌧pe↵ (A4)

On the other hand, if the bare short range interaction islarge and repulsive,

`p

⇡ ⇡pe↵

, eV �pe↵ (A5)

17

i.e. the magnitude of the gap is reduced by a factor oftwo on the logarithmic scale compared to the case (A4),Finally, if the bare short range interaction is large andattractive,

`p

⇡ 1

|eV |, eV < 0, |eV | �

pe↵ (A6)

which is just the usual BCS result. In any case, e↵`p

<

⇡pe↵ ⌧ 1 hence the running of e↵ can, indeed, be ne-

glected in estimating the size of the gap. Moreover,`p

< ⇡/pe↵ ⌧ `

nFL

⇠ 1/e↵, hence the non-Fermi-liquidphysics is preempted by pairing.

If we turn on a finite ✏, the flow of eV to �1 persists.Let us estimate the size of the gap. In the present casee↵ and v

F

run as,

e↵(`) =e↵(0)e✏`/2

1 +2e↵(0)

✏(e✏`/2 � 1)

vF

(`) =v

F

(0)

1 +2e↵(0)

✏(e✏`/2 � 1)

. (A7)

Note that for e↵ ⇠ e↵⇤ = ✏/2 the scale at which non-Fermi-liquid e↵ects become manifest is `

nFL

⇠ 1/✏. Fore↵ � ✏, one first observes logarithmic running of v

F

for` & `

nFL

⇠ 1/e↵, and then power law running for ` & 1/✏.Finally, if e↵ ⌧ ✏, `

nFL

⇠ 2

✏

log(✏/e↵). Proceeding to

the flow of eV , we observe that if e↵ � ✏2, the flow ofe↵ can be ignored for the purpose of estimating the pair-ing scale and previous results Eqs. (A2) and (A3) hold.Comparing the pairing scale and the non-Fermi-liquidscale, we find that the former is always parametricallyhigher in this regime. Indeed, for e↵ ⇠ O(✏) and e↵ � ✏,`p

< ⇡/pe↵ ⌧ `

nFL

⇠ 1/e↵, while for ✏2 ⌧ e↵ ⌧ ✏,

`p

< ⇡/pe↵ ⌧ `

nFL

⇠ 2

✏

log ✏

e↵ . In the remaining regimee↵ . ✏2, the flow of e↵ cannot be ignored. However, thisregime can be e↵ectively analyzed as a part of a widerrange e↵ ⌧ ✏. As is already clear from the argumentsabove, if we start with e↵ ⌧ ✏, e↵ will remain in this rangethroughout the evolution. Hence, in this regime, we mayapproximate,

de↵d`

=✏

2e↵ (A8)

e↵deVde↵ = �2

✏(e↵ + eV 2) (A9)

We may eliminate the ✏ dependence from Eq. (A9) bydefining e↵ = ✏2↵, eV = ✏V . Then,

↵dV

d↵= �2(↵ + V 2) (A10)

The solution to Eq. (A10) is,

V (↵) = �p

↵(J1

(4p

↵) + CY1

(4p

↵))

J0

(4p

↵) + CY0

(4p

↵)(A11)

x01 2 3 4 5 6x ! 4 Α

#

#6

#4

#2

2

4

6

Y0!x"#J0!x"

FIG. 7: Determination of the pairing scale `p

and the corre-sponding value ↵

p

in the regimes e↵ ⇠ O(✏2) and e↵ ⌧ O(✏2)(see Eqs. (A13), (A15)).

where initial conditions fix the constant C to be,

C = �p

↵(0)J1

(4p

↵(0)) + J0

(4p

↵(0))V (0)p↵(0)Y

1

(4p

↵(0)) + Y0

(4p

↵(0))V (0)(A12)

Observe that V has a divergence at ↵ = ↵p

with

J0

(4p

↵p

)

Y0

(4p

↵p

)= �C (A13)

As is clear from Fig. 7, irrespective of the value of C, if ↵(↵) is of O(✏2) (of O(1)) or less, the above equation hasa solution with ↵

p

at most of O(1). Hence,

`p

=2

✏log

✏2↵p

e↵ ⌧ `nFL

⇠ 2

✏log

✏

e↵ (A14)

Thus, in this regime pairing also always occurs beforenon-Fermi-liquid e↵ects become significant. Having es-tablished this, we will not analyze the full behaviour ofthe pairing scale as a function of initial e↵ and eV in thisregime, but will only discuss the case of smallest coupling,e↵ ⌧ ✏2 (↵ ⌧ 1). Then, Eq. (A13) may be rewritten as,

Y0

(4p

↵p

)

J0

(4p

↵p

)⇡ Y

0

(4p

↵)

J0

(4p

↵)� 1

2⇡V(A15)

The function Y0

(x)/J0

(x) is increasing wherever it iscontinuous (see Fig. 7). Hence, as we need a solution with↵

p

> ↵, for V > 0 we conclude that 4p

↵p

> x0

, withx

0

⇡ 2.405 - the first zero of J0

(x). Moreover, as Y0

(x) ⇡2

⇡

log x for x ! 0, the right-hand-side of Eq. (A15) tendsto �1, hence,

↵p

⇡ x2

0

16⇡ 0.361, V > 0, e↵ ⌧ ✏2, (A16)

and the pairing scale can be obtained from Eq. (A14).Now, if V < 0 but |V | ⌧ (log 1

↵

)�1, the right-hand-side

18

of Eq. (A15) tends to +1 and Eq. (A16) still holds.Finally, if V < 0 and |V | � (log 1

↵

)�1, ↵p

⌧ 1 and fromEq. (A15) we obtain

↵p

= ↵e1/2| ¯V |, V < 0, |V | �✓

log1

↵

◆�1

, e↵ ⌧ ✏2

(A17)which gives the standard BCS result `

p

= 1

|eV |.

Appendix B: Solution of RG equations: HLR phasewith Coulomb interactions

In this appendix we provide a detailed solution of theRG equations, (2.12), (2.26), for the HLR phase withCoulomb interactions. We, thus, set ⇣ = �1 in Eq. (2.26)and ✏ = 0 in Eq. (2.12).

The flow in the (eV , ↵) plane takes the form shown inFig. 6. Note that part of the phase space is controlled bythe fixed point eV = 0, ↵ = 0, while the rest of the flowtrajectories are towards eV = �1. We will show belowthat the two regions are separated by the line eV = �

pe↵.

Let us proceed to solve the flow equations. The flowof e↵ is the same as for the Ising-nematic case, Eq. (A1).Defining eV =

pe↵g,

dg

d`=

pe↵(1 � g2) +

1

2e↵g (B1)

In the limit e↵ ! 0, we can neglect the last term inEq. (B1). Then,

dg

de↵ = �e↵�3/2(1 � g2) (B2)

which gives,

g(`) =(g(0) + 1) exp

⇥4(e↵(`)�1/2 � e↵(0)�1/2)

⇤+ (g(0) � 1)

(g(0) + 1) exp⇥4(e↵(`)�1/2 � e↵(0)�1/2)

⇤� (g(0) � 1)

(B3)As e↵ flows to zero the following cases are possible. Ifg(0) > �1, i.e. eV (0) > �

pe↵(0), then g flows to 1,

eV (`) !pe↵(`) ⇡ 1p

`, eV > �

pe↵ (B4)

If one starts exactly on the transition line, g(0) = �1,then g remains fixed at g = �1,

eV (`) = �pe↵(`) ⇡ � 1p

`, eV = �

pe↵(0) (B5)

Finally, if g(0) < �1, i.e. eV (0) < �pe↵(0), eV (`) flows to

�1 and diverges at ` = `p

, with

`p

=1

16

log

eV �pe↵

eV +pe↵

+ 4e↵�1/2

!2

� e↵�1 (B6)

In particular, as eV approaches the transition line, �V =eV +

pe↵ ! 0�, the pairing gap � ⇠ ⇤

!

e�`p vanishes as,

� ⇠ ⇤!

exp

✓� 1

16log2 |�V |

◆= |�V |

116 log

1|�V | (B7)

1 H. v. Lohneysen, A. Rosch, M. Vojta and P. Wolfle, Rev.Mod. Phys. 79, 1015 (2007).

2 L. Taillefer, Ann. Rev. of Cond. Matt. Phys. 1, 51 (2010).3 T. Shibauchi, A. Carrington, Y. Matsuda, Ann. Rev. ofCond. Matt. Phys. 5, 113 (2014).

4 J. A. Hertz, Phys. Rev. B 14, 1165 (1976).5 P. Coleman, C. Pepin, Q. Si, and R. Ramazashvili, J.Phys. Condens. Matter 13, R723 (2001) .

6 T. Senthil, S. Sachdev, and M. Vojta Phys. Rev. Lett.90, 216403 (2003); T. Senthil, M. Vojta, and S. SachdevPhys. Rev. B 69, 035111 (2004).

7 I. Paul, C. Pepin, and M. R. Norman Phys. Rev. Lett. 98,026402 (2007).

8 T. Senthil, Phys. Rev. B 78, 045109 (2008).9 D. Podolsky, A. Paramekanti, Y. B. Kim, and T. Senthil,Phys. Rev. Lett. 102, 186401 (2009).

10 For a recent phenomenological treatment of this problem,see J.-H. She and J. Zaanen, Phys. Rev. B 80, 184518(2009); P. W. Phillips, B. W. Langley and J. A. Hutasoit,Phys. Rev. B 88, 115129 (2013).

11 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128(2010).

12 E. Berg, M. A. Metlitski and S. Sachdev, Science 338,

1606 (2012).13 D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev. B

55, 9452 (1997).14 D. Belitz, T. R. Kirkpatrick, and T. Vojta, Phys. Rev.

Lett. 82, 4707 (1999).15 A. V. Chubukov, C. Pepin, and J. Rech, Phys. Rev. Lett.

92, 147003 (2004).16 J. Rech, C. Pepin, and A. V. Chubukov, Phys. Rev. B 74,

195126 (2006).17 D. L. Maslov and A. V. Chubukov, Phys. Rev. B 79,

075112 (2009).18 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127

(2010).19 While an analysis of the e↵ective field theory governing

the Ising-nematic QCP shows that fluctuations do notdrive the transition first order, many microscopic modelsbelieved to harbor such a QCP display a first order tran-sition already at the mean field level.25,26 However, withband-structure and interaction parameters in the appro-priate region, the first order transition can be avoidedin the mean field treatment.26 Moreover, even when thetransition is first order at the mean field level, fluctuationsmay drive it second order as found in Ref. 30.

19

20 E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisensteinand A. P. Mackenzie, Ann. Rev. Cond. Matt. Phys. 1, 153(2010).

21 H. Yamase and H. Kohno, J. Phys. Soc. Jpn. 69, 2151(2000).

22 C. J. Halboth and W. Metzner, Phys. Rev. Lett. 85, 5162(2000).

23 V. Oganesyan, S. A. Kivelson, and E. Fradkin, Phys. Rev.B 64, 195109 (2001).

24 W. Metzner, D. Rohe, and S. Andergassen, Phys. Rev.Lett. 91, 066402 (2003).

25 H.-Y. Kee, E. H. Kim, and C.-H. Chung, Phys. Rev. B68, 245109 (2003).

26 H. Yamase, V. Oganesyan, and W. Metzner, Phys. Rev.B 72, 035114 (2005).

27 M. J. Lawler, D. G. Barci, V. Fernandez, E. Fradkin, andL. Oxman, Phys. Rev. B 73, 085101 (2006).

28 M. J. Lawler and E. Fradkin, Phys. Rev. B 75, 033304(2007).

29 P. Jakubczyk, P. Strack, A. A. Katanin, and W. Metzner,Phys. Rev. B 77, 195120 (2008).

30 P. Jakubczyk, W. Metzner, and H. Yamase, Phys. Rev.Lett. 103, 220602 (2009).

31 D. L. Maslov and A. V. Chubukov, Phys. Rev. B 81,045110 (2010).

32 D. F. Mross, J. McGreevy, H. Liu, T. Senthil, Phys. Rev.B 82, 045121 (2010).

33 S. C. Thier and W. Metzner, Phys. Rev. B 84, 155133(2011).

34 C. Drukier, L. Bartosch, A. Isidori, and P. Kopietz, Phys.Rev. B 85, 245120 (2012).

35 D. Dalidovich and Sung-Sik Lee Phys. Rev. B 88, 245106(2013).

36 S. Sur and S.-S. Lee, Phys. Rev. B 90, 045121 (2014).37 Y. Ando, K. Segawa, S. Komiya, and A. N. Lavrov, Phys.

Rev. Lett. 88, 137005 (2002).38 V. Hinkov, D. Haug, B. Fauque, P. Bourges, Y. Sidis, A.

Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Science319, 597 (2008).

39 Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien,T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi,S. Uchida, and J. C. Davis, Science 315, 1380 (2007).

40 R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choiniere, F.Laliberte, N. Doiron-Leyraud, B. J. Ramshaw, R. Liang,D. A. Bonn, W. N. Hardy, and L. Taillefer, Nature 463,519 (2010).

41 M. J. Lawler, K. Fujita, Jhinhwan Lee, A. R. Schmidt,Y. Kohsaka, Chung Koo Kim, H. Eisaki, S. Uchida, J. C.Davis, J. P. Sethna, and Eun-Ah Kim, Nature 466, 347(2010).

42 A. Mesaros, K. Fujita, H. Eisaki, S. Uchida, J. C. Davis, S.Sachdev, J. Zaanen, M. J. Lawler, Eun-Ah Kim, Science333, 426 (2011).

43 Y. Kohsaka, T. Hanaguri, M. Azuma, M. Takano, J. C.Davis, H. Takagi, Nature Physics 8, 534 (2012).

44 T.-M. Chuang, M. P. Allan, Jinho Lee, Yang Xie, NiNi, S. L. Budko, G. S. Boebinger, P. C. Canfield, andJ. C. Davis, Science 327, 181 (2010).

45 J.-H. Chu, J. G. Analytis, K. De Greve, P. L. McMahon,Z. Islam, Y. Yamamoto, and I. R. Fisher, Science 329,824 (2010).

46 C.-L. Song, Y.-L. Wang, P. Cheng, Y.-P. Jiang, W. Li,T. Zhang, Z. Li, K. He, L. Wang, J.-F. Jia, H.-H. Hung,C. Wu, X. Ma, X. Chen, and Q.-K. Xue, Science 332,

1410 (2011).47 M. Yi, D. H. Lu, J.-H. Chu, J. G. Analytis, A. P. Sorini,

A. F. Kemper, B. Moritz, S.-K. Mo, R. G. Moore,M. Hashimoto, W. S. Lee, Z. Hussain, T. P. Devereaux,I. R. Fisher, Z.-X. Shen, PNAS 108, 6878 (2011).

48 J.-H. Chu, H.-H. Kuo, J. G. Analytis, I. R. Fisher, Science337, 710 (2012).

49 S. Kasahara, H. J. Shi, K. Hashimoto, S. Tonegawa, Y.Mizukami, T. Shibauchi, K. Sugimoto, T. Fukuda, T.Terashima, A. H. Nevidomskyy, and Y. Matsuda, Nature486, 382 (2012).

50 M. P. Allan, T. -M. Chuang, F. Massee, Y. Xie, N. Ni, S.L. Bud’ko, G. S. Boebinger, Q.Wang, D. S. Dessau, P. C.Canfield, M. S. Golden, and J. C. Davis, Nature Physics9, 220 (2013).

51 H.-H. Kuo, M. C. Shapiro, S. C. Riggs, I. R. Fisher,Phys. Rev. B 88, 085113 (2013).

52 R. A. Borzi, S. A. Grigera, J. Ferrell, R. S. Perry, S. J.S. Lister, S. L. Lee, D. A. Tenant, Y. Maeno, and A. P.Mackenzie, Science 315, 214 (2007).

53 For a conceptually di↵erent example of superconductivityappearing in the vicinity of a phase with ~Q = 0 order, see:O. Vafek, J. M. Murray and V. Cvetkovic, Phys. Rev.Lett. 112, 147002 (2014); J. M. Murray and O. Vafek,Phys. Rev. B 89, 205119 (2014).

54 D. T. Son, Phys. Rev. D 59, 094019 (1999).55 R. Shankar, Physica A 177, 530 (1991).56 J. Polchinski, arXiv:hep-th/9210046 (1992).57 B. L. Altshuler, L. B. Io↵e and A. J. Millis, Phys. Rev. B

50, 14048 (1994).58 J. Polchinski, Nucl. Phys. B 422, 617 (1994).59 S.-S. Lee, Phys. Rev. B 80, 165102 (2009).60 C. Nayak and F. Wilczek, Nucl. Phys. B 417, 359 (1994);

430, 534 (1994).61 M. A. Metlitski, to appear.62 Ar. Abanov, B. Altshuler, A. Chubukov, and

E. Yuzbashyan (unpublished).63 For a review, see P. A. Lee, N. Nagaosa and X.-G. Wen,

Rev. Mod. Phys. 78, 17 (2006).64 P. A. Lee and N. Nagaosa, Phys. Rev. B 46, 5621 (1992).65 Y. B. Kim, A. Furusaki, X.-G. Wen and P. A. Lee, Phys.

Rev. B 50, 17917 (1994).66 O. I. Motrunich, Phys. Rev. B 72, 045105 (2005).67 S.-S. Lee and P. A. Lee, Phys. Rev. Lett. 95, 036403

(2005).68 D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Phys.

Rev. B 79, 205112 (2009).69 H.-Y. Yang, A. Laeuchli, F. Mila, K. P. Schmidt, Phys.

Rev. Lett. 105, 267204 (2010).70 Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato and

G. Saito, Phys. Rev. Lett. 91, 107001 (2003).71 T. Itou, A. Oyamada, S. Maegawa, M. Tamura and

R. Kato, Phys. Rev. B 77, 104413 (2008).72 D. Watanabe, M. Yamashita, S. Tonegawa et. al., Nat.

Commun. 3, 1090 (2012).73 S. Yamashita, Y. Nakazawa, M. Oguni, et. al., Nature

Physics 4, 459 (2008).74 S. Yamashita, T. Yamamoto, Y. Nakazawa, M. Tamura

and R. Kato, Nature Comm. 2, 275 (2011).75 M. Yamashita, N. Nakata, Y. Senshu et. al., Science 328,

1246 (2010).76 M. Yamashita, N. Nakata, Y. Kasahara et. al., Nature

Physics 5, 44 (2009).77 Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda and

20

G. Saito, Phys. Rev. Lett. 95, 177001 (2005).78 R. S. Manna, M. de Souza, A. Bruhl, J. A. Schlueter and

M. Lang, Phys. Rev. Lett. 104, 016403 (2010).79 T. Itou, A. Oyamada, S. Maegawa and R. Kato, Nature

Physics 6, 673 (2010).80 S.-S. Lee, P. A. Lee and T. Senthil, Phys. Rev. Lett. 98,

067006 (2007).81 T. Grover, N. Trivedi, T. Senthil and P. A. Lee,

Phys. Rev. B 81, 245121 (2010).82 In this paper, we only consider conventional BCS pairing

of spinons with total pair momentum ~P = 0. A di↵erentpossibility, encouraged by the Amperian current-currentinteraction, where the total pair momentum |~P | = 2k

F

was considered in Ref. 80. However, in the ✏-expansionemployed in the present paper, the spinon FS is stable tosuch pairing.

83 M. U. Ubbens and P. A. Lee, Phys. Rev. B 49, 6853(1994).

84 S. B. Chung, I. Mandal, S. Raghu, and S. ChakravartyPhys. Rev. B 88, 045127 (2013).

85 W. Witczak-Krempa, E. Sorensen and S. Sachdev, Na-ture Physics 10, 361 (2014); S. Gazit, D. Podolsky,A. Auerbach, D. P. Arovas, Phys. Rev. B 88, 235108(2013); J. Smakov and E. Sorensen, Phys. Rev. Lett. 95,180603 (2005); M.-C. Cha, M. P. A. Fisher, S. M. Girvin,M. Wallin, and A. P. Young, Phys. Rev. B 44, 6883(1991).

86 T. Senthil and M. P. A. Fisher, Phys. Rev. B 62, 7850(2000).

87 B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47,7312 (1993).

88 R. L. Willett, Adv. Phys. 46, 447 (1997).89 Y. B. Kim, P. A. Lee, X.-G. Wen and P. C. E. Stamp,

Phys. Rev. B 51, 10779 (1995).90 Y. B. Kim, P. A. Lee, and X.-G. Wen, Phys. Rev. B 52,

17275 (1995).91 G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991).92 M. Greiter, X. G. Wen, and F. Wilczek, Nucl. Phys. B

374, 567 (1992).93 R. H. Morf, Phys. Rev. Lett. 80, 1505 (1998).94 E. H. Rezayi and F. D. M. Haldane, Phys. Rev. Lett. 84,

4685 (2000).95 G. Moller, A. Wojs and N. R. Cooper, Phys. Rev. Lett.

107, 036803 (2011).96 Z. Papic, F. D. M. Haldane and E. H. Rezayi, Phys. Rev.

Lett. 109, 266806 (2012).97 N. E. Bonesteel, Phys. Rev. Lett 82), 984 (1999).98 Strictly speaking, it is not known whether higher loop

terms in the RPA resummed expansion are suppressedby powers of ↵. Thus, the validity of the perturbativeexpansion in ✏ with N - finite, as well as of the results at✏ = 0 both with N finite and with N ! 1, has not beenfirmly established. On the other hand, the calculationsin the double-scaling limit ✏ ! 0, N ! 1, N✏ ⇠ O(1)are certainly controlled.32 In the present paper, we will

assume that the first expansion is controlled, as well.99 Strictly speaking, this assumption is unphysical for the

Ising-nematic QCP, however, it is not expected to changemost of our results qualitatively (see end of section III forfurther discussion).

100 A. V. Chubukov and J. Schmalian, Phys. Rev. B 72,174520 (2005).

101 N. E. Bonesteel, I. A. McDonald, and C. Nayak, Phys.Rev. Lett. 77, 3009 (1996).

102 E.-G. Moon and A. V. Chubukov, J. Low Temp. Phys.161, 263 (2010).

103 T. A. Maier and D. J. Scalapino, Phys. Rev. B 90, 174510(2014).

104 S. Lederer, Y. Schattner, E. Berg and S. A. Kivelson,arXiv:1406.1193.

105 Y. Huh and S. Sachdev, Phys. Rev. B 78, 064512 (2008).106 S A. Parameswaran, S. A. Kivelson, S. L. Sondhi and

B. Z. Spivak, Phys. Rev. Lett. 106, 236801 (2011).107 S. A. Parameswaran, S. A. Kivelson, E. H. Rezayi,

S. H. Simon, S. L. Sondhi and B. Z. Spivak, Phys. Rev.B 85, 241307(R) (2012).

108 O. I. Motrunich, Phys. Rev. B 73, 155115 (2006).109 A. L. Fetter and J. D. Walecka, Quantum Theory of Many-

Particle Systems, Dover, New York (2003).110 Strictly speaking, the HLR phase is only defined at zero

temperature, where it occurs at a fixed electron density(⌫ = 1/2 for the clean, Gallilean invariant system). For ⌫deviating from 1/2 and T = 0, composite fermions see abackground magnetic field Be↵ = B�2(2⇡)n, and so formLandau levels. At low enough temperature, one, thus, ex-pects to see a sequence of integer QH states of compositefermions, which correspond to a sequence of FQH statesof physical electrons with ⌫ = p/(2p± 1).87 In our discu-sion, we assume that the temperature T is much largerthan the cyclotron frequency !

c

of composite fermions,so that Landau-level quantization can be neglected. Atthe same time, for our estimates to be correct, we needthe temperature to be much lower than the gap, �, ofthe paired CF phase. We expect the cyclotron frequency,!c

= Beffm

⇤ = vFBeffkF

. The Fermi-velocity in the HLR phase

runs with energy, (2.16), vF

⇠ !1�1/zf . When Landau-levels of composite fermions form, we expect that therunning of v

F

will be cut-o↵ at ! = !c

. We, thus, ob-tain, !

c

⇠ (Be↵)zf . At the first order transition from the

paired CF phase to the HLR phase, Be↵ ⇠ �nc

⇠ �5/6, so!c

⇠ �5/4. Thus, our calculations are legitimate as longas �5/4 ⌧ T ⌧ �.

111 Y. Nakai, T. Iye, S. Kitagawa, K. Ishida, H. Ikeda et al.,Phys. Rev. Lett. 105, 107003 (2010).

112 V. M. Apalkov and T. Chakraborty, Phys. Rev. Lett. 107,186803 (2011).

113 Z. Papic, D. A. Abanin, Y. Barlas, R. N. Bhatt, Phys.Rev. B 84, 241306(R) (2011).