A Solution of RG equations: Ising-nematic QCP

# Are non-Fermi-liquids stable to Cooper pairing?

## Abstract

States of matter with a sharp Fermi-surface but no well-defined Landau quasiparticles arise in a number of physical systems. Examples include: (i) quantum critical points associated with the onset 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 this work, 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 critical point as an example of an ordering phase transition in a metal, and demonstrate that the attractive interaction mediated by the order parameter fluctuations always leads to a superconducting instability. Moreover, in the regime where our calculation is controlled, superconductivity preempts the destruction of electronic quasiparticles. On the other hand, the spinon Fermi-surface and the Halperin-Lee-Read states are stable against Cooper pairing for a sufficiently 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) and spin-liquid states; (ii) Halperin-Lee-Read and Moore-Read states.

## I Introduction

It is well-known that ordinary metals described by Fermi-liquid (FL) theory are unstable to an arbitrarily weak attractive interaction in the BCS channel, which leads to Cooper pairing of electrons and drives the system into a superconducting phase. The purpose of the present paper is to examine the stability of certain non-Fermi-liquid (nFL) states in two dimensions to Cooper pairing. We study systems where the non-Fermi-liquid behavior arises as a result of the interaction of a gapless bosonic mode with fermions in the vicinity of the Fermi-surface (FS). Specific examples we analyze are described in the following subsections.

### i.1 Quantum critical points in metals

Many correlated metals appear to possess quantum critical points (QCPs) with fascinating properties.(1); (2); (3) Frequently, there is a striking breakdown of Fermi liquid theory in the vicinity of the QCP. Equally strikingly superconductivity is often but not always strengthened near the QCP. Indeed, a fairly common phase diagram (see Fig. 1, top), shared for instance by cuprate, pnictide and certain heavy-fermion materials, has a superconducting dome around the putative ‘metallic’ QCP with ‘optimal’ transition temperature right at the QCP. On the other hand, there are prominent quantum critical heavy electron metals such as CeCuAu and YbRhSi where superconductivity does not appear down to very low temperatures.(1) It thus appears that superconductivity is enhanced at some but not all quantum critical points in metals. Despite this there is currently limited understanding of the interplay between the quantum criticality, the non-Fermi liquid ‘normal’ state, and the possible superconductivity. Clearly, a theory of the relationship of superconductivity and quantum criticality has to accommodate the absence of superconductivity at some and enhancement at other quantum critical points.

It is important right away to recognize that there are two fundamentally distinct classes of quantum criticality in metallic systems. They are distinguished by the fate of the electron Fermi surface as the metal undergoes the quantum phase transition. In one class, the electron Fermi surface evolves continuously through the critical point but is distorted in some way. These QCPs are typically associated primarily with the onset of a broken symmetry characterized by a Landau order parameter in a metal. Examples include the onset of ferromagnetism or antiferromagnetism in a paramagnetic metal. The proper theoretical framework to describe such a phase transition is through coupling the low energy electronic degrees of freedom at the Fermi surface to fluctuations of the Landau order parameter.(4) An alternate class of quantum phase transitions involves a more violent transformation of the electronic structure where the electron Fermi surface (or a sheet of it) disappears completely on crossing the critical point.(5) Surprisingly, such a discontinuous evolution of the electron Fermi surface can happen through a continuous phase transition. Examples include the so-called Kondo breakdown transition in Kondo lattices(6); (7) and continuous Mott metal-insulator transitions in two(8) or three dimensions.(9) There is currently only one known theoretical framework that yields such a phase transition: this is based on slave-particle methods and inevitably leads to a description in terms of fractionalized slave-particles coupled to fluctuating emergent gauge fields.

In this paper we will consider examples of both kinds of quantum critical points as case studies for the relationship between quantum criticality, superconductivity, and non-Fermi liquid physics. In the example studied of the first class, where the Fermi surface is distorted through the development of a broken symmetry, we show that superconductivity is strongly enhanced near the critical point. We suggest that this may be more generally true: order parameter fluctuations enhance superconductivity. In the example studied of the second class where the entire electron Fermi surface is annihilated, we argue that superconductivity is suppressed. This dichotomy may explain the phenomenology described above where some but not all QCPs show an enhancement of superconductivity.

We begin with QCPs associated with the onset of a symmetry breaking order. Strong fluctuations of the order parameter present at the QCP tend to decohere the electronic quasiparticles: as the system is tuned to the critical point, the residue and the Fermi-velocity of quasiparticles approach zero. A common feature of such QCPs is that there exists some pairing channel in which the order parameter fluctuations mediate attraction. The strength of the attraction increases as one approaches the QCP, yet the same order parameter fluctuations, which provide the pairing glue, also destroy the very quasiparticles that are trying to pair. The central question is which of these two competing effects wins. In particular, is such a QCP in a metal inherently unstable to superconductivity, as empirical observations suggest?(10)

In the present paper we address the above question for the class of metallic QCPs, where the order parameter carries a wave-vector (for recent progress on the case, see Refs. (11); (12)). The most familiar example of such a phase transition is the Stoner instability associated with the development of ferromagnetic order. Modern developments show that due to fluctuation effects the Stoner transition is likely modified at low temperature and becomes first order (or develops an intermediate spiral ordered phase).(13); (14); (15); (16); (17) A different example which does not suffer from these complications(16); (18) (see, however, footnote (19)) is the transition associated with the onset of Ising-nematic order, characterized by spontaneous breaking of a four-fold rotational symmetry of the lattice to a two-fold subgroup.(20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (31); (18); (32); (33); (34); (35); (36) The order parameter in this case is just a real Ising field . Growing evidence for such order has been found in a number of physical systems including cuprate,(37); (38); (39); (40); (41); (42); (43) pnictide(44); (45); (46); (47); (48); (49); (50); (51) and ruthenate(52) materials. From a theoretical viewpoint, the Ising-nematic QCP is perhaps one of the simplest phase transitions in metals. It, thus, provides a convenient setting for studying the interplay between quantum criticality, nFL and pairing physics.(53)

We perform a systematic renormalization group (RG) analysis of the Ising-nematic QCP. Our approach utilizes an idea introduced by D. T. Son in his study of quark pairing by the color gauge field in dense baryonic matter.(54) We combine the conventional Fermi-liquid RG treatment of Refs. (55); (56) with the so-called “two-patch” scaling approach of Refs. (57); (58); (59); (18). Analytical control is gained through the -expansion introduced in Ref. (60) and its subsequent large- improvement.(32) We find that the Ising-nematic QCP is always unstable to superconductivity. In particular, attractive pairing interaction mediated by the order parameter fluctuations dominates over other residual short range interactions (even if they are repulsive) and drives a pairing instability as the QCP is approached. However, the residual short range interactions determine the angular momentum/spin channel in which the pairing instability occurs; as a result, the pairing symmetry is non-universal. The usual weak coupling BCS formula, , relating the superconducting to the strength of the short-range interaction clearly does not hold in the vicinity of the QCP. Rather the superconductivity is strongly enhanced, and at the QCP scales in a power-law manner with the coupling between order-parameter fluctuations and the electrons. Thus, in this example we clearly demonstrate the importance of quantum criticality in optimizing the superconducting . Moreover, in the regime where our calculation is controlled (small ), the energy scale at which superconductivity sets in is parametrically larger than the energy scale at which electronic quasiparticles are destroyed. Thus, the superconducting instability is so strong that it preempts the nFL physics (see the bottom figure in Fig. 1). The above results of our RG analysis are in exact agreement with a direct solution of Eliashberg-like integral equations, as is shown elsewhere by one of us.(61); (62)

Next, we proceed to the class of QCPs associated with annihilation of the Fermi-surface. We take as an example the Mott transition from a Fermi-liquid to an insulating spin-liquid with a spinon Fermi-surface.(8) Applying the RG procedure described above, we show that spinon pairing is suppressed both in the spinon Fermi-surface phase and at the Mott transition itself. As a result, the Mott transition and the FL phase in its vicinity will be stable to superconductivity. We expect similar conclusions to hold for the Kondo breakdown transition in the Kondo lattice. First, however, we review the construction and properties of the spinon Fermi-surface phase.

### i.2 Spinon Fermi-surface phase

The spinon Fermi-surface phase is an exotic Mott-insulating spin-liquid with emergent spin- fermionic spinon excitations, , .(63) The spinon dispersion is such that they form a Fermi-surface. This phase may be accessed in the slave-particle (parton) treatment of a lattice spin model, where electron spin operators are represented as , subject to the local constraint, . While the spinons are neutral under the physical electromagnetic field, they carry a charge under an emergent gauge field , hence this phase is also often referred to as a spin-liquid. An effective Lagrangian of the spinon FS phase may be written as,

 Lf=f†α[∂τ−iaτ+ϵ(−i∇−→a)]fα+12g2(ϵμνλ∂νaλ)2+… (1)

where is the spinon dispersion and the ellipses denote additional perturbations, such as four-spinon interactions.

The spinon FS phase is expected to naturally arise in so-called “weak” Mott insulators - ones proximate to a metal-insulator transition. In this situation the spinon FS state may be conveniently described within a slave particle description of an electronic Hubbard model. We write the electron operator as , where is a charge- boson with zero spin, and is the spinon as described above. This representation introduces a gauge redundancy under which carries gauge charge and carries gauge charge . We consider a state in which form a Fermi surface. If in addition the boson is condensed, we obtain the usual metallic Fermi-liquid. If, however, is gapped, we obtain an electrical insulator but with a spinon Fermi-surface coupled to a fluctuating gauge field. This is the spinon Fermi-surface state introduced above. We note that right at the Mott metal-insulator transition, the boson is critical while the 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 the next section.

There is numerical evidence for the presence of the spinon FS spin-liquid phase in the triangular lattice Hubbard model in the intermediate range of .(66); (67); (68); (69) Moreover, it has been proposed as a candidate for the quasi-2d triangular lattice organic insulators and (abbreviated ET and DMIT below).(66); (67) These materials have an estimated spin-exchange coupling K, yet display no magnetic order down to mK temperature. Moreover, these electrical insulators, surprisingly, show metallic behavior in their low temperature spin-susceptibility ()(70); (71); (72) and specific-heat ().(73); (74) DMIT also exhibits metallic thermal transport at low temperature, ,(75) while ET shows activated thermal transport, albeit with a rather small gap K.(76) Both materials can be driven metallic by an application of a moderate pressure of GPa, with ET developing superconductivity below K on the high-pressure side.(77) At ambient pressure, ET displays a phase transition (or a very rapid crossover) at K,(78) resulting in partial loss of low-energy excitations as evidenced by specific heat.(73) It has been suggested that this low-temperature anomaly may be due to a pairing instability of the spinon FS.(80); (81)

Current theoretical understanding of the spinon FS phase is based on the following observations. The presence of gapless spinon excitations in the vicinity of the FS strongly affects the gauge field dynamics. The longitudinal fluctuations of the emergent electric field are Debye screened by the spinon FS and become gapped. The fluctuations of the emergent magnetic field are Landau-damped by the FS, but remain gapless. The coupling of these Landau-damped magnetic field fluctuations to spinons is expected to lead to “non-Fermi-liquid” behavior of the spinon FS,(64); (65); (57); (58) e.g. the anomalous scaling of specific heat .

In this paper we analyze whether the spinon FS phase is stable to BCS pairing of spinons. We first observe that the gapless fluctuations of the magnetic field mediate a long-range repulsive interaction in the BCS channel and hence are not expected to cause spinon pairing. Indeed, fluctuations of the magnetic field mediate a current-current interaction. The spinons in a BCS pair have opposite momenta and opposite currents and hence, by Ampere’s law, repel. Therefore, gapless gauge field fluctuations suppress spinon pairing.(82) However, in addition to gauge field mediated long-range interactions, short-range interactions between the spinons will generally be present. Depending on the microscopic details of the system, such short range interactions may be attractive in the BCS channel with some angular momentum and spin. If the short range attraction is sufficiently strong, we expect the spinons to pair, developing a condensate (we leave the angular/spin structure of the pair wave-function implicit for now). As in an ordinary superconductor, the spinon excitations acquire a gap, except possibly at symmetry dictated (or accidental) point nodes on the FS. The pair condensate spontaneously breaks the emergent gauge symmetry down to a subgroup. As a result, the gauge field becomes gapped through the Higgs mechanism. Gauge excitations now take the form of gapped vortices carrying a magnetic flux . Such excitations are often referred to as visons. Visons and spinons possess mutual semionic statistics. Thus, the paired phase of spinons is just a spin-liquid.

We confirm the above intuitive picture with a systematic RG calculation. We show that the spinon FS phase, is, indeed, stable as long as the strength of the short-range attractive BCS interactions is smaller than a critical value for all angular momentum channels (we employ a sign convention where represents an attractive interaction). However, once for some , the spinon FS develops an instability to pairing in angular momentum channel . , thus, marks the quantum phase transition between the spin-liquid and the spin-liquid. We find the phase transition to be continuous and calculate the critical exponents using the -expansion of Refs. (60); (32). Our findings are contrary to previous claims(83) that this phase transition is driven first order by gauge field fluctuations. We discuss the properties of the paired phase in the vicinity 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 (specific heat, uniform and finite wave-vector spin-susceptibility, spin-chirality correlations) are not modified from those in the spinon Fermi-surface phase itself. Our findings are in exact agreement with an Eliashberg-like treatment of the problem.(61)

Previously, the pairing quantum phase transition from the spinon Fermi-surface state was considered in 3 dimensions by Chung et al.(84) within an Eliashberg-like approximation. Our paper presents an RG analysis directly in 2 dimensions, although there are some qualitative similarities with the results of Chung et al.(84) In particular, Chung et al. have also concluded that a continuous pairing transition is possible. However, we believe that some of the results of Chung et al. are not generic. In particular, Chung et al. find that pairing can only occur in angular momentum channels . In contrast, we believe that both in 2d and 3d pairing with arbitrary angular momentum can be induced by tuning the appropriate . Furthermore, we expect the power-law onset of the pairing gap found by Chung et al. in 3d to be modified by the renormalization of spinon quasiparticle residue and Fermi-velocity. In fact, we anticipate that the precise critical properties of the pairing transition in 3d will be very similar to those of the 2d Halperin-Lee-Read phase in the presence of long-range Coulomb interactions, discussed in section I.4.

### i.3 Mott transition from a Fermi-liquid to a spinon Fermi-surface phase

The Mott transition from a Fermi-liquid to a spinon FS phase is an example of a QCP where the entire electron FS disappears. As noted in the previous section, this transition is driven by condensation of the slave boson . The transition may be described by the effective theory,

 L=Lb+Lf (2)

where the Lagrangian for the complex scalar field is

 Lb=|(∂τ−iaτ)b|2+v2b|(∇−i→a)b|2+t|b|2+u|b|4 (3)

and is still given by Eq. (1). Note that here we are considering a Mott transition occurring at fixed electron density. When is large and positive, the boson is gapped and can be integrated out, so the system is in the spinon FS phase. On the other hand, when is large and negative, is condensed, . As a result, the gauge field becomes gapped via the Higgs mechanism; furthermore, the electron and the spinon are identified, . Thus, the system is in the ordinary FL phase.

We now discuss the fate of the system when is tuned to a critical value where is gapless (for more details, see Ref. (8), whose findings we summarize here). If the fluctuations of the gauge field are ignored then the spinon and boson sectors in Eq. (2) decouple, and the boson sector undergoes a transition in the XY universality class, while the spinon sector remains a “spectator” Fermi-liquid across the transition. Proceeding to include gauge field fluctuations, we note that the longitudinal electric field is again Debye screened by the spinon Fermi-surface and so can be ignored. The fluctuations of the magnetic field are again Landau-damped by the spinon Fermi-surface, but remain gapless. It turns out that such Landau-damped gauge fluctuations do not affect the -sector of the theory, which remains decoupled from the spinon sector and continues to be described by the XY critical theory. On the other hand, the -sector does affect the low energy gauge fluctuations. Integrating the gapless boson at the XY critical point out, one obtains the following effective action for the magnetic field fluctuations,

 Sa=12∫d2→xd2→x′dτ(∇×→a)(→x,τ)Π(→x−→x′)(∇×→a)(→x′,τ) (4)

where , and is the universal conductivity of the model.(85) Thus, the gauge-spinon sector of the theory is described by the action

 S=∫d2xdτLf+Sa (5)

which will be the starting point for our theoretical analysis in this paper. We note that this action coincides with that of the Halperin-Lee-Read state with Coulomb interactions, discussed in section I.4. Studying the theory (5), one finds that gauge field fluctuations turn the spinon FS at the Mott transition into a marginal Fermi-liquid with a specific heat , which dominates the overall specific heat of the system. We remind the reader that since the physical electron is a product of the boson and the spinon , the actual physical electron Green’s function displays a deviation from Fermi-liquid theory at the Mott transition.

One may now ask whether the spinon FS at the Mott transition is stable to BCS pairing of spinons. Before we address this question, we would like to stress that independent of whether the spinons pair, we expect no long-range superconductivity exactly at the Mott transition. After all, at the Mott transition charge degrees of freedom are on the verge of becoming localized so long-range phase coherence will be suppressed. Instead, spinon pairing should be interpreted as a local tendency of electrons to pair. Let us discuss the scenario where spinon pairing does occur at the transition. In this case, as we tune the system away from the Mott transition to , a condensate appears, i.e. the compressible phase adjacent to the Mott transition is a superconductor rather than a Fermi-liquid. On the other hand, the phase with , where the boson is gapped, is a spin-liquid insulator as discussed in the previous section. Thus, if the spinons are paired the Mott transition occurs between a superconductor and a spin-liquid insulator.(86) As we approach the transition from the superconducting side, both the superconducting and the superconducting condensate will vanish, however, the gap to a single electron will remain finite across the transition. In contrast, if the spinon FS is stable against pairing then the single electron gap at the transition will vanish.

With the above remarks in mind, we now summarize the conclusions of our RG analysis. As with the spinon FS phase, we show that repulsive current-current interactions mediated by the gauge field suppress spinon pairing at the Mott transiton. As a result, as long as the strength of short-range attraction between spinons is below a critical value , the spinon FS at the Mott transition is stable. We believe that in this regime a stronger statement actually holds: no spinon pairing occurs on either side of the Mott transition, in particular, no superconductivity develops in the FL phase adjacent to the Mott transition. Thus, the Mott transition is an example of a QCP in a metal, which is stable to superconductivity.

On the other hand, once for some , the spinons at the Mott transition pair, developing a condensate . Thus, in this parameter regime the Mott transition occurs between a superconductor and a spin-liquid insulator, and the single electron gap remains finite across the transition.

The RG formalism developed in this paper can be applied to analyze the stability of yet another exotic phase: the Halperin-Lee-Read (HLR) phase. The HLR phase is a compressible phase of the quantum Hall (QH) fluid at a filling fraction .(87) It is believed to be experimentally realized by the conventional 2DEG in the lowest 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, the composite fermions will, on average, see no magnetic field and form a Fermi-surface. Technically, flux attachment is performed with an aid of a Chern-Simons (CS) gauge field , leading to the action

 S = ∫d2xdτ(Lf+LCS)+SU, Lf = f†[∂τ−iaτ−12m(∂i−iai+iAi)2]f (6) LCS = i2(4π)ϵμνλaμ∂νaλ (7) SU = 12∫d2→xd2→x′dτf†f(→x,τ)U(→x−→x′)f†f(→x′,τ)

Here, is the composite fermion operator, is the vector potential for the external magnetic field and is the microscopic electron-electron interaction potential. Integration over produces the constraint,

 ∇×→a=2(2π)f†f (9)

linking the magnetic flux density of the CS field to the electron density . This constraint can be used to rewrite in terms of .

At , the flux of the CS gauge field on average cancels the external magnetic field, however, fluctuations of about the average flux persist. The dynamics of are nearly the same as in the spinon FS phase with the longitudinal electric field Debye screened and gapped, and the magnetic field Landau damped and gapless. As the electric field is gapped, the CS term in Eq. (7) is irrelevant in the RG sense (more precisely, it generates a charge-current interaction of composite fermions which is suppressed in the small momentum limit compared to the current-current interaction). Therefore, the low-energy effective theory of the HLR phase is nearly identical to that of the spinon FS phase when the microscopic electron interaction in the QH fluid is short ranged. For a power law interaction,

 U(→x)∼1|→x|1+ϵ, (10)

with , the electron density fluctuations and hence the gauge field fluctuations are suppressed.(87) In fact, for , the composite fermion quasiparticles remain sharply defined, while for Coulomb interaction, , the HLR phase is believed to be a marginal Fermi-liquid with a specific heat .(87); (60) For , the HLR phase is a true nFL,(87); (65); (89); (90) with a power law specific heat , however, the theory is under analytical control in the limit .(60); (32).

In passing, we note that the HLR phase may alternately be described within a slave particle approach that exposes the conceptual similarity to a spin-liquid Mott insulator with a spinon Fermi-surface discussed above. We represent the electron operator as a product of a charge- boson and a charge neutral fermion : . Then the bosons are at filling factor and we take them to be in the bosonic Laughlin state at that filling. Being neutral, the fermions see no magnetic field, and form a Fermi-surface. This slave particle description introduces a gauge redundancy, with and carrying opposite charges under an emergent gauge field . The corresponding gauge constraint fixes the number density of the bosons to equal that of the fermions. Being electrically charged, the boson density is simply equal to the physical electron density. Thus, the density of fermions also equals the physical electron density. Consequently, the size of the Fermi-surface is set by the physical electron density. Since the bosonic Laughlin state is gapped, we can integrate the boson degrees of freedom out, generating a Chern-Simons term (7) for the emergent gauge field . Thus, the slave particle description is completely equivalent to the familiar flux-attachment picture described above.

In this paper, we address the stability of the HLR phase to BCS pairing of composite fermions. As with the spinon FS phase, the long-range current-current interaction mediated by gapless gauge field fluctuations suppresses pairing in the BCS channel. Thus, we find that the HLR phase is stable as long as the strength of the short-range attractive BCS interaction is smaller than a critical value . However, once , pairing of composite fermions will occur, giving rise to an incompressible QH phase with a Hall conductivity . A possible “microscopic” source of an attractive BCS interaction is the short-distance part of the charge-current interaction mediated by the CS gauge field, which produces attraction in the channel.(92) In fact, if pairing occurs in the channel, the resulting phase is just the familiar Moore-Read (MR) “Pfaffian” state.(92) After the pairing transition, composite fermions become gapped neutral fermion excitations of the MR phase. Gauge excitations are also gapped through the Higgs mechanism and appear in the form of vortices carrying magnetic flux of , which via Eq. (9), translates into physical electric charge . Furthermore, these vortices support Majorana zero modes of composite fermions in their core and, therefore, can be identified with non-Abelian quasiparticles of the MR state. We find the phase transition between the HLR and the MR phases to be continuous, consistent with numerical simulations (94); (95); (96), but contrary to previous theoretical claims.(97) We describe how the neutral fermion gap and the charge gap vanish as one approaches the QCP from the MR side, and discuss the phenomenology of the MR phase in the vicinity of the transition.

## Ii Renormalization Group analysis

Although various nFL states described above arise in very different physical systems, they admit a unified theoretical treatment involving a gapless boson interacting with the FS. We denote the boson as : it represents the order parameter in the case of the Ising-nematic QCP and the transverse component of the vector potential in the case of the spinon FS phase, the Mott critical point, and the HLR phase. We denote the fermions (physical electrons in the Ising-nematic case, spinons in the spinon FS phase/Mott transition case and composite fermions in the HLR case) as . We take the flavor index to run from to . Physically, (two spin flavors) for the Ising-nematic QCP and spinon FS phase/Mott transition, and for the spin-polarized HLR phase.

Due to Landau-damping, boson fluctuations with wave-vector interact most strongly with fermions in the regions of the FS to which is nearly tangent.(57); (58); (59) We divide the FS into pairs of antipodal patches, labelled by an index , with

 width~{}Λy≪kF and thickness~{}Λx∼Λ2ykF≪Λy≪kF, (11)

where is the Fermi momentum; see Fig. 2. For simplicity, we assume that the Fermi-surface is connected and convex, and furthermore, that the local Fermi-surface curvature and Fermi momentum are comparable.

Antipodal points on the FS are chosen in patch pair and directions perpendicular () and tangent () to the FS at defined. The fermion operator is then expanded in terms of patch fields as,

 fα(x)=∑j(fj+α(x)ei→kj⋅→x+fj−α(x)e−i→kj⋅→x). (12)

We also define boson patch fields to include only momenta nearly tangent to the FS in patch :

 |qx|<|qy|ΛykF,|qy|<Λy. (13)

The effective action describing the fermion-boson interaction then breaks up into decoupled actions for each patch pair,

 S=∑jSj, (14)

with(59); (18); (32)

 Sj=∫d2xdτ(Lf[fj]+Lint[fj,ϕj])+Sϕ[ϕj], (15)

The Lagrangian densities are

 Lf = f†+α(∂τ+vF(−i∂x−∂2y2K))f+α (16) + f†−α(∂τ+vF(i∂x−∂2y2K))f−α Lint = vFϕ(f†+αf+α+ζf†−αf−α) (17) Sϕ = N2g2∫d2→qdω(2π)3|qy|1+ϵ|ϕ(→q,ω)|2 (18)

Here, we have suppressed the patch index . The Fermi-surface curvature , the Fermi-velocity and, in the case of the Ising nematic transition, the coupling constant , will generally vary along the Fermi-surface (i.e. will be patch-dependent). The constant for the nematic QCP and for the spinon FS phase/Mott transition and the HLR phase.

For general , the action is non-local. For the HLR state, this term encodes the long-range microscopic electron-electron interation, . The important case of a Coulomb interaction corresponds to , while for a short-range interaction, , and the term (18) is local. In case of the nematic QCP or spinon FS phase, the physical value of is . However, one may be able to access via an expansion around .(60); (32) We, thus, work in the regime below. Proceeding finally to the case of the Mott transition, the screening of the gauge field by the gapless boson also generates a non-local term (4) in the gauge action, i.e. the effective action for the gauge-spinon sector is described by Eqs. (16)-(18) with .

In the case of the Ising nematic transition, the Lagrangian also allows for a perturbation , which tunes the system across the QCP. Below, we will work directly at the QCP, setting . We also perform all our RG calculations at temperature . As usual, we treat finite as an infra-red cut-off when running the RG equations.

As already noted, distinct pairs of patches are decoupled in the above description and can be treated independently. We will shortly discuss the crucial role played by the inter-patch interactions in the pairing physics, however, for now, let us ignore such couplings and review the RG analysis of the two-patch theory (16) - (18).(60); (32) The two-patch theory is described by a single dimensionless coupling constant,

 α≡g2vFΛ−ϵy(2π)2. (19)

The fermion part of the action (16) dictates the scaling of frequency and momenta:

 ω→e−zfℓω,qx→e−ℓqx,qy→e−ℓ/2qy, (20)

with the bare dynamical exponent, . As we will see below, will generally be renormalized by interactions, however, the “anisotropic” momentum scaling, , is exact due to the non-renormalization of the FS curvature .(18) The full interacting fermion Green’s function depends only on the distance to the FS, , so we may identify with the fermion dynamical exponent. On the other hand, the full boson propagator of the two-patch theory depends only on the momentum tangent to the FS, , so the above scaling fixes the relationship(18) between the boson dynamical exponent and the fermion dynamical exponent ,

 zb=2zf. (21)

Under the above scaling with bare , flows as . Hence, the fermion-boson interaction is irrelevant for , relevant for and marginal at tree-level for . To compute quantum corrections to the RG flow one can utilize either a perturbative expansion in (Ref. (60); however, see footnote (98)) or a expansion (Ref. (32)). At leading order both expansions give the same result. To one loop order (first order in ), and run as,

 dαdℓ = ϵ2α−α2N (22) dvFdℓ = −αNvF (23)

and the fermion field acquires an anomalous dimension,

 f(ω,qx,qy)→[1+(74−ηf2)dℓ]f(edℓω,edℓqx,edℓ/2qy) (24)

with . For , flows logarithmically to zero, and the system is a marginal Fermi-liquid with the fermion self-energy,

 Σ(ω)∼−iαNωlogΛω|ω| (25)

with - the energy cut-off. For , the flow (22) has an infra-red stable fixed point at . If is of then ensures that the fixed-point occurs at weak coupling. On the other hand, if , we take to make and obtain a well-defined large- limit. In either case, at the fixed point,

 dvFdℓ=−ϵ2vF,ηf=ϵ2 (26)

implying a fermionic dynamical exponent

 zf=1+ϵ2, (27)

and a fermion self-energy

 Σf(ω)∼ω1−ηf. (28)

The exponent directly manifests itself in the nFL specific heat,

 C∼T1/zf. (29)

We note that the expression for in Eq. (27) holds to all orders in : this is tied to the non-analytic nature of the dependence in , which undergoes no renormalization. On the other hand, for , is analytic in and, in principle, can undergo renormalization. Our ability to access the physically important point through an expansion around is, thus, tied to such renormalizations being absent. No renormalization of in the theory has been found up to three loop order, (18) however, a general proof of this statement is currently lacking.

We next return to consider the effect of inter-patch interactions, which have been mostly ignored in previous studies. However, as we demonstrate below, such couplings must be included in the theory, as they are automatically generated in the RG process. This fact was first noted in Ref. (54) in the context of 3d QCD at finite quark density, and here we closely follow the RG treatment proposed by Ref. (54). So far, we have left the precise RG procedure somewhat implicit. Recall that under the scaling we advocated for the two-patch theory, and , so in the RG process we reduce both the fermion momentum cut-off perpendicular to the FS, , and the cut-off tangent to the FS, . While can be, as usual, shrunk by integrating out gapped fermion excitations away from the FS, reducing in the same manner would require integration over gapless fermions on the FS, which is illegal. Instead, during each RG step we re-partition the FS into smaller patches with width , while the reduction in the patch thickness is still performed by integrating out gapped fermions away from the FS. Simultaneously, in each RG step we integrate out boson fluctuations with momenta ; see Fig. 3. Before the RG step, such boson fluctuations mediate non-local intra-patch interactions between the fermions. However, after the RG step, these generate a local four-fermion inter-patch interaction, as shown in Fig. 3 (bottom).

As is well known from ordinary FL theory, a very restricted set of four-fermion inter-patch couplings on the FS is kinematically allowed.(55); (56) Only forward-scattering and BCS scattering interactions survive as the shell of fermion states around the FS is shrunk in the RG process. As we are interested in the physics of pairing, in the present paper we concentrate only on four-fermion interactions in the BCS channel, which can be described by the action,

 SBCS = −14∫4∏i=1d2→kidωi(2π)3f†α(k1)f†β(k2)fγ(k3)fδ(k4)(2π)3δ3(k1+k2−k3−k4) (30) ×((δαγδβδ+δαδδβγ)Va(→k1,→k2;→k3,→k4)+(δαγδβδ−δαδδβγ)Vs(→k1,→k2;→k3,→k4))

Here, / are, respectively, symmetric/antisymmetric under exchanging , and . Only the values of the interaction for BCS-matched momenta, , play a role; furthermore, can be taken to lie on the FS. From now on, we assume that the system is rotationally invariant,(99) so we may write , with - angles on the FS. Performing an expansion in angular harmonics,

 Vs,a(θ1−θ2)=∞∑m=−∞Vs,ameim(θ1−θ2) (31)

involves only even angular momentum components and - odd. It is convenient to define dimensionless BCS interaction constants,

 ˜Vs,am≡kF2πvFVs,am. (32)

In the absence of the coupling to the gapless boson (i.e. in a Fermi-liquid), the RG flow of the BCS interaction (30) can be determined as in Refs. (55); (56). The RG in their work involves only the rescaling of , which is the same as that in Fig. 3 (top). Our rescaling of plays no role in the renormalization of the BCS interaction, and so we can read off the renormalization of from their results: this interaction is marginal at tree level, and acquires the following flow at one-loop level (see Fig. 4),

 d˜Vs,amdℓ=−(˜Vs,am)2 (33)

Thus, in a Fermi-liquid, if the initial value of the BCS interaction is repulsive, , then flows logarithmically to zero, while if the initial value of the interaction is attractive, , runs away to at an energy scale, , signaling an instability to fermion pairing.

Next, we study how the flow of the four-fermion BCS interactions (33) is modified by the presence of the gapless boson . In the limit (or ), , the leading modification comes from the diagram in Fig. 3 (bottom, left), which represents the one-boson exchange contribution to the four-fermion BCS amplitude. As already noted, integration over intermediate large-momentum modes in Fig. 3 generates an inter-patch four-fermion interaction,

 δVs,a(→k1,−→k1;→k2,−→k2)=−ζ2v2FD>(0,→k1−→k2) (34)

where is the boson propagator and the subscript “” indicates that only modes in the momentum shell should be kept. We remind the reader that the constant distinguishes between the different nFLs: we have for the Ising-nematic case, and for the spinon Fermi-surface phase/Mott transition and HLR cases. Note that the frequencies of the external fermions and, hence, of the boson in Fig. 3 (bottom) can be set to 0. Eq. (34) gives for the case of small angle scattering, ; the result for is determined by symmetry. The static boson propagator is given by . Computing the angular harmonics corresponding to (34),

 δ˜Vs,am = −2(kF2πvF)ζ2v2F∫dθ2πD>(0,kFθ)e−imθ = −ζg2vF2π2N∫Λye−ℓ/2Λydqq1+ϵcos(mq/kF)=−ζαNℓ

In the last step, we have dropped the factor as . Thus, the process in Fig. 3 (bottom) contributes a term to the RG flow of , which combines with Eq. (33) to give,

 d˜Vs,amdℓ=−ζαN−(˜Vs,am)2 (36)

There are also terms of order which arise from vertex corrections and the flow of in the definition (32), but we have dropped them because they are are higher order in . Note that the flow (36) is independent of the angular momentum and spin channel; hence we drop the angular momentum/spin indices on below. The flow equation (36) for the inter-patch BCS interaction in conjunction with the flow of the intra-patch coupling constant in Eq. (22) determines the physics of the nFL states considered. We next analyze these RG equations and discuss their consequences. However, we first point out that in the regime of analytical control , all the conclusions of our RG treatment can be reproduced by solving the Eliashberg equation for the pairing vertex, as has been shown elsewhere.(61) This lends further support to our results.

## Iii Results: Ising-nematic QCP

We first discuss the solution to RG Eqs. (22), (36) for the nematic QCP. In this case, the constant in Eq. (36), so the fluctuations of the order parameter captured by the first term in Eq. (36) drive the short-range interaction negative (attractive), as expected. In fact, as discussed in appendix A, we find that independent of the initial values of and , flows to at a finite , indicating an instability of the Ising-nematic QCP to superconductivity. Thus, we expect both the zero temperature electron pairing gap and the superconducting critical temperature to be proportional to . Unlike in the ordinary Fermi-liquid, the run-away flow occurs even if the initial value of is repulsive: gapless order parameter fluctuations eventually drive attractive. However, the magnitude of and hence the pairing gap does depend on the initial value of : the smaller the initial - the larger the gap. As already noted, the flow equations for in different angular momentum/spin channels decouple and are identical. We, thus, expect pairing to occur in the channel where diverges first, i.e. one which has the smallest initial . Hence, the pairing symmetry is non-universal.

It is interesting to compare the pairing scale with the energy scale at which electronic quasiparticles get destroyed. Here, we identify as the energy at which the Fermi-velocity , whose flow is determined by Eq. (23), starts to deviate significantly from its bare value. We find that as long as our calculation is controlled, (i.e. ), , so the superconducting instability preempts the destruction of quasiparticles and associated nFL behavior. This is quite distinct from the physics of many materials where nFL behavior is observed at energies/temperatures well above the superconducting . As we take the artificial control parameter to its physical value , the two scales and approach each other, however, at this point we lose analytical control.

We now briefly illustrate the above conclusions for several regimes of , , (see appendix A for more details). First, consider the case . Here, we find

 Δ=Λωexp[−1√˜α(π2+tan−1˜V√˜α)] (37)

with

 ˜α≡αN. (38)

If the bare short-range interaction is small compared to the long-range interaction, , then Eq. (37) reduces to . On the other hand, if the bare short-range interaction is large and repulsive, , , i.e. the gap is reduced by a factor of two on the logarithmic scale compared to the case of small . Finally, if the bare short-range interaction is large and attractive, , the gap takes the standard BCS form, . The scale at which nFL effects become appreciable is . Thus, as long as , the pairing gap is parametrically larger than the nFL scale . We note in passing that the result (37) is identical to one obtained for the problem of quark pairing by color gauge fields in 3d dense baryonic matter,(54) and for electron pairing near a ferromagnetic QCP in 3d.(100)

Proceeding to the case (which may be continuously connected to the physical case ), we find that the nFL scale is still given by for , as well as for , while for , . The pairing scale is still given by the expression in Eq. (37) for , so the relation holds. For and (or , but ), we obtain

 Δ∼Λω(˜αϵ2)2/ϵ (39)

so depends on the coupling constant in a power-law manner and . Eq. (39) has been previously obtained within an Eliashberg-like treatment in Ref. (62). Naive extrapolation of the above result to the physically relevant value gives, , i.e. the pairing and nFL scales become parametrically equal. This conclusion is again supported by the direct solution of Eliashberg-like equations.(101); (102)

Note that in the above analysis, we have ignored the -wave dependence of the coupling between the Ising-nematic order parameter and electrons on the angle around the FS. We don’t expect the -wave form-factor to affect the maximum magnitude of the pairing gap strongly, however, it will certainly affect its angular dependence. In fact, recent results of Maier and Scalapino(103) and Lederer et al.(104) suggest that the angular dependence of the gap at the QCP may be quite singular. These authors study the regime where the system is tuned sufficiently away from the Ising-nematic QCP that the standard weak coupling BCS machinery can be applied. They find that as the QCP is approached, the superconducting gap becomes strongly peaked around the angle where the coupling between the order parameter and the electrons is maximal (i.e. around the anti-node). It is interesting whether this result survives all the way to the QCP. In the future, we hope to settle this question by extending our RG analysis to the physical case with no rotational symmetry.

While our RG analysis is performed exactly at the metallic critical point, superconductivity will survive when one tunes the system slightly away from the QCP with the perturbation . Recall that induces a finite correlation length for the order parameter, , , with the corresponding energy scale . Away from the QCP, serves as an IR cut-off on the RG equation (36). Therefore, the pairing gap will be essentially unmodified from its QCP value , as long as . We may then estimate the characteristic width of the superconducting dome as (here, we consider the most interesting regime when pairing at the QCP is dominated by order parameter fluctuations rather than the bare short-range BCS attraction ). As always, the precise shape of the dome for cannot be determined from RG considerations alone. The dome will generally have tails extending to , where the gap () will be strongly suppressed compared to . The precise form of in these tails can be obtained by running the RG equation (36) for the BCS coupling up to the energy scale . It is easy to see that for , , i.e. the system at energy is in the weak-coupling regime. Below the energy , is not critical, and the system is described by Fermi-liquid theory, so the BCS coupling continues to flow according to Eq. (33). Therefore, if , then the system will develop superconductivity, with . On the other hand, if , no superconducting instability will occur. Thus, if the bare is attractive, the tails of the superconducting dome will extend to all (as long as one remains in the regime of applicability of the critical theory). On the other hand, if the bare is repulsive, then the dome will terminate at a finite , corresponding to .

We would like to note that in the above discussion, denoted the deviation from the “metallic” QCP. As we saw, this QCP is unstable to superconductivity, so the true Ising-nematic QCP will occur inside the superconducting dome. In addition, its location will generally shift away from that of the putative metallic QCP at to . On general scaling grounds, we expect . The universality class of the true QCP at depends on whether pairing gives rise to a fully gapped or a nodal superconductor. If the superconductor is fully gapped, this transition will be in the classical 3D Ising univerality class. The character of this transition in a nodal -wave superconductor has been discussed in Ref. (105). The critical behavior associated with the true QCP will only be observable for , and the system will cross over to the metallic critical behavior discussed in this paper for .

## Iv Results: Spinon FS phase and HLR phase

We now turn to the solution of the RG equations Eqs. (22), (36) for the spinon FS phase and HLR phase. The constant in Eq. (36) now takes the value , hence gauge field fluctuations drive repulsive, in accordance with intuition. We first solve Eqs. (22), (36) when (with an eye to describing the physical spinon FS phase and the HLR phase with short-range interactions, where ). Here, the coupling flows to the fixed point , and we may substitute this fixed point value into the RG equation for , (36). We then find two perturbatively accessible fixed points for : , see Fig. 5. The fixed point is infra-red stable; as long as the initial value of is greater than , flows to . Thus, the spinon FS and HLR phases are controlled by the fixed point and are to fermion pairing. However, if the initial value of in some angular momentum/spin channel is smaller than , runs away to , and fermion pairing occurs. , thus, marks the phase transition between the and spin-liquid phases (HLR and incompressible QH phases). Note that unlike in a Fermi-liquid, a finite strength of the attractive short-range interaction is needed to overcome the long-range repulsion mediated by the gauge field and cause fermion pairing. Pairing in a given angular-momentum/spin channel can be driven by tuning the corresponding . The pairing transition is continuous and the spinon (neutral fermion) gap onsets in a power law fashion, , where

 1zν=dd˜V(d˜Vdℓ)∣∣∣˜V=˜V−∗=√2ϵ (40)

This is, again, distinct from an ordinary FL where the electron gap has the familiar exponential form .

We note that to the leading order in discussed above, the presence of inter-patch interactions does not affect the flow of the intra-patch coupling constant , Eq. (22), and the Fermi-velocity , Eq. (23). As a result, most physical properties (fermion and boson dynamical exponents , ; specific heat; exponents(32) etc.), at the two fixed points are identical. This conclusion may be true to all orders in , since, perturbatively, BCS interactions do not influence the single particle properties (, ) in a FL.

We next discuss the marginal case , which describes the QH fluid with Coulomb interactions. Here, the coupling constant logarithmically flows to . The combined flow of , is shown in Fig. 6 (see appendix B for details). The flow is characterized by a single fixed-point , and features an attractor line and a separatrix . As long as the initial values of , satisfy , the couplings flow to the attractor line and then into the fixed point , . So, the HLR phase with Coulomb interactions is stable in a finite region of parameter space. On the other hand, if the initial , runs away to and fermion pairing occurs. Thus, the separatrix describes the transition between the HLR phase and the paired QH phase. Note that this separatrix also logarithmically flows into the fixed point , , so the stable and the unstable fixed points , found for , merge into a single fixed-point here. The pairing transition is continuous and the fermion gap turns on as the separatrix is crossed in an unusual super-power law fashion,

 Δ∼exp[−116log2(Vc−V)] (41)

with .

## V Results: Mott transition

As already noted, the theory also describes the QCP between the spinon FS phase and a Fermi-liquid phase. Thus, the results in section IV imply that the spinon Fermi-surface at the Mott transition is stable as long as the initial values of lie to the right of the separatrix in Fig. 6. On the other hand, if the initial values of lie to the left of the separatrix, the spinon acquires a gap, and the Mott transition occurs between a spin-liquid and a superconductor.

In the former regime , where the spinon FS at the Mott transition is stable, we expect that an even stronger statement holds: the spinon Fermi-surface remains stable as one tunes the system slightly away from the Mott transition. Indeed, if one tunes the system into the compressible phase, in Eq. (3), the gauge field becomes Higgsed by the condensate below a momentum scale , where is the correlation length exponent of the XY universality class. The corresponding energy scale will serve as an IR cut-off on the RG equations for the flow of (, ), (22), (36). Below this energy scale, gauge fluctuations become non-critical and the spinon FS will be described by FL theory. Now, as we discussed in section IV, for energies above , the flow of tends to the attractor line . Thus, at the crossover scale , so no spinon pairing will occur as one further lowers the energy into the Fermi-liquid regime. Consequently, the Fermi-liquid phase adjacent to the Mott transition will not develop superconductivity.

Likewise, if one tunes the system into the insulating phase, in Eq. (3), the screening (4) of the gauge field by will cease at a momentum scale , with the corresponding energy scale . Below this energy scale the system is effectively in the spinon FS phase. Again, by the time the scale is reached, will approach the attractor line . Since, as we discussed in section IV, a finite strength of attraction