Deconfined Metallic Quantum Criticality: a $ U (2) $ Gauge-Theoretic

Deconﬁned metallic quantum criticality: A U(2) gauge-theoretic approach

Liujun Zou1 and Debanjan Chowdhury2 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 2Department of Physics, Cornell University, Ithaca, New York 14853, USA We discuss a new class of quantum phase transitions — Deconfined Mott Transition (DMT)— that describe a continuous transition between a Fermi liquid metal with a generic electronic Fermi surface and an electrical insulator without Fermi surfaces of emergent neutral excitations. We construct a unified U(2) gauge theory to describe a variety of metallic and insulating phases, which include Fermi liquids, fractionalized Fermi liquids (FL*), conventional insulators and quantum spin liquids, as well as the quantum phase transitions between them. Using the DMT as a basic building block, we propose a distinct quantum phase transition — Deconfined Metal-Metal Transition (DM2T)— that describes a continuous transition between two metallic phases, accompanied by a jump in the size of their electronic Fermi surfaces (also dubbed a ‘Fermi transition’). We study these new classes of deconfined metallic quantum critical points using a renormalization group framework at the leading nontrivial order in a controlled expansion, and comment on the various interesting scenarios that can emerge going beyond this leading order calculation. We also study a U(1)×U(1) gauge theory that shares a number of similarities with the U(2) gauge theory and sheds important light on many phenomena related to DMT, DM2T and quantum spin liquids.

Contents E. BCS coupling 28

F. RG ﬂows for the U(1) × U(1) gauge theory at I. Introduction1 = 0 28

II. U(2) gauge theory4 A. Parton construction4 I. Introduction B. Metallic phases4 C. Insulating phases5 Studying the zoo of quantum phases, realized as D. Fractionalized Fermi liquids6 ground states of many-body systems, and the phase transitions between them is one of the central themes of III. Deconfined Mott and Metal-Metal Transitions7 physics. Quantum phase transitions (QPT) offer an in- A. Deconfined Mott Transition7 teresting window not just into the universal low-energy B. Deconfined Metal-Metal Transition8 physics associated with the criticality, but also on the intricate interplay between the relevant degrees of free- IV. Renormalization-group analysis9 dom in the nearby phases. For conventional QPT in in- A. U(1) × U(1) gauge theory9 sulating phases of matter [1], the critical theory can be B. U(2) gauge theory 12 described using the classic Landau-Ginzburg-Wilson paradigm developed originally for studying classical V. Discussion 15 (thermal) phase transitions and formulated in terms of the long-wavelength/low-energy fluctuations of a lo- References 16 cal order-parameter field. A striking example of QPT in insulators that does not fit into this paradigm is A. Details of the U(2) parton construction 19 offered by the example of ‘deconfined’ quantum criticality [2,3]. B. Topological nature of some examples of Metallic quantum criticality comes in many avatars insulating phases 20 and they are inherently more complicated to describe 1. Type-I insulator: short-range entangled arXiv:2002.02972v4 [cond-mat.str-el] 17 Jun 2020 due to the abundance of gapless electronic excitations insulator 20 near the Fermi surface (FS). Arguably one of the most 2. Chiral spin liquid as a type-II insulator 20 common forms of metallic criticality is associated with 3. Z2 topological order as a type-III insulator 21 the onset of some form of broken-symmetry in a metal, 4. An example of a type-IV insulator 22 which is typically described in terms of the critical fluctuations of a bosonic order-parameter coupled to a C. Framework for Wilsonian Renormalization Fermi surface. There are numerous strongly-correlated Group 22 materials where such forms of criticality might be relevant for the experimental phenomenology [4–6]. The D. Correlation functions and beta functions from standard Hertz-Millis-Moriya (HMM) framework [4] Wilsonian RG 25 for dealing with this problem runs into serious difficul- 2 ties in (2+1)-dimensions [7]; much theoretical progress One of our motivations to study DMT is clearly their has nevertheless been made in recent years [8]. conceptual novelty. From a more practical point of One of our main focus in this paper will be on conti- view, one of our other motivations comes directly from nous metal-insulator transitions. In this context, there experiments on numerous strongly correlated materi- are simpler and relatively well understood mechanisms. als, where the same framework for a DMT may offer For example, in clean systems, an example of such a us a useful route to understanding continuous QPT QPT is through a Lifshitz transition into a band insu- between different metals. In numerous rare-earth el- lator, where the electronic Fermi surface can be shrunk ement based compounds that display heavy-fermion to a point (e.g. by changing the chemical potential). physics, the Kondo-breakdown transition out of the In disordered systems, a localization transition into an heavy FL into a metal with broken symmetries display Anderson insulator can localize the electronic states phenomenology at odds with the expectations of HMM near the Fermi level in the presence of strong disorder. theory (see discussion in Ref. [13] for some examples). A conceptually distinct and somewhat novel form Similarly, in the hole-doped cuprates, there is growing of metallic criticality is associated with the disappear- evidence for a transition between a FL metal at large ance of an entire Fermi surface as a result of either doping to an unconventional ‘pseudogap’ metal with a a Kondo breakdown transition in heavy Fermi liquids markedly different density at small values of the dop- [9–12], or, a Mott transition to a quantum spin liq- ing (without any additional exotic FS); see Ref. [21] uid insulator [13, 14]. In both of these examples, a and references therein. Inspired by these examples, we description of the transition requires the introduction are interested in studying a direct continuous QPT be- of fractionalized degrees of freedom (d.o.f.) coupled tween two metals that have electronic FS with finite to an emergent dynamical gauge field, and the elec- but different sizes on either side of the transition. We tronic Fermi surface evolves into a ghost Fermi surface dub such QPT as Fermi Transition (FT), or, Decon- of the electrically neutral fractionalized d.o.f. across fined Metal-Metal Transition (DM2T). We note that the transition. The resulting critical point has a ‘criti- within our organizing principle, the two metallic phases cal’ Fermi surface—a sharp Fermi surface of electrons separated by a DM2T can either be ordinary FL met- without any long-lived low-energy electron-like quasi- als, or be of a more exotic nature, such as a fraction- particles. See Refs. [13–17] for some examples of such alized Fermi liquid (FL*) [22]. transitions. A key question that remains unanswered is We remark that in discussing DMT and DM2T, we as follows: Can we describe continuous quantum phase require that immediately across the transition, the to- transitions from a metal to an insulator (or to another tal size of the metallic FS is finite, which is allowed to distinct metal) without any remnant Fermi surfaces of be a multiple of the size of the Brillouin zone (BZ). fractionalized d.o.f. on the insulating side (or the other Later we will see an explicit example of a metal that metallic side)? has two FS whose total size is the same as the BZ In this paper, we introduce a new gauge-theoretic size. The DMT or DM2T out of such a metallic phase formulation to study the exotic quantum phase tran- possesses all of the essential characteristics of related sitions out of a metal introduced above. As already transitions in metals where the FS size is not a multiple emphasized, our main focus will be on a distinct class of the BZ size. of continuous QPT between a Fermi liquid (FL) metal In the remainder of our discussion, we exclude elec- and a Mott insulator (MI), where the size of the electronic transitions purely driven by the onset of various tronic Fermi surface is finite on one side of the transi- 1 forms of translation-symmetry-breaking order, such as tion within the FL phase , and the MI has no FS of any spin or charge density waves [23]. The latter can be fractionalized d.o.f. We dub such metal-insulator tran- described within the conventional HMM framework. sitions as Deconfined Mott Transition (DMT). Notice Infinitesimally close to the critical point on the or- that the insulating phase across the DMT can either dered side of such continuous transitions (i.e., when be short-range entangled (e.g., a conventional magnet) the order-parameter is infinitesimally small in magni- or long-range entangled (e.g., a quantum spin liquid tude), the Fermi surfaces are nearly identical to the (QSL)), and the only requirement of the insulator is one on the disordered side of the transition, except for the absence of any ghost Fermi surfaces. We note, in in the narrow vicinity of the ‘hot-spots’. We are in- passing, that numerical evidence for a DMT between a terested in DM2T, where the electronic Fermi surface FL and a chiral spin liquid has been reported recently 2 undergoes an abrupt change across the critical point, [18]. which necessarily goes beyond the HMM framework. It might already be apparent to the reader that the framework for a DMT can be readily generalized 1 This is different from a Lifshitz-type transition, where the size to describe a DM2T . For example, consider a two- of the Fermi surface on the metallic side is infinitesimal immediately across the transition. orbital electronic model with a globally conserved den- 2 In the context of semi-metals (rather than metals), transitions sity, where each orbital has a FS. We denote these FS of a somewhat similar flavor between a Dirac semi-metal and as ‘hot’ and ‘cold’, respectively, for reasons that will a featureless insulator have been discussed [19, 20]. be clear momentarily. From Luttinger’s theorem, the 3

FIG. 2. Some possible landscapes of the allowed phases. The parameter m2 indicates the gap of the bosonic parton FIG. 1. Examples of deconfined metallic quantum criti- (see Eq. (12)), and Ve represents the short-range interac- cality: (a) deconfined Mott transition (DMT) between a tion among the fermionic partons (see Sec.IVA). Here Fermi liquid (FL) and an insulator without a ghost Fermi FL, SC, INS and NFS stand for a Fermi liquid metal, a su- surface. The solid circle represents an electronic Fermi sur- perconductor, an insulator without a ‘ghost’ Fermi surface face. (b) Mott transition between a FL and an insulator of fractionalized d.o.f., and a quantum spin liquid with an with a ghost Fermi surface, as described in Refs. [14]. The emergent neutral Fermi surface, respectively. The red lines dashed circle represents a neutral ghost Fermi surface. (c) are schematic phase boundaries. A direct DMT without A Fermi transition (FT) between two distinct FL metals fine-tuning can occur only in case (a) and (b). with FS of different size (also dubbed as deconfined metal- metal transition (DM2T )). The FL on the left has FS of ‘hot’ electrons (which undergo a DMT as in (a)) and ‘cold’ theory of coupled critical bosons, fermions with FS, a electrons (which remain spectators). (d) a FT between a dynamical (2) gauge field, and possibly a FS of cold FL and a fractionalized Fermi liquid (FL*), driven by the U hot electrons undergoing a transition into a neutral ghost electrons. A landscape of the possible phases and the Fermi surface, as in (b) (see Ref. [13]). QPT between them in terms of some of the parameters in our theory appears in Fig.2. Exploring the nature of all of the QPT that appear on Fig.2 is beyond total size of the FS is the sum of the sizes of the in- the scope of the present manuscript and will appear in dividual FS from the two orbitals. Suppose now that future work. as a result of strong orbital-selective interactions, the We apply a renormalization group (RG) analysis to hot electrons undergo a DMT, while the cold electrons critically examine whether these putative critical theo- remain in a FL phase with a fixed FS size. If the cou- ries describe continuous DMT and DM2T. We find that pling between the hot and cold electrons is irrelevant in the U(2) gauge field is effectively ‘quasi-Abelianized’, a renormalization-group (RG) sense, the cold electrons i.e., many of its non-Abelian features do not actually decouple and act as spectators to the more interesting play an important role in the universal low-energy limit DMT of the hot electrons. Therefore, such a transition that is of physical interest to us. Our calculations can be viewed as a DM2T of the entire system. It is provide considerable insight and some preliminary ev- worth emphasizing that this is only one possible route idence that the sought-after QPT can be described by to a DM2T , and it does not imply that all DM2T can these critical field theories. However, we stress that necessarily be viewed as a DMT on top of a dynami- corrections beyond those considered here can poten- cally decoupled FL. A schematic representation of the tially alter the nature of these transitions. A more different metallic QPT of interest to us in this paper elaborate investigation in the future is required to fully appears in Fig.1. understand the intricate low-energy dynamics of these The focus of this paper is to provide a possible theories. mechanism for a DMT, and a related mechanism for Along the way, we also study a U(1) × U(1) gauge a DM2T. Our key strategy will be to formulate the theory that shares many common aspects as the U(2) problem in terms of an (emergent) U(2) gauge theory. gauge theory. As we discuss, our results on this More precisely, we will view the electron as a bound U(1) × U(1) gauge theory have important implications state of a boson and a fermion, where the boson car- on various interesting phenomena related to DMT, ries all of the quantum numbers of the electron and the DM2T and QSLs. fermion carries only the fermionic statistics. Further- The rest of this paper is organized as follows: In Sec. more, both the boson and the fermion are coupled to a II we provide a concrete parton construction that nat- dynamical U(2) gauge field. We will show that by tun- urally leads to a U(2) gauge theory. We discuss the ing a few parameters of the resulting low-energy theory, routes to obtaining different phases, including a metal- this powerful unified framework is capable of describing lic FL, various insulating phases (including short-range multiple quantum phases, including FL metals, orthog- entangled insulators and long-range entangled insula- onal metals, ordinary insulators, different topologically tors) and metallic FL* phases starting with the above ordered states and FL* metals. The same framework U(2) gauge theory. In Sec.III, we discuss possible will offer new insights into the putative critical theo- mechanisms for a DMT and a DM2T based on the U(2) ries for a DMT and a DM2T , which are described by a gauge theory. In Sec.IV, we apply a renormalization 4 group analysis to the U(2) gauge theory of a DMT, Under the SU(2) spin rotation symmetry, as well as a similar U(1) × U(1) gauge theory. While we will provide preliminary and compelling evidence B(r) → VB(r), f(r) → f(r), (4) that the theory can describe a continuous transition of where V is an SU(2) matrix. Under time reversal T , this type, there are technical challenges that prevent us from addressing some aspects of this transition. Fi- B(r) → B(r), f(r) → f(r), i → −i, (5) nally, in Sec.V, we conclude with a discussion of the where is the rank-2 anti-symmetric matrix with = possible scenarios for the global landscape of the un- 12 − = 1. Under lattice symmetries, derlying U(2) gauge theory. 21 B(r) → B(r0), f(r) → f(r0) (6)

II. U(2) gauge theory where r0 is the transformed position of r under the relevant lattice symmetry. 4 The parton construction given by Eq. (1) can be In this section we present a formalism based on a generalized to a many-body system in a straightfor- U(2) gauge theory and apply it to describe various ward fashion. In terms of these partons and the emer- quantum phases of interest to us. We will see that gent U(2) gauge ﬁeld, the low-energy dynamics of the within our framework these phases can be obtained by 3 system can be described by an action of the following tuning a few parameters. form:

SDMT = S[B,a] + S[B,f] + S[f,a] + S[a], (7) A. Parton construction where S[B,a] (S[f,a]) represent the actions of B (f) that are both minimally coupled to a, S represents the We start by presenting a parton decomposition of [B,f] coupling between B and f, and S[a] represents the the electron: action of the U(2) gauge field, a. In the above ex- c B B f pressions, the couplings to external gauge fields corre- 1 = 11 12 · 1 (1) c B B f sponding to the global symmetries are implicitly dis- 2 21 22 2 played, according to the symmetry actions specified or, more schematically, above. The precise forms of these actions are unimportant at this stage, and they will be specified later. c = Bf, (2) Below we apply this U(2) gauge theory to describe different quantum phases. where c1 (c2) is the annihilation operator of an electron with spin-up (-down). The B’s are bosonic operators and the f’s are fermionic. This parton construction has B. Metallic phases a U(2) gauge redundancy, under which f → Uf and B → BU †, with U a U(2) matrix. This means that at The above parton construction can be applied to de- low energies, the system is described by a theory where scribe a Fermi liquid in any dimension (> 1) and on the partons, B and f, are coupled to a dynamical U(2) any lattice geometry. For concreteness, we consider a gauge field, denoted by a. We will refer to the index of two dimensional triangular lattice with two electronic f and the second index of B as the color index, while orbitals per site at half-filling (i.e., two electrons per the first index of B is the physical spin index. site on average). Throughout this paper, we will as- We leave details of this parton construction to Ap- sume that the mean-field part of the fermion action, pendixA, where we discuss the generators of the (2) U S[f,a], corresponds to a Hamiltonian that gives rise to gauge transformations, the gauge constraints of a phys- a generic FS (i.e., a FS without any special nesting ical state and the procedure for constructing physical property). A particular choice of the mean field Hamil- electronic states using B and f. Below we only sum- tonian for f is marize the actions of the relevant physical symmetries X X that we will use in this paper. Hf = − t1 fαi† (r)fαi(r0) Under the U(1) charge conservation symmetry, rr0 α,i=1,2 h i (8) X X iθ − † (r) (r) + h c B(r) → e B(r), f(r) → f(r). (3) t2 fα1 fα2 . ., r α=1,2

3 We note in passing that U(2) gauge theories have been studied 4 Notice the above is only a particular choice of how symmetries recently in a different context in other interesting settings [24], act on B’s and f’s, and in general there are different projective but our current framework is different from the previous U(2) representations of these symmetries [25], which will not be gauge theory. discussed in this paper. 5 where α = 1, 2 represent the color indices, and i = 1, 2 ing state is determined by the state B is in, as well represent the two orbitals. The first term above rep- as the interaction between both partons and the dy- resents the nearest-neighbor hopping within the same namical U(2) gauge field, a. In order to approach this orbital, and the second term represents the hybridiza- problem in a systematic fashion, below we first ignore tion between the two orbitals at the same site. When the coupling of B to f and a, and discuss the possible |t2| & |t1|, there is no FS. However, when |t1| & |t2|, at gapped states that B can be in. We then combine each half filling, there will be two FS sheets with different of these states with the f and a to address the possi- sizes that add up to the size of the BZ. We will focus ble insulators that can be accessed by the U(2) gauge on the latter case here. It is possible to have a metal- theory. insulator transition by tuning t1/t2, without changing Generically, there can be the following four types of the total size of the Fermi surface up to the BZ size. gapped states of B. Notice that in all of the gapped However, later we will discuss a more exotic possibility states that we will discuss here, the global symmetries of metal-insulator transition, i.e., a DMT where on the of the system may or may not be broken spontaneously, metallic side the sizes of the two FS are finite in the depending on the microscopic details (which we do not immediately vicinity of the transition. specify). For our particular setup of the lattice prob- The bosons, B, can be condensed or gapped out by lem, none of the symmetries has to be broken sponta- tuning appropriate parameters in S[B,a]. If B is con- neously. However, depending on the particular setup of densed, the dynamical U(2) gauge field gets Higgsed the lattice problem, there can be a Lieb-Schultz-Mattis generically and does not play an important role at low (LSM) type theorem that forbids a featureless state of energies. According to the original relation in Eq. (1), B [28–31]. The general discussion below applies to all the FS of f are then identified with the FS of the origi- lattice setups, and we will explicitly point out the fea- nal electron with a renormalized quasiparticle residue, tures that are specific to our particular setup. Z ∼ h|B|2i. The resulting state is simply a Fermi liquid metal. Furthermore, by choosing appropriate interac- 1. Type-I: A short-range entangled (SRE) state tion terms in S[B,a], B can be condensed in a channel with a trivial response to the dynamical U(2) where all physical symmetries are preserved [24, 26]. gauge field, a. A particular example of such a For example, the condensate of B can have the form state is where the B’s within the two orbitals at hB11i = hB12i = hB21i = −hB22i = B0, where B0 6= 0 each site form a dimer: is independent of the site or orbital. It is straightfor- Y ward to check that this condensate is invariant under αα0 ββ0 (σx)ijBαβi† (r)Bα† 0β0j(r)|0i, (9) all physical symmetries discussed above (up to a U(2) r gauge transformation). Finally we note that it is straightforward to realize where |0i represents a state with no bosons, other metallic phases in this formalism. For example, i, j = 1, 2 represent the two orbitals at a site, by choosing the parameters in S[B,a] such that B is and r labels the position of the site. This is a condensed in a channel with hB11i = 2hB22i = B0 6= 0 product state that preserves all global symme- and hB12i = hB21i = 0 realizes a ferromagnetic FL tries as well as the U(2) gauge symmetry. Notice metal if B0 is independent of the site or orbital index. the existence of such a featureless state is due On the other hand, if B0 oscillates from site to site, to our particular choice of lattice. If we were it realizes a spin-density wave metal. As another ex- to consider instead another lattice system with ample, when B is gapped but a bound state of B is a LSM-type constraint, the SRE state needs to 2 2 spontaneously break some symmetry. condensed, e.g., by taking hBi = 0, hB11i = hB22i 6= 0 and hB2 i = hB2 i = 0, the U(2) gauge structure can 12 21 2. Type-II: A SRE state with a nontrivial response be broken down to Z2, which realizes an orthogonal metal [27]. to the dynamical U(2) gauge field, a. That is, B forms a nontrivial symmetry-protected topologi- Therefore, the above U(2) gauge theory can describe cal (SPT) state under the (2) gauge symmetry. a wide variety of metallic states, depending on whether U It is known that in two dimensions if B is in such B or its bound states are condensed in an appropriate a nontrivial SPT, integrating out generates a fashion. B Chern-Simons action for a [24, 26, 32–34], which can potentially modify the low-energy dynamics of the entire system. C. Insulating phases 3. Type-III: A long-range entangled (LRE) state Having constructed various metallic phases from the without a nontrivial Chern-Simons response to above U(2) gauge theory by condensing B, we are now the dynamical U(2) gauge field, a. An example of interested in analyzing the phases that one obtains such a state is a Z2 topological order [25]. It will when B is gapped while the mean field Hamiltonian be useful later on to notice that since B is already for f describes a generic FS. The nature of the result- in the fundamental representation of the SU(2) 6

gauge symmetry, the SU(2) gauge charge of the that the resulting topological order is equivalent topological order can be completely screened by to a chiral spin liquid state [25], also known as B, i.e., the sector of the topological order can al- the Laughlin-1/2 state. Notice in the usual ter- ways be made into a singlet sector of the SU(2) minology, a chiral spin liquid refers to a quantum gauge symmetry. spin liquid emergent from a bosonic system, but in our case gapped local electrons are also present 4. Type-IV: A LRE state with a nontrivial Chern- in the system. Simons response to the dynamical U(2) gauge field, a. Such a state can be obtained by forming 3. Type-III: In this case, the SU(2) gauge field will a nontrivial SPT of B on top of a type-III state confine. Interestingly, according to the discus- of B. sion above, because the topological order of B can be made completely in the singlet sector of Next we discuss the dynamics of the f and a sector. the SU(2), this topological order is intact when According to the above discussion, in this case, at low the SU(2) gauge field confines. As a result, the energies the system is described by FS of f that are resulting state of the entire coupled system of B, coupled to the dynamical U(2) gauge field, a. The U(2) f and a is again generically a topologically or- gauge field can be decomposed as a = a +a ˜1, where dered state, whose precise nature is determined a is a 2-by-2 SU(2) gauge field, anda ˜ is a U(1) gauge by both the topological order of B and the sym- field. The meaning that a is a U(2) gauge field is that metry fractionalization pattern of the U(2) gauge an excitation with an odd (even) charge undera ˜ must symmetry on this topological order. In Appendix carry a half-odd-integer (integer) spin under a, and B we give an example where the system is in a Z2 vice versa. Generically, there is no relation between the topologically ordered state (supplemented with SU(2) gauge coupling and the U(1) gauge coupling— local electrons). this will be an important consideration for us later. The leading instability of the f-FS will be to pairing. 4. Type-IV: In this case, the SU(2) gauge field does In our setting, as a result of the difference in the size not confine due to its Chern-Simons action, and of the two FS, pairing is likely to occur within the the resulting state is also generically a topolog- same FS. The coupling between f and a is expected ically ordered state. In AppendixB, we give a to have significant influence on the pairing instability. relatively simple example of such a state. However, independent of the details of this coupling (which we will discuss in great detail later in Sec.IV), In summary, our U(2) gauge theory can describe in- we can always explicitly add a four-fermion attractive sulators that are either SRE or LRE, depending on interaction to the theory in Eq. (7) so that there is the parameters of Eq. (7). Notice that all of the above an s-wave pairing in the singlet channel of the SU(2) insulating phases have no FS of any emergent excita- gauge symmetry. The f-FS is then fully gapped out tions (as long as the attractive four-fermion interaction and the U(2) gauge structure will be broken down to is reasonably strong). an SU(2) gauge structure. The nature of the final state In passing, we note that one can also construct gap- then depends on the dynamics of the remaining SU(2) less phases of B (e.g., a Dirac spin liquid [25]) that gauge field. The advantage of studying the fate of the respect the U(2) gauge symmetry. However, we do not theory in the absence of gapless FS of the f is that explore such scenarios any further in this paper. the dynamics of the remaining SU(2) gauge field only depends on the nature of the insulating state B is in (i.e., Type-I,..,IV). We discuss these in detail below. D. Fractionalized Fermi liquids We shall return to the full problem in the presence of the f-FS in Sec.III, where we will also investigate the Our formalism can also describe fractionalized Fermi tendency to pairing of the f-fermions. liquids (FL*) [22], upon including additional d.o.f. in the problem. Consider including additional electrons 1. Type-I: In this case, it is generally believed that that form a FS. Let us denote the creation and annihi- the SU(2) gauge field confines [35]. It turns out lation operators for these electrons by d†, d. Then the that the system is in an ordinary SRE insulator modified theory is given by (see AppendixB).

SDM2T = SDMT + S[d] + S[d,c], (10) 2. Type-II: In this case, the SU(2) gauge field will not confine due to its Chern-Simons action, and where SDMT is as originally defined in Eq. (7), S[d] the resulting state is generically a topologically is the action of the d-electrons. The coupling between ordered state. For example, if the SPT of is R 2 B the d- and c-electrons, S[d,c] = dτd xL[d,c], takes the a U(2) symmetric bosonic integer quantum Hall familiar form state discussed in Ref. [33], using the method in Ref. [24] (see AppendixB), we can determine L[d,c] = λd†Bf + h.c. + ··· (11) 7 with λ the hybridization strength between the d- and assumed a relatively strong short-ranged attractive in- c-electrons. teraction between the f-fermions in the singlet channel If the c-electrons form a FL (i.e., if B is condensed), of the SU(2) gauge symmetry to induce pairing. How- the entire system is just a FL made of both the d- and ever, in this regime across the condensation transition c-electrons (with a single globally conserved density). for the boson, a relatively strong short-ranged attrac- On the other hand, when B is gapped, the resulting tion may be present for the electrons as well, thereby metallic phases can be potentially interesting. If B is inducing pairing of the FL. Thus, we need to relax the gapped and the c-electrons form an insulator of Type-I requirement of strong attraction in order to describe a discussed above, the system can be viewed as a FL of DMT transition from a SRE insulator to a FL metal the d-electrons on top of an ordinary insulating state (rather than to a superconductor). We are thus im- of the c-electrons. However, if B is gapped and the mediately forced to revisit the question of whether the c-electrons form an insulator of Type-II, -III or -IV SRE insulator can be obtained in the regime without described above, the system can be viewed as a FL of strong attraction in the first place (i.e., whether the f the d-electrons coexisting with an essentially decoupled FS is still unstable to pairing, as a result of the coupling (in an RG sense) topologically ordered state of the c- to a = a +a ˜1, when the B’s are gapped)? electrons. In other words, this provides us with a route It is well known that a U(1) gauge field,a ˜, coupled towards realizing FL*; by changing the nature of the minimally to a neutral FS suppresses pairing [36], while gapped state of B, different FL* ground states can be the SU(2) gauge field mediates a long-range attractive obtained. interaction among the fermions in the SU(2) singlet To conclude this section, we remark that our single channel, thereby enhancing the tendency towards pair- unified framework is able to describe a myriad of inter- ing. Therefore, in the regime without a strong short- esting quantum phases. They include various metallic ranged attraction between the f-fermions, the fate of phases with or without broken symmetries/topological the FS to pairing depends on the competition between order as well as fully gapped insulators. The latter are the U(1) and SU(2) gauge fields. If the U(1) gauge obtained when both B and f are gapped (at least when field wins over the SU(2) gauge field and the FS re- the attractive interaction between the f’s is reasonably main stable, we obtain a new type of quantum spin strong). Moreover, letting at least one of them be in liquid that has stable neutral FS of fractionalized d.o.f. a nontrivial SPT under the U(2) gauge symmetry al- coupled to a dynamical U(2) gauge field. The transi- lows us to describe many types of topologically ordered tion from a FL to such an exotic state is clearly in- phases [24]. The full technical advantage of our setup teresting, but it is not an example of a DMT, as the will become apparent in the following discussion. above insulator has a neutral FS. On the other hand, if the SU(2) gauge field wins over the U(1) gauge field and the FS become unstable to pairing, then this ap- III. Deconfined Mott and Metal-Metal proach can potentially describe a DMT from a FL. In Transitions Sec.IV we will provide suggestive evidence that the SU(2) gauge field wins over the U(1) gauge field in this Based on the above descriptions of various phases competition, although our results at this stage cannot in terms of our U(2) gauge theory, in this section we account for all of the intricate complexities and much present possible mechanisms for a deconfined Mott more remains to be understood. In the remainder of transition and a deconfined metal-metal transition. this Sec.III, we assume that the SU(2) gauge field In this section, we will only restrict ourselves to indeed wins over the U(1) gauge field and continue to making rather general remarks on these mechanisms. discuss the associated critical theory. In Sec.V we will The quantitative analysis using a formal machinery of discuss other possible scenarios where the U(1) gauge renormalization group for these critical theories ap- field wins over the SU(2) gauge field in this competi- pears in Sec.IV. tion. We show a few plausible phase diagrams in Fig. 2, where a direct DMT without fine-tuning can only occur in cases (a) and (b). A. Deconfined Mott Transition Based on the above discussion, we now flesh out the details of the boson (B) induced order-disorder tran- In this section, we present a possible mechanism for sition, which potentially describes a DMT between a a DMT between a FL metal and an insulator without a FL metal and a SRE insulator. The critical theory is FS of any fractionalized d.o.f., starting from our U(2) described by the action SDMT in Eq. (7), where B is gauge theory. For the sake of concreteness, the insula- tuned to be critical. For the transition into a Type-I tor we focus on here will be SRE (i.e., Type-I in Sec. insulator (with a fixed electron density), S[B,a] can be R 2 IIC); similar mechanisms can be applied to other types taken to be S[B,a] = dτd xL[B,a] with a relativistic of insulators. Lagrangian, We begin by reminding the readers that in our previ- µ µ 2 ous discussion of the possible insulators in Sec.IIC, we L[B,a] = Tr Da B†Da B + m Tr(B†B) + ··· ,(12) 8

µ µ µ where Da = ∂ − ia is the covariant derivative with if the scaling dimensions of gauge invariant operators respect to a (we have set the boson velocity to unity), made of B’s are larger than 3/2, the coupling between m2 is the tuning parameter of the transition, and “··· ” B and f is also irrelevant. Then f and a are dynam- represents all other terms that are allowed by global ically decoupled from the B sector. We now turn to symmetries and gauge invariance. the f and a sector. Even though this sector is dynam- The coupling between B and f, S[B,f] = ically decoupled from the B sector, the latter plays an R 2 dτd xL[B,f], needs to be gauge invariant, so direct important role at low energies. In particular, the B- coupling between B and bilinears of f is forbidden, sector changes the dynamical critical exponent associ- and their leading coupling is of the form ated with the f and a sector from what is expected naively at the RPA-level, and can thereby modify the L[B,f] = λ1f †B†Bf + λ2f †f · Tr(B†B) + ··· ,(13) conclusion regarding pairing instabilities. Specifically, = 1 when is gapped (see Eqs. (19) and (43) for where λ1 and λ2 are coupling constants. B The coupling between f and a has the form the definition of ), but = 0 when B is at its own Z transition described by a 2 + 1 dimensional conformal f field theory. S[f,a] = f † −iω − µf + ia0 + k+a f, (14) ω,k To summarize, in this subsection we have presented 5 where f above is in its frequency-momentum represen- a possible mechanism for a continuous DMT, and tation, and f is the mean-field dispersion of f. The the key ingredient is that the f-pairing is dangerously R 2 irrelevant at the above critical point. In passing, we action of a, S[a] = dτd xL[a], is of the usual Maxwell- Yang-Mills form: remark that the purpose of considering a U(2) gauge theory (rather than just an SU(2) gauge theory) is to 1 ˜ ˜ 1 have the possibility that pairing can be irrelevant at L[a] = fµν fµν + Tr (fµν fµν ) , (15) 2e2 2g2 the critical point, due to the U(1) gauge field. where e and g are respectively the U(1) and SU(2) ˜ gauge couplings (not necessarily the same), fµν = B. Deconfined Metal-Metal Transition ∂µa˜ν − ∂ν a˜µ and fµν = ∂µaν − ∂ν aµ − i[aµ, aν ] are the U(1) and SU(2) field strengths, respectively. We have assumed that the speeds for the U(1) and SU(2) gauge Building on the above mechanism for a continuous fields are the same, and will show later that this causes DMT, we can present a mechanism for a continuous 2 no loss of generality as far as the universal low-energy DM T , which is described by SDM2T in Eq. (10) (with physics is concerned. SDMT as defined in Sec.IIIA). Using the terminology In the above theory, when B is condensed, the re- introduced in Sec.I, we will view the c-electrons in sulting state is a FL metal (or possibly more exotic SDMT as ‘hot’ and the d-electrons as ‘cold’. Because of variants thereof, as discussed in Sec.IIB). On the the strong correlations at the criticality, B and f can other hand, when B is gapped, the resulting state can be very incoherent, which may potentially render the be a SRE insulator (see Sec.IIC). For this theory to coupling in Eq. (11) irrelevant. Then the cold electrons indeed describe a continuous DMT, the only relevant are just spectators at the DMT associated with the hot perturbation should be the one parameterized by m2, electrons and we obtain a continuous DM2T between the tuning parameter of this transition. We have al- two metals, where the size of the FS on either side ready assumed (without any justification for now) that of the transition (i.e., in the two metallic phases) are pairing of f is relevant in a RG sense when B is gapped. different and finite. Moreover, there are only electronic The important question that remains to be addressed FS in the two metallic phases and no neutral FS of any is whether pairing is (ir)relevant at the critical point. other emergent excitations. Only if pairing is (dangerously) irrelevant at the critical As we have repeatedly emphasized, we have not ex- point, can the above critical theory describe a contin- amined some of our crucial assumptions behind the uous DMT. above mechanisms critically. One of our central as- A framework for analysing the critical theory of a sumptions has been related to the nature of the pair- simpler version of the above model has been devel- ing instability associated with the f-fermions and the oped in earlier works [14, 36, 37]. We summarize the possibility of them being dangerously irrelevant at the key ideas borrowed from previous work and associated critical point. In the next section, we will turn to a with the setup for studying the above transition here; RG framework to examine these issues in a lot more see Ref. [14] for a detailed discussion. We begin by fo- critical detail. cusing on the criticality associated with the B-sector while ignoring its coupling to a and f, which is expected to be described by a Wilson-Fisher type fixed point. However, due to the presence of the f-FS, a will be Landau-damped, which drives the coupling be- 5 Based on certain assumptions, which we discuss at length in subsequent sections. tween a and B in S[B,a] to be irrelevant . Furthermore, 9

IV. Renormalization-group analysis The fermion fields, f1 and f2, carry charges under the even and odd combinations of the gauge fields, ac + as and − , respectively. We will also assume the This section will be devoted to a study of the f- ac as following additional exchange symmetry: ↔ , FS coupled to a dynamical U(2) gauge field, where B f1 f2 is taken to be gapped. In addition to its potential as → −as. We will now consider the case where f1 relevance for the mechanism for a DMT as explained and f2 form Fermi surfaces, and our goal is to under- previously, it might also play an important role in the stand the infrared (IR) properties of this system. physics of Kitaev quantum spin liquids [24]. As a result of minimal coupling to the fermion density, the temporal components of the gauge fields, [ac]0 and [as]0, are screened out in a manner similar to the screening of Coulomb interactions in a metal, and they A. U(1) × U(1) gauge theory will be ignored in the following discussion. By work- ing in the Coulomb gauge, ∇ · ac,s = 0, we can focus Before directly tackling the technically challenging on the transverse components of the gauge fields cou- problem of a FS coupled to a U(2) gauge field, we pled minimally to the FS. Previous studies focusing first study a simpler version of the problem. Consider on the problem of a single U(1) gauge field coupled to two (2 + 1)-dimensional FS coupled to two U(1) gauge a Fermi surface [7, 37, 42] identified the importance fields, ac and as. Both FS carry the same charge under of coupling of the small-momentum (q) fluctuations of ac, but they carry opposite charges under as. This im- the transverse components of the gauge field to the low- plies that ac (as) mediates repulsive (attractive) long- energy excitations of the fermions near pairs of antipo- range Amperean interaction between fermions on an- dal patches with nearly parallel Fermi velocities (see tipodal regions of the FS, i.e., as favors pairing of the Fig.3). We will adopt the same strategy below and fermions while ac suppresses this tendency towards consider a “patch” description of the low-energy the- pairing [36, 37]. The fate of the FS at low-energies ory. towards pairing thus depends on the competition between ac and as, which is a simpler version of the full y problem of FS coupled to the U(2) gauge field. More- over, even within this simpler setting, arguments analogous to those in Sec.II can be used to construct interesting quantum phases. Furthermore, the introduction x of an additional boson that carries opposite charges as the fermions under the above gauge fields can be used q to study DMT and DM2T. We note that the problem with the two independent U(1) gauge fields has appeared in a variety of different contexts in previous works, most notably in the study + y of the bilayer quantum Hall problem [38–41]. How- ever, in the context of bilayer quantum Hall systems, long-range Coulomb interactions play a defining role in the problem, while in our setting due to the charge x gap they can be ignored in the low-energy regime of our interest. This distinction makes the physics in the FIG. 3. A pair of antipodal patches on the FS that have two settings different. We will develop a RG frame- nearly collinear Fermi velocities. The momentum of the work that goes beyond the considerations of previous gauge mode, q, is nearly perpendicular to these Fermi ve- work. At the leading nontrivial order, we find that the locities. result of the competition between ac and as and the stability of the FS against pairing are determined by non-universal microscopic details. The extensions of these results beyond leading order will be the subject Patch theory: symmetries and scaling of future work; we will outline some of the interesting scenarios in Sec.V. Let us introduce the fields, ψαp, which denote the The action of the U(1) × U(1) problem is given by low-energy fermionic fields near the antipodal patches R 2 S = dτd xL, where the Lagrangian has the form, labeled by p = ± and color index, α = 1, 2 (Fig.3). Based on Refs. [7, 37, 42], the Lagrangian in an explicit

L = L[f1,ac+as] + L[f2,ac as] + L[ac] + L[as]. (16) form is given by − 10

L = Lf + L[ac,as], (17) with

X 2 X h i Lf = ψαp† η∂τ + vF −ip∂x − ∂y ψαp − vF λp (ac + as)ψ1†pψ1p + (ac − as)ψ2†pψ2p (18) p= ,α=1,2 p= ± ± and N N L = | |1+| |2 + | |1+| |2 (19) [ac,as] 2 ky ac 2 ky as , 2ec 2es

where vF denotes the Fermi velocity, and λ = ±1. which equivalently in momentum space reads The time-derivative terms of the gauge fields± are irrelevant [7, 37, 42], so they have been dropped off. ac,s(ω, qx, qy) → ac,s(ω, qx − θqy, qy) Then the difference in the speed of “photon” corre- θ2 θ sponding to ac and as can be absorbed into the gauge ψαp(ω, qx, qy) → ψαp ω, qx − θqy − p 4 , qy + p 2 . couplings, ec and es, whose flows will be the subject (21) of subsequent discussion. Notice that we have introduced two additional parameters — N, the number of A direct consequence of the above rotational flavors of fermions f1,2, and — with an eye for doing a controlled double-expansion in small and large N symmetry is that the propagators satisfy the fol- [37, 43]. Notice, in principle, that we can choose the ’s lowing properties: for ac and as to be different, but we will take them to have the same value in accord with the physical case, Dc,s(ω, qx, qy) = Dc,s(ω, qy) (22) where, for example, = 1 corresponds to the usual 2 Gαp(ω, qx, qy) = Gαp(ω, pqx + qy), example of a spinon Fermi surface coupled to a U(1) gauge field [44], and, = 0 corresponds to the ν = 1/2 Landau level problem for the composite Fermi liquid i.e., the gauge field propagator does not depend with Coulomb interactions [45]. The physical case we on qx, and the Fermi surface curvature does not flow under RG. (In more technical terms, focus on corresponds to = 1. 2 A few general remarks are in order here. ipψ†∂xψ and ψ†∂y ψ renormalize in the same way). • Starting with the above low-energy field theory, Because of the forms of these propagators, the Eq. (17), we can either (i) fix v = 1 and study F relation between the dynamical exponents of ac, the flow of η, or, (ii) fix η = 1 and study the flow as and ψ, denoted za and zf , respectively, are of v , in addition to the flow of the coupling con- c,s F also fixed: za = za = 2zf . Notice here za is stants e2, e2. It is easy to see that we can go back c s c,s c s the relative scaling between ω and qy in the gauge and forth between the two schemes by appropri- field propagators, while zf is the relative scaling ate rescalings of the ψ-fields. For the remainder between ω and qx in the fermion propagator. of our discussion, we will adopt scheme (i) above. In this case, the physical Fermi velocity is 1/η. • When < 1, due to the non-analytic nature of the kinetic terms of the gauge fields, z = z = • The above Lagrangian only describes a particular ac as 2+ exactly. This implies a great simplification in pair of antipodal patches on the Fermi surfaces. the discussion of pairing instability, as discussed In order to describe the whole system, we need below. However, because = 1 in the absence to include all the patches defined along the entire of additional gapless bosons that carry opposite contour of the Fermi surface; we will consider charges as the fermions under a and a , we can- the effects of inter-patch interactions below after c s not directly utilize the properties associated with analyzing the scaling structure and key features the case with < 1. Nevertheless, the case with associated with a single pair of patches. = 0 is pertinent to the critical points of DMT 2 • Let us review the general structure of the theory and DM T, which will be discussed briefly at the defined in Eqn. (17). As noted in Ref. [42], the end of this subsection. theory has an emergent rotational symmetry: • Consider now the following scaling transformations for the frequency and momenta, ac,s(τ, x, y) → ac,s(τ, x, y + θx)

θ θ2 (20) ip 2 y+ 4 x zf l l l/2 ψαp(τ, x, y) → e− ψαp(τ, x, y + θx) ω → e ω, qx → e qx, qy → e qy, (23) 11

2 2 which leaves the bare kinetic energy for ψαp in That is, the ratio of the gauge couplings, ec /es = Eq. (17) invariant with zf = 1 (at tree-level). αc/αs = r, is RG invariant to this order. Therefore, Under the above scaling, the ﬁelds transform as the beta functions at this order predict that in the IR the ﬁxed point values for αc and αs are given by z ( 1 + f )l l ψαp → e 4 2 ψαp, ac,s → e ac,s, (24) N r r αc = · = , ∗ 2 r + 1 2(r + 1) and by simple power counting, in order to make (30) N 1 1 the whole Lagrangian, Eq. (17), invariant, the αs = · = , ∗ 2 r + 1 2(r + 1) scaling transformation for ec,s should be which is determined by the non-universal ratio, r. No- z l 2 (1+ 2 f ) 2 tice that the physical values of N = 1 and = 1 have ec,s → e − ec,s. (25) been substituted in the last step.

Because zf = 1 at the tree level, the above scaling indicates that both gauge couplings are relevant as long as > 0, which is the case of our interest.

RG analysis and pairing instability

In order to study the low-energy properties of this two-patch theory, we now apply an RG analysis to go beyond the tree level scaling considerations. In Ap- pendixC, we develop a framework for Wilsonian RG that can be used to calculate the flows of these coupling constants systematically. Our detailed derivation also FIG. 4. The leading-order RG flow of αc and αs in the clarifies a number of conceptual issues that are poten- U(1) × U(1) gauge theory, where the blue arrows represent tially beneficial for the readers. We apply this frame- the direction of the flow. The red line represents the non- work to calculate the beta functions of the couplings trivial fixed line, where the solid (dashed) part represents the regime stable against (unstable to) pairing. Replacing in AppendixD. αc byα ˜ and αs by α, this figure represents the leading-order We focus on dimensionless gauge coupling constants, RG flow of the U(2) gauge theory. 2 2 ec es αc = , αs = . (26) 4π2ηΛ 4π2ηΛ Now we use the above results to analyze the pairing instability, via the approach in Ref. [36]. We fix N = 1 where Λ is the cutoff on qy. Their beta functions (de- while still take to be a small expansion parameter. We fined explicitly in AppendixC) to the next leading or- focus on the RG flow of the dimensionless four-fermion der are 6 interactions in the BCS s-wave color-singlet channel, denoted by Ve (see AppendixE for a precise definition αc + αs β(αc) = − αc of this dimensionless coupling constant). Results for 2 N other channels can be obtained similarly, and we ignore (27) αc + αs them here for simplicity. β(αs) = − αs. 2 N There are two contributions to the flow of Ve. The first is the usual FL contribution [36, 46–48]: From these beta functions, we get a nontrivial fixed line at: 2 β(Ve) = −Ve . (31) N FL α + α = . (28) c s 2 The second contribution comes from the coupling between the fermions and the gauge fields. In particular, We also notice that the above beta functions imply that as already pointed out in previous work [36], integrating out the fast gauge modes along with shrinking and e2 α c = c = 0 (29) breaking each patch of the FS generates inter-patch in- β 2 β es αs teractions. Generalizing these calculations to our setting in a straightforward fashion, we find that at leading order, the gauge fields contribute in the following way to the flow of V , 6 To this order, these beta functions can be viewed as a result e of either an expansion in terms of small but finite N, or a double expansion of small and large N. β(Ve) = αc − αs. (32) gauge 12

As expected, the first (second) term arises from the corrections. These beta functions are given by minimal coupling to ac (as), which suppresses (en- hances) pairing. Taken together, the full beta function β(αc) = −(αc + αs) · αc at this order is given by, β(αs) = −(αc + αs) · αs (34) 2 2 β(Ve) = αc − αs − Ve β(Ve) = αc − αs − Ve . (33) The flows of these couplings are discussed in detail in Combining the above flow equation for Ve with Eqs. AppendixF. It is shown that when r < 1, the FS is (27)-(30), we see that at the leading nontrivial order, unstable, while the FS is perturbatively stable when the propensity of the FS towards pairing depends on r > 1. Notice when r = 1, one has to go to the next non-universal microscopic details, parametrized by r leading order of the third equation above to analyze the (introduced around Eq. (30)). More precisely, if r < 1, effects of gauge fluctuations on pairing, as discussed in as wins in the competition, Ve flows to −∞ in the IR AppendixF. and the FS are unstable to pairing. On the other hand, This result is useful for constructing a DMT and 2 if r > 1, ac wins in the competition, as long as the DM T critical point, because if additional bosons with q r 1 opposite charges as the fermions under ac and as are UV value of Ve > − 2(r−+1) . In the IR Ve flows UV introduced, the mechanism discussed in Sec.III can q r 1 to Ve = 2(r−+1) > 0, and the FS are perturbatively also be applied to this U(1) × U(1) gauge theory to ∗ 2 stable against pairing. If r = 1, the effects of the two describe a DMT and DM T. Because these additional bosons are the analog of the B bosons discussed in Sec. gauge fields exactly cancel each other and Ve flows to −∞ if it starts with a negative value in the UV, in III, here we also refer to them as B bosons. Recall that in this mechanism, the DMT and DM2T are driven by which case the FS are unstable. If V starts with a e the condensation transition of these B bosons, and it is positive value in the UV, it flows to 0, and the FS are important that pairing is irrelevant at the transition, stable. 7 These results are summarized in the flow in order for these transitions to be continuous with- diagram in Fig.4. out fine-tuning. When the condensation transition of When higher order corrections to the beta functions the B bosons is described by a 2 + 1 dimensional con- in the two-patch theory are included, the fixed line formal field theory, Ref. [14] pointed out that = 0 described by Eq. (28) might collapse into interesting right at the transition. From our discussion above, we fixed points. Furthermore, in order to be internally see that pairing can indeed be irrelevant at the transi- self-consistent, including higher order corrections to tion where = 0. This observation provides more ev- β(αc) and β(αs) requires one to include the corrections idence that this theory can potentially describe DMT to β(Ve) to the same order. While clearly interesting, and DM2T, if one can further show that pairing be- this is beyond the scope of the present work; we will comes relevant immediately after B becomes gapped. instead discuss the possible scenarios in Sec.V and Suppose this pairing instability indeed exists when B is leave a detailed systematic study of these corrections gapped, then one concrete example of DMT can be re- to future work. alized in a bilayer system, where each layer undergoes the bandwidth-tuned Mott transition described in Ref. [14], and the two layers are weakly coupled in a way 2 Possible DMT and DM T critical points that preserves a layer-exchange symmetry. In fact, as shown in Ref. [39], in this case right at the DMT the Before finishing the discussion on this U(1) × U(1) two layers are effective decoupled because interlayer gauge theory, we note that an interesting simplification couplings are irrelevant, and the critical point hosts occurs when = 0, which may be relevant for DMT sharply defined FS without any long-lived quasiparti- and DM2T critical points. As discussed in Appendix cles. C, due to the non-analyticity of the kinetic terms of the gauge fields and the associated absence of an anomalous dimension, the ratio between the two gauge cou- B. U(2) gauge theory plings is exactly RG invariant when = 0 (not just to the leading nontrivial order). In this case, it is suffi- We now return to the problem of FS coupled to cient to only consider the beta functions at the leading a U(2) gauge field. Compared to the U(1) × U(1) nontrivial order without worrying about higher order problem, there are now potential complications due to the non-Abelian nature of this gauge group. We will show that many of the anticipated complications do not play an essential role as a result of the underlying 7 Until the Kohn-Luttinger mechanism for pairing eventually FS, and the problem reduces to a form similar to the becomes important [48, 49], which we will ignore throughout U(1) × U(1) problem studied above. this paper. The action now has the seemingly more complicated 13 form, that at finite density forms FS. The auxiliary field, Φ, and SFP, corresponding to the action for the Faddeev- S = S[f,a] + S[a,Φ] + SFP, (35) Popov (FP) ghosts, are introduced for the sake of gauge fixing in this non-Abelian gauge theory [50]. As before, where a = a+ã1 is a U(2) gauge field, with a its SU(2) we will use the Coulomb gauge, i.e., ∇ · a = 0, where part anda ˜ its U(1) part. The field f is the fermion field the action has the explicit form,

Z Z f S[f,a] = fx†,τ (∂τ − µf + ia0)fx,τ + fk†,ωk+afk,ω, (36) x,τ k,ω

Z 1 1 2 ξ = ˜2 + α + (Φα)2 + Φα∇ · aα (37) S[a,Φ] 2 fµν 2 fµν , x,τ 2e 4g 2

and coupled to FS becomes similar to the Uc(1) × Us(1) Z gauge theory coupled to FS. α αβ β SFP = c¯ −∇ · Dadj c , (38) Before moving to a patch formulation of the theory, x,τ it is useful to examine the remaining gauge symmetry ˜ and the BRST symmetry of this theory [50]. In the where the field strengths are fµν = ∂µa˜ν − ∂ν a˜µ and α α α αβγ β γ αβγ Coulomb gauge, there is a remaining gauge symmetry: fµν = ∂µaν − ∂ν aµ + aµaν , with the fully anti-symmetric tensor satisfying 123 = 1. We choose f → Uf the generators of SU(2) in the fundamental represen- ˜ → ˜ − ˜( ) (39) tation to be T α = σα/2, with σα the standard Pauli aµ aµ ∂µθ t α β αβγ γ matrices, such that T ,T = i T . The disper- aµ → UaµU † + iU∂µU †, sion f is taken to exhibit FS at the mean-field level. ˜ α The covariant derivative in the adjoint representation with U = ei(θ(t)+θα(t)T ). Notice that the θ’s depend αβ only on time but not on space. In its infinitesimal form, is Dαβ ≡ ∂ δαβ − i T γ aγ ≡ ∂ δαβ − αβγ aγ . adj,µ µ adj µ µ µ the BRST symmetry reads, The Grassman fields, c andc ¯ are the Faddeev-Popov α α αβ β ghost and anti-ghost, respectively. The field Φ is an δaµ = θBDadj,µc auxiliary field to implement the gauge fixing, and the δf = iθ cαT αf Coulomb gauge is achieved in the limit ξ → 0. B As we can see, the present (2) gauge theory is very α 1 αβγ β γ U δc = − θB c c (40) similar to the previous Uc(1) × Us(1) problem, where 2 α α the the SU(2) gauge field a plays a role analogous to as δc¯ = θBΦ in the latter problem. One therefore expects that the δΦα = 0 stability of the FS will be determined by the competition between the U(1) gauge field,a ˜, and the SU(2) with θB an infinitesimal Grassmann variable. Con- gauge field, a, where the former tends to suppress pair- strained by these symmetries among with other ing and hence stabilize the FS, while the latter tends symmetries in the system, we are able to write down to enhance the tendency towards pairing. the most general effective action. In particular, There are a few notable differences between the two we notice that no other bilinears of the ghost and setups, which we now highlight. The first difference anti-ghost fields with less than two spatial derivatives arises from the self-interaction of a due to its non- can be written down (e.g., a “chemical potential” type Abelian nature. However, in the low-energy limit we termc ¯αcα is forbidden). will see that within the patch formulation the self- interaction of a becomes irrelevant in an RG sense [51]. Another notable difference for the U(2) problem is related to the presence of the Faddeev-Popov ghosts. In Patch theory the literature on color superconductivity, the ghosts are often argued to be unimportant using the “hard- As before, due to the finite density of f fermions, the dense loop approximation” [52, 53]. We will also show temporal components of the gauge field are screened that the ghosts become irrelevant for the low-energy and we ignore them in the following treatment. More- physics within the patch formulation. Thus, in spite over, the Coulomb gauge allows us to keep only the of these differences, ultimately the U(2) gauge theory transverse components of the spatial components of the 14 gauge fields. Expanding the fermion operators in the with vicinity of the Fermi surface and focusing only on pairs of antipodal regions with nearly parallel Fermi velocities, we obtain the patch formulation of the theory Eq. (35):

L = L[ψ,a] + L[a] + Lirre, (41)

X 2 X L[ψ,a] = ψp† η∂τ + −ip∂x − ∂y ψp − λpψ[†pα]aαβψ[pβ], (42) p= p= ± ±

and applied here, once we (schematically) replace ac bya ˜ N N and as by a. The differences between the two problems L = |k |1+|a˜|2 + |k |1+|aα|2, (43) in the low-energy limit arise mainly from the difference [a] 2 2 y 4 2 y e g in the structure of the interaction vertices. where ψ is the low-energy fermionic operator in the patch formulation, and a now only represents the transverse components of the spatial components of the RG analysis and pairing instability gauge field. Notice we have already introduced N and , just as in the Uc(1) × Us(1) case, with an eye for Now we apply the RG framework developed in Ap- doing a controlled double-expansion. pendixC to the U(2) problem with appropriate cou- A useful starting point, as before, is to consider the pling constants and interaction vertices. Our cal- scaling structure of the patch theory: culations will be performed to the leading order in

` O() ∼ O(1/N) (see AppendixD for details). zf ` ` ω → e ω, kx → e kx, ky → e 2 ky, Focusing on the dimensionless gauge couplings, z ( 1 + f )` ψ(x, τ) → e 4 2 ψ(x, τ), e2 3g2 (44) α˜ = , α = , (45) a(x, τ) → eà(x, τ) 4π2ηΛ 8π2ηΛ 2 (1+ zf ) 2 2 (1+ zf ) 2 8 e → e 2 − e , g → e 2 − g the beta functions are given by with z = 1 at the tree level. Notice that according to α˜ + α f β(˜α) = − α,˜ the action of the ghosts, Eq. (38), the ghosts should 2 N transform as → 3`/4 under the above scaling. (46) c e c α˜ + α From the above scaling structure, it is straightfor- β(α) = − α. 2 ward to verify that the self-interaction of a and the N coupling between the ghosts with the fermion-gauge We find once again that at this order there is a non- sector are irrelevant. For example, consider the 3-gluon trivial fixed line described by 3 self-interaction, which schematically has the form kya . N This term has a larger scaling dimension compared to ˜ + = (47) 1+ α 2 α α . |ky| |a | , so it is irrelevant. As another example, 2 consider the coupling between the ghosts and the gauge The ratio between the gauge couplings is RG invariant, field, which schematically has the form kyacc¯ . This i.e., β (˜α/α) = 0. The values ofα ˜ and α in the IR are coupling has a larger scaling dimension compared to given by, vF aψ†ψ, so it is also irrelevant. We include all of the irrelevant terms in L , which will be ignored in the N r r irre α˜ = · = , following discussion. ∗ 2 r + 1 2(r + 1) (48) The main lesson we draw from the above scal- N 1 r ing analysis is that as a result of the scaling struc- α = · = ∗ 2 r + 1 2(r − 1) ture that is tied to the presence of the underlying f- FS, the SU(2) gauge field in the current U(2) problem is effectively “quasi-Abelianized”. The problem thus effectively reduces to the previous setup involving 8 As in the U(1) × U(1) gauge theory, to this order these beta Uc(1) × Us(1) gauge fields. In particular, the features functions can be viewed as the result of either an expansion of the patch theory discussed in Sec.IVA and the RG of small but finite N, or a double expansion of small and framework developed in AppendixC can be directly large N. 15 where r is the ratioα/α ˜ at the cutoff scale, which is de- fixed points) at low energies in the two-patch theories, termined by non-universal microscopic details. Again, where based on the UV values of the gauge coupling the physical values of N = 1 and = 1 have been constants, part of the line remains stable against pair- substituted in the last step. ing while the complementary part becomes unstable to Just like in the case of the Uc(1) × Us(1) problem, pairing. It is important to remain cautious as correc- we can now study the flow of the dimensionless BCS tions that are higher order in the expansion parame- coupling in the s-wave color-singlet channel [36] in the ters can potentially alter the true infrared properties current setting, which at leading order of for N = 1 of these field theories. is given by We have also studied a U(1)×U(1) gauge theory that shares many common aspects with the U(2) gauge the- 2 β(Ve) =α ˜ − α − Ve . (49) ory. This U(1) × U(1) gauge theory has many important implication on various phenomena. For example, Not surprisingly, we find the nature of the pairing in- if long-range interactions are included (as opposed to stability in the current setting to be similar to the purely local interactions considered in this paper), we U (1) × U (1) problem: the FS are stable against c s are able to propose a concrete setup where a continu- pairing depending on non-universal microscopic details ous DMT can arise, and we can describe the universal parametrized by r. If r > 1, the U(1) gauge field wins physics associated with that DMT in a controlled man- in the competition and the FS are perturbatively sta- ner [54]. This U(1)×U(1) gauge theory is also directly ble. If on the other hand r < 1, the SU(2) gauge field related to the possible ground state of the intensely wins in the competition and the FS are unstable to studied QSL candidates, including κ-(ET) Cu (CN) pairing. If r = 1, the FS are stable against (unstable 2 2 3 and EtMe Sb[Pd(dmit) ] [55, 56]. These are layered to) pairing if the microscopic short-range interactions 3 2 2 materials, and are argued to host QSL with emergent among the fermions are repulsive (attractive). Once neutral Fermi surfaces. Our results on the (1)× (1) again, these results for the RG flows can be depicted U U gauge theory clearly raise concern on the stability of schematically as in Fig.4. such QSL states as a true ground state in these layered We emphasize here once again that higher order cor- materials. rections to the above beta functions can potentially We end this paper by discussing possible scenarios cause the fixed line of (˜α, α) to collapse into (undeter- that may arise after higher order corrections are taken mined) fixed points, which can have an effect on the into account in the presence of only local interactions, nature of the pairing instability. These considerations and leave a detailed analysis of their effects for the are beyond the scope of the present work but we dis- future. All of these scenarios are depicted pictorially cuss some of the interesting scenarios in the final Sec. in Fig.5. V, leaving a detailed analysis for future work. We also stress that the argument leading to the simplification in the discussion of the pairing instability in the U(1) × U(1) gauge theory with = 0 does not directly apply to the U(2) gauge theory, because the SU(2) gauge field may acquire a nonzero anomalous dimension (see AppendixC).

V. Discussion FIG. 5. Different scenarios of the U(1)×U(1) gauge theory after taking into account higher-order contributions. No- tice in Scenario A the shape and position of the fixed line In this paper, we have shown that a U(2) gauge the- can potentially be modified compared to the leading-order ory based approach can describe a myriad of quan- result. Replacing αc byα ˜ and αs by α gives rise to the tum phases, including conventional phases (e.g., con- analogous scenarios for the U(2) gauge theory. ventional insulators and FL metals with/without broken symmetries) and more exotic phases (e.g., quantum spin liquids, orthogonal metals and fractionalized 1. Scenario A: The fixed lines for the two-patch the- Fermi liquids). We have addressed the possible routes ories discussed in Sec.IV remain robust upon towards describing interesting quantum phase transi- the inclusion of higher order corrections. If so, tions between these phases, which include the decon- on part of the fixed line the FS remains stable fined Mott transition and Fermi (or, deconfined metal- against pairing. Then we can have an interesting metal) transition. scenario where two dimensional translationally Using a renormalization group formalism, we have invariant critical systems are described by fixed outlined and identified possible mechanisms (at leading lines. On the fixed line, the system can be viewed order in our expansion parameters) through which such as FS coupled to two dynamical U(1) gauge fields transitions can occur. In particular, we have noticed or a dynamical U(2) gauge field, which is a new the existence of fixed lines (instead of the more familiar type of QSL. Furthermore, these act as parent 16

states for other interesting phases, obtained by whether they remain continuous. tuning different parameters in the system (see Fig.2). However, in this case, it may be difficult to describe a continuous DMT or DM2T. On the 3. Scenario C: After higher order corrections are other portion of the fixed lines the FS is unsta- included, there is no fixed point at all. This ble to pairing, which may potentially be relevant suggests that the two-patch theory actually does for DMT or DM2T. Finally, if the above scenario not describe a critical system, and the system is is true, it is worth investigating what underly- gapped for reasons other than pairing. For exam- ing feature associated with the theory protects ple, it is possible that one of the gauge couplings the fixed line even upon including higher order flows to infinity, which suggests that this gauge corrections. field confines within the two-patch theory. In this scenario, the nature of the resulting state is de- 2. Scenario B: Upon including higher order correc- termined by other microscopic details, which are tions, the fixed lines of the two-patch theory col- beyond the purview of the present discussion. lapse into fixed points. Broadly speaking, there are two distinct types of fixed points, depending on their tendency towards pairing (i.e., stable vs. Acknowledgments unstable). The stable fixed points characterize new exam- We thank Zhen Bi, Sung-Sik Lee, Max Metlitski, T. ples of quantum spin liquids, as discussed above Senthil, Dam Thanh Son and Chong Wang for use- in ‘scenario A’. While interesting in their own ful discussions. LZ is supported by the John Bardeen right, they are not relevant for describing conti- Postdoctoral Fellowship at Perimeter Institute. Re- nous DMT/DM2T. search at Perimeter Institute is supported in part On the other hand, the nature of the unstable by the Government of Canada through the Depart- fixed points are determined by what state the ment of Innovation, Science and Economic Develop- boson B is in, as elaborated in Sec.IIC. These ment Canada and by the Province of Ontario through states can potentially be relevant for describing the Ministry of Colleges and Universities. DC is sup- continuous DMT/DM2T from metals into vari- ported by startup funds at Cornell University. ous insulating states without any neutral FS, or, Note added: While this manuscript was being pre- to other distinct metals. However, we empha- pared for submission, an independent work appeared size that even if such fixed points are found, the on the arXiv [57], which also attempts to propose a re- criticality associated with the putative critical lated but distinct critical theory in a different setting points need to be examined carefully to ascertain to describe a transition between a FL and FL*.

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A. Details of the U(2) parton construction

In this appendix we provide additional details of the U(2) parton construction given by Eq. (1). Here we focus on the case of a single electronic orbital, and it is straightforward to generalize the discussion to a lattice of orbitals. Just as a reminder, we denote the electron operator by ca, where a = 1, 2 stands for the two spin d.o.f. We will consider the following parton construction:

c B B f 1 = 11 12 · 1 (A1) c2 B21 B22 f2 where f1,2 are complex fermion operators and B’s are canonical boson operators. Schematically, this equation reads c = Bf, where c is a 2-by-1 matrix, B is a 2-by-2 matrix and f is a 2-by-1 matrix. Apparently this parton construction has a U(2) gauge redundancy, and the gauge transformation reads B11 B12 B11 B12 f1 f1 → · U †, → U · (A2) B21 B22 B21 B22 f2 f2 where U is a U(2) matrix. The U(2) transformation has 4 generators: Q, the generator of the U(1) part, and Tx,y,z, the generators of the SU(2) part. The explicit expressions of these generators can be written in terms the f and b operators:

Q = f1†f1 + f2†f2 − B11† B11 − B12† B12 − B21† B21 − B22† B22 = f †f − Tr(B†B) 1 1 1 T = f †f + f †f − B† B − B† B − B† B − B† B = f σ f − Tr(B Bσ ) x 2 1 2 2 1 11 12 12 11 21 22 22 21 2 † x 2 † x i 1 1 (A3) T = − f †f − f †f + B† B − B† B + B† B − B† B = f σ f − Tr(B Bσ ) y 2 1 2 2 1 11 12 12 11 21 22 22 21 2 † y 2 † y 1 1 1 T = f †f − f †f + B† B + B† B − B† B − B† B = f σ f − Tr(B Bσ ) z 2 1 1 2 2 12 12 22 22 11 11 21 21 2 † z 2 † z with σ’s the standard Pauli matrices. Demanding that the theory be in a gauge neutral sector, or equivalently, demanding that Q = Tx = Ty = Tz = 0, yields the gauge constraints:

B11† B11 + B21† B21 − f1†f1 = 0

B† B12 + B† B22 − f †f2 = 0 12 22 2 (A4) B12† B11 + B22† B21 − f1†f2 = 0

B11† B12 + B21† B22 − f2†f1 = 0.

It is straightforward to check that on a single orbital only 4 states satisfy all these constraints, and they are |0i, |1i ≡ |B11f1i + |B12f2i, |2i ≡ |B21f1i + |B22f2i and |di ≡ |B11B22f1f2i − |B12B21f1f2i, where |0i is a state with no or particle, and | i ≡ † †|0i for = 1 2, and | i ≡ † † † †|0i. Not B f Bijfk Bijfk i, j, k , Bi1i2 Bj1j2 fkfl Bi1i2 Bj1j2 fk fl surprisingly, if we identify |0i as the empty site, then |1i = c1†|0i has a single spin-up electron, |2i = c2†|0i has a single spin-down electron, and |di = c1†c2†|0i is a doubly-occupied state that has both spin-up and spin-down electrons. Therefore, the parton construction Eq. (A1) supplemented with gauge constraints Eq. (A4) is a faithful representation of all of the physical electronic states. Furthermore, as noted in the main text, various global symmetries can be implemented as

U(1) : B → eiθB, f → f SU(2) : B → V B, f → f (A5) T : B → B, f → f where V is an SU(2) matrix and is the rank-2 anti-symmetric tensor with 12 = −21 = 1. For later convenience, we write down the physical U(1) charge,

ν = B11† B11 + B12† B12 + B21† B21 + B22† B22 = Tr(B†B) (A6) 20 from which we see that the density of B is the same as the density of the physical electrons. The generators of the SU(2) spin rotational symmetry: 1 1 S = B† B + B† B + B† B + B† B = Tr(B σ B) x 2 11 21 21 11 12 22 22 12 2 † x −i 1 S = B† B − B† B + B† B − B† B = Tr(B σ B) (A7) y 2 11 21 21 11 12 22 22 12 2 † y 1 1 S = B† B − B† B + B† B − B† B = Tr(B σ B). z 2 11 11 21 21 12 12 22 22 2 † z

B. Topological nature of some examples of insulating phases

In this appendix we apply the method in Ref. [24] (see Appendix A therein) to derive the topological nature of a few examples of the insulating phases described in Sec.IIC.

1. Type-I insulator: short-range entangled insulator

Let us start with the type-I state. Recall that in this state B is in a completely trivial state, and f is paired in the singlet s-wave channel of the SU(2) gauge group. The low-energy topological quantum ﬁeld theory of the system is given by 1 L = − adb˜ (B1) π

1 wherea ˜ is the original dynamical U(1) gauge field, and b is a gauge field such that 2π db is the current of the singlet Cooper pairs of f. This topological action comes from pairing of the fermions in the singlet channel of the SU(2) gauge field, a. To understand the topological properties of such a theory, it is crucial to understand the possible charged excitations of these gauge fields. In our case, any excitation with an odd (even) charge undera ˜ must carry a half-odd-integer (integer) spin under a, and vice versa. Also, due to the presence of local electrons, without loss of generality, in analyzing the topological properties of this theory, we can assume that the charges of these gauge fields are all bosonic. Now we note that because of the confinement of the SU(2) gauge field, all excitations should be a singlet under SU(2), which must carry even charge undera ˜. More precisely, in terms ofa ˜0 = 2ã, the charges of all the excitations take all integers. Therefore, the above topological theory is more appropriately written as 1 L = − a˜0db (B2) 2π where there is no restriction on the charges of the excitations undera ˜0 and b. This theory is topologically trivial, i.e., short-range entangled [25].

2. Chiral spin liquid as a type-II insulator

Next, we turn to the example of the type-II state described in Sec.IIC. Recall that in this insulator B is in a U(2) symmetric bosonic integer quantum Hall (BIQH) state discussed in Ref. [33], and f are gapped out due to pairing in the singlet s-wave channel of the SU(2) gauge ﬁeld. The low-energy topological quantum ﬁeld theory of this state is given by

2 1 2i 1 L = ad˜ a˜ − Tr ada − a3 − adb˜ (B3) 4π 4π 3 π where, again,a ˜ is the original dynamical U(1) gauge field, a is the dynamical SU(2) gauge field, and b is a gauge 1 field such that 2π db is the current of the singlet Cooper pairs of f. The first two terms represent the the response of the BIQH of B to the dynamical U(2) gauge field [24, 26, 32–34], and the last term is due to the pairing of fermions in the singlet channel of a. 21

It is convenient to split the TQFT Eq. (B3) into two parts, i.e., L = L1 + L2, where

1 2i L = − Tr ada − a3 1 4 3 π (B4) 2 1 1 L = ad˜ a˜ − adb˜ = (˜a, b) · K · (da,˜ db)T 2 4π π 4π with the K-matrix: 2 −2 = (B5) K −2 0

The theory of L1 is SU(2)1, which has a single anyonic excitation that is a semion, which has topological spin i. This semion can be created by creating an excitation in the spinor representation of a. Excitations obtained by creating a linear representation of a are all local bosons. Now we move to determine the topological properties of the excitations in L2, which turns out to be equivalent to a double-semion topological order. The excitations in L2 can be labeled by (l1, l2), where l1 and l2 are the charges of this excitation undera ˜ and b, respectively. Consider only L2, the topological spin of this excitation is given by (0) 1 T iπ 2 θ = exp −iπ(l1, l2) · K− · (l1, l2) = exp (2l1l2 + l ) (B6) (l1,l2) 2 2

Let us enumerate a few of these topological spins:

(0) (0) (0) (0) θ(1,0) = 1, θ(0,1) = i, θ(1,1) = θ(1, 1) = −i (B7) − In fact, (1, 0), (0, 1) and (1, 1) are just the bosonic, semionic and anti-semionic bosons of the double-semion topological order, respectively. Because the charges under a anda ˜ are not independent, these two sectors are not decoupled. In fact, because an excitation with odd (even) l1 carries a half-odd-integer (integer) spin under a, the topological spin of this excitation should be further multipied by i (1). So the true topological spins of these excitations are

θ(1,0) = i, θ(0,1) = i, θ(1,1) = θ(1, 1) = 1 (B8) − It appears from the above topological spins that there are two semions in the theory, (1, 0) and (0, 1). However, these two excitations should actually be identiﬁed. To see it, consider the excitation (1, −1), which is the bound state (1, 0) and the anti-particle of (0, 1). Its braiding statistics with any excitation (l1, l2) is

(1, 1) 1 T θ − = exp (iπl ) · exp −2πi(1, −1) · K− · (l , l ) (B9) (l1,l2) 1 1 2 where the ﬁrst and second factors come from the L1 and L2 sectors, respectively. It is straightforward to check (1, 1) that θ − = 1 for any l and l . That is, (1, −1) braids trivially with all excitations. So it should be a local (l1,l2) 1 2 excitation, which also means that (1, 0) and (0, 1) are actually in the same topological sector [58], and they represent a single semionic excitation in the system. One can further check that this semion is the only anyonic excitation in the theory Eq. (B3). The above discussion has established that the bulk topological order of the theory Eq. (B3) is identical to a chiral spin liquid (also known as a Laughlin-1/2 state) [25]. Now we examine the edge modes of this theory, which can diﬀer by chiral Majorana modes for a given bulk topological order. Integrating out a,a ˜ and b generates a gravitational Chern-Simons term CSg, which means that the system has a single chiral edge mode with chiral central charge c = 1. Therefore, this topological order is indeed a chiral spin liquid state, with no additional − chiral Majorana edge modes which will shift the gravitational Chern-Simons term by a multiple of CSg/2.

3. Z2 topological order as a type-III insulator

Now we give an example of a type-III state. Consider the case where B itself is in the Z2 topological order (with no additional nontrivial Chern-Simons response to the dynamical U(2) gauge ﬁeld), and f is paired in the singlet s-wave channel of the SU(2) gauge group. Furthermore, let us assume that the Z2 topological order is 22 completely neutral under U(2) (i.e., corresponding to the trivial fractionalization pattern of U(2)). The low-energy topological quantum ﬁeld theory of the system can be written as

1 1 L = − adb˜ + a da (B10) π π 1 2 As in the case of the type-I state, only even charge ofa ˜ is allowed, while there is no restriction on the charges of other gauge ﬁelds. Therefore, this topological theory can be more appropriately written as

1 1 L = − a˜0db + a da (B11) 2π π 1 2 witha ˜0 = 2˜a. This is still a Z2 topological order.

4. An example of a type-IV insulator

Finally, we give a relatively simple example of a type-IV insulator. We simply stack the type-II and type-III states of B described above. Similar arguments as above indicate that the system will be in a topologically ordered state that can be viewed as a chiral spin liquid stacked on top of a Z2 topological order.

C. Framework for Wilsonian Renormalization Group

In this appendix, we generalize the methods developed in Refs. [37, 42] to describe the framework we use to carry out a Wilsonian RG analysis to the two-patch theory described by Eq. (17). Our derivation of the framework will also help clarify some of the conceptual points that arise in these calculations. Suppose the theory is defined originally with all of the coupling constants at a cutoff scale, Λ, by which we mean that all modes with |ky| < Λ are included in the√ theory, while other modes are not included. Suppose that after integrating out the fast modes with |ky| ∈ (Λ/ b, Λ), the effective action becomes (with the generated irrelevant perturbations discarded):

√ b " Z Λ X 2 Seff = dτdxdy (1 + δf ) ψαp† (−ip∂x − ∂y )ψαp + (η + δη) ψαp† ∂τ ψαp α=1,2;p= ± # X X (C1) − (1 + δc) λpac ψ1†pψ1p + ψ2†pψ2p − (1 + δs) λpas ψ1†pψ1p − ψ2†pψ2p p= p= ± ± Z √Λ 1+ b dωdkxdky N|ky| 1 2 1 2 + |ac| + |as| , (2 )3 2 2 + 2 2 + 2 π ec δec es δes √ √ √ where the superscript b/Λ or Λ/ b indicates that only modes with ky < Λ/ b are included in the theory, and , , , , 2 and 2 arise as a result of integrating out the fast modes. Clearly, the ’s depend on and they δf δη δc δs δec δes δ b approach zero when b → 1. Furthermore, all these δ’s are expected to have no explicit dependence on Λ. In order to obtain the beta functions of various couplings, we need to rescale the frequency and momenta so that the cutoff is Λ again, which requires us to determine the relative scaling between imaginary time and real space, specified by a “dynamical exponent”, zΛ. The value of zΛ can be taken to be arbitrary for now, and this value determines the RG scheme we use, as will be clear soon. For now, we leave the value of zΛ unspecified until we decide on the RG scheme. We stress that this zΛ should not be confused with the physical dynamical exponent, which is obtained from the scaling structure of the correlation functions. zΛ 1 1/2 We define τ 0 = τb− , x0 = xb− and y0 = yb− , and we rewrite the effective action given by Eq. (C1) as

1 " # Z Λ X 2 X Seﬀ = dτ 0dx0dy0 ψαp0† (η0∂τ 0 − ip∂x0 − ∂y0 )ψαp0 − λp (ac0 + as0 )ψ10†pψ10 p + (ac0 − as0 )ψ20†pψ20 p α=1,2;p= p= ± ± (C2) Λ Z dω0dk0 dk0 N N + x y | |1+| |2 + | |1+| |2 3 2 ky0 ac0 2 ky0 as0 . (2π) 2ec0 2es0 23

Comparing this eﬀective action and Eq. (C1) yields

zΛ 1 η + δη p 2 + 4 1 zΛ ψαp0 = 1 + δf · b · ψαp, η0 = · b − , 1 + δf 2 1 + 1 + δc 2 2 δc zΛ+1+ = · · = ( + 2 ) 2 ac0 b ac, ec0 ec δec b− , (C3) 1 + δf 1 + δf 2 1 + 1 + δs 2 2 δs zΛ+1+ = · · = ( + 2 ) 2 as0 b as, es0 es δes b− . 1 + δf 1 + δf 2 2 Therefore, the beta functions for η, ec and es are

η0 − η δη − ηδf β(η) = lim = (1 − zΛ)η + lim , b 1 b − 1 b 1 b − 1 → → 2 2 δ 2 2 ec0 − ec 2 ec 2 δc − δf β(ec ) = lim = 1 + − zΛ ec + lim + 2ec lim , (C4) b 1 b − 1 2 b 1 b − 1 b 1 b − 1 → → → 2 2 δ 2 2 es0 − es 2 es 2 δs − δf β(es) = lim = 1 + − zΛ es + lim + 2es lim . b 1 b − 1 2 b 1 b − 1 b 1 b − 1 → → → From the above expressions, we see that the beta functions depend on the choice of zΛ, which is determined by the RG scheme. One choice is to take zΛ = 1 and another choice is to take the value of zΛ so that η does not 2 2 ﬂow, i.e., β(η) = 0. Notice for all choices, ec /η and es/η have a zΛ-independent beta functions: 2 2 2 2 e e 1 δe2 + 2e (δc − δf ) e (δ − ηδ ) β c = · c + lim · c c − c η f η 2 η b 1 b − 1 η η2 → (C5) 2 2 2 2 e e 1 δe2 + 2e (δc − δf ) e (δ − ηδ ) β s = · s + lim · s s − s η f . η 2 η b 1 b − 1 η η2 →

Notice 2 = 2 =0 if 1, due to the non-analytic nature of the kinetic term of the gauge fields. δec δes < Often it is simpler to evaluate the δ’s first and obtain the associated beta functions, as above. However, this approach has the disadvantage that the properties of the correlation functions of the theory are obscure. Therefore, we apply a different method which allows us to read off the beta functions directly from the correlation functions. Within this approach, the structure of the correlation functions is more transparent. To this end, we will first derive a Callan-Symanzik type equation. For this purpose, let us first define

1 ˜ 2 p ψα ≡ Zf ψα ≡ 1 + δf ψαp η + δ η˜ ≡ η 1 + δf

1 2 1 + δc a˜c ≡ Zc · ac ≡ · ac 1 + δf

1 2 1 + δs (C6) a˜s ≡ Zs · as ≡ · as 1 + δf 2 2 2 1 + δc 2 ˜ ≡ 2 · ≡ · + 2 ec Zec ec ec δec 1 + δf 2 2 2 1 + δs 2 ˜ ≡ 2 · ≡ · + 2 es Zes es es δes , 1 + δf so that the eﬀective action, Eq. (C1), in terms of these ﬁelds and quantities with tildes, reads

˜ " # Z 1/Λ X ˜ 2 ˜ X ˜ ˜ ˜ ˜ Seff = dτdxdy ψαp† (˜η∂τ − ip∂x − ∂y )ψαp − λp (ãc +a ˜s)ψ1†pψ1p + (ãc − a˜s)ψ2†pψ2p α=1,2;p= p= ± ± (C7) ˜ Z Λ dωdk dk N N + x y | |1+|˜ |2 + | |1+|˜ |2 3 2 ky ac 2 ky as , (2π) 2˜ec 2˜es √ with Λ˜ ≡ Λ/ b. Notice that this form of the effective action differs from Eq. (C2) in that the space-time is not rescaled to match the new cutoff with the original one, while the forms of the Lagrangian in Eqs. (17), (C2) and (C7) are the same. 24

n ,nc,ns Next, consider a physical correlation function, Γ f , made of nf fermion fields, nc ac-fields and ns as-fields. ˜ nf ,nc,ns Denote the corresponding correlation function calculated in terms of the fields ψ,a ˜c anda ˜s by Γ˜ , which is related to Γnf ,nc,ns via nf n n n ,n ,n c s n ,n ,n f c s − 2 − 2 − 2 ˜ f c s Γ = Zf Zc Zs · Γ . (C8)

nf ,nc,ns ˜ 2 2 ñf ,nc,ns ˜ In general, Γ is a function of Λ,e ˜c ,e ˜s,η ˜, Zf , Zc and Zs. Similarly, Γ is in general a function of Λ, 2 2 ñf ,nc,ns e˜c ,e ˜s andη ˜. Notice, however, there is no explicit dependence of Γ on Zf , Zc or Zs. When the cutoff varies (e.g., from Λ to Λ),˜ in order to keep the physical correlation function, Γnf ,nc,ns , invariant, 2 2 nf ,nc,ns the values ofe ˜c ,e ˜s,η ˜, Zf , Zc and Zs also need to be adjusted accordingly. The invariance of Γ yields a 2 constraint equation between the changes of Λ,e ˜c , etc. This equation is the Callan-Symanzik equation that will be derived below. For notational convenience, let us define 2 ˜ 2 2 2 2 + 2 ( − ) Λ de˜c Λ e˜c − ec 2 δec ec δc δf bc = = lim = − lim e˜2 Λ˜ b 1 e˜2 Λ − Λ e2 b 1 b − 1 c d → c √b c → 2 ˜ 2 2 2 2 + 2 ( − ) Λ de˜s Λ e˜s − es 2 δes es δs δf bs = = lim = − lim e˜2 Λ˜ b 1 e˜2 Λ − Λ e2 b 1 b − 1 s d → s √b s → η+δη − 1 d logη ˜ Λ 1+δf 2 δη − ηδf bη = Λ˜ = lim = − · lim dΛ˜ b 1 η Λ − Λ η b 1 b − 1 → √b → (C9) Λ˜ dZf Λ δf δf γf = = lim Λ = −2 · lim Zf Λ˜ b 1 Zf − Λ b 1 b − 1 d → √b →

Λ˜ dZc Λ δc − δf δc − δf γc = = 2 · lim Λ = −4 · lim Zc Λ˜ b 1 Zc − Λ b 1 b − 1 d → √b →

Λ˜ dZs Λ δs − δf δs − δf γs = = 2 · lim Λ = −4 · lim . Zs Λ˜ b 1 Zs − Λ b 1 b − 1 d → √b → Notice that using some of the above quantities, the various beta functions introduced before can be written as b β(η) = 1 − z − η · η Λ 2 b β(e2) = 1 + − z − c e2 c 2 Λ 2 c b β(e2) = 1 + − z − s e2 (C10) s 2 Λ 2 c + b − b e2 β(e2/η) = η c · c c 2 η + b − b e2 β(e2/η) = η s · s . s 2 η

From these equations we see immediately that in order to obtain the beta functions, all we need are bc, bs and bη. The invariance of Γnf ,nc,ns implies that Λ˜ d Γnf ,nc,ns = 0. More explicitly, dΛ˜ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Λ˜ + b e˜2 + b e˜2 + b η˜ + γ · Z + γ · Z + γ · Z · ˜ c c 2 s s 2 η f f c c s s ∂Λ ∂e˜c ∂e˜s ∂η˜ ∂Zf ∂Zc ∂Zs (C11) nf ,nc,ns 2 2 ˜ Γ {py}, {px}, {ω}, e˜c , e˜s, η,˜ Zf ,Zc,Zs, Λ = 0.

n ,nc,ns Now plugging Eq. (C8) into the above equation and noting that Γ˜ f has no explicit dependence on Zf , Zc and Zs, we ﬁnally arrive at the Callan-Symanzik equation: ∂ ∂ ∂ ∂ n n n Λ˜ + b e˜2 + b e˜2 + b η˜ − f γ − c γ − s γ · ˜ c c 2 s s 2 η f c s ∂Λ ∂e˜c ∂e˜s ∂η˜ 2 2 2 (C12) ˜nf ,nc,ns 2 2 ˜ Γ {py}, {px}, {ω}, e˜c , e˜s, η,˜ Λ = 0, 25

This Callan-Symanzik equation can be a lot more illuminating if the correlation function Γ˜ is expressed as a product of a dimensionful part and a dimensionless part. Using dimensional analysis, we ﬁnd that Γ˜ can be written as

2 2 py px ωe˜ ωe˜ ηω˜ ω Γñf ,nc,ns = Λ˜ δ · g , , c , s , , (C13) Λ˜ Λ˜ 2 Λ˜ 2+ Λ˜ 2+ Λ˜ 2 Λ˜ z with z + 5 δ = z + 3 − · n − (z + 1) · (n + n ). (C14) 2 f c s Notice that the value of z can only be determined after obtaining the explicit expression of the correlation function and writing it in the above form, Eq. (C13). Furthermore, a priori, the value of z can be different for different correlation functions. Plugging Eq. (C13) into Eq. (C12) yields another scaling form of the Callan-Symanzik equation py px g1 + 2 · g2 Λ˜ Λ˜ 2 2 2 ωe˜c ωe˜s ηω˜ ω + (2 + − bc) g3 + (2 + − bs) g4 + (2 − bη) g5 + z g6 (C15) Λ˜ 2+ Λ˜ 2+ Λ˜ 2 Λ˜ z n γ + n γ + n γ = δ − f f c c s s g, 2

∂g(x1,x2,x3,x4,x5,x6) with gi the partial derivative of g with respect to its i-th argument, i.e., gi ≡ . From here we ∂xi also see that γf , γc and γs are the anomalous dimensions of f, ac and as, respectively. Because of the Ward identities, = = 0 for this (1) × (1) problem [36]. Combining this, Eq. (C4) and that 2 = 2 = 0 γc γs Uc Us δec δes when < 1 yields β(e2) = 1 + − z e2 c 2 Λ c (C16) β(e2) = 1 + − z e2 s 2 Λ s

2 2 2 2 which implies, when < 1, β ec /es = 0 to all orders, i.e., ec /es is exactly RG invariant. On the other hand, although in the main text we argue that the RG framework developed here can be mostly applied to the U(2) gauge theory as well, we stress that for the U(2) gauge theory one cannot directly conclude that the ratio of the U(1) and SU(2) gauge couplings is exactly RG invariant when < 1, due to a possible nonzero anomalous dimension of the SU(2) gauge field [50]. n ,nc,ns We remark that Γ˜ f is calculated from the theory defined with cutoff Λ˜ and coupling constantse ˜c,e ˜s and η˜. By calculating it and requiring that it satisfy Eq. (C12) or Eq. (C15), we can obtain the values of bc, bs, bη and γf,c,s. Then we can substitute these quantities into Eq. (C10) and get the corresponding beta functions. n ,nc,ns Notice, however, in order to obtain the physical correlation function, Γ f , one still needs to substitute γf,c,s into Eq. (C9) to solve for Zf,c,s, and use Eq. (C8).

D. Correlation functions and beta functions from Wilsonian RG

Based on the framework for Wilsonian RG presented in AppendixC, here we give examples of some speciﬁc correlation functions, Γ˜nf ,nc,ns , and obtain the beta functions to O() ∼ O(1/N). The propagator for ac at this order is [1]

1/N Dc(ω, ky) = 1+ , (D1) ω ky γ | | + | |2 ky e | | c with γ = 1/(4π). This is obtained by adding the bare propagator and the self-energy shown in Fig.6. Requiring that this propagator satisfy Eq. (C12) yields bc = 0. Similar considerations for the propagator of as at this leading order will give bs = 0. 26

FIG. 6. The self-energy diagram of the gauge ﬁeld. The external wavy line can represent any one of the gauge ﬁelds (ac and as in the U(1) × U(1) gauge theory, or, a anda ˜ of the U(2) gauge theory) described in this paper.

FIG. 7. The fermion self-energy diagram due to one of the gauge ﬁelds. In the U(1) × U(1) gauge theory, this diagram represents contribution from either ac or as. In the U(2) gauge theory, it represents contribution from either a ora ˜. The total self-energy includes contributions from both gauge ﬁelds.

Next consider the self-energy of the fermion (see Fig.7 for the Feynman diagram and the end of this appendix for computational details): Z (0) Σp(ω, kx, ky) = (Dc(Ω, qy) + Ds(Ω, qy))Gp (ω − Ω, kx − qx, ky − qy) Ω,q ,q x y (D2) 2 2 i 2 2 2 2 1 2π 2 i(ec + es) Λ− ω = e (γe )− 2+ + e (γe )− 2+ sin− |ω| 2+ sgn(ω) − · . 4πN c c s s 2 + 2π2N Substituting this self-energy into the fermion propagator and using the Callan-Symanzik equation, Eq. (C12), 2 2 ec +es yield bη = − 2π2NηΛ + O(/N) and γf = 0 + O(/N). In passing, we note that the vertex correction to the fermion-gauge coupling vanishes at this order [37], as expected from the momentum-independence of fermion self-energy in Eq. (D2) and the Ward identity of this U(1) × U(1) gauge theory [42]. In summary, at O() ∼ O(1/N), the relevant RG factors are

2 2 ec + es bη = − , bc = bs = 0, 2π2NηΛ (D3)

γf = γc = γs = 0.

Combining these results with Eq. (C10) and choosing zΛ = 1 as in Ref. [36] yield beta functions

e2 + e2 β(η) = c s · η, 4π2ηΛ e2 + e2 β(e2) = − c s · e2, (D4) c 2 4π2NηΛ c e2 + e2 β(e2) = − c s · e2. s 2 4π2NηΛ s

2 2 ec es Deﬁning the dimensionless gauge coupling constants, αc = 4π2ηΛ , αs = 4π2ηΛ , their corresponding beta functions are αc + αs β(αc) = − αc, 2 N (D5) αc + αs β(αs) = − αs. 2 N 27

Analogous calculations can be carried out for the U(2) problem in the “quasi-Abelianized” limit, where the propagator ofa ˜ is

˜ 1/N D(ω, ky) = 1+ , (D6) ω ky γ | | + | |2 ky e | | and the propagator of aα is

αβ 2/N αβ D (ω, ky) = 1+ · δ . (D7) ω ky γ | | + | |2 ky g | | αβ αβ The self-energy of the fermions is Σp (ω, kx, ky) = Σp(ω)δ , with 2 3 2 i 2 2 3 2 2 1 2π 2 i e + 2 g Λ− ω Σp(ω) = e (γe )− 2+ + g (2γg )− 2+ sin− |ω| 2+ sgn(ω) − · . (D8) 4πN 2 2 + 2π2N Moreover, the vertex correction to the fermion-gauge coupling vanishes at this order [37]. Therefore at this order, the U(2) analog of the quantities in Eq. (D3) are 2 + 3 2 ec 2 g ˜ bη = − , b = b = 0, 2π2NηΛ (D9)

γf =γ ˜ = γ = 0.

Similar to the Uc(1) × Us(1) case, if we take zΛ = 1, we obtain e2 + 3 g2 β(η) = 2 · η, 4π2NηΛ e2 + 3 g2 β(e) = − 2 · e2, (D10) 2 4π2NηΛ e2 + 3 g2 β(g2) = − 2 · g2. 2 4π2NηΛ

e2 3g2 In terms of the dimensionless gauge couplings,α ˜ ≡ 4π2ηΛ , α ≡ 8π2ηΛ , the beta functions are α˜ + α β(˜α) = − α,˜ 2 N (D11) α˜ + α β(α) = − α. 2 N Before we end this appendix, we present some additional details for the calculation of the fermion self-energy, Eq. (D2) (or Eq. (D8)), given by Z (0) Σp(ω, kx, ky) = (Dc(Ω, qy) + Ds(Ω, qy))Gp (ω − Ω, kx − qx, ky − qy) Ω,qx,qy Z 1 = ( (Ω ) + (Ω )) Dc , qy Ds , qy 2 (D12) Ω,qx,qy −i(ω − Ω) + p(kx − qx) + (ky − qy) Z i · sgn(ω − Ω) = (Dc(Ω, qy) + Ds(Ω, qy)), Ω,qy 2 where only the modes with |qy| < Λ should be integrated over. 2 2 Now we carry out the integral over qy for one of the two terms in the bracket [59], and denote either ec or es by e2:

Z 2 Z 2 ∞ dqy e ∞ qydqy e π 2 2+ 1 2π D(Ω, qy) = = · γe |Ω| − sin− , (D13) 2 2+ 2π πN 0 γe |Ω| + qy πN 2 + 2 + −∞ and, for large Λ |ω|, Z 2 Z 2 Z 2 ∞ dqy e ∞ qydqy e ∞ 1 e Λ− 2 · D(Ω, qy) = ≈ q− − dqy = , (D14) 2 2+ y Λ 2π πN Λ γe |Ω| + qy πN Λ πN 28 so

Z Λ 2 dqy e π 2 2+ 1 2π Λ− D(Ω, qy) = γe |Ω| − sin− − . (D15) Λ 2π πN 2 + 2 + − Finally the integral over Ω leads to

Z 2 dΩ i · sgn(ω − Ω) e π 2 2+ 1 2π Λ− · γe |Ω| − sin− − 2π 2 πN 2 + 2 + (D16) 2 2 ie 2 1 2π 2 ie Λ− ω = (γe )− 2+ sin− |ω| 2+ sgn(ω) − · , 4πN 2 + 2π2N which leads to the expressions for the above self-energy.

E. BCS coupling

For the sake of completeness, in this appendix we review the deﬁnition of the dimensionless BCS coupling [36]. For this purpose, we will consider the physical case with N = 1 and focus on the pairing between the two colors of the fermions. The dimensionless BCS coupling is extracted from the four-fermion interaction given by the following action:

Z 4 2 BCS Y d kidωi 3 3 S = ψ†(k )ψ†(k )ψ (k )ψ (k ) · (2π) δ (k + k − k − k ) V (k , k ; k , k ) (E1) (2π)3 1 1 2 2 2 3 1 4 1 2 3 4 1 2 3 4 i where k = (ω, k) collectively denotes the frequency and momentum, and subscripts 1 and 2 are color indices. The patch indices are not shown explicitly and they can be inferred from the requirement of momentum conservation. Due to the kinematic constraints resulting from a Fermi surface, only couplings among the BCS-related momenta V (k1, −k1; k2, −k2) are important [46–48]. For simplicity, we assume that the system is rotationally invariant, such that the BCS couplings can be written as V (k1, −k1; k2, −k2) = V (θ1 − θ2), where the angles θ1,2 parameterize the contour of the Fermi surface. They can be further decomposed into angular harmonics:

X im(θ1 θ2) V (θ1 − θ2) = Vme − . (E2) m The dimensionless BCS coupling is then deﬁned as

kF kF η Vem = Vm = Vm, (E3) 2πvF 2π where kF is the Fermi momentum. In the main text, we denote Ve ≡ Ve0 to be the dimensionless BCS s-wave coupling constant.

F. RG ﬂows for the U(1) × U(1) gauge theory at = 0

Using the method in Ref. [36], in this appendix we discuss the ﬂows described by Eq. (34), which is reproduced here for clarity:

dαc = − (αc + αs) · αc d` dαs = − (αc + αs) · αs (F1) d`

dVe 2 = αc − αs − Ve d` Notice that the physical value of N = 1 has been substituted here. Denoting αt ≡ αc + αs and adding the ﬁrst two equations in Eq. (F1) yield dα t = −α2 (F2) d` t 29 whose solution is given by

αt(0) αt(`) = (F3) 1 + αt(0)` with αt(0) a non-universal constant determined by microscopic details. Recall from AppendixC that

d α c = 0 (F4) d` αs holds exactly in this case, we set r ≡ αc , which is also determined by non-universal microscopic details, and αs obtain the solutions for αc and αs:

αt(0) r αc(`) = · 1 + α (0)` r + 1 t (F5) αt(0) 1 αs(`) = · 1 + αt(0)` r + 1

r 1 Next, we discuss the flow of Ve. To solve for the flow of Ve when r > 1, define rn ≡ − , αn ≡ αtrn = αc − αs √ r+1 and g ≡ V/e αn. Now solving for Ve amounts to solving for g, and the flow equation of g is

dg √ 2 gαn = αn(1 − g ) + (F6) d` 2rn

As both αn → 0, the last term of the above equation can be ignored. Then we get

3 dg 2 2 = −αn− (1 − g )rn (F7) dαn which yields

− 1 − 1 2 2 4rn αn (`) αn (0) (g(0) + 1) e − + g(0) − 1 g(`) = (F8) − 1 − 1 2 2 4rn αn (`) αn (0) (g(0) + 1) e − + 1 − g(0)

Physically, there are 3 distinct cases. p 1. When g(0) > −1, i.e., Ve(0) > − αn(0), g → 1 as ` → ∞, that is,

p Ve(`) → αn(`) (F9)

In this case, the FS is stable against pairing. p 2. When g(0) < −1, i.e., Ve(0) < − αn(0), the FS is unstable, and g(`), as well as Ve, diverges at

" p #2 1 (0) − (0) 2 1 Ve αn 1 ` = rn αn− (0) + ln p − αt− (0) (F10) 4rn Ve(0) + αn(0)

p 3. When g(0) = −1, i.e., Ve(0) = − αn(0), g(`) = −1, so

p p Ve(`) = − αn(`) = − αt(`)rn (F11)

This is a transition line between the two other cases. 30

Next, if r < 1, Ve → −∞ along the RG flow, and the FS is unstable. In this case, since αn flows slowly, we treat it as a constant when analyzing the flow of Ve, and obtain ! √ √ 1 Ve(0) Ve(`) = −αn tan − −αn` + tan− √ (F12) −αn

Finally, if r = 1, the contributions from the gauge ﬁelds to the RG equation for Ve in Eq. (F1) cancel, and the beta function of Ve to this order is the same as that of a Fermi liquid:

dVe = −Ve 2. (F13) dl Therefore, to analyze the effect of gauge fields, we need to go to the next-leading order, and we will see that the result at this next-leading order further suppress pairing. To this end, it is more convenient to to consider the effective Lagrangian given by Eqs. (17), (18) and (19), but in terms of a1,2, rather than ac,s:

L = Lf + L[a1,a2], (F14) with

X 2 X Lf = ψαp† η∂τ + vF −ip∂x − ∂y ψαp − vF λp a1ψ1†pψ1p + a2ψ2†pψ2p , (F15) p= ,α=1,2 p= ± ± and

N 1+ 2 2 L = |ky| |a | + |a | , (F16) [a1,a2] 4e2 1 2 with e = ec = es. Importantly, when r = 1 and < 1, there is no direct coupling between a1 and a2 in this effective Lagrangian, and such a coupling cannot be generated in the RG process due to the non-analytic nature of Eq. (F16). This is actually another way to see why the contribution from the gauge fields to the flow of Ve vanishes at the leading order, given by Eq. (F13).

1 1 1 1

a2 2 2 2 2

(a) (b)

FIG. 8. The next-leading-order contribution to the flow of Ve coming from vertex correction. Notice when r = 1, there is no gauge field line that connects the fermions with the two different colors.

The next-leading order contribution comes from vertex correction in Fig.8 and from the flow of η in the definition of Ve, Eq. (E3)[36]. This vertex correction vanishes as a result of the pole-structure of the fermionic propagators [37]. Notice that there is no analogous vertex correction with an internal gauge field line connecting two fermion lines with two different colors (i.e. connecting ‘1’ and ‘2’), because of the absence of a coupling between a1 and a2 when r = 1. The physical implication of the flow of η is that quasiparticles are destroyed, which is expected to further suppress pairing. Indeed, according to Eqs. (D4) and (E3), the flow of η changes the beta function of Ve into

dVe 2 = αtVe − Ve . (F17) dl

The first term, coming from the flow of η, further suppresses pairing. In the above equation, because αt flows slowly, we view it as a constant, and obtain a “fixed point” of Ve at Ve = αt if Ve(0) > 0, which then flows to zero ∗ 31 according to Eq. (F2) 9. In this case, the FS is stable against pairing. If Ve(0) < 0, Ve → −∞ along the RG flow, which means the FS is unstable to pairing. Taken together, we conclude that when r = 1, gauge fields suppress pairing, and the FS are perturbatively stable. In summary, when r < 1, the FS is unstable to pairing, while it is stable when r > 1.

9 There can be higher order corrections that can modify how the result that αt will slowly ﬂow to zero. exactly αt ﬂows to zero, but they are not expected to change