PHYSICAL REVIEW X 7, 031009 (2017)

Pairing in Luttinger Liquids and Quantum Hall States

Charles L. Kane,1 Ady Stern,2 and Bertrand I. Halperin3 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 2Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel 3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 8 February 2017; published 18 July 2017) We study spinless electrons in a single-channel quantum wire interacting through attractive interaction, and the quantum Hall states that may be constructed by an array of such wires. For a single wire, the electrons may form two phases, the Luttinger liquid and the strongly paired phase. The Luttinger liquid is gapless to one- and two-electron excitations, while the strongly paired state is gapped to the former and gapless to the latter. In contrast to the case in which the wire is proximity coupled to an external superconductor, for an isolated wire there is no separate phase of a topological, weakly paired superconductor. Rather, this phase is adiabatically connected to the Luttinger liquid phase. The properties of the one-dimensional topological superconductor emerge when the number of channels in the wire becomes large. The quantum Hall states that may be formed by an array of single-channel wires depend on the Landau-level filling factors. For odd- denominator fillings ν ¼ 1=ð2n þ 1Þ, wires at the Luttinger phase form Laughlin states, while wires in the strongly paired phase form a bosonic fractional quantum Hall state of strongly bound pairs at a filling of 1=ð8n þ 4Þ. The transition between the two is of the universality class of Ising transitions in three dimensions. For even-denominator fractions ν ¼ 1=2n, the two single-wire phases translate into four quantum Hall states. Two of those states are bosonic fractional quantum Hall states of weakly and strongly bound pairs of electrons. The other two are non-Abelian quantum Hall states, which originate from coupling wires close to their critical point. One of these non-Abelian states is the Moore-Read state. The transitions between all of these states are of the universality class of Majorana transitions. We point out some of the properties that characterize the different phases and the phase transitions.

DOI: 10.1103/PhysRevX.7.031009 Subject Areas: Condensed Matter Physics, Superconductivity, Topological Insulators

I. INTRODUCTION analyze instabilities towards other phases, including super- conductors or spin-gapped states, it does not find such Luttinger liquid theory is a powerful tool for the instabilities for the minimal one-dimensional system: a theoretical study of interacting systems in one dimension, single-channel quantum wire of spinless fermions. allowing for detailed analysis of their low-energy properties In this work, we use the Luttinger liquid as a starting [1,2]. Many attempts have been made to extend the theory point for exploring phases and phase transitions of a single- for the study of systems of higher dimensions. A particu- channel wire of spinless electrons that attractively interact. larly successful approach employs an array of quantum We have two goals: Within the realm of one dimension, we wires, each being described as a Luttinger liquid of spinless are interested in the possible superconducting phases that electrons, to construct two-dimensional topological states are constructed in finite one-dimensional systems, where of matter. This coupled wire construction was originally the number of electrons is conserved. In particular, we are formulated to describe Abelian fractional quantum Hall interested in the distinction between a Luttinger liquid, a states [3] and has been generalized to describe non-Abelian topological superconductor, and a nontopological super- quantum Hall states [4], as well as other two- and three- conductor in such systems. dimensional topological phases [5–15]. Beyond the one-dimensional realm, we are interested in While for systems of spinful electrons, or systems of using wires of attractively interacting electrons as building more than one channel, Luttinger liquid theory is able to blocks for two-dimensional fractional quantum Hall states. Here, we are motivated by the understanding that the most prominent series of non-Abelian quantum Hall states, the Published by the American Physical Society under the terms of Read-Rezayi series [16,17], is intimately related to clusters the Creative Commons Attribution 4.0 International license. ν ¼ 5 2 Further distribution of this work must maintain attribution to of electrons. In particular, the = Moore-Read state is the author(s) and the published article’s title, journal citation, a paired state. We seek a coupled wire construction of this and DOI. state that incorporates the pairing physics and simplifies the

2160-3308=17=7(3)=031009(17) 031009-1 Published by the American Physical Society KANE, STERN, and HALPERIN PHYS. REV. X 7, 031009 (2017) construction of Ref. [4], which involved an unnatural construction of quantum Hall states. Finally, Sec. V con- spatial modulation of the magnetic field. cludes the paper. Our study begins with a single-channel quantum wire, continues with wires of many modes, and then focuses on II. PHYSICAL PICTURE AND SUMMARY quantum Hall states formed by an array of single-mode OF RESULTS wires. For single-mode wires, we analyze the strong- pairing and weak-pairing superconducting phases that A. Single wire occur for attractively interacting electrons. Interestingly, We begin by considering a single-channel quantum wire we find that the weakly paired phase is adiabatically with attractive interactions. Bardeen-Cooper-Schrieffer connected to the Luttinger liquid phase formed by repul- (BCS) MFT predicts two topologically distinct supercon- sively interacting electrons, but it is separated by a phase ducting states, which both have a single-particle energy transition from the strongly paired phase. The two phases gap. In the paradigmatic Kitaev chain [19], the transition differ in their spectrum of single-electron excitations. While between the trivial and the topological superconducting the strong-pairing phase is gapped to single electrons and phases is driven by tuning the chemical potential across the gapless to pairs of electrons, we find that the weak-pairing band edge. In the topological phase, there exist zero-energy phase, like the Luttinger liquid phase, is gapless to both. Majorana modes at the ends of the wire. Domain walls These results are different from those obtained in the mean between the trivial and topological phases also host field theory (MFT) of superconductivity, which is appli- Majorana modes, and the presence of four or more domain cable when the superconductivity in the wire is induced by walls leads to a topological degeneracy in the ground state. proximity coupling to an external bulk superconductor. Recent work has examined the role of fluctuations in For multimode wires, we discuss even-odd modulations the superconducting order parameter [20,21] in number- of the ground-state energy as a function of the number of conserving one-dimensional superconductors [22,23]. electrons, as well as single-electron tunneling density of Emphasis was placed on the case of quantum wires with states; we contrast the results of the multimode wire to the several channels. It was found that in the weak-pairing case of a single-mode wire coupled by proximity to an phase, the finite-size splitting in the topological ground- external superconductor. We examine the way by which the state degeneracy is suppressed only as a power law of two systems become similar in the limit of a large number the distance between Majorana modes, rather than the of modes. exponential behavior predicted by MFT. The power-law For quantum Hall states formed of arrays of single- dependence is a consequence of backscattering amplitudes mode wires, we focus on states in which the edge is that are introduced by impurities or by nonuniformities composed of a single charged mode. We find two distinct (e.g., the nonuniformity that gives rise to the existence of phase diagrams as a function of the strength of the pairing. topological-trivial interfaces). The exponent in the power For odd-denominator filling factors ν ¼ 1=ð2n þ 1Þ (with law depends on the interactions as well as the number of n ¼ 0; 1; 2; …), there are two possible states—a strong- channels, and when the number of channels is large, the pairing state, which is essentially a bosonic fractional exponent becomes large, effectively recovering the MFT quantum Hall state of filling 1=½4ð2n þ 1Þ, and a behavior. These works did not explicitly address the fate Laughlin state. The two states differ in topological proper- of the single-particle energy gap, the low-energy single- ties such as the quasiparticle charge and the ground-state particle tunneling density of states, or the nature of the degeneracy on a torus. The transition between the two is of transition to other one-dimensional phases. the 3D Ising universality class. For even-denominator In our analysis, we go beyond MFT by describing the filling factors ν ¼ 1=2n, there are three types of states— wire as a coexistence of two coupled fluids: charge e strong-pairing states, non-Abelian states of the Moore- fermions and charge 2e bosonic Cooper pairs. The two are Read type, and anisotropic Abelian quantum Hall states, coupled by a nonquadratic pairing term that breaks a with defects that carry non-Abelian localized Majorana bosonic pair into two electrons and pairs two electrons fermions. In addition to the conventional Moore-Read state, into a bosonic pair. This term makes the Hamiltonian our construction allows a related state that has counter- charge conserving and couples the low-energy fluctuations propagating charge and Majorana modes at the edge. This of the gapless mode, whose existence is guaranteed by state has the same topological order as the particle-hole translational invariance, to the superconducting pairs. We symmetric Moore-Read state that has recently been dis- show that this model has two phases, which we refer to as cussed in the context of the theory of the particle-hole strong- and weak-pairing phases. We find that in the strong- symmetric half-filled Landau level [18]. pairing phase, the single-electron fluid is gapped as in the We present the physical picture that emerges from our MFT. In contrast, the weak-pairing phase is gapless to the study and a summary of our results in the next section, introduction of single electrons. Sec. II. Following that, in Sec. III, we present our analysis The weak-pairing phase can be viewed in two ways: It of the single-mode wire. In Sec. IV, we discuss the wire can be considered as a one-dimensional topological

031009-2 PAIRING IN LUTTINGER LIQUIDS AND QUANTUM HALL … PHYS. REV. X 7, 031009 (2017) superconductor with a fluctuating phase, or it can be charge e excitations. The relation between the pairing considered as an ordinary single-channel Luttinger liquid transition and the Ising transition has previously been with attractive interactions. These two pictures are in fact explored in the context of molecular and atomic superfluids equivalent: The topological superconductor is adiabatically [25,26]. connected to the single-channel Luttinger liquid and, As in the conventional theory of bosonization, there are ultimately, to noninteracting electrons. Despite this equiv- additional composite charge-e fermion operators that alence, we continue using the name “weak-pairing phase” involve 2kF backscattering, leading to operators similar when pairing aspects are the focus of our attention. to Eq. (1) with all odd integer multiples of θρ. However, our In MFT, there is a gap for adding single electrons to the generalized theory includes an additional class of charge-e bulk of a topological superconductor. The same is true in operators with an even number of θρ ’s. For example, the our theory for “bare” single-electron operators that do not “bare” fermion operator has the form couple to the bosonic sector. However, we show that when † 2 iφρ fluctuations in the charge e sector are accounted for, there Ψ0 ∼ ðγ1 þ iγ2Þe : ð2Þ exist “composite electron” operators, which carry a single- electron charge and commute with the pairing term, leading While this operator has the “wrong” number of θρ ’s, it is to gapless charge e excitations. The composite electron accompanied by Majorana fermion operators γ1;2 acting in operators involve scattering off of the finite wave-vector the neutral sector. These are related to σx;y by a Jordan- density fluctuations in the charge 2e fluid and transfer Wigner string. These operators are usually unimportant at momentum to the bosonic fluid. Alternatively, adding a low energies because γ1;2 have a gap in both the ordered composite electron is equivalent to adding a bare electron, and disordered phases of the Ising model. However, γ1;2 are along with tunneling a superconducting vortex across the gapless at the transition, as well as near a boundary that wire. The vortex tunneling leads to a 2π phase slip in the hosts a Majorana zero mode in the weak-paired phase. superconducting order parameter. In a topological super- The picture we outline above also applies to a wire with conductor, this process by itself leads to a change in the multiple channels when an attractive interaction leads to a local fermion parity of the ground state, resulting in an superconducting energy gap in all but a single collective excited quasiparticle above the gap [19,24]. Adding the charge mode. For a weak attractive interaction, an odd bare fermion then couples the system back to the low- number of channels form a weakly paired superconductor, energy sector with no quasiparticles excited. with the set of properties described above. In contrast, an We establish the equivalence between the weak-paired even number of channels form a strongly paired super- superconductor and the Luttinger liquid by employing a conductor. However, this even-odd dependence on the unitary transformation that transforms the original charge-e number of channels does not hold when the attractive fermion mode and charge-2e boson mode into a bosonic interaction is not weak. In fact, the weak- to strong-pairing charge mode and a fermionic neutral mode. In doing transition that occurs with the opening of new channels so, we generalize the conventional bosonization formulas when the interaction is weak is a particular case for which to account for the possibility of a strong-paired phase. our analysis applies, but it is not the general case. In either The bosonic charge mode resembles the bosonic mode of phase, however, the wire is described by the generalized an ordinary single-channel Luttinger liquid, while the single-channel Luttinger liquid theory described above, neutral mode describes the degrees of freedom of a 1 þ 1- with a Luttinger parameter Kρ that depends on the number dimensional transverse-field Ising model, where the of channels. When the number of channels is large, Kρ is ordered and disordered phases of the Ising model corre- large. In the limit Kρ → ∞, we recover the classical MFT spond to the weak and strong-paired phases, respectively. behavior. In Sec. III B, we show that composite electron operators In MFT, the transition between the weak and strong- may be written in the form paired phases is a transition between two gapped states, and † it is in the 2D Ising universality class. At the transition, a Ψ ∼ ðσx þ σyÞ iðφρθρÞ ð Þ R=L i e ; 1 pair of counterpropagating Majorana modes described by γ1;2 become gapless. For a charge-conserving wire, the where φρ and θρ are the bosonic fields in the charge sector, transition is between two states that each have a gapless and σx;y are spin variables characterizing the transverse- collective charge mode. One may wonder whether the field Ising neutral sector. In the ordered phase of the Ising nature of the transition changes. This problem has been model (the weak-paired phase), σx has long-range order and studied in Refs. [27,28]. The coupling between the Ising may be replaced by its expectation value hσxi. We therefore variables and the gapless charge mode is found to be recover the conventional bosonization of a single-channel marginally irrelevant but nonetheless changes the nature of Luttinger liquid. In the disordered phase of the Ising model the critical behavior at the transition because it leads to a (the strong-paired phase), σx;y have short-range correlations strong renormalization of the velocities of the modes. in space and time, which implies that there is a gap for Depending on the coupling, there are two possible

031009-3 KANE, STERN, and HALPERIN PHYS. REV. X 7, 031009 (2017) behaviors: Either the transition is converted to a discon- pattern in the ground-state energy as a function of the tinuous first-order transition, or the transition exhibits a number of electrons. This degeneracy is absent in a wire continuous transition that resembles the Ising transition, but with open boundary conditions. with a logarithmic renormalization of the velocity. The distinction between the strong- and weak-paired B. Array of wires as a fractional quantum Hall state phases can be probed experimentally in two ways: (1) the The coupled wire model constructs quantum Hall states tunneling density of states at the bulk and at the end of the by connecting neighboring wires by tunneling in a way that wire and (2) the dependence of the ground-state energy of a mutually gaps a right-moving mode on one wire with a finite system on the parity of the number of particles. In left-moving mode on its neighbor. Provided the tunneling the strong-pairing phase, both of these properties follow the conserves momentum, this leads to a bulk gap, while chiral MFT prediction. There is a single-particle energy gap in the modes are left gapless on the edge. The balance of tunneling density of states, and the ground-state energy momentum equates the momentum transferred to the exhibits an even-odd effect, in which the energy difference tunneling charge by the Lorentz force with the total change for an odd and even number of electrons remains finite in in the electrons’ momentum as a consequence of the the thermodynamic limit, as is observed in mesoscopic tunneling. For tunneling of charge q across the interwire superconductors. spacing d in magnetic field B, the Lorentz force provides For the weak-paired topological superconductor, it is momentum qBd. Fermionic Laughlin states are stabilized interesting to compare these properties with those of a by a process in which this momentum is balanced by the single-channel Luttinger liquid, as well as with noninter- momentum 2k ð2n þ 1Þ associated with an electron tun- acting electrons. Because of the composite electron oper- F neling from momentum state kF on one wire to momentum ator, the tunneling density of states in the bulk of the state −k on the neighbor, while backscattering n electrons weak-paired phase vanishes like a power of energy. This F from þkF to −kF on each wire (here, n ¼ 0; 1; 2; …). behavior is identical to an ordinary Luttinger liquid. For a The balance is satisfied at the Laughlin filling factors finite wire, the tunneling density of states at the end differs ν ¼ 2kF=qBd ¼ 1=ð2n þ 1Þ. When the constituent par- from that of the bulk. As argued in Refs. [20,21], the ticles are bosons, the Laughlin states at ν ¼ 1=ð2n þ 2Þ δ -function peak at zero energy due to the Majorana mode are formed if the tunneling boson applies a density operator predicted by MFT is replaced by a power-law divergence in of momentum ðn þ 1Þ2πnb in the two wires involved. the tunneling density of states. The same behavior arises in In the present work, we consider a wire construction in a single-channel Luttinger liquid, where tunneling density which the wires are in (i) the weak-paired phase, (ii) the of states at the end of the wire is different from that in the strong-paired phase, and (iii) the critical point between middle [29]. When the interactions are strong or the number them. For the weak-paired phase, we simply reproduce the of modes in the wire is large, the exponent in the bulk construction described above for the Laughlin states [3]. becomes large, and we effectively recover the gap predicted This is consistent with the equivalence we draw for the by MFT. In parallel, the exponent at the end approaches weakly paired phase with the Luttinger liquid. In the −1, and we recover the sharp peak characteristic of a strong-paired phase, the single electrons are frozen, and Majorana mode. the wires may be regarded as composed of bosons of charge In MFT, the even-odd effect in the weak-pairing phase 2e. Tunneling then leads to a sequence of strong-paired depends on the boundary conditions. For open boundary quantum Hall states of charge-2e bosons at (electron) conditions, the even-odd effect is exponentially small in the filling factors ν ¼ 4=ð2nÞ, which have quasiparticles with system size L because the odd electron can be added to the charge 2e=2n with statistics angle 2π=ð2nÞ and a degen- Majorana end mode. For periodic boundary conditions eracy 2n on a torus. Notable examples include ν ¼ 2 with (a ring), however, there are no Majorana modes, and the semionic charge-e quasiparticles, ν ¼ 1 with charge e=2 even-odd effect is finite for large L. Since fluctuations quasiparticles, and ν ¼ 1=2 with charge-e=4 quasiparticles eliminate the single-particle gap, we expect this to be [30], known in the literature as the K ¼ 8 state [31]. modified. We predict that for open boundary conditions, the For odd-denominator filling factors ν ¼ 1=ð2n þ 1Þ, even-odd effect remains exponentially small in the system’s there exist both ordinary Laughlin states (for weak-paired size, while for periodic boundary conditions, it vanishes wires) and strong-paired states (for strong-paired wires). only as 1=L. This distinction between periodic and open We argue that the transition between them is in the 3D Ising boundary conditions can be understood by considering the universality class. ground-state energy of a single-channel Luttinger liquid, For even-denominator filling factors ν ¼ 1=ð2n þ 2Þ, and it persists even for noninteracting electrons. The “composite electrons” cannot tunnel in a way that con- difference between the two boundary conditions is most serves momentum. The wires are then coupled by pair easily understood at the noninteracting level. For a wire tunneling, which gaps the gapless modes in the bulk. In the with periodic boundary conditions, the single-particle strong-pairing regime, a bosonic fractional quantum Hall spectrum is doubly degenerate, leading to an even-odd effect (FQHE) state is obtained. On the weak-pairing side,

031009-4 PAIRING IN LUTTINGER LIQUIDS AND QUANTUM HALL … PHYS. REV. X 7, 031009 (2017) gapless Majorana zero modes are left at the wires’ ends. two-particle bound states. For V2 > −2V1 and small t, The ends may be coupled by single-electron tunneling, in the system will form a gas of charge-2e bosons with which case the localized Majorana end modes disperse into repulsive interactions. Thus, at low energies, the system a pair of counterpropagating Majorana modes. Defects in will be a Luttinger liquid of charge-2e particles, and there the array, e.g., wires that break or terminate at the bulk of will be a gap jV1j for creating charge-e particles. the sample, carry localized zero-energy Majorana modes We wish to develop a low-energy theory that is capable that do not hybridize with the gapped bulk. These modes of describing both of these phases, as well as the behavior are static non-Abelian defects, of the type found at the ends of a transition between them. Our approach is to consider a of 1D topological superconductors. low-energy “two-fluid model” consisting of a channel of When the individual wires are at the transition between charge-e fermions coexisting with charge-2e bosons. weak and strong pairing, each wire carries both a gapless charged pair of modes and a gapless neutral pair of A. Two-fluid model Majorana modes. It is then possible to construct an electron operator that allows for single-electron tunneling with Consider a single wire described by a single fermionic 2 momentum conservation. This operator couples both to channel with dispersion EðkÞ¼ϵ0 þ k =2m and average the charge and to the neutral modes. Together with pair density ρ¯f coupled to a Luttinger liquid of charge-2e tunneling that couples only to the charge mode, the bulk is bosons with average density ρ¯b. The Hamiltonian density H ¼ H0 þ H0 gapped, and the edge remains to carry chiral charge and is f b, where Majorana modes. The precise form of the single-electron tunneling determines the relative direction of motion of the v 1 H0 ¼ Kð∂ φÞ2 þ ð∂ θÞ2 − 2μð∂ θ=2π þ ρ¯ Þ; ð4Þ charge and Majorana modes on the edge. When they b 2π x K x x b copropagate, the resulting state is topologically identical to the Moore-Read state, which is a px þ ipy supercon- 0 † 2 2iφ H ¼ ψ ðϵ0 − ∂x=2m − μÞψ þ uðψ∂xψe þ H:c:Þ: ð5Þ ductor of composite fermions. When the two edge modes f − counterpropagate, the resulting state is a px ipy super- Here u is the pair tunneling between the fermion channel conductor of composite fermions. Its edge structure is and the boson Luttinger liquid channel, which is neces- symmetric to a particle-hole transformation. For electrons sarily a p wave. Note that ∂xθ=2π describes the fluctuations on a plane, the p − ip state is believed to be inferior in 0 x y in the boson density about ρ¯b, and ½φðxÞ; θðx Þ ¼ energy to the Moore-Read state since the latter may be iπΘðx − x0Þ. The chemical potential μ couples to the total associated with a trial wave function (the “Pfaffian” wave † charge density ρe ¼ ψ ψ þ 2ρb. function) that is entirely within the lowest Landau level, MFT treats φ as a classical variable, in which case H0 while the former necessarily involves states from higher f is a one-dimensional version of the Read-Green model of a Landau levels. For weakly coupled wires, a limit that is far p-wave superconductor. It describes a transition between a from that of electrons on a plane, it is possible to have trivial and a topological one-dimensional superconductor, situations in which the two states are comparable in energy. as a function of ϵ0, where ϵ0 < 0 is the strong-paired phase, The starting point in which each wire is in a critical state ϵ 0 also allows for construction of the anti-Pfaffian, the while 0 > is the weak-paired, or topological, phase, particle-hole conjugate of the Moore-Read state. Since this which exhibits Majorana modes at the end. In mean field state carries more than one charged edge mode, its con- theory, there is a gap for adding a charge-e particle in both struction requires interwire tunnel coupling that goes the strong and weak-paired phases. In order to correctly further than nearest-neighboring wires. describe the correlation functions for a charge-e particle in both phases, it is necessary to go beyond the mean field III. PAIRED STATES IN 1D theory. The low-energy fluctuations in the boson density have We begin with a simple model of spinless charge-e important contributions near wave vectors q ¼ 2πnρ¯ , ρ ≪ 1 n b fermions with density e on a one-dimensional lattice. X Consider the Hamiltonian ρbðxÞ¼ρ¯b þ ρnðxÞ: ð6Þ X X n H ¼ − ð † þ † Þþ ð Þ t ci ciþ1 ciþ1ci Vpniniþp 3 i i;p The long-wavelength boson density fluctuation is ρ0ðxÞ¼∂xθðxÞ=π. The density wave at q ∼ 2πρ¯n has a with first and second neighbor interactions V1 and V2, and phase modulated by θ, Vp>2 ¼ 0.ForV1, V2 > 0, this system will be a Fermi gas in(2πρ¯bxþθðxÞ) with repulsive interactions, and at low energy, it will be a ρnðxÞ ∝ e : ð7Þ Luttinger liquid with gapless charge-e excitations. For V1 < 0, however, the ground state will favor forming Importantly, the operators ρn are local operators.

031009-5 KANE, STERN, and HALPERIN PHYS. REV. X 7, 031009 (2017) ψ 1 The bare fermion operator is a local operator, but it is H ¼ v ð∂ φ Þ2 þ ð∂ θ Þ2 − μð∂ θ π þ ρ¯ Þ ϵ ≠ 0 ρ Kρ x ρ x ρ x ρ= e ; gapped when 0 because of the pairing term. However, 2π Kρ the local composite electron operators ð15Þ Ψ† ð Þ ∝ ψ †ρ ð Þ þ;n x n 8 where we identify Kρ ¼ K. The term describing the neutral and degrees of freedom is

† † 2 † † 2iφ Hσ ¼ ψ σðϵ0 − ∂ 2 Þψ σ þ ðψ σ∂ ψ σ þ ψ σ∂ ψ σÞðÞ Ψ−;nðxÞ ∝ ψe ρn ð9Þ x= m iu x x 16 are not necessarily gapped because einθ introduces a 2πn and the interaction term is phase slip into the superconducting phase 2φ.Itiswell 2 ð∂ φρÞ i∂ φρ v∂ θρ known that when the phase across a topological-insulator H ¼ x ψ †ψ − x ψ †∂ ψ − x ðψ †ψ − ρ¯ Þ 2π int σ σ σ x σ σ σ f : Josephson junction is advanced by , there is a level 2m m Kρ crossing that results in an excited quasiparticle, along with ð Þ a shift in the fermion parity of the ground state. It follows 17 that when n is odd and the fermions are in the topological H superconducting state, the fermion operator can annihilate The second term, σ, can be recognized as the mean this extra quasiparticle, returning the system to the low- field Read-Green model, which has a second-order topo- ϵ ¼ 0 energy sector with no gapped quasiparticles. Therefore, in logical transition at 0 that is in the 2D Ising univer- ψ 1 2 the weak-paired phase, the operators Ψ 1 create gapless sality class. Since has dimension = at the transition, the ; H excitations and have power-law correlations characteristic first two terms in int have dimension 3 and are strongly of a charge-e Luttinger liquid. irrelevant. The third term, however, with dimension 2 is marginal, and it affects the critical behavior [27,28]. Before describing the critical behavior, we examine the correspon- B. Bosonization near the Ising transition dence between the phases and critical behavior of our H ¼ 0 In order to make contact with the more familiar model with int and the corresponding behavior of the bosonization of the charge-e Luttinger liquid, it is useful Ising model. Our motivation for doing this is that, away ψ H to introduce a transformation that decouples the fermions from the transition, the gap in σ renders int irrelevant. H ¼ 0 and bosons by effectively transferring the charge of the Moreover, we find the int limit a useful starting point fermions to the bosons. We perform a canonical trans- in Sec. IV when we consider two-dimensional gapped formation H → UHU† generated by phases constructed from coupled wires, which have a R gapped charge mode. i dxðψ†ψ−ρ¯ ÞφðxÞ The connection between Eq. (16) and the Ising model is U ¼ e f ; ð10Þ well known. It is convenient to view the fermion fields ψ σ as the continuum limit of lattice fermion fields ψ , which where ρ¯ is the average fermion density. Identifying the j f may be written as the Jordan-Wigner transformation of new boson and fermion fields as φρ, θρ, and ψ σ, this has the lattice spin variables σ ¼ σx iσy via effect of transforming j j j P þ − −iφρ iπ ψ ψ ψð Þ → ψ ð Þ ð Þ σ ¼ ψ i>j i i ð Þ x σ x e ; 11 j j e : 18

φð Þ → φ ð Þ ð Þ þ † − x ρ x ; 12 Here, we write ψ ≡ ψ σ and ψ ≡ ψ σ. As reviewed in Z Appendix A, the Hamiltonian (16) is then the continuum ∞ † limit of a 1 þ 1-dimensional transverse-field Ising model, θðxÞ → θρðxÞþπ ρ¯fx þ dxψ σψ σ : ð13Þ x X H ¼ hσz − Jσxσx ; ð19Þ ∂ θ π I j j jþ1 Thus, x ρ= now describes the fluctuations in the total j electron density about the average value ρ¯e ¼ ρ¯f þ 2ρ¯b. The fermion field ψ σ is neutral. with ϵ0 ¼ 2ðh − JÞ. The weak-paired phases ϵ0 < 0 cor- The transformed Hamiltonian has three terms, respond to the ordered phase of the Ising model with a nonzero order parameter hσxi ≠ 0. The strong-paired phase H ¼ H þ H þ H ð Þ x ρ σ int: 14 ϵ0 > 0 is the disordered phase with hσ i¼0. The local composite electron creation operators given in The term describing the charged degrees of freedom is Eqs. (8) and (9) can now be written as

031009-6 PAIRING IN LUTTINGER LIQUIDS AND QUANTUM HALL … PHYS. REV. X 7, 031009 (2017) X † φ ð þθ Þ † † † Ψ ∝ ξ i ρ in kFx ρ ð Þ ψ ¼ Ψ þ Ψ ð Þ ;n n e e ; 20 e aþ;n þ;n a−;n −;n; 24 ≡ πρ¯ where kF e comes from the total electron density and with coefficients a;n that depend on the interactions R between the charge-e and charge-2e fluids. It follows that ∞ inπ ψ †ψ ξ ¼ ψ x ð Þ n e : 21 X 1 GðτÞ¼ ðτÞ ð Þ 1ð −1þ 2 Þ gn;σ : 25 τ2 Kρ n Kρ From Eq. (18), it is clear that when n is even, the Jordan- n Wigner string has no effect, while when n is odd, the Jordan-Wigner string converts the fermion variable into the Each term factorizes into a Luttinger liquidlike power law ðτÞ Ising variable. We thus have in the charge sector times a contribution gn;σ from the neutral sector. The latter depend crucially on whether n is σ ¼ðσx iσyÞ=2 n odd even or odd. The behavior in the weak and strong-paired ξ ¼ ð Þ n pffiffiffi 22 phases can be analyzed by considering the low-energy ψ ¼ðγ1 iγ2Þ= 2 n even; behavior near the Ising transition ϵ0 ∼ 0, where the neutral sector should exhibit scaling behavior. where σx;y are the continuum limit of Pauli spin matrices For even n, g σ describes the correlation function of σx;y defined on each lattice site, while γ1;2 are Majorana neven; j 1D Majorana fermions with a mass gap jϵ0j. Since ψ has fermion operators that form the continuum limit of the dimension 1=2 at the transition, we expect ψ lattice fermion operators j . Equations (20) and (22) can be viewed as a generaliza- 1 g σðτ; ϵ0Þ ∼ F ðϵ0τÞ: ð Þ tion of the conventional bosonization formulas to the neven; τ e 26 vicinity of the pairing transition. Deep in the weak-paired ð → 0Þ ∼ 1 phase, the Ising spins have long-range order, so in the terms The scaling function satisfies Fe x , reflecting the with odd n, they may be replaced by their nonzero gapless Majorana mode at the transition. In the opposite x −jxj expectation value hσ i¼hσ i. These terms give the con- limit, Feðx → ∞Þ ∼ e , reflecting the mass gap propor- ventional bosonization formula for a charge-e Luttinger tional to jϵ0j on either side of the transition. liquid, in which the integer coefficients of φρ and θρ have For odd n, gnodd;σ describes the correlation function of σ σ 1 8 the same parity. The usual left- and right-moving fermions the Ising spin operator x. Since x has dimension = at ΨR and ΨL are given by the n ¼1 operators. Note, the transition, we expect however, that in the strong-paired phase, hσi¼0, and the 1 Ising spins have exponentially decaying correlations. It ðτ ϵ Þ ∼ ðϵ τÞ ð Þ gn odd; σ ; 0 1 4 Fo 0 : 27 follows that in the strong-paired phase, there are no charge- τ = e operators with long-range correlations, reflecting an ð → 0Þ ∝ 1 energy gap for charge-e particles. Charge-2e particles, The scaling function satisfies Fo x , describing the 2 φ behavior at the transition exactly. The behavior for x → ∞ created by e i ρ, do not have a gap. reflects the spin correlations on either side of the transition. The even n terms, which have the “wrong” number of ϵ 0 hσ ðτÞσ ð0Þi ∼ θρ’s, involve ψ σ, which has a gap everywhere except in the On the ordered side, for 0 > , we expect x x ϵ2β τ β ¼ 1 8 ð → ∞Þ∼ vicinity of the Ising transition. Near the transition, Ψ;0 0 independent of , with = . Thus, Fo x 1=4 couples to the gapless Majorana modes present at the x . On the disordered side, ϵ0 < 0, the spin correlations −ν transition. Expressed in terms of the Majorana operators, decay exponentially in τ with a correlation length ξ ∝ jϵ0j −jxj Eq. (16) becomes with ν ¼ 1. Therefore, Foðx → −∞Þ ∼ e . The behavior described above can be summarized as T z 2 y GðτÞ ∼ −ϵ0τ Hσ ¼ γ ½−iu∂xτ þðϵ0 − ∂x=2mÞτ γ; ð23Þ follows: In the strong-paired phase, e , reflecting a gap, while in the weak-paired phase, the electron operator is a T ð þ −1Þ 2 where the Pauli matrices τ act on γ ¼ðγ1; γ2Þ . At the dominated by n ¼1, giving GðτÞ ∼ 1=τ Kρ Kρ = . At the transition ϵ0 ¼ 0, γ1 and γ2 can be recognized as right- and Ising critical point, the n ¼ 0 and n ¼1 terms compete, −1 −1 left-moving chiral Majorana modes. Nonzero ϵ0 introduces and we have GðτÞ ∼ 1=τMinðKρ =2þ1=4;Kρ=2þKρ =2þ1Þ. Thus, for γ γ a mass term that couples 1 and 2, leading to an Kρ < 3=2, the n ¼ 1 term dominates, while for Kρ > 3=2, energy gap. the n ¼ 0 term dominates. As discussed in the following The spectral properties of single electrons can be section, however, this critical behavior will be modified by summarized by the imaginary-time, local, single-particle the interaction with the charge mode. ’ GðτÞ¼h ½ψ ð τÞψ †ð 0Þi Green s function Tτ e x; e x; , where, in Finally, we note that in the weak-paired state, the fermion † general, the electron operator ψ e is a sum over all possible sector has a topologically nontrivial gap. This means that composite operators, there exist zero-energy Majorana zero modes at the ends of

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† φ Ψ ∼ ψ i ρ expect Kρ ∝ N , such that the initial value of λ → 0. In this the wire. Therefore, at the ends, ;0 e is not ch gapped, and it exhibits long-range temporal correlations case, Eqs. (31) imply that the effect of renormalization −1 with GðτÞ ∼ τ−Kρ . This reflects the well-known fact that the becomes significant only at exponentially low energy. exponent for tunneling into the end of a Luttinger liquid However, this analysis is modified if the interchannel differs from the exponent for tunneling into the middle of a Coulomb interaction is included. In that case, we find that 1=2 → ∞ Luttinger liquid [29]. Kρ, vare both proportional to Nch in the limit Nch [33]. The initial values of the parameters are then not C. Critical behavior necessarily small, which complicates the determination of † the RG flow. The effect of the marginal interaction λ∂xθρψ σϕσ (with λ ¼ v=Kρ) on the critical behavior of the Ising transition D. Distinguishing characteristics of the two phases has been analyzed in Refs. [27,28]. These works performed a renormalization group (RG) analysis that showed how λ, We now discuss two physical characteristics that dis- as well as the velocities u and v, is renormalized at low tinguish the weak-paired/Luttinger liquid phase from the energies at the transition. Here, we write the RG equations strongly paired phase in a one-dimensional wire. We also in a scheme in which space and time are renormalized at explore the case in which the wire has a large number of 2 channels. identical rates and the coefficient of the ð∂xφρÞ term in Eq. (15), namely, the product vKρ, is held fixed. The RG 1. Even-odd modulation of the ground-state energy equations are then as a function of the electron number N 2 Δ ð Þ d log v Kρλ We define the energy alternation E L , for a system of ¼ − ; ð28Þ dl 4uv length L,by

2 X∞ d log Kρ Kρλ ¼ ð Þ ΔEðLÞ¼2 eiπnE wðNÞ; ð32Þ 4 ; 29 N dl uv N¼0 2 d log u Kρλ ¼ − ; ð30Þ where EN is the ground-state energy of a system with N dl ðu þ vÞ2 fermions in the length L, and wðNÞ is a smooth weight function, with a maximum at a value N ¼ n0L and a width 2 d log λ Kρλ ð2u þ vÞ that is large compared to unity but small compared to n0L. ¼ − : ð31Þ dl 2uðu þ vÞ2 For example, wðNÞ might be a Gaussian with a second moment equal to n0L. We are interested in the behavior of These equations were found to result in two distinct types ΔE as L becomes large, with the mean fermion density n0 of flows. For the first type, u, v, and λ flow to zero with held fixed. u=v ¼ 1. This fixed point describes a continuous transition Note that ΔEðLÞ samples the even-odd modulation of in which the fermions and bosons are decoupled but their the ground-state energy over the range of wðNÞ, and it is velocities both vanish logarithmically at low energy. In the rather insensitive to the smooth variation of EN. In MFT, second type of flow, the boson velocity v renormalizes to for a wire with Majorana end modes, it may be interpreted zero faster than u, λ, and it flows to zero at a finite l. This as the energy splitting due to interactions between the signals a first-order transition that resembles a transition Majorana modes at the two ends of the wire, when such between the degenerate ground states associated with the modes exist. It should not be confused with the second Peierls instability. The separatrix between the two types of difference, flow is not known precisely. 2 When the initial conditions are such that λ ≪ v and Δ ðENÞ ≡ ENþ1 − 2EN þ EN−1; ð33Þ u ≪ v, the transition is expected to be continuous, with a flow of the first type. While we do not provide a detailed with N ≈ n0L. For a particle-conserving system in the estimate of the microscopic parameters, we make two topological superconducting state, with short-range inter- 2 observations. First, in the situation where superconductivity actions, the value of Δ ðENÞ will only go to zero as 1=L, for is induced into a semiconducting wire by proximity to large L, regardless of the number of channels, as the second metallic one-dimensional channels (see, e.g., Albrecht et al. difference samples the smooth dependence of EN on N.For [32]), we expect v ≫ u since v originates from the metal all phases, there is a smooth contribution to the ground- 2 while u originates from the superconductivity induced in state energy of the form EcðN − N0Þ =2 which accounts for 2 the semiconductor. Second, in our analysis above, the charging energy. This contribution determines Δ ðENÞ¼ λ initial value of is given by v=Kρ. As we review in Ec ∝ ð1=LÞ but does not lead to an even-odd effect and Δ ð Þ Sec. III D 2, when the number of channels Nch is large, we gives a negligible contribution to E L .

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Focusing on ΔEðLÞ, we note that for a strong-paired MFT, if the position x is close to the end of a semi-infinite superconductor, there is another contribution ð−1ÞNΔ, wire in the weak-pairing topological state, there will be a originating from the superconductor having an unpaired peak in spectral density at very low energies, which is electron. This leads to ΔEðLÞ, which is independent of L. associated with a localized Majorana mode at the wire end. The weak paired state, however, does not have a gap for In the case of the hybrid wire, coupled to an infinite single electrons, but rather, it behaves as a Luttinger liquid superconductor, this peak should be a delta function at zero described by Eq. (15). The dependence of the ground-state energy. The amplitude of the delta function will fall off energy on parity depends on the boundary conditions. For exponentially as x moves away from the end of the wire, open boundary conditions, there is just a single chiral and there will be an energy gap about zero energy, where channel that reflects back and forth from the ends. For a the spectral density is zero for any position x. segment of length L, the energy is then For a wire with conserved particle number, for x near the end of the wire, Aðx; εÞ will have only power-law diver- 2π ε ¼ 0 ε ≠ 0 segment ¼ v ð − Þ2 ð Þ gence at . The finite spectral density at occurs EN N N0 : 34 LKρ because the injected electron will necessarily produce phononlike excitations of the charge density in the wire, This gives a negligible contribution to ΔEðLÞ, reminiscent which exist down to arbitrarily low energies in a semi- of the absence of an even-odd effect in a topological infinite wire. Although the amplitude of the zero-energy superconductor due to the Majorana end modes. In con- singularity will decrease with increasing x, there should not trast, for a ring geometry, there are independent left- and be a hard gap in the spectral density, even for x in the right-moving chiral modes with integer charges NR=L, middle of an infinite wire. In the weak-pairing phase, there with N ¼ NR þ NL. The energy for a ring of circumference will always be some weight at low energies, in the middle L is then of the wire, though this weight will typically decrease rapidly with the number of channels in the wire. 2π 1 ring ¼ v ð þ − Þ2 þ ð − Þ2 The tunneling density of states is closely related to the EN NR NL N0 Kρ NR NL : ’ L Kρ single-particle Green s function discussed in Sec. III B.We ð Þ find that the spectral density for tunneling an electron into a 35 point near the end of a semi-infinite wire should have the form, for ε → 0, The ground state therefore has an even-odd modula- tion ΔEðLÞ¼2πvK=L. β Aðx; εÞ ∼ CðxÞjεj ; ð36Þ We note that this behavior of the even-odd effect persists even for noninteracting electrons. In this case, in the β ¼ −1 − 1 ð Þ segment geometry, the single-particle states are nondegen- Kρ ; 37 erate and evenly spaced, leading to Eq. (34), while in the Kρ ring geometry, every state (except at k ¼ 0) is doubly where is the Luttinger parameter defined in Eq. (15) degenerate, leading to Eq. (35). above. For a multichannel wire, the prefactor C should fall It is interesting to compare these conclusions to the off exponentially with distance from the end for short conclusions of MFT, in which a quantum wire is proximity distances, and it should fall off as a power law for large coupled to an ideal infinite bulk superconductor. In that distances. By contrast, the spectral density for tunneling at case, the electron number N is not a good quantum number. a point in the bulk of the wire, infinitely far from an end, Then, the ground state of the wire, at a given chemical should have the form potential, will be a superposition of states with different ðεÞ ∼ jεjα ð Þ values of N but a fixed number parity. When the wire is in a Abulk D ; 38 topological state and the geometry is that of a segment, the number parity of the ground state will alternate between where even and odd as L is increased, but the energy difference −1 Kρ þ Kρ between even and odd parity should decrease exponentially α ¼ − 1; ð39Þ with L. This would not happen for a ring, in which ΔEðLÞ 2 would stay independent of L, since the MFT weakly paired D phase is gapped to single electrons. and is a constant of proportionality. For a wire with attractive interactions, we expect to find Kρ > 1, so both α and β are positive; thus, the spectral density at the end of the 2. Tunneling density of states wire should diverge for ε → 0, while it vanishes in this limit A quantity of particular interest is the spectral density in the bulk. The expressions for these power laws have Aðε;xÞ for tunneling an electron into the wire at a position x precisely the same form as in the well-known case of and energy ε, measured relative to the Fermi energy. Within single-channel spinless Luttinger liquid, where, typically,

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Kρ is not very different from unity. In contrast, for a Both the weak- and strong-pairing states are one-dimen- multichannel wire with weak attractive interactions, we sional versions of a superconductor, in the sense that they expect the Luttinger parameter of the charge mode, Kρ,to have superconducting order parameters with quasi-long- increase with the number of channels, Nch, as may be seen range order. Specifically, the correlation functions for by the method presented in Ref. [33]. The bosonic variables operators that inject a pair of electrons at a point x and 0 j − 0j describing the chargeP channel are the total density remove a pair at point x will fall off as a power of x x , ½ð∂ θ Þ 2π¼½1 ð2πÞ ∂ θ with an exponent that becomes small as the number of x ρ = P= x i and the conjugate variable φ ¼½1 ð Þ ϕ channels becomes large. ρ = Nch i (where the sums are over the chan- In a multichannel wire with weak interactions between nels). The total density scales like Nch. In the absence of interaction between the channels, both the energy associ- the electrons, there is a difference between an even and an ated with density-density interactions and the kinetic odd number of channels. For an odd number of channels, energy should scale like N , requiring the velocity v to with weak attractive interactions, the ground state is ch predicted to be a one-dimensional version of a topological be independent of Nch and the Luttinger parameter to follow superconductor, with low-energy properties that coincide with the weak-pairing state discussed above [20,21]. For a wire with an even number of channels and weak attractive Kρ ∼ κN ; ð40Þ ch interactions, the ground state is expected to be effectively a one-dimensional version of a nontopological superconduc- where Nch is the number of channels and, for weak attractive interactions, κ is a number slightly larger than tor, with a finite energy difference between states of even α and odd number parity, and an energy gap for adding a unity. Thus, the exponent will become large, when Nch is large, and β will be close to −1; thus, the right-hand side of single electron to the wire. The low-energy properties of Eq. (36) will be close to a delta function at zero energy. In this state coincide with those of the strong-pairing situation the presence of an interchannel density-density interaction, discussed above. 1=2 both v and Kρ scale like Nch [33]. We remark that there will be a constant of proportionality IV. PAIRED QUANTUM HALL STATES on the right-hand side of Eq. (38), which should, itself, IN THE COUPLED WIRE MODEL become small in the case of a large number of channels We now apply the formalism developed in Sec. III to with weak interactions. This is because the bare fermion describe paired quantum Hall states. Strong-paired quan- † creation operator ψ ðxÞ does not couple directly to low- tum Hall states can occur when electrons are strongly energy excitations. Instead, we must employ an operator of bound into pairs, which themselves form fractional quan- the form Ψ† ∼ ψ †η, similar to the operators in Eqs. (8) and tum Hall states of bosonic charge-2e particles. The simple (9), where η is an operator that produces a phase slip of bosonic Laughlin states occur at electron filling factors 2π φ strength in the superconducting phase . For wire with ν ¼ 4ν2e ¼ 4=m, where m is an even integer and ¼ 2 þ 1 η Nch n , the operator will involve excitation of n ν ¼ ρeh=ðeBÞ¼4ρ2eh=ð2eBÞ.Forν ¼ 1=2, the strong- particles and n holes in 2n different channels. Thus, the paired state is equivalent to charge-2e bosons at filling † operator Ψ will appear only at order n in perturbation ν2e ¼ 1=8. Strong-paired phases can also occur at other theory, and it should therefore be small, proportional to the filling factors, such as ν ¼ 1 (ν2e ¼ 1=4). nth power of the ratio between the interaction strength and Here, we show how these states can be formulated within the Fermi energy, for weak interactions. (In Ref. [20], these the coupled wire model. We begin by reviewing the higher-order terms were overlooked, and it was incorrectly coupled wire model and then show how it is modified suggested that there should be a hard gap in the spectral by incorporating pairing into the Luttinger liquids. We then weight for a wire with three or more channels.) demonstrate two applications of this technique. First, we In contrast to the weak-paired phase, the strong-paired show that the transition between the conventional ν ¼ 1 phase has a gap for single-particle excitations. The coef- state and the strong-paired ν ¼ 1 state is in the 3D Ising ficient D in Eq. (38) is zero, and AðεÞ¼0 for ε < Δ. universality class. Second, we provide an explicit con- However, since two-particle tunneling is allowed in the struction of the Moore-Read quantum Hall state, as well as strong-paired phase, there will remain a nonzero tunneling related weak-paired states at ν ¼ 1=2, and a new, intrinsi- conductance G, due to a process analogous to Andreev cally anisotropic, quantum Hall state at ν ¼ 1=2. reflection, which is suppressed by both a higher power of energy and a higher power in the bare electron tunneling A. Coupled wire model ∼ matrix element. While in the weak-paired phase G The coupled wire model provides an explicit formulation 2 2 2þ −1 2−1 t AðεÞ ∼ t εKρ= Kρ = (where ε is the larger of temper- of fractional quantum Hall states that takes advantage of the ature or voltage), we find that in the strong-paired power of Abelian bosonization for describing strongly −1 phase G ∼ t4ε2ðKρþKρ Þþ2. interacting quantum systems. The power of studying the

031009-10 PAIRING IN LUTTINGER LIQUIDS AND QUANTUM HALL … PHYS. REV. X 7, 031009 (2017) anisotropic limit of quantum Hall states has long been Θρ;jþ1=2 jumps by 2π corresponds to a charge-e=m known in the theory of the integer quantum Hall effect [34]. Laughlin quasiparticle. A coupled wire construction for Abelian fractional quan- To describe a strongly paired quantum Hall state, it is tum Hall states was introduced in Ref. [3] and later tempting to consider the momentum-conserving tunneling generalized in several directions [4,5,7–12]. of pairs of electrons between the wires, described by We begin by considering an array of coupled wires that X are each single-channel Luttinger liquids described by H ¼ Hρ þ t2 cos 2Θρ;jþ1=2: ð44Þ conventional bosonization. This is equivalent to consider- j ing Eqs. (20) and (22) for n odd and setting σ to a Θ π constant. In this case, there is only a single fermion This term pins jþ1=2 at a multiple of . A kink in which Θ π ð2 Þ operator, so we will omit the subscript on Ψj in ρ;j jumps by has a charge e= m that is expected for a Eq. (20). It is convenient to choose a periodic boundary Laughlin quasiparticle of charge-2e bosons at filling condition of circumference L for the 2D system in which 1=ð4mÞ. wire j at x ¼ L connects to wire j þ 1 at x ¼ 0. Thus, the However, having t2 flow to strong coupling, with t1 ¼ 0, 2D system consists of a single wire that is “wrapped on a does not describe the strong-paired phase because it is spool” with ΨjðxÞ ≡ Ψðx þ jLÞ. In this model, the coupled unstable to the perturbation t1, which is still a local wires resemble the grooves on a phonograph record, with a operator. When t1 ≠ 0, the π kink no longer connects single wire that begins on the outside and ends on the degenerate ground states, so the charge-e=ð2mÞ quasipar- inside. In this case, since there is only a single wire, the ticles are confined. This phase is just the conventional anticommutation between fermion operators is automati- ν ¼ 1=m Laughlin state. cally built in, so additional Klein factors are not necessary. To describe the paired quantum Hall states, we need to Suppose there is a magnetic flux per unit length augment the single- and pair-tunneling terms with intrawire b ¼ ρ¯ ϕ0=ν between the wires [here, ρ ≡ 2k =ð2πÞ is pairing. We employ our generalized bosonization approach, e e F σ the total electron density per wire, and ϕ0 ¼ h=e is the flux Eqs. (20) and (22), which includes the sector and quantum]. Tunneling an electron between neighboring incorporates strong pairing into the 1D wires. We thus wires is associated with a phase 2πbx=ϕ0 ¼ 2kFx=ν. consider The allowed momentum-conserving tunneling terms are X H ¼ H þ H þ þ ð Þ determined by comparing the magnetic field phase to the ρ σ V1j V2j; 45 phase due to the momentum in the electron operators. j Consider electron tunneling operators of the form where Hρ and Hσ are the single-wire Hamiltonians (15) † Ψ Ψ 2πibx=ϕ0 þ ð Þ and (16), and the tunneling terms have the form j;m jþ1;−me H:c: 41

V2 ¼ t2 cos 2Θρ þ1; ð46Þ for odd integer m, where j and j þ 1 enumerate the wires. j ;j 2 When the filling factor is ν ¼ 1=m, the oscillating factor 8 P 2π ϕ 2 > a b ibx= 0 imkFx Ψ’ > t1 σ σ cos Θ 1 m odd e is exactly canceled by the phase e in the s. < ;ab j jþ1 ρ;jþ2 a;b Expressed in bosonized form, the tunneling term then has V1 ¼ P ð47Þ j > a b the form > t1 γ γ cos Θ 1 m even: : ;ab j jþ1 ρ;jþ2 X a;b H ¼ Hρ þ t1 cos Θρ;jþ1=2; ð42Þ j Here, we have chosen to express the four allowed tunneling terms involving Ψ;j;m and Ψ;jþ1;m in Eq. (20) in terms of where the Hermitian operators σx;y or γx;y. We again assume that pair tunneling between the wires, described by Eq. (44),is Θjþ1=2 ≡ φρ;j − φρ;jþ1 þ mðθρ;j þ θρ;jþ1Þ: ð43Þ relevant and leads to the pinning of Θρ;jþ1=2 at multiples of π. The choice between the two lines of Eq. (47) is dictated Here, Hρ is a sum of terms of the form (15), as well as by the need to cancel the magnetic field phase e2πibx=ϕ0 . forward scattering interactions that couple the wires. In the When m is odd, in addition to the conventional ν ¼ 1=m spirit of the coupled wire model, we assume that the Laughlin state, we describe the strong-paired Laughlin state forward scattering interactions are such that t1 is relevant as well as the critical behavior of the transition between the under the renormalization group and that it flows to strong weak and strong-paired phases. When m is even, strong coupling. It then follows that the set of mutually commut- pairing on the wires leads to a strong-paired ν ¼ 1=m ing variables Θρ;jþ1=2, which are defined on each link Laughlin state, while weak pairing on the wires leads to a between wires, are pinned at an integer multiple of 2π, novel anisotropic quantum Hall state. When the wires are resulting in a gapped quantum Hall state. A kink in which near the transition between the weak and strong-paired

031009-11 KANE, STERN, and HALPERIN PHYS. REV. X 7, 031009 (2017) phases, we find that coupling the wires leads to states we is a single-particle energy gap—even at the edge. However, can identify with the Moore-Read state as well as gener- the charge sector has a gapless chiral Luttinger liquid edge alizations of it. For both even and odd m, the V2j terms gap mode, which allows the low-energy tunneling of pairs of the charge modes at the bulk and hence suppress the effect electrons. Bulk quasiparticles correspond to kinks where H Θ of the intrawire interaction Hamiltonian int in Eq. (17). ρ;jþ1=2 jumps from one minima to another in Eqs. (46) and Thus, the modifications to the critical behavior that we (47). In the high-temperature phase, the single-particle found for a single wire in Sec. III C do not occur in arrays of tunneling term in Eq. (47) is suppressed by the disordered coupled wires. As we see below, the phase transitions in the Ising spins, so π kinks in Θρ;jþ1=2, with charge-e=ð2mÞ,have two-dimensional arrays of wires are continuous second- a finite energy. It follows that the charge-e=ð2mÞ quasipar- order transitions and involve a closure of the gap to neutral ticle is deconfined, as expected in the strong-paired phase. In excitations. addition, in the “spool model”, the channel in which Θρ;jþ1=2 We consider the cases of odd and even m separately in is defined begins on the outside edge and ends on the inside the following two sections. edge like the groove in a phonograph record. A charge- e=ð2mÞ quasiparticle can thus be transported from the outer B. Pairing transition for ν = 1 (or m odd) to the inner edge. It follows that the inner and outer edges ð2 Þ In this section, we study the strong-paired ν ¼ 1=m state support charge-e= m quasiparticles. for m odd, as well as the critical behavior of the transition to In the low-temperature phase of the Ising model, hσxi ≠ 0 the conventional ν ¼ 1=m Laughlin state. The simplest . There remains an energy gap in the bulk, but case is ν ¼ 1. To this end, we consider the strong coupling now the single-particle propagator is gapless at the edge. ν ¼ 1=m limit t2 → ∞. In the ground-state sector in which no This is the ordinary Laughlin state. In this case, a π ð2 Þ quasiparticles are present, we may set Θρ þ1 2 ¼ 0.We pair of kinks with charge e= m are confined because ;j = Θ first introduce a mapping to a 2 þ 1-dimensional trans- between them ρ;jþ1=2 is not at the minimum of Eq. (47). verse-field Ising model, which allows us to show that the There is thus an energy cost proportional to their separation transition is in the 3D Ising universality class. We then leading to confinement. Similarly, the edges do not support ð2 Þ explore the scaling behavior near the transition, which can charge-e= m quasiparticles. be probed by transport and thermodynamic measurements. The other tunneling terms t1;xy and t1;yy will not modify the phases or critical behavior—at least when they are weak. At the 3D Ising critical point, there is a single 1. 3D Ising transition relevant operator, and all remaining operators are irrelevant. When m is odd, the Hamiltonian in the neutral sector Therefore, adding these terms as perturbations can only involves Eq. (16) on each wire, as well as the tunneling shift the location of the transition but not modify it. terms due to t1;ab [the top line of Eq. (47)]. Since the tunneling term is best described by the Ising variables, it is 2. Critical behavior convenient to also employ the Ising lattice-regularized description for the neutral sector of the wires’ At the transition between the strong-paired and conven- Hamiltonians, as seen in Eq. (19) and reviewed in tional ν ¼ 1=m states, the charge gap remains finite, so this Appendix A. This then leads to an anisotropic 2 þ 1D transition will not exhibit scaling in the longitudinal dc lattice model of the form conductance that occurs in conventional quantum Hall X transitions. To access the critical behavior, it is necessary ¼ σz þ σx σx þ σa σb ð Þ to probe the neutral degrees of freedom that acquire long- H h i;j J i;j i;jþ1 t1;ab i;j iþ1;j; 48 i;j ranged correlations at the critical point. In this section, we discuss two quantities: the heat capacity and the edge σa where i;j describes the spin on the jth site of the ith chain. tunneling conductance. We argue that both quantities If we consider the anisotropic limit t1;ab → 0 and bosonize exhibit scaling behavior that is sensitive to the bulk 3D the weakly coupled chains using the “spool” boundary Ising transition. conditions discussed at the beginning of this section, then The scaling of the heat capacity CðT;δÞ with temperature we arrive at a model with precisely the same Jordan-Wigner T and distance to the critical point δ can be deduced from strings as the fermion model. dimensional analysis. With an appropriate linear rescaling For simplicity, consider first the case where t1;xx is the of space and time, the critical point exhibits a Lorentz only nonzero term. Then, we have precisely an anisotropic invariance. The only length scale is the correlation length, −ν 2 þ 1D transverse-field Ising model. This model has two which diverges as ξ ∼ δ , where ν ¼ 0.630 is the corre- phases that are separated by a critical point in the 3D Ising lation length exponent of the 3D Ising model. The only ν model universality class. energy scale is Δ ∼ v=ξ ∼ δ . Since the heat capacity per The “high-temperature phase” of the Ising model, unit area has units L−2, precisely at the transition, we must hσai¼0, is the strong-paired phase. In this phase, there have CðT;δ ¼ 0Þ ∝ T2, which is characteristic of any 2D

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G(T, ) G(T, ) Lorentz invariant system. More generally, it is expected to 1.5 >0 .20 exhibit scaling behavior, 1.0 .10 .05 2 1=ν <0 CðT;δÞ¼T fðδ=T Þ: ð49Þ 0.5 T>0 .02 = 0 T = 0 T Since for fixed δ ≠ 0 there is an energy gap Δ, we can –2. –1. 1. 2. .01 .02 .05 .10 .20 deduce the asymptotic behavior FIG. 1. Critical behavior of the edge tunneling conductance G ν δ fðX → ∞Þ ∝ e−jXj : ð50Þ as a function of temperature T and tuning parameter near the 3D Ising critical point. Panel (a) shows G as a function of δ for several values of T, highlighting the zero-temperature behavior in ð δÞ 2β At a fixed low temperature, we therefore expect C T; to which Gð0; δÞ ∼ θðδÞδ . Panel (b) shows G as a function of T for exhibit a peak as a function of δ near δ ¼ 0. The peak is several values of δ, highlighting the behavior at the critical point predicted to sharpen as the temperature is lowered, and data GðT;0Þ ∼ Tmþη. from different temperatures should collapse to a single curve in a scaling plot. ð → ∞Þ¼ 2β ¼ νð1þηÞ ð → −∞Þ∼ Tunneling into the edge is another probe of the pairing where h X X X and h X −cXν transition. On the weakly paired side, the edge tunneling is e . ¼ 1 described by the conventional Luttinger liquid theory, In Fig. 1, we show the predicted behavior for m ð Þ which, for filling 1=m, predicts a low-temperature tunnel- using a simple approximate scaling function h X that ¼∞ ing conductance due to tunneling of single electrons that interpolates between the known limits at X . 2 m−1 Figure 1(a) shows G as a function of δ for several values scales as G1ðTÞ ∼ t T , where t is the electron tunneling δ2β matrix element. On the strongly paired side, there is a gap of T.AtlowT, it shows a sharp transition and grows as δ 0 for tunneling single electrons. The dominant contribution for > . At higher temperatures, the transition is rounded. comes from tunneling pairs of electrons. This leads to a Figure 1(b) shows a log-log plot of G as a function of T for 4 4mþ2 several values of δ. For δ < 0, the low-temperature con- tunnel conductance G2ðTÞ ∼ t T . In the critical region, m−1 the single-electron tunneling G1ðT;δÞ will exhibit critical ductance approaches a constant (or, more generally, T ), δ 0 behavior, while the two-particle contribution will behave while for > , it goes to zero exponentially. Precisely at δ ¼ 0 ð Þ ∝ mþη smoothly. Here, we focus on G1ðT;δÞ, which dominates in the transition , G T T . the large barrier limit t → 0. In the above analysis, we assumed that the critical The single-electron tunneling density of states follows behavior for correlation function (52) at the Ising transition from the single-particle Green’s function, GðτÞ¼ is the same on the boundary of the system as it is in the bulk. † † þ iϕρ iϕρ The boundary exponents can, in general, be different. This hTτ½ΨðτÞΨ ð0Þi, where Ψ ∝ σ e . Here, e adds a charge e in the charge sector, while in the neutral sector, it will modify the exponents and scaling functions, but the involves the Ising spins. The charge-sector operators general form of the scaling behavior will remain the same. exhibit the usual Luttinger liquid behavior, C. Paired phases for ν =1=2 (or m even) h iϕρðτÞ −iϕρð0Þi ∼ 1 τm ð Þ e e = ; 51 We now consider the case of even m, for which the simplest example is ν ¼ 1=2. In this case, there is no when the filling is 1=m. On the ordered side of the Laughlin state for fermions, but we expect to describe a hσ i ∝ δβ transition, we expect x , while on the disordered strong-paired state, as well as a non-Abelian Moore-Read hσ i¼0 side, x . In the critical region, the correlation state. Our starting point is to have each wire at the pairing function exhibits scaling, transition so that there exist gapless charge and Majorana modes on each wire. We then consider perturbations that 1 hσ ðτÞσ ð0Þi ¼ gðδτ1=νÞ; ð52Þ couple the wires and lead to paired quantum Hall states. x x τ1þη When m is even, the single-electron tunneling term is shown in the second line of Eq. (47), and we can work with 2β ν where gðX → 0Þ ∼ X and gðX → ∞Þ ∼ X and the expo- the 1D continuum fermion theory description of the wires’ nents β ¼ 0.326 and η ¼ 0.036 are related by neutral sectors. We again consider the strong coupling limit 2β ¼ νð1 þ ηÞ . This in turn leads to scaling behavior in t2 → ∞, and in the absence of quasiparticles, we set the tunneling conductance. We find that near the transition, Θρ;jþ1=2 ¼ 0. The Hamiltonian for the neutral sector then the single-particle tunneling conductance has the form becomes H ¼ H0 þ Ht, X δ 2 mþη H ¼ γT½− τz∂ þðϵ − ∂ 2 Þτyγ ð Þ ð δÞ¼ ð Þ 0 i iu x 0 x= m i 54 G1 T; T h 1=ν ; 53 T i

031009-13 KANE, STERN, and HALPERIN PHYS. REV. X 7, 031009 (2017) and (a) Strong Paired: = 1/2, c = 1 (d) Anti-Pfaffian: = 1/2, c = -1/2 X H ¼ γaT γb ð Þ t i i t1;ab iþ1: 55 i

0 Fourier transforming, we may write t2

¼ γTð− Þhð Þγð ÞðÞ (b) Pfaffian: = 1/2, c = 3/2 H k k k 56 (3) t1,RL t2 with the 2 × 2 matrix

z hðkÞ¼T1 sinkyI þðukx þ Δ1 sinkyÞτ (1) 2 y x t 1,LR t2 þðϵ0 þ kx=2m þ T2 coskyÞτ þ Δ2 sinkyτ ; ð57Þ (e) C-SP: = 1/2, c = 0 where T1 ¼ t1;11 þ t1;22, T2 ¼ t1;12 − t1;21, Δ1 ¼ t1;11− t1;22, and Δ2 ¼ t1;12 þ t1;21. Here, T ¼ T1 þ iT2 can be ψ †ψ (c) PH-Pfaffian: = 1/2, c = 1/2 interpreted as the complex tunneling amplitude T j jþ1, while Δ ¼ Δ1 þ iΔ2 can be interpreted as the complex pairing term with py symmetry, Δψ jψ jþ1. y The T1 term violates C2 symmetry, and Δ1 is a p pairing term with the same phase as the px pairing term on the (2) t (1) t 1,RL 2 wires. To get a gapped phase in the bulk, the important t 1,RL t2 y pairing term is Δ2, which gives ip pairing. We therefore consider the simplified model

¼ γT½ τz þðϵ þ 2 2 þ Þτy þ Δ τxγ H ukx 0 kx= m T2 cosky 2 sinky : FIG. 3. Schematic diagrams for quantum Hall states at ν ¼ 1=2. ð58Þ Each shaded region depicts a wire at the pairing transition with right- and left-moving Majorana modes (dashed lines) as well as right- and left-moving charge modes (solid lines) describing As a function of ϵ0 and Δ2, this model exhibits several R=L ϕ ¼ φρ mθρ with m ¼ 2. The charge modes are coupled phases indicated in Fig. 2 and discussed below. ρ;i ;i ;i by t2 in Eq. (46), leaving a single unpaired chiral charge mode on each edge. The Majorana modes are coupled in different ways as 1. Strong-paired phase described in the text, resulting in topologically distinct states with ϵ different numbers of chiral Majorana edge modes. In panel (d), When 0 is large and positive, each wire is in a strong- C-SP refers to the particle-hole conjugate of the strong-paired state. paired phase, so the resulting phase is simply a strong- paired quantum Hall state of charge-2e bosons at filling 1=ð4mÞ. This phase describes the region of the phase As was the case when m was odd, it is a quantum Hall diagram with ϵ0 > jT2j. state with a bulk energy gap and a gapless chiral charge mode on the boundary. There is a single-particle energy gap

2 everywhere, including the edge. This phase is pictorially represented in Fig. 3(a) for the case where t1;ab ¼ 0 and ϵ0 > 0 Pfaffian , where the Majorana modes couple within each wire to open a gap in the neutral sector. px+ipy Anisotropic Strong Paired 0 2. Pfaffian and PH-Pfaffian phases -|T2| |T2| jϵ0j < jT2j PH-Pfaffian For , the neutral sector fermions exhibit a 2D topological phase with a nonzero Chern number px-ipy N ¼ sgnΔ2. The neutral sector has the structure of a weakly paired px ipy superconductor, and there exists a chiral Majorana edge mode, in addition to the charge mode at the edge (which is identical to that in the strong-paired phase). FIG. 2. Phase diagram for m odd as a function of parameters ϵ0 Interestingly, there are two distinct phases depending on and Δ2 in Eq. (58). the sign of Δ2, which correspond to interactions that

031009-14 PAIRING IN LUTTINGER LIQUIDS AND QUANTUM HALL … PHYS. REV. X 7, 031009 (2017) produce either a px þ ipy phase or a px − ipy phase in the In principle, there is a hierarchy of different states with neutral sector. The px þ ipy state has the same topological all possible values of N. States with even N will be Abelian order as the Moore-Read Pfaffian state, which has cop- states. For instance, the state with N ¼ −2, depicted in ropagating charge and neutral modes at the edge, leading to Fig. 3(d), with c ¼ 0, is topologically equivalent to the a chiral central charge c ¼ 1 þ 1=2 ¼ 3=2. In contrast, for particle-hole conjugate of the strong-paired state. When N the px − ipy state, the neutral Majorana mode propagates in is even, quasiparticles will bind an even number of ðp¼jNjÞ the opposite direction from the charge mode, leading to a Majorana zero modes. In principle, a kink where tab chiral central charge c ¼ 1 − 1=2 ¼ 1=2. This state has the changes sign can bind an unpaired Majorana mode. same topological order as the variant of the Moore-Read However, unless there is an extra symmetry that forbids ð ¼ Þ state, dubbed the PH-Pfaffian [18], which has recently been odd-neighbor coupling, t p odd , the unpaired Majorana discussed in connection with the particle-hole symmetric ab modes will, in general, be confined. Finally, we note that half-filled Landau level. Though our coupled wire model particle-hole conjugate states can also have an alternative does not have particle-hole symmetry, the topological order simpler construction by considering “antiwires” inside a of this state is compatible with particle-hole symmetry. ν ¼ 1 quantum Hall state or, equivalently, “shifted wires” These states can most easily be pictured in the limit made from the right-moving electron modes of wire i ϵ0 ¼ 0, and only t1 (or t1 ) is nonzero, as shown in ;RL ;LR paired with the left mover on wire i þ 1. However, we will Figs. 3(b) and 3(c). In this case, the chiral Majorana modes not pursue that direction here. on neighboring wires couple and open a gap, leaving a single chiral Majorana mode at the edge. 4. Anisotropic phase

3. Anti-Pfaffian phase and generalizations When ϵ0 is large and negative, we have an anisotropic Though it is outside of the scope of the simple model in quantum Hall state that, to our knowledge, has not been Eqs. (54)–(57), the coupled wire model also allows the discussed before. Both the charged and neutral sectors are construction of additional paired quantum Hall states if gapped in the bulk. However, the neutral sector has a further neighbor interactions are included. Here, we keep topological structure similar to that of the decoupled limit the pair tunneling in the charge sector to be given by t2 in tab → 0. The neutral sector of each wire has the structure of Eq. (46), and we consider the effect of further neighbor a 1D topological superconductor, and for ϵ0 < −jT2j, these hopping of single electrons. 1D topological superconductors are coupled together to A momentum-conserving electron tunneling term that form a “weak topological superconductor”. connects wire j to wire j þ p can be written as This phase does not have unpaired Majorana modes bound to the charge-e=2m quasiparticles. However, at the X X ðpÞ ðpÞ a b ends of the wires in the decoupled limit, each wire has a V ¼ t γ γ cos Θρ þ −1 2 : ð59Þ 1j 1;ab j jþp ;j l = Majorana zero mode at the end. When the wires are coupled a;b l¼1;p together, these Majorana modes broaden to form a band. This term involves both tunneling of an electron from wire Provided the lattice of wires has the symmetry under j to wire j þ p and backscattering of electrons on the wires translation by one lattice constant, this band of Majorana in between. When Θρ þ1 2 is pinned by Eq. (46), this leads modes is necessarily gapless. ;j = γ to a Hamiltonian for the Majorana fermions γa that couples If we label the Majorana mode on each wire as j, then j the low-energy Hamiltonian will be pth neighbors. When expressed in momentum space, as in Eq. (56), hðkÞ can be characterized by a more general X ¼ γ γ ð Þ Chern number N, such that jNj ≤ p. This leads to phases H it j jþ1: 60 with N chiral Majorana modes, where N>0 (N<0) j indicates they are right (left) movers. Taking into account the charge mode, this leads to a quantum Hall state with This has precisely the structure of Kitaev’s 1D Majorana chiral central charge c ¼ 1 þ N=2. The cases N ¼ 0 chain. The discrete translation symmetry by one lattice (strong paired), N ¼ 1 (Pfaffian), and N ¼ −1 (PH- constant guarantees that it is precisely at criticality. The Pfaffian) were discussed above. Majorana modes will have dispersion The anti-Pfaffian state [35,36] fits into this more general construction. This state is the particle-hole conjugate of the Ek ¼ t sin k; ð61Þ Pfaffian state, with ν ¼ 1=2 and c ¼ 1–3=2 ¼ −1=2. This corresponds to Chern number N ¼ −3, which is depicted in ¼ 0 ð3Þ which exhibits a pair of helical Majorana modes at k Fig. 3(e) when t1;RL couples right- and left-moving and k ¼ π. Furthermore, defects in the wires’ array, e.g., Majorana modes on third neighbors, leaving three upstream dislocations, would bind static non-Abelian defects in the Majorana modes at the edge. form of isolated Majorana modes.

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V. DISCUSSION AND CONCLUSION Council under the European Unions Seventh Framework Program (FP7/2007–2013)/ERC Project MUNATOP, the This paper has developed a framework for incorporating DFG (CRC/Transregio 183, EI 519/7-1), Minerva the physics of pairing into the traditional Luttinger liquid Foundation (A. S.), and a Simons Investigator grant from formulation of interacting fermions in one dimension. We the Simons Foundation (C. L. K.). argued that for sufficiently strong attractive interactions, even a single-channel Luttinger liquid composed of spin- less fermions can exhibit a strong-paired phase that is APPENDIX: MAPPING TO ISING MODEL qualitatively distinct from a weakly paired Luttinger liquid. Here, we review the fermionization of the 1 þ 1D It is distinguished by the existence of a single-particle transverse-field Ising model, which demonstrates the iden- energy gap, despite the existence of gapless two-particle tification of the spin operator above. excitations and a gapless collective charge mode. On a 1D lattice, the transverse-field Ising model has These two phases of a Luttinger liquid are in one-to-one Hamiltonian correspondence with the topological and trivial super- X conducting phases of a one-dimensional superconductor. H ¼ σz − σxσx ð Þ I h j J j jþ1: A1 The fluctuating phase of the one-dimensional supercon- j ductor describes a Luttinger liquid, and the topological and trivial phases of the superconductor are indistinguishable This can be fermionized by introducing lattice fermion from the strong and weak-paired phases of the Luttinger operators liquid. We have shown that the hallmarks of 1D topological Y — † ¼ σþ σz ð Þ superconductors associated with Majorana end modes cj j i ; A2 including the zero-energy peak in the tunneling density of iquantum Hall effect using the coupled I 0 x= m iu x H:c: ; A5 wire model. In addition to allowing the formulation of a ϵ ¼ 2ð − Þ 1 2 ¼ ¼ 2 new (and much simpler) coupled wire model for the with 0 h J , = m J, and u J. This result has Moore-Read state, it allows us to describe a number of precisely the form of Hf. Introducing Majorana operators ψ ¼ γ þ γ additional phases and critical points at filling ν ¼ 1=m, 1 i 2, we may write this as including the particle-hole symmetric PH-Pfaffian phase, ¼ γT½− τz∂ þðϵ − ∂2 2 Þτyγ ð Þ an intrinsically anisotropic phase when m is even, and a HI iu x 0 x= m : A6 strongly paired state when m is odd. By undoing the Jordan-Wigner transformation, we can express the spin operators in terms of fermions, ACKNOWLEDGMENTS P † þ † iπ c c σ ¼ i>j m i ð Þ We thank Ehud Altman for helpful discussions and for j cj e : A7 pointing us to Ref. [28]. We also thank Nick Read and Chetan Nayak for helpful discussions. This work was In the continuum, we thus obtain supported in part by grants from the Microsoft R ∞ þ † iπ dx0ψ †ψ Corporation and the US-Israel Binational Science σ ðxÞ¼ψ ðxÞe x : ðA8Þ Foundation (B. I. H. and A. S.), the European Research

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