An Effective Field Theory for Fractional Quantum Hall Systems near $\nu=5/2$

We propose an effective field theory (EFT) of fractional quantum Hall systems near the filling fraction $\nu=5/2$ that flows to pertinent IR candidate phases, including non-abelian Pfaffian, anti-Pfaffian, and particle-hole Pfaffian states (Pf, APf, and PHPf). Our EFT has a 2+1$d$ O(2)$_{2,L}$ Chern-Simons gauge theory coupled to four Majorana fermions by a discrete charge conjugation gauge field, with Gross-Neveu-Yukawa-Higgs terms. Including deformations via a Higgs condensate and a fermion mass term, we can map out a phase diagram with tunable parameters, reproducing the prediction of the recently-proposed percolation picture and its phase transitions. Our EFT captures known features of both gapless and gapped sectors of time-reversal-breaking domain walls between Pf and APf phases. Moreover, we find that Pf$\mid$APf domain walls have higher tension than domain walls in the PHPf phase. Then the former, if formed, may transition to the energetically-favored PHPf domain walls; this could, in turn, help further induce a bulk transition to PHPf.


Introduction
One of the first non-abelian topologically ordered candidate states was observed experimentally in 1987 [1]. It is the filling fraction ν = 5/2 fractional quantum Hall (fQH) state of an interacting electron gas in 2+1 spacetime dimensions (denoted as 2+1d). It has a fractional quantized Hall conductance σ xy = 5/2 in units of e 2 /h where e is the electron charge and h is the Planck constant. There have been many proposed candidate states to describe the underlying topological orders of this system: the major non-abelian candidates include Moore-Read's Pfaffian state [2] (see also [3]), its particle-hole conjugate known as the anti-Pfaffian state [4,5], and a particle-hole symmetric state known as the particle-hole Pfaffian state [6]. The particle-hole Pfaffian state [6] was originally proposed to be a particle-hole symmetric version of a composite fermion theory for the half-filled Landau level system [7].
In 2017, a remarkable experimental measurement by Banerjee et al [8] suggested that the thermal Hall conductance of the ν = 5/2 fQH state is κ xy = 5/2 in units of π 2 k 2 B T /3h, where k B is the Boltzmann constant and T is the temperature. 1 In this work, we propose a unified bulk effective field theory (EFT) that give rise to various topological field theories (TQFTs) and their edge modes pertinent to the ν = 5/2 fQH system. We map the EFT parameters to experimental quantities to produce a phase 1 The edge modes of the quantum Hall system can be understood via the bulk-boundary correspondence of 2 + 1d Chern-Simons theory. In fact, the thermal Hall conductance κ xy = (c L − c R ) π 2 k 2 B 3h T is proportional to the chiral central charge c − ≡ c L − c R , which is the difference between the left/right central charges c L and c R . It counts the degrees of freedom of chiral modes of the (1 + 1)d edge conformal field theory (CFT) living on the boundary of a bulk-gapped 2 + 1d topological state [9]. For non-abelian fQH states, the half-integer κ xy is attributable to an odd number of (1+1)d chiral real Majorana-Weyl fermions on the boundary [10], in addition to (1 + 1)d chiral bosons or chiral complex fermions. diagram in terms of the filling fraction (or the magnetic field) vs. the disorder strength. In the following, we first recall pertinent proposals from the literature.
Ref. [23] argued that the numerical simulations are simplified systems lacking both disorder (say, induced by impurities of experimental samples) and Landau-level mixing (LLM), which occur in real laboratory experiments. Ref. [23] further suggested that the particle-hole Pfaffian may be stabilized by disorder, i.e. LLM and impurities that break particle-hole symmetry. However, Ref. [23] did not provide analytic details on how disorder can help realize this possibility in practice.
We recall that: 1. Neither the Pf nor the APf state has particle-hole (PH) symmetry [27]. Both Pf and APf have their lower Landau levels fully occupied with spin-polarized electrons (which contribute σ xy = 2). However, in the absence of LLM, if we assume that spin-polarized electrons in the highest, half-filled Landau level (so there is another contribution of σ xy = 1/2 and ν = 5/2 in total) interact only through two-body interactions, then exact PH symmetry is present in the idealized Hamiltonian. 4 With 2 The particle-hole Pfaffian is analogous to the T -Pfaffian or CT -Pfaffian that occur on the surface of 3 + 1d topological superconductors, see [11,12]. 3 For the sake of brevity, below we abbreviate Pfaffian state as Pf, anti-Pfaffian state as APf, and particle-hole Pfaffian as PHPf. See Appendix A of Ref. [26] for the systematic list of data of the pertinent ν = 5/2-quantum Hall liquids in terms of 2 + 1d bulk topological quantum field theories (TQFTs) and (1 + 1)d edge theories. 4 In the literature, there are two conventions for naming the Landau levels. One convention is to call the lowest level the zeroth Landau level (which here is fully occupied, with spin-up and spin-down polarized electrons contributing σ xy = 2), and call the next the first Landau level (which here is half-filled with The question about the nature of the percolating phase becomes the question of understanding whether the domain wall degrees of freedom are localized in the bulk or delocalized through the whole bulk-boundary system (see the picture illustration in Fig. 3 in [26]).
Ref. [24,25] modeled the ν = 5/2 system in terms of a checkerboard network (of alternating Pf and APf in each chequered pattern) known as a Chalker-Coddington network model [29] (previously used in modeling the integer quantum Hall plateau transition). Ref. [26] performed perturbative and non-perturbative analyses of the (1 + 1)d edge theory on the domain wall between Pf and APf states at different disorder energy scales, with particular focus on the emergent symmetries

Comparison of three related proposals on disordered percolating systems
We compare the results of Ref. [24][25][26], which we also summarize pictorially in Fig. 1 and Fig. 2 below: 7 (1): Ref. [24] proposed that a single first-order-like transition between Pf and APf occurs at ν c and at zero disorder, due to an O(4) symmetry rotating four gapless chiral Majorana modes. The presence of these Majoranas induce a jump ∆κ xy = ∆c − = 2.
In the presence of any nonzero disorder, which weakly perturbs the first-order critical Wen's long-range entangled topological order (beyond Landau). Disorder here is mainly used to mean quenched disorder caused by impurities or a spatially non-uniform chemical potential, inducing puddles of Pf or APf near ν c .
• Percolation: When we say that a phase percolates, we mean that the phase can extend through the whole bulk-boundary system (e.g., see Fig. 3 (a) and (c) of [26]). When we instead say the domain walls percolate, we mean that Pf|APf domain walls can extend through the whole bulk-boundary system (e.g., see Fig. 3 (b) of [26]).
• Localized vs. delocalized: When we say the neutral Majorana modes are delocalized, we mean that the Majorana modes can diffuse freely on the network of domain walls. The delocalization happens at the percolation transition (approximately near a percolation critical point). When the neutral Majorana modes are delocalized, the thermal Hall κ xy is unquantized, thus either causing a percolation transition or a thermal metal phase. When neutral Majorana modes are localized (on the domain walls), we have a quantized κ xy .
"Percolation" is used to indicate when a spatial subregion (e.g. Pf, APf, or domain walls) spreads in the spatial sample, whereas "(de)localization" is used to indicate when zero energy modes or energetic modes in the energy spectrum are de/localized in the spatial sample. 7 There is an alternative interpretation from [30][31][32][33] favoring the anti-Pfaffian state (see also the criticism [34] of this interpretation), which proposes that partial-or non-thermal equilibrium of anti-Pfaffian edge modes can explain the κ xy = 5/2 measurement [8], even though the anti-Pfaffian bulk state has κ xy = 3/2 at equilibrium. We shall not discuss this scenario [30][31][32][33], since we wish to obtain an effective bulk field theory motivated by the scenario of [26]. Regions with unquantized κ xy Λ > Λ TH Λ 2 < Λ < Λ TH Λ 1 < Λ < Λ 2 Λ = 0 ν c 5/2 Figure 1: Thermal Hall conductance κ xy vs filling fraction ν for the scenario proposed in Ref. [26]; see also Fig. 3 for a phase diagram. At different disorder energy scales Λ, we plot several curves of (ν, κ xy ). At Λ = 0, the κ xy (drawn as a dotted line) jumps at ν c under a first order phase transition. From 0 < Λ < Λ 1 , the jump can become smoother due to disorder. In the regime Λ 1 < Λ < Λ 2 , drawn as a dashed line, an intermediate κ xy = 5/2 plateau phase appears. Finally, when Λ 2 < Λ < Λ TH , there are multiple plateau phases at κ xy = 3, 5/2, and 2. Notice that when Λ > 0, all transitions between different quantized κ xy can have broadening, where the jumps at transitions become smoother slopes. On the top panel, we show different line intervals which represent the extent of broadening over ranges of ν demarcated on the horizontal axis, for the values stated of Λ on the top right corner. When Λ > Λ TH , the slope is smooth enough to become a thermal metal so there is no quantized κ xy between 7/2 and 3/2. See Remark (3).
(2): Ref. [25] suggested that for a finite range of ν ν c , the Pf|APf domain walls percolate. 8 If the charge neutral Majorana edge modes can diffuse freely in the network of Ref. [24] (left) and scenario (II) from Ref. [25] (right). We use the same legend for drawing curves at different scales Λ as in Figure 1. At Λ = 0, κ xy (drawn as a dotted line) jumps at ν c under a first order phase transition. For Λ > 0, scenarios (I) and (II) differ. Scenario (I)'s κ xy has four jumps at the plateau for any 0 < Λ < Λ TH , and κ xy becomes smooth with non-quantized values for Λ > Λ TH .
domain walls in the bulk-boundary system, Ref. [25] proposed a thermal metal phase with an unquantized thermal Hall κ xy but a divergent κ xx (and, as usual, a quantized Hall conductance σ xy = 5/2, and σ xx = 0 at zero temperature). If the neutral Majorana modes are localized, Ref. [25] proposed a quantized σ xy = 5/2 phase with a quantized thermal Hall conductance κ xy = 5/2. Ref. [25] suggested that between the Pf and APf phases, there is a possible wide range of thermal metal behavior, even at low disorder. By tuning ν, in the absence of disorder, there is a first-order-like transition between Pf → APf. At low disorder, there is a sequence of transitions from Pf → thermal metal → APf. At larger disorder, there is a sequence of transitions from Pf (κ xy = 7/2) → thermal metal → κ xy = 5/2 → thermal metal → APf (κ xy = 3/2). The intermediate thermal metal phase is a distinct key feature of [25]'s proposal. See the phase diagrams in Fig. 1 and Fig. 8 of [25]. We illustrate Ref. [25]'s thermal Hall prediction in Fig. 2's (II).
(3): Ref. [26] performed perturbative and non-perturbative analyses on the (1 + 1)d edge theory, and studied emergent symmetries on the domain wall between Pf and APf states at different disorder energy scales Λ = v/ 0 (which is related to the inverse of the puddle size 0 but proportional to the mean value of the edge state velocity v). Then in Ref. [26]'s language, not only the Pf|APf domain walls percolate, but also both Pf and APf percolatebecause some regions of Pf or APf extend through the whole bulk-boundary.
Ref. [26] proposed a more specific phase diagram of the ν = 5/2 disordered system, schematically shown in Fig. 3. An example of Ref. [26]'s thermal Hall prediction is illustrated in Fig. 1.
By the perturbative renormalization group (RG) analysis on disorder and scattering, Ref. [26] finds different emergent symmetries at different disorder energy scales. By a Berezinskii-Kosterlitz-Thouless (BKT)-type RG analysis, Ref. [26] finds for weak disorder we have four transitions all of which can be (broadened) first order transitions, or second order.
Finally, for Λ > Λ TH , when the disorder is very strong, the κ xy becomes unquantized and we enter into the thermal metal (TH) regions (the light red area on the top of Fig. 3). The percolation transition to the thermal metal phase guarantees the divergence of the correlation length, which therefore guarantees that the transitions from all topological orders to the thermal metal (drawn as the red solid curves in ν (filling fraction) Figure 3: A schematic phase diagram similar to Ref. [26]'s proposal. To see all these phases with varying κ xy requires that the ν c 5/2 plateau spans a sufficient range around ν c . Previous work [24][25][26] can obtain various quantized values of κ xy but cannot directly derive the bulk topological orders via the percolation transition argument. In this work, we propose a bulk effective field theory (EFT) not only consistent with [24][25][26] but can reproduce all the implicated bulk topological orders. At zero disorder, Λ = 0, the transition at ν c is first order. For Λ > 0, there are different possibilities for transitions or crossovers, depending on the microscopic details of samples. One scenario in [26] suggests that there are second order phase transitions (drawn in solid black lines) between topological orders for Λ > 0. Another scenario in [26] suggests that there can be first order phase transitions between topological orders for Λ > 0, but that disorder broadens these first order transitions to regions (light red shaded regions) with unquantized κ xy . These broadened regions cannot merely be crossovers, because the topological orders and global symmetries are distinct on the two sides. The boundaries of these broadened regions (drawn as dash-dotted red curves) could also be second order phase transitions. At larger Λ Λ 2 , a percolation transition from topological order to a thermal metal, also with unquantized κ xy , is known to be a second order phase transition (drawn as solid red curves). We propose a unified EFT in eqn. (2.1) in Sec. 2 and an upgraded version in Sec. 2.3 to describe all phases in the phase diagram. Note that the aforementioned disorder-broadening regions have unquantized κ xy and hence can behave similarly to a thermal metal as an intermediate phase. However, to be a precise thermal metal, one needs to check that κ xx diverges at zero temperature.
We expect the first-order disorder-broadening spreads to a region of size that is exponentially suppressed by e −f (Λ 1 ,Λ 2 )/Λ 2 with some functional form f of Λ 1 and Λ 2 [25], which grows wider as the disorder increases (i.e. the light red area becomes wider in Fig. 3 along the phase boundaries) [28]. What might be the outcomes of this phase boundary broadening?
• One possibility is that the broadening region becomes a new intermediate phase, such as a thermal metal, with unquantized κ xy , while the split phase boundaries (the dotted red lines in Fig. 3 along the phase boundaries) become two new second order phase transitions.
• Another possibility is that the percolation transition of the domain walls can be induced within the broadening region. Since at the percolation transition critical point, the domain wall size and correlation length diverge (at least for an infinite-sized system), this induces a new single second order transition within the broadening. 10 Broadening regions cannot become crossovers between neighboring phases, because the bulk phases have different topological orders and/or global symmetries.
In fact, Remark (3), following the scenario from Ref. [26], can be regarded as a general scenario that recovers both of the two scenarios from Remarks (1) and (2) in certain limits.
The key point for us is that Ref. [24][25][26] suggested that a κ xy = 5/2 plateau may be induced when Pf|APf domain walls percolate. However, Ref. [24][25][26] have not directly demonstrated that the resulting bulk order is indeed PHPf. Although PHPf has κ xy = 5/2, it remains an open question to show the bulk PHPf induces this κ xy = 5/2. In this work, we propose an effective field theory that can be viewed as a parent quantum field theory at some higher energy scale 11 , which at low energies can give rise to all the relevant IR TQFT phases listed in Fig. 3, including K = 8, PHPf and 113-state, etc.

Outline
In the previous subsections, we have summarized several proposed phase diagrams in the literature for the ν = 5/2 fractional quantum Hall state. We will focus on reproducing the phase diagram of [26], illustrated in Fig. 3. Our 2 + 1d EFT will also be able, in special limits, to reproduce phase diagrams arising from the other proposals [24,25], as will become clear in the subsequent sections. The EFT description also reproduces the 1 + 1d domain wall worldvolume theory predicted by [26], and it additionally fixes the type of phase transitions at the various phase boundaries (i.e. first order vs. second order). We also begin a preliminary study of the energetics of our EFT by performing computation of the domain wall tension, valid in a semiclassical limit, in the relevant phases. The tension of the walls differs in the Pf|APf and PHPf phases of the theory due to the presence of the chiral Majorana fermions in the former regime.
We conclude this introduction by summarizing the plan for the rest of this article.
In Sec. 2, we introduce our effective field theory, discuss its various IR phases, and describe in detail how it maps to the phase diagram in Fig. 3.
In Sec. 3, we describe the anyon spectra in the various IR phases of our EFT in terms of TQFTs and their quantum numbers, which will be matched to the many-body wavefunctions later (in Appendix F).
In Sec. 4, we analyze the domain wall theory and excitations in some detail. In particular, we study the gapless sectors and evaluate the tension of the walls.
In Sec. 5, we conclude, make final remarks, and point out several future directions.
Several appendices contain additional background and some technical details used in the body of the paper. In Appendix A, we review the relation between the gravitational Chern-Simons term and the thermal Hall response. In Appendix B, we describe abelian and non-abelian versions of Z 2 gauge theory in 2 + 1d. In Appendix C, we clarify some details about the fermion path integral and counterterms. In Appendix D, we discuss the procedure for gauging a one-form symmetry in a 2 + 1d TQFT. In Appendix E, we systematically introduce O(2) 2,L Chern-Simons theories, their Hall conductance, and other high-energy lattice cutoff scale a −1 lattice in the far UV. The length scales run from small to large as follows: lattice cutoff a lattice < magnetic length B < phase-coherence length ϕ ξ < sample size L.
The corresponding energy scales, the inverse of the length scales, run from large to small accordingly. The fluctuation length ξ is the length scale of the chemical potential fluctuation due to the impurity/doping in the system and it is roughly Λ −1 .
The 0 is the puddle linear size which is the link size for the Chalker-Coddington network model [29]. The disorder energy Λ = v/ 0 is tunable and set by the inverse of the tunable puddle size 0 [26]. When the length 0 is large compared to the domain wall thickness w, the domain walls tend to expand and the energetics of the system warrant a more careful analysis [25] to determine if the system prefers Pf or APf percolation instead of domain wall percolation. We discuss the tension of the domain walls in Sec. 4 relevant physical properties. In Appendix F, we review the wavefunction descriptions of the IR TQFTs relevant for our study. In Appendix G, we provide some additional details regarding our one-loop computation of the domain wall tension.
2 Effective field theory near the critical filling fraction in 2 + 1d We now present our effective field theory. The 2+1d effective field theory consists of three sectors: • O(2) gauge field with Chern-Simons (CS) term O(2) 2,1 (in the notation of [35]).
• A real scalar φ coupled to the Dirac fermions by a Yukawa term.
To make the connection with fQH, the Chern-Simons gauge field is coupled to a background U(1) EM gauge field. We begin by considering a particular mass term for the Dirac fermions and an even scalar potential so that the EFT preserves particle-hole symmetry and captures the phase transition at the critical filling fraction. More general mass terms and particle-hole-breaking scalar potentials will be considered in subsequent sections to produce the entire phase diagram of Fig. 3.
Explicitly, the theory is a 2+1d gauged Gross-Neveu-Yukawa theory with an O(2)×O(2) global symmetry Despite the appearance of the Chern-Simons term, the theory in fact has a particlehole symmetry, also called the time-reversal CT symmetry 12 (possibly with an anomaly): Ψ i → ij γ 0 Ψ j and φ → −φ, so that φ transforms as a pseudoscalar. To see this, we express the theory in (2.1) as where the quotient denotes gauging the Z 2 one-form symmetry generated by the composite line given by the product of the O(2) Wilson line in the non-trivial one-dimensional representation and the Z 2 electric line. 13 We briefly review the notion of gauging one-form symmetries in Appendix D. The O(2) 2,1 Chern-Simons theory is time-reversal invariant by level/rank duality [36,35]. Each of the two theories in the numerator has time-reversal zero-form symmetry and Z 2 one-form symmetry, where the zero-form and one-form symmetries do not have a mixed anomaly (since gauging the one-form symmetry reduces the two theories to SO(2) 2 and the Gross-Neveu-Yukawa theory, respectively, both of which are time-reversal invariant). Therefore, the quotient theory is also time-reversal invariant.
As discussed in Appendix E.1, the theory can couple to a background U(1) EM electromagnetic gauge field A to have a fractional quantum Hall conductivity σ xy = 5/2 under the U(1) EM electromagnetic charge's transverse conductivity measurement. The U(1) EM electromagnetic gauge field only couples to the Chern-Simons gauge field, and hence all the phases we discuss have the same Hall conductivity. 14 We will consider phases with µ 2 > 0, which implies that the real (pseudo-)scalar field φ condenses with a vacuum expectation value (vev): This spontaneously breaks the time-reversal CT symmetry, and there can be two symmetrybreaking vacua exchanged by the (broken) symmetry transformation in the 2+1d bulk. The spontaneously broken time-reversal symmetry CT leads to a 1+1d domain wall that interpolates between the two vacua. We will investigate the domain walls in Section 4.
Let us elaborate more on the global symmetries and gauge group: CT indeed corresponds to the anti-unitary particle-hole conjugation transformation in the condensed matter literature of ν = 5/2 quantum Hall systems. We will see in Appendix E that among the O(2) 2,L gauge theories with L ∈ Z 8 classes, only L = 1 and L = 5 produce time-reversal invariant theories. The O(2) 2,1 gauge theory will be later used for the particle-hole Pfaffian (PH-Pfaffian). 13 Gauging this one-form symmetry identifies the Z 2 gauge field in (Z 2 ) 0 with the first Stiefel-Whitney class w 1 of the O(2) gauge field. Namely, the Z 2 gauge field in (Z 2 ) 0 is w 1 (E), with E the O(2) gauge bundle. The one-form symmetry involved is different from the center one-form symmetry of O(2). Note the Z 2 electric line in the Z 2 gauge theory with matter is topological, while the magnetic line is not topological.
14 As discussed in Appendix E.1, the Hall conductivity only depends on the first Chern-Simons level in O(2) 2,L , while integrating out massive fermions in the sign representation only changes the second level L [35]. • Discrete global symmetries - 1. Z f 2 fermion parity symmetry: This is not to be confused with the Z C 2 charge conjugation acting by Ψ j → −Ψ j , which is dynamically gauged due to / D C . (Neither Ψ 1 nor Ψ 2 are gauge-invariant local fermionic operators). In fact, the Z f 2 acts not on Gross-Neveu-Yukawa sector, but only on the O(2) 2,1 CS and −3CS grav . Note that eqn. (2.1) is an intrinsically fermionic theory (defined on spin manifolds) because both O(2) 2,1 CS and −3CS grav are spin TQFTs, whose UV completion, say on a lattice, requires some gauge-invariant local fermionic operators.

Z CT
4 -symmetry: This is the particle-hole (PH) symmetry, also known as the CTsymmetry. This is an anti-unitary symmetry. Its normal subgroup is the Z f 2 fermion parity, since (CT x). We will further explain how the CT acts on the CS theories in TQFT sectors in Sec. 3. (There are other time-reversal symmetries given by composing it with Z 2 subgroup unitary symmetries. There are also other discrete charge C and parity P symmetries for the Lorenz invariant QFT.) 3. Z 4 one-form global symmetry [37] from the O(2) Chern-Simons theory: The Z 2 subgroup is generated by the O(2) Wilson line in the sign representation. We will not focus on this global symmetry in the paper. (The 't Hooft anomaly of the one-form symmetry is related to the fractional part of the Hall conductivity; see Appendix E.1).
4. Z 2 magnetic 0-form symmetry of the O(2) gauge field [35]. We will not discuss this symmetry in the paper.
• Gauge sector - The second line rewrites Ising and spin-Ising TQFTs as CS theories. 16 For background information on this sector, see Sec. E around eqn. (E.10).
In the following subsections, we discuss several deformations of our theory (2.1): 1. In Sec. 2.1, we consider a CT -preserving mass deformation, but the CT -symmetry turns out to be spontaneously broken. The deformation explicitly breaks O 2. In Sec. 2.2, we add an odd polynomial in φ to our action, which explicitly breaks 3. In Sec. 2.3, we add additional Majorana mass terms that break the entire O(4) symmetry, but preserve the CT -symmetry.

Particle-hole (time-reversal CT )-preserving deformation
Let us turn on the deformation m in eqn. (2.1).
0 < m < gv : When m increases from zero to gv, the theory has the following phases in the two vacua: • At the vacuum φ = −v, Ψ 1 has mass m − gv, while Ψ 2 has mass −m − gv. For m below gv, at low energies we can integrate out both negative mass Dirac fermions, and the theory becomes the gapped TQFT The O(2) 2,L Chern-Simons gauge theory (see Appendix E) contains a U(1) 8 Chern-Simons theory that contributes a net chiral central charge c − = c L − c R = 1, while the matter sectors do not contribute any net chiral central charge. The theory has Hall conductivity σ xy and thermal Hall conductivity κ xy matching those of the Pfaffian state: 17 σ xy = 5/2, κ xy = 1 + 5/2 = 7/2.
The two different regimes capturing our time-reversal-symmetric deformations are depicted in Figure 4.
With m treated as a proxy for disorder strength (the precise relation is discussed in Section 2.4), the gapped phases in the above discussion are precisely those that appear in 17 The spin gravitational Chern-Simons term has chiral edge modes contributing to the thermal Hall conductivity κ xy by a chiral central charge c − = −1/2; see Appendix A. 18 For m = gv, when φ = −v, the fermion Ψ 1 becomes massless; when φ = +v, Ψ 2 becomes massless. the scenario of [25,26]: for small disorder strength, the microscopic theory is at a first orderlike phase transition with coexisting Pfaffian and anti-Pfaffian phases, while increasing the disorder strength produces the PH-Pfaffian phase. From the above discussion, it is thus natural to identify the parameter m (or its magnitude |m|) in the effective phenomenological theory with the disorder strength in the microscopic material.
We remark that the first-order phase transition with distinct gapped vacua persists for a range of the parameter m ∈ [0, gv), which is consistent with the phase diagram proposed in [25,26]. We may identify Λ 1 in [25,26] with gv, with v controlled by the scalar mass µ in the effective theory. When gv is small, the phase diagram approaches that described in [24,25].

Particle-hole (time-reversal CT )-breaking deformation
In this section, we investigate the effect of adding a time-reversal-breaking deformation that preserves the O(2) × O(2) symmetry (O(4) when m = 0). In the experiment, this corresponds to applying an additional time-reversal-breaking magnetic field that changes the filling fraction ν slightly. In this discussion, we set the Yukawa coupling to g = 1 for simplicity.
Since φ is a time-reversal-odd pseudoscalar field, we consider the simple time-reversal-

Pfaffian
Anti-Pfaffian The green line in the middle represents a first order phase transition with spontaneously broken time-reversal symmetry (i.e., anti-unitary particle-hole symmetry) that gives rise to domain wall excitations that interpolate between the Pfaffian and anti-Pfaffian phases. The phase diagram is in qualitative agreement with the schematic phase diagram discussed in [25,26] with the time-reversal breaking deformation m odd identified with the external magnetic field in the experiment and the time-reversal preserving deformation |m| identified with the microscopic disorder strength. In experiment, it is so far undetermined whether Pfaffian or anti-Pfaffian is favored at ν < ν c 5/2 (or ν > ν c 5/2); we can easily flip our phase diagram by defining the sign of m odd to match the tuning parameter for the filling fraction ν.

PH-Pfaffian
breaking deformation given by an odd polynomial of φ. The most relevant deformation will be δV (φ) ∝ −m odd φ. This modifies the scalar potential and lifts the degenerate vacua.
In the lowest-order approximation, we can take the effect to be such that the original vacua shift to the locations φ = v + m odd + O(m 2 odd ) and −v + m odd + O(m 2 odd ). Depending on the sign of m odd , one of the above is the true vacuum: for m odd > 0 it is the former and for m odd < 0 it is the latter. In other words, the true vacuum has a vev: The two Dirac fermions then have masses given by (with g = 1) There are critical lines when any of the fermions become massless. 19 The phase diagram whose coordinates are our two parameters (m, m odd ) for a fixed v is given in Figure 5. Given by the mass deformation formula (2.9), the left critical line in Figure 5 has m 2 0, while the right critical line in Figure 5 has m 1 0. It is in qualitative agreement with the schematic phase diagram discussed in [25,26], and suggests that the corresponding Pf|PHPF and APf|PHPf phase boundaries are given by second-order phase transitions.

K = 8 and 113 states from O(2) × O(2)-breaking masses
In our earlier discussion, we mainly focused on mass deformations preserving the O(2)×O(2) symmetry that transforms the two Dirac fermions. If we allow Majorana masses that break the O(2) × O(2) symmetry, the effective theory can also describe the K = 8 state and the 113 state (the two states are related to one another by the particle-hole CT symmetry). Denote the four Majorana fermions by η ia where i = 1, 2 labels the Dirac fermions and a = 1, 2 labels the Majorana components. Consider the Majorana mass deformation: 20 where is a small number, Θ is the step function: Θ(x) = 0 for x ≤ 0 and Θ(x) = 1 for x > 0. Then the deformation is only nonzero when m > m * , where we take m * > gv. The deformation preserves the time-reversal CT symmetry. The four Majorana fermions have masses The situation is similar to eqn. (2.7). One might worry that the critical line can receive a quantum correction; however, since the scalar field has a mass of the order m φ ∝ µ around the vacuum, in the vicinity of the critical line with distance less than m φ there is a light fermion. 20 We take (γ 0 ) αβ = (−i σ y ) αβ = αβ with the spinor indices α, β. Note that the Dirac mass term Ψ j Ψ j = Ψ j † γ 0 Ψ j . In the Majorana basis, we write Ψ j = η j1 + iη j2 , and Ψ j Ψ j = αβ (η j1,α η j1,β + η j2,α η j2,β ) ≡ η 2 j1 + η 2 j2 . We define the Majorana mass term as αβ η α η β ≡ η 2 .
where the vev φ = v sgn(m odd ) + m odd + O(m 2 odd ) depends on the CT symmetry-breaking deformation m odd .
What becomes of the phase diagram under the deformation? For m ≤ m * it is the same as before, while for m > m * there are new gapped phases: • m 21 , m 22 < 0, and m 12 < 0, m 11 > 0: the theory flows to as a U(1) 8 CS theory or equivalently the abelian K = 8 K-matrix CS theory. If we write the U(1) 8 CS 1-form gauge field as b, and the U(1) EM gauge field as A, then the action is 8 4π bdb + 2b 2π dA − 4CS grav , up to a trivial spin-TQFT to represent a fermionic gapped sector (see Appendix A of [26]). It has quantum Hall conductivity and thermal Hall conductivity σ xy = 5/2, κ xy = 1 + 2 = 3.
• m 11 , m 12 > 0, and m 21 < 0, m 22 > 0: the theory flows to where we used the duality O(2) 2,2 ↔ O(2) −2,0 − 4CS grav [35] and O(2) 2,0 ↔ U(1) 8 . This phase is called the 113 state, since it can be described by the 3d Abelian Chern-Simons theory action, with 1-form gauge field b, as It has quantum Hall conductivity and thermal Hall conductivity It is related to the previous phase by the anti-unitary particle-hole symmetry (up to an anomaly).
In addition, there are critical lines separating the gapped phases where some of the fermions become massless. The phase diagram is in Figure 6. In the rest of the discussion we will focus on the case without the deformation M (m).

Random coupling and the thermal metal phase
In the theory (2.1), we can further choose the parameter m to be a random coupling with Gaussian distribution m = m 0 , m 2 = δ 2 . (2.14) The theory depends on the average m 0 and the fluctuation δ. In [39], it is found that for strong fluctuations δ → ∞, the system of free Dirac fermions becomes a thermal metal. We will set the magnitude of fluctuation to be for some non-negative, monotonically increasing function h that grows faster than a linear function (for instance, h(m 0 ) = m 2 0 ). Then m 0 controls the disorder strength of the system. At large enough m 0 , i.e. strong disorder, the fluctuation becomes sufficiently strong and the model (2.1) with random coupling m enters a thermal metal phase. Since in our model the electromagnetic background field only couples to the O(2) gauge field and does not couple to the fermions, the Hall conductivity does not depend on the mass of the fermions and remains the same value σ xy = 5 2 . This is consistent with the proposal in [25,26]. Near the critical lines of the phase diagram, the physical mass of one of the fermion becomes close to zero. If the disorder strength is nonzero for zero mass, h(0) > 0, the disorder will cause the region sufficiently near the critical lines to have thermal metal behavior, which accommodates the behavior described in [25] and illustrated in Fig. 3.

Anyonic excitations in the topological phases
Let us spell out the key properties of the TFT phases and their anyonic excitations. 21 We assume standard knowledge from the Chern-Simons (CS) description of fQHE. We will delineate the following: • Fractionalized anyon statistics, i.e. the spin or exchange statistics exp(i2πs) of anyons with spin s.
• Fractionalized U(1) EM electromagnetic charge Q/e (e is the electron charge).
They are summarized in Tables 1, 2, 3, 4, and 5, for the Pfaffian, anti-Pfaffian, PH-Pfaffian, K = 8, and 113 states respectively, in the notation For the PH-Pfaffian, since it enjoys PH-symmetry (time-reversal CT ), we also specify the (CT ) 2 quantum number for the appropriate anyons, and write We will first examine the non-abelian states, i.e. the Pfaffian in eqn. (2.5), anti-Pfaffian in eqn. (2.6), and PH-Pfaffian in eqn. (2.8). They can be written as the following Chern-Simons theories (see Appendix A of Ref. [26], and Appendix E): anti-Pfaffian : with their chiral central charges c − = κ xy . These TQFTs are obtained from gauging a diagonal one-form Z 2 symmetry in the (U(1) 8 CS theories) and the (ν ∈ Z 8 -class spin-TQFTs) 22 in 2+1d. More generally, these TQFTs are U(1) 8 ×T L Z 2 for L = −1, +1, +3, and one gauges a diagonal Z 2 one-form symmetry generated by the composite line given by the tensor product of the charge 4 Wilson line of U(1) 8 and a non-transparent fermion line in T L . See Appendix E for details on the T L theories. When gauging a diagonal Z 2, [1] symmetry, we identify their charged objects (the line with odd U(1) charge in U(1) 8  1. Spin statistics. The spin of an anyon is given by where K is the level of abelian CS theory, and q is the integer labeling the abelian anyon associated with the line operator e i q b of 1-form gauge field b. 23 Here, s nab means the spin from the non-abelian sector of the TQFT. For the Ising, Ising, and SU(2) −2 TQFTs in eqn.
2. Electromagnetic charge. For the anyon's U(1) EM charge Q/e, we can look at the coupling q of the electric current to the U(1) EM gauge field A. The charge can be changed by an integer by tensoring the line with a classical Wilson line A. 24 The U(1) EM charge Q and the Hall conductance σ xy can be computed via (see Appendix E for details): Based on the experimental constraint of σ xy = ν = 1/2, we have to introduce the appropriate U(1) EM coupling 2b This is a coupling with charge q = 2. Indeed, this gives half-filled ν = σ xy = qK −1 q = 2 2 /8 = 1/2. The anyon with U(1)-charge 2 is identified with the non-abelian σ anyon in the gauged U(1) 8 ×T L Z 2 CS theory. This non-abelian anyon has U(1) EM charge Q/e = K −1 q = 2/8 = 1/4. 22 Here the 2+1d ν ∈ Z 8 -class spin-TQFTs are obtained from gauging the Z 2 internal "Ising" symmetry of the 2+1d fermionic Z 2 × Z f 2 -SPTs, with fermion parity symmetry Z f 2 [38,40]. From this class of TQFTs, we will use the Ising, Ising, and SU(2) −2 cases. 23 In the K-matrix CS theory, we replace q 2 2K → q T K −1 q 2 where q is a charge vector in the second expression. 24 If we demand the spin/charge relation with spin c connection A, then the isolated A is not well-defined and the transparent fermion line in all theories is charged under U(1) EM . Then the charge is instead taken modulo 2 from tensoring with 2 A.
We can obtain all 12 anyons' U(1) EM charges by the same argument, with the results shown in Tables 1, 2, and 3. 3. PH-symmetry. In the PH-Pfaffian theory, PH-symmetry (or time-reversal CT ) is preserved, so to those anyons not permuted by the time-reversal symmetry, whose spin statistics exp(i2πs) are real-valued, 25 we can assign (CT ) 2 = ±1 quantum numbers. For those anyons α a whose spin statistics exp(i2πs) are complex valued, the spin statistics are mapped to their complex conjugates exp(−i2πs) under the CT transformation.
On the other hand, the Pfaffian and anti-Pfaffian states do not have CT symmetry. Instead, they map into each other under the CT transformation as follows.
The 12 anyons, and their spin statistics exp(i2πs), U(1) EM charges, and CT properties are organized in Tables 1, 2, and 3. 26 The list of anyons in the Tables contains not only quasiparticles but also quasiholes of quantum Hall liquids, to be explained in Appendix F. 27 Although there are 12 anyons, the number of ground states on a spatial 2-torus T 2 25 In other words, exp( i 2πs) = ±1 for such anyons, so they are self-bosonic or self-fermionic. 26 Note that the sigma anyon σ n notation in our present work is actually the σ −n in Ref. [26].  CS. We provide (exp(i2πs), Q/e) for 12 anyons. The σ −1 anyon has e i π 8 statistics.
Anti-Pfaffian U(1) 8 CS CS. We provide (exp(i2πs), Q/e) for 12 anyons. The σ 3 anyon has e − i 3π 8 statistics. CS. We provide (exp(i2πs), Q/e) or (exp(i2πs), Q/e) (CT ) 2 for 12 anyons. The σ 1 anyon has e − i π 8 statistics. In comparison, notice that Ref. [12] represents related TQFT data in terms of U(1) −8 ×Ising   theory, related by a GL(2,Z) transformation [26]: (We omit an electron charge normalization factor e.) The quantum numbers for the abelian states are shown in Tables 4 and Table 5. The spin statistics can be obtained from eqn. Although there are 16 anyons in each the abelian state, the GSD is only 8. The corresponding 8 ground states depend on the spin structure of the spin manifold T 2 . 28 28 Since all five theories are fermionic spin-TQFTs, we can specify various spin structures on the T 2 to characterize the GSD. There are 4 choices corresponding to the periodic (P) or anti-periodic (A) boundary conditions along each of two 1-cycles of T 2 : (P,P), (A,P), (P,A), and (A,A). The Hilbert space up to an isomorphism only depends on the fermionic parity Z f 2 (the Z 2 value of the Arf invariant). The fermionic parity Z f 2 is odd for (P,P), and the Z f 2 is even for (A,P), (P,A), (A,A). We denote the corresponding spin 2-tori T 2 as T 2 o for odd and T 2 e for even. The ground states on T 2 o or on T 2 e can come from different states. The 6 ground states on T 2 in Table 1 Table 4 and 5, depending on T 2 o or T 2 e , are chosen differently among 16 line operators. See more discussions about the spin structure dependence in [40][41][42].

Domain wall theory and tension
As reviewed in the introduction, the proposal of [26] suggests a percolation transition involving puddles of Pf and APf phases separated by domain walls. To this end, we consider the model (2.1) on the slice of parameter space with time-reversal symmetry preserved, i.e. m odd = 0. We would like to study some basic properties of the domain walls, from the EFT point of view, that result when time-reversal symmetry is spontaneously broken.
Let us ignore the discrete gauge field which couples to the fermions, for now, and write the Lagrangian as (in the mostly positive Lorentzian signature) The vacua are doubly degenerate, with the vevs given by ±v where v ≡ µ/ √ λ. Throughout this section, we assume without loss of generality that m, g ≥ 0.
The classical solution for a static domain wall is, as usual, with z 0 the center-of-mass coordinate. 29 We assume that the effective perturbative expansion parameter in the scalar sector, λ/µ, is small to validate the semiclassical analysis that we perform presently. The classical action evaluated on the domain wall saddle is where d 2 x is over the parallel directions to the domain wall, and the transverse z direction has already been integrated over. Divided by the area worldvolume area d 2 x, this famously gives the classical domain wall tension [43,44] For nonzero fermion mass, the two vacua are gapped. At energies smaller than the 2 + 1d bulk gap, we have well-defined 1+1d domain wall theories. To derive the domain wall theories, we first analyze the fermionic zero modes (which survive the low energy limit) in section 4.1, and then proceed to quantize the zero modes to obtain the domain wall theories in section 4.2. We then study another aspect of the domain walls -their tension, and we do so at one-loop order.

Fermionic zero modes in the domain wall background
In the semiclassical approximation, the transverse profile of fermion modes solves the Dirac equation in the domain wall background: We have, for the moment, suppressed dependence on the spatial direction parallel to the domain wall. We use the Majorana basis for Γ-matrices γ 0 = −iσ 2 , γ 1 = σ 3 , γ 2 = σ 1 , and write the two-component spinors explicitly as Ψ j (z) = (u j r (z), v j r (z)) T , j = 1, 2 (so the top component is of definite chirality and the bottom component has the opposite chirality). With these conventions, the above equation becomes We are interested in the zero-modes, which survive the low energy limit. For 0 = 0, we can solve these equations in the classical domain wall background: and Let us discuss the properties of the zero modes in the Pfaffian/anti-Pfaffian regime m < gv and the PH-Pfaffian regime m > gv. These properties will be the key in our subsequent determination of the respective domain wall theories in section 4.2.
When m < gv, since the solution for u j 0 (z), j = 1, 2 is not normalizable, we set both ψ j 0,+ = 0 and are therefore left with two complex parameters ψ j 0,− , which constitute our expected four real fermionic zero modes of a single chirality (thus, they correspond to four chiral Majorana fermions). In the extreme limit of m gv, the zero-modes satisfy ΨΨ ∼ Ψ † σ 2 Ψ = 0, and hence do not backreact on the scalar via the equations of motion.
When m > gv, the fermions delocalize and are essentially described by plane wave solutions. For each Dirac fermion, the normalizable edge modes of opposite chiralities survive on different sides of a half-space: The semiclassical limit µ/λ 1 is also a "hard-wall" limit, in which the soliton solution tends towards a steep step-function at z = z 0 with an insurmountable height barrier. Then we can indeed consider the normalizable edge modes on two half-spaces that can only interact via possible couplings on the interface. Among the relevant interactions, a 1+1d Majorana mass term for each fermion species, induced from the bulk mass term, can survive precisely on the wall, and gaps out the fermionic degrees of freedom at low energies. This is rather analogous to wall-localized supersymmetric couplings that appear in [45].

Domain wall worldvolume theory in 1 + 1d
There is a natural proposal for the domain wall worldvolume theory following from simple anomaly considerations. It is the O(2) WZW model coupled to two massless complex Dirac fermions by a common Z 2 orbifold that acts as the charge conjugation in O(2). The chiral anomaly accounts for the relative shift of the Chern-Simons level in the two bulk vacua. Since the U(1) part of the gauge field is confined in 1+1d, the theory naturally flows to Z 2 coupled to two complex fermions. The domain wall theory has O(4)/Z 2 symmetry which rotates the four massless real fermions. This is consistent with the proposal in [26].
Let us now derive the domain wall worldvolume theory from first principles to verify this intuition. For the moment, we will ignore the presence of the discrete gauge field, and reinstate its effect at the end. The vacua, which spontaneously break the time-reversal invariance, occur at φ = ±v. The fermions in each of these vacua have tree-level masses ±m + g φ = ±m ± gv.
To get the 1+1d domain wall theories, we wish to quantize the zero-modes in the two regimes of interest, m < gv and m > gv. We first describe a sector of the worldvolume theory without fermions, and then describe the interesting fermionic sector alluded to above. In the following, all quantities are the renormalized versions, as we imagine having already integrated out the bulk massive modes.
Goldstone mode Since the domain wall breaks translational invariance, there is an effective action for the bosonic Goldstone center-of-mass mode. It arises from promoting the modulus 30 z 0 ∈ R adiabatically to functions of the worldvolume directions m ∈ (t, x). Integrating over z and dropping a standard additive constant (hence our use of ∼ below) gives where the bosonic tension is in agreement with the tension (4.4) derived from evaluating the classical (effective) action on the domain wall solution. We neglect irrelevant higher-derivative terms in the fluctuation z 0 (x, t).

Wess-Zumino-Witten (WZW) models
Since the Chern-Simons sector of the bulk theory does not interact with the degrees of freedom on the wall except via the Z 2 gauging of the fermions, the domain wall is transparent to the continuous gauge degrees of freedom. As is well known, the 1+1d theory that furnishes a trivial interface for a Chern-Simons gauge field is the corresponding WZW theory.
This bulk Chern-Simons term on the two sides of the wall contributes a diagonal CFT on the wall due to the opposite orientations with respect to the bulk. The theory on the wall can be constructed as follows. First we start with the bulk theory SO(2) 2 = U(1) 2 on both sides of the wall, so that the theory on the wall is naturally a compact boson at the self-dual radius. Then we deposit additional L units of Z 2 SPT phases in the bulk, which induce additional fermions on the wall. The amount L of Z 2 SPT phases appropriate for each phase was discussed in Section 2.1, which we summarize here for the convenience of the reader: Finally, we gauge the diagonal Z 2 symmetry of the entire configuration that acts as charge conjugation on the SO(2) gauge field. This introduces a single Z 2 gauge field throughout the bulk and on the wall. In other words, at the interface we identify the Z 2 gauge field on the left side of the wall with the gauge field on the right. We may employ the relation among Chern-Simons theories where the theories T L , L mod 8 are described in Appendix E.
In the PH-Pfaffian regime, we have L = 1, and the contribution from the bulk on one side is given by gauging a diagonal Z 2 symmetry in the product of a left-moving compact boson ϕ at the self dual radius and a right-moving Majorana fermion ζ R . The contribution from the other side of the wall is the same with left exchanged with right. Of course, the chiral anomaly of this sector from both sides of the wall is trivial.
In the Pfaffian/anti-Pfaffian regime, an interface interpolating between the Pfaffian and anti-Pfaffian WZW theories differs from this basic one (L = 1) by precisely four additional Majorana fermions of the same chirality. On the Pfaffian side of the wall we have a leftmoving compact boson and a left-moving Majorana fermion, while on the anti-Pfaffian side we have a right-moving compact boson and five right-moving fermions as appropriate for the theories with L = −1, (3 ≡ −5), respectively. Both sides are again gauged by a single Z 2 gauge field. We denote the discrete Z 2 gauge field below as a, which implements a projection on the spectrum -on the Ψ j as well as ϕ, ζ.
Therefore, the domain wall theory before the contribution of the SPT-induced fermions is S G [z 0 ] + S W ZW [a, ζ, ϕ] (4.13) in the obvious notation, where the superscript W ZW denotes the appropriate WZW model for a given phase. The theory for the fermions that the SPT phases deposit on the wall will now be derived using our previous analysis of bulk fermionic zero modes in Section 4.1.
Fermionic sector Let us study the Pfaffian/anti-Pfaffian regime m < gv in the extreme limit of m = 0. We take the normalizable zero-modes ψ 0,− and promote them to worldvolume fields. We substitute the corresponding solutions in terms of two complex Weyl fermions (4.6, 4.8) into the Lagrangian (4.1) to obtain where all derivatives only run over the worldvolume coordinates k = x, t, and we have used the superscript to indicate that this is the domain wall theory that interpolates between the Pfaffian and anti-Pfaffian vacua. The coefficient of the kinetic term σ 31 is given by the integral , (4.15) which in the limit of small Yukawa coupling becomes The gauge field couples to the fermions on the wall exactly as it did in the bulk. Note that the WZW sector is almost decoupled from the fermions except for the Z 2 gauging.
In the PH-Pfaffian regime m > gv, let us set g = 0 for simplicity, and use the plane wave solutions on opposite sides of the walls. Doing the respective integrals for the surviving zero-modes over the two half-spaces (z ≤ z 0 , z ≥ z 0 ) then gives where now the superscript indicates that the domain wall theory is for the PH-Pfaffian phase. 32 The mass term gaps out the fermions at low energies, hence only the Goldstone and WZW sectors of the domain wall theory survives on the wall.
The analysis of the zero modes in the two extreme regimes also suggests a natural candidate domain wall theory (in the universality class of the theory) that describes the wall's phase transition: a 1+1d Z 2 -gauged Gross-Neveu-Yukawa theory (suppressing the dependence of the fields on the worldvolume coordinates (x, t)) 33 Although we call this coefficient σ, due to its formal similarity with σ as computed in eqn (4.10), we stress that it is not to be confused with the tension. The fermionic contribution to the tension will be computed in later subsections. 32 The appearance of 1/µ is not only expected by dimensional analysis. Recall from Section 4.1 that the plane wave solutions on opposite sides of the wall overlap in the vicinity of the wall, where a mass coupling is possible. On the domain wall, the mass term is therefore proportional to the width of the wall, which is 1/µ. 33 Analogous studies and proposals of domain wall worldvolume theories were made in the context of domain walls in four-dimensional QCD at θ = π [46], or four-dimensional SU(2) Yang-Mills gauge theory at θ = π [47].
Here, the condensation of the scalar σ as we tune the scalar mass term implements the phase transition between the two regimes. If we canonically normalize the fermions in L PHPf , then the coefficient of the mass term becomes m 2 µ , so that we set m 2 µ ∼ g 3 g 2 √ g 4 , which naively suggests g 4 ∼ µ 2 , g 3 ∼ g 2 ∼ m. We defer a more detailed analysis for future work.

One-loop effective action and tension
Let us return to our 2 + 1d bulk theory and study the (Euclidean) effective action and the domain wall tension from integrating out fermions at one-loop. 34 We ignore the Z 2 gauging and revisit its effect towards the end.
Consider expanding the theory in transverse fluctuations around a saddle φ = φ 0 + χ, where φ 0 could be either the vacuum saddle φ 0 = ±v or the domain wall saddle (4.2). The matter part of the action then takes the form where (suppressing the 2 + 1d spacetime dependence of the fields) is the action for the fluctuations. We will study the counterterms S ct below.

Effective action at φ 0 = v
At one-loop order around the vacuum saddle φ 0 = v, there are terms in the fluctuation action that contribute to χ via tadpole diagrams: We need to include counterterms to cancel the tadpole so that the location of the vacuum remains fixed, φ = v. Explicitly, where δ b and δ f denote counterterms that arise from consideration of bosonic χ and fermionic Ψ i loops, respectively, and Λ is a UV cutoff. The mass of the fluctuating χ field is given by √ 2µ at tree level, but gets corrected at one-loop, with the Feynman diagram given in Figure 7. Since we want to focus on the effect of the fermions, let us ignore the bosonic loop corrections for now. The one-loop effective action from integrating out two Dirac fermions with masses ±m and coupled to the scalar with Yukawa coupling g is where D φ m,g is the effective Dirac operator When m = gv, the leading terms in the derivative expansion amount to treating χ as a constant, (4. 25) We see that the mass of χ is renormalized as In what regime can we trust this result? Let us estimate this by computing some higher derivative terms in the one-loop effective action. For a single Dirac fermion with mass m, the first few higher derivative corrections quadratic in χ are with some computational details given in Appendix G.1. 35 Estimating the ∂ 2 for bosonic fluctuations by the mass √ 2µ, we find that the higher derivative corrections are suppressed by factors of |m − gv|/µ. Thus our result is a reasonable approximation in the regime We will come back to this at the end of Section 4.3.2.

Domain wall tension
To evaluate the one-loop corrections to the tension, we will closely follow the method of [48] (see also [49]). First, we formulate the theory in a Euclidean box with half-length L in the z direction and area V || in the worldvolume directions, so that the energy density is given 35 To apply the results of Appendix G.1, make the replacement m → m − gv.
in terms of the effective action Γ as under a scheme such that the expectation values of the vacua ±v are unrenormalized, and the effective action is normalized to vanish when evaluated on the vacua ±v.
Formally, the full one-loop correction to this quantity receives contributions from the classical term σ cl (taking the form of (4.4) with µ renormalized), the quantum correction σ qu , and the counterterms σ ct : The operators ∆, ∆ (0) are the inverse propagators of a fluctuating field in the soliton background and in the vacuum (trivial background), respectively. The central idea of [48] is that the fluctuations are independent of the worldvolume coordinates and may therefore be partially diagonalized by a Fourier transform in those directions. Then, the ratio of functional determinants can be related to a ratio of solutions of ordinary differential equations, which is then (numerically) integrated over the transverse coordinates. 36 First, we express the one-loop tension in terms of the renormalized parameters of the theory. As is standard [50,56,48], the renormalized scalar mass µ can be related to the bare mass µ bare by a one-loop computation in the perturbative sector of the fluctuation theory, i.e. in one of the two degenerate ground states. We follow [48] and use the MS scheme to fix the counterterms, and require that, as discussed above, the tadpole diagrams are cancelled by the counterterms. This coincides with the condition to fix the renormalized mass by requiring φ = ± 6µ 2 /λ. 37 The full one-loop tension can be broken up into a sum of the classical tension and the bosonic and fermionic one-loop contributions, of the form 38

32)
36 This bypasses numerous technical complications appearing in more traditional methods, and in particular provides a convenient way to deal with the regularization of sums of zero-point energies in different topological sectors. See, however, [43,[50][51][52] for results in 1+1d using analytic solutions of the fluctuation spectra and [53][54][55] for other approaches based on making successive Born approximations for scattering phase shifts. 37 The quartic coupling is only renormalized by a finite amount. 38 The coefficient δ b σ qu /µ 2 is also, in general, a function of the dimensionless scalar coupling λ/µ. We take λ/µ to be small throughout our analysis for the semiclassical approximation, and just consider the leading order λ-independent contribution.
where µ is the renormalized scalar mass, [δ b σ qu /µ 2 ] is a dimensionless constant, and [δ f σ qu /µ 2 ] is a dimensionless quantity with the following functional dependence on a dimensionless fermion mass w and a dimensionless Yukawa coupling γ, (4.33) The normalization of w is chosen such that at w = 1, the effective mass m − gv of one of the Dirac fermions vanishes. The normalization of γ is chosen so that γ = 1, ν = 0 corresponds to the N = 1 supersymmetric point in the case of a theory with a single Majorana fermion.
The classical piece of the domain wall tension in (4.32) with one-loop renormalized scalar mass is where the domain wall tension is renormalized at one-loop by the bosonic χ fluctuations alone. In relative terms, this correction is which is small in the semiclassical approximation with γ ∼ O(1). There is a first order transition in the domain wall tension at the critical point m = gv, but we expect it to be smoothed out by higher order corrections.
We can now determine [δ b σ qu /µ 2 ] , [δ f σ qu /µ 2 ] using the technology of [48]. Since χ only self-interacts at one-loop, and since the relevant computation was performed in [48], we can simply borrow their result, which was computed in dim-reg in terms of the analytically continued dimension n, and take n → 3. The result is It remains to determine the integral encapsulating the quantum fermionic contributions to the tension, following [48]. As always, we would like to keep λ/µ small. In addition, we also want the Yukawa coupling to be small to suppress large backreaction by the fermions, but we can keep the ratio g 2 /λ finite. Of course, when g = 0, δ f σ qu = 0. 39 Including the counterterm (4.22), the formula for the quantum one-loop tension from 39 An analogous computation performed in a supersymmetric theory with a single Majorana fermion and g = √ 2λ in [49] gives δ b σ SU SY qu + δ f σ SU SY qu = −µ 2 /4π. We reproduce this result.
integrating out the fermions is and This formal expression (4.37) can be evaluated explicitly as outlined in Appendix G.2. The results are shown in Figure 8, expressed in terms of the dimensionless variables ν and γ defined in (4.33). Some notable features are • The quantum one-loop correction to the tension δ f σ qu is of the same order as the correction from one-loop mass renormalization, namely: • Both are monotonically decreasing with respect to the fermion mass m.
• The effect diminishes rapidly once the mass m increases past the critical point m = gv.
Note that there is no mass renormalization at all for m > gv. Note: The dimensionless fermion mass w and Yukawa coupling γ are defined in (4.33). Notice that we have divided out by µ 2 γ 3 . The region near w = 1 (m = gv) is blocked out by dashed lines because the fermions become light and higher derivative corrections become important.
The validity of the lowest order approximation in the derivative expansion was analyzed earlier, and the estimate (4.28) translated to dimensionless quantities becomes As long as γ is O(1), the one-loop tension to leading order in the derivative expansion is a valid approximation when m is sufficiently large. Furthermore, if the Yukawa coupling g is large enough relative to λ, then we can also trust our results in some neighborhood of small m. 40 Finally, we have assumed that the couplings λ and g are small relative to the masses w and m, therefore the higher loop corrections and corrections involving more powers of χ are suppressed.

The effect of gauging
Let us discuss the effect of the Z 2 gauge field on the one-loop tension. Recall that the Z 2 gauge field acts as Ψ j → −Ψ j and leaves the scalar untouched. In the path integral, having a discrete gauge field amounts to summing over its holonomies. For a domain wall interpolating between two vacua, the Euclidean spacetime is R 3 with no boundary or nontrivial cycle, so it is unclear whether the gauging has any effect on the tension at all, even non-perturbatively. On a spacetime with nontrivial cycles, it is logically possible that the sum over holonomies introduces new saddles that dominate over the original saddle (trivial holonomy), but such effects go beyond perturbation theory.
In fact, for the sake of argument, let us imagine that the Z 2 is a subgroup of a continuous U(1) gauge symmetry acting on the fermions as Ψ j → e iα Ψ j , with associated gauge connection A. In the one-loop effective action from integrating out the fermions, to lowest order in the derivative expansion there in principle is a coupling of the form (∂ µ χ) A µ . However, we find that the coefficient of this term is zero by an explicit computation in Appendix G.1. Thus the U(1) gauging has no effect at the order of our approximation.

Conclusion and future directions
In this work, we have presented an effective field theory that captures the qualitative features of the phase diagram proposed in [26] to describe the ν = 5/2 fractional Quantum Hall system. We also studied some simple properties of domain walls present at the timereversal-symmetric locus (or, in condensed matter terminology, the particle-hole-symmetric locus), including their effective worldvolume theory and their tension, computed to one-loop order in a semiclassical approximation. The tension computed with the EFT is lower in the PH-Pfaffian phase than in the Pfaffian/Anti-Pfaffian phase, suggesting that the former phase may in fact be energetically favored over the latter in the presence of domain walls. This may explain the percolation transition, and serve as a resolution of the dilemma between the experiment [8] (favoring PH-Pfaffian) and bulk energetics studies [13][14][15][16][17][18][19][20][21][22] (favoring Pfaffian/Anti-Pfaffian). We leave a more complete study of bulk/domain wall energetics to future work.
We conclude with an incomplete list of additional questions and future directions raised by this study. Of course, most interesting is whether the proposal of [26] indeed provides the correct microscopic description of the ν = 5/2 state. If so, we hope our EFT provides a useful conceptual framework for studying aspects of this system.

Our effective field theory is a standard relativistic QFT, though various non-relativistic
EFTs have been proposed to study quantum Hall systems (see e.g. [6,7]). Is there a useful non-relativistic bulk EFT description of this system?
It is worthwhile to note that our EFT is a (super-)renormalizable QFT in 2+1d, and it is UV complete by itself. Although our EFT does not require a further UV completion at higher energy, it may still be helpful to understand how this relativistic EFT can be obtained from RG flow from a non-relativistic EFT, the electron wavefunctions, or a lattice model at the condensed matter UV cutoff scale.

2.
We computed the tension of the domain walls in an approximation where λ/µ << 1.
Roughly speaking, λ and µ respectively govern the height and width of the domain walls, so that the limit corresponds to studying rigid and thick walls. It would be interesting to determine if the domain walls, assuming they are indeed realized in the ν = 5/2 system, actually satisfy this limit so that our tension result can be used reliably to understand energetics of the system.
3. It may be that the particle-hole symmetry is explicitly, weakly broken in the experimental setup. If so, the domain walls would be metastable. It would be instructive to compute the decay rate for these walls in our EFT when one turns on our small but non-vanishing m odd deformation.
4. It would be very instructive to compute the spin-structure-dependent ground state degeneracy by performing an explicit path integral in our EFT. Such a quantity could potentially be measured in a real experimental setting if one fixes the boundary conditions of the lab sample, i.e. periodic or anti-periodic boundary conditions, similar to those on a spatial 2-torus.

Acknowledgements
The authors are listed in the alphabetical order by a standard convention. We thank Bert Halperin for a conversation. JW thanks Biao Lian for a previous collaboration on [26] and

A Gravitational Chern-Simons term and thermal Hall response
Any 3-manifold has a spin connection ω. 41 The fermion spinor field of spin 1/2 couples to the spin connection as ∇ = ∂ − 1 2 ω. Integrating out one massive Majorana fermion ψ gives 41 Explicitly, in terms of the frame metric η ab = g µν e µ a e ν b and the coframe e a µ e µ b = δ a b ω i jµ = e i ν Γ ν λµ e λ j + e i ν ∂ µ e ν j , where Γ is the Christoffel symbol.
the gravitational spin Chern-Simons term for positive mass m compared to negative mass: More explicitly, Tr ωdω + 2 3 ω 3 . For an even L, we can use the property SO(2) 1 = U (1) 1 to express Lf (B) = (L/2)·2f (B) as the U(1) × U(1) Chern-Simons term − L/2 4π BdB + 2 2π Bdu with u constrains B to be a Z 2 gauge field. By the field redefinition u → u + B, we find that for L = 8 the SPT phase is the same as L = 0. This reproduces the Z 8 classification of the SPT phases Gauging the Z 2 symmetry with a dynamical gauge field by summing over B gives rise to 8 different Z 2 gauge theories. For L = 0 it is the untwisted Z 2 gauge theory (the Z 2 toric code), while for L = 4 it is the Dijkgraaf-Witten twisted Z 2 gauge theory (the so-called double semion theory). See the list of 8 different Z 2 gauge theories (where L ∈ even yields an abelian TQFT and L ∈ odd yields a non-abelian TQFT) in Table 2 of [40].

C Fermion Path Integral and Counterterms
Consider a 2 + 1d Majorana fermion coupled to a Z 2 gauge field B ∈ H 1 (M, Z 2 ), and give it a large mass. The fermion path integral depends on the sign of the mass, given by

D Gauging one-form symmetry in 2 + 1d TQFT
Here we review some rules for gauging a one-form symmetry in 2+1d TQFT. For gauging a Z 2 one-form symmetry generated by the symmetry generator charge line a of integer spin, the rules are (see e.g. [57][58][59][60]): • Discard the lines that transform non-trivially under the one-form symmetry. These are the lines (of objects charged under the one-form symmetry) that braid non-trivially with a.
• Identify every remaining line W with its fusion with a: W ∼ W · a.
• For the remaining lines that are fixed points under fusion with a, there are two copies of the line.
In the corresponding chiral algebra, the procedure is equivalent to extending the chiral algebra by a simple current that obeys, together with the identity, the Z 2 fusion algebra. To compare our present work to Ref. [26], we note that Ref. [26] writes the TQFTs for Pfaffian, PH-Pfaffian, and anti-Pfaffian states as: Here T L can be written as another Z 8 class of fermionic spin TQFTs [38,40] as the Spin(L) −1 ×SO(L) 1 Chern-Simons gauge theory in 2+1d [35]. Explicitly, we can express the relations with the following CS theories: T 7 ↔ Ising × (spin-Ising), (E.8) and T −L = T L where bar denotes its time-reversal CT (i.e., particle-hole conjugate) image. The Spin(L) −1 × SO(L) 1 theories have a net zero chiral central charge c − = c L − c R = 0, and they are equivalent to 2+1d Kitaev spin liquids [61] tensored with suitable invertible spin TQFTs (with only {1, f }, a trivial operator and a transparent spin-1/2 fermionic line operator) to cancel the chiral central charge. Here the K-matrix CS theories have a gauge group given by products of U(1) × U(1) × . . . groups, with a symmetric-bilinear integer matrix K. In our case, only for an even integer L, we have the K matrix = 0 2 2 L/2 mod 4 which corresponds to an abelian CS theory. The Z 8 class of fermionic spin TQFTs can be obtained by gauging the Z 2 -internal symmetry of fermionic symmetry-protected topological states generated by the spin bordism group Ω Spin 3 (BZ 2 ) = Z 8 . For more details, see Table 2 of [40]. Moreover, if we disregard the thermal Hall conductance (the chiral central charge c − ) difference, the T L can also be related to the Spin(L) −1 Chern-Simons gauge theory in 2+1d, which is a bosonic non-spin TQFT with a Spin(L) gauge group at the level −1 [35]  In T L , the Z 2, [1] symmetry acts on the magnetically charged line, which is the line operator associated to the so-called σ anyon. In the condensed matter terminology, this is called the magnetic vortex line or vison loop. Z 2, [1] takes this line to minus itself (−σ). In the U(1) 8 Chern-Simons theory, the Z 2, [1] symmetry also transforms the lines with odd U(1) charges to minus themselves. In other words, the line operators of the σ anyon in T L and of odd U(1) charge in U(1) 8 are both charged objects under the diagonal Z 2,[1] 1-form symmetry. As already mentioned in the main text, the Z 2, [1]

E.1 Hall conductivity
The theory also has a Z 4 one-form global symmetry generated by the line with U(1) charge 2 in the U(1) 8 Chern-Simons theory. One can use this one-form symmetry to couple to a background electromagnetic U(1) gauge field A at level-1 as 1 2π (2b)dA = 2b 2π dA. The U (1) 8 action, including the coupling to the probe electromagnetic background, is: where b is the gauge field of U(1) 8 Chern-Simons theory, and A couples to the properly quantized U(1) gauge field 2b. This system gives the Hall conductance σ xy = 2 2 /8 = 1/2.
In addition, we can add a Chern-Simons action for the background gauge field A where the coefficient r ∈ Z is quantized to be an integer in fermionic systems. Then the system has a quantum Hall conductance measured in units of e 2 /h. In the application to the experiment with quantum Hall conductivity σ xy = 5 2 (see [26] and the references therein), we take r = 2.
Here the r = 2 corresponds to the lowest (zeroth) Landau levels with both spin-up and spin-down complex fermions, which contribute σ xy = 2 quantum Hall conductance. The first Landau level contributes another σ xy = 1 2 from the half-filled first Landau level with spin-polarized fermions.
The discussion does not change when there are fermions in the nontrivial one-dimensional representation of Z 2 (i.e. they are coupled to T L but not U(1) 8 ) that do not couple to the background A.
There is another way to see the quantum Hall conductance using the Z 4 one-form symmetry. The line generating that symmetry has spin 1/4, and thus the one-form symmetry has the anomaly 8 4π 4d where B 2 is the two-form background field of the Z 4 one-form symmetry. The anomaly then implies that the coupling to A by fixing the value B 2 = 1 4 dA has a half-integer Hall conductance (see Appendix E of [62]). The one-form symmetry is present in massive or massless theories with the same 't Hooft anomaly, since the one-form symmetry is preserved by mass deformations, and thus the quantum Hall conductance is the same across the phase diagram.

E.2 Quantum numbers of quasi-excitations
Using (E.10) and (E.18), we see that a line with odd charge in the U(1) 8 theory is identified with the σ line in T L theory. It is therefore the line operator of a non-abelian anyon (when L is odd) with quantum dimension 2. Let us label this line operator as (odd, σ).
Moreover, we can determine the U(1) electromagnetic charge of this anyonic quasi-excitation from the level K = 8 and the charge vector q = 2 in (E.18) via This means our theory has fractional U(1) electromagnetic charges ± 1 4 from quasiparticle (odd, σ) and quasihole excitations.
The semions have fractional spin statistics with spin 1 4 , and also have fractional electromagnetic charges: Therefore, there are fractional U(1) electromagnetic charges in the theory ± 1 2 from the quasiparticle semion (even, s) and the corresponding quasihole.

F Many-body wavefunctions
In this Appendix, we recall and examine the electron wavefunctions for the non-abelian Pf/PHPf/APf states and also abelian states. In contrast to the EFT language used in the bulk of our work (which uses the second-quantization language), this section is formulated in a many-body quantum mechanics picture (the first quantization language). This Appendix can be a companion to the Sec. 3. The Pfaffian state wavefunction Ψ Pf was introduced by Moore-Read [2,63] for a ν = 1/2 fractional quantum Hall state in the zeroth Landau level. It is a rotationally invariant state.
The wavefunction is (F.1) In particular, we look at k = 2, for N electrons, and for a magnetic length B = c/(|e|B) for magnetic field B. Here z * i ≡ z i is the complex conjugate of the coordinate z i = x i + iy i ∈ C. The Pf is the Pfaffian of the rank-N antisymmetric matrix M ij = 1/(z i − z j ), so (Pf(M ij )) 2 = det(M ij ). Namely, for an even positive N , we have a degree-N/2 polynomial with the symmetry group S N and sgnσ = ±1 the signature of the element σ ∈ S N , so The Pf factor is crucial to obtain an antisymmetric wavefunction, as appropriate for a fermionic electron system. The Laughlin-like factor N 1=i<j (z i − z j ) 2 with second-order zeros dictates that there are repulsive interactions between electrons. The Pf 1 z i −z j factor cancels some of the zeros present in the Laughlin-like factor , making the electrons less repulsive on net. This implies that electrons in Ψ Pf are closer together than those in a purely Laughlin-like state. All the electrons are spin polarized in the Ψ Pf state.
Filling fraction: To determine the filling fraction ν of Ψ Pf , we compute the angular momentum operator where k(N − 1) − 1 is from the Laughlin factor and −1 is from the Pf factor. This gives rise to the angular momentum k for the i-th electron, which encircles the larger area of the droplet with a radius r k = 2(k(N − 1) − 1) B (at the location where the wavefunction density is maximal). Recall Φ 0 = 2π( B ) 2 B = hc/|e|, so we verify that Ψ Pf has the ν = number of particles number of flux quanta (i.e., ν = 1/2 and σ xy = 1/2 for k = 2 for the Moore-Read Pfaffian). To employ this wavefunction to the study of ν = 5/2, we employ the Pf state for the first half-filled, spinpolarized Landau level, while we also include spin up and down electrons fully occupying the zeroth Landau levels. This gives a total filling fraction of ν = 5/2; also, σ xy = 5/2. The interaction produces an energy gap the order of the Coulomb interaction energy e 2 / B , so this state is incompressible.

Quasi-excitations:
We can obtain a quasihole by adding a hole excitation in a complex coordinate ζ, We could view the z i as dynamical variables (that should be integrated over to obtain the density), while viewing ζ as a background, or probe, parameter. Because the additional factor N i =1 (ζ − z i ) introduces more zeros into the wavefunction, the system becomes less repulsive, so also less dense -this is a hallmark of a hole excitation. If ζ is instead a dynamical variable, then the (ζ − z i ) k factor introduces an electron at position ζ. For ζ a background parameter, this has the interpretation of removing an electron at ζ. Putting this together, a k-fold factor (ζ − z i ) k removes an electron at ζ and so, given the electron charge −|e|, we have produced a quasihole of charge |e|/k. The second line in eqn. (F.2) is a rewriting of the first line, by absorbing the quasihole into the Pf factor: Pf | ζ 1 =ζ 2 =ζ which can be regarded as the quasihole splitting into two further fractional quasiholes at ζ 1 and ζ 2 , each with charge |e|/(2k). For the Moore-Read Pfaffian at k = 2, we have a quasihole of charge |e|/2 which further fractionate to a quasiholes of charge |e|/4.
For each quasihole, there is a corresponding quasiparticle excitation with opposite global symmetry quantum numbers, but with the same spin statistics. The quasiparticles/quasiholes may be regarded as vortices/anti-vortices because the phase of the wavefunction winds when a particle winds around the quasiexcitation at ζ. For example, the fractionalized charge |e|/4 or −|e|/4 excitations are in fact the ±π-vortices, which we shall identify as the non-abelian σ anyons in our EFT and TQFTs.
Chiral central charge c − = c L −c R (the degrees of freedom of 1+1d left-moving minus right-moving edge modes) can be determined from two parts of the wavefunctions: first, the Laughlin sector (z i − z j ) k corresponds to U(1) k CS theory. It has an edge theory which can be described as a complex chiral boson or fermion, which yields c − = 1; we will describe the edge theory more in Sec. ??. Second, the Pf factor corresponds to the angular momentum L z = 1 between the composite fermion with a chiral p-wave (p x + ip y ) pairing [63]. This gives rise to c − = 1/2 corresponding to an edge theory given by a real-valued chiral Majorana mode. The total c − for the Pfaffian state eqn. (F.1) is c − = 3/2.
Composite fermion pairing: The above discussion is consistent with the fact that the Ising TQFT contributes c − = 1/2 in eqn. (3.3), which can be induced from the (p x + ip y )wave pairing of composite fermions (CF), with angular momentum ∝ (k x + ik y ) for L z = 1. In the Dirac composite Fermi liquid (CFL) picture, the Dirac CF gains a π-Berry phase around the Fermi surface. For Dirac CF, the pairing becomes the (d x + id y )-wave pairing ∝ (k x + ik y ) 2 with L z = 2.
F.2 Anti-Pfaffian state for κ xy = 3/2 Anti-Pfaffian (APf) state wavefunction: The bulk system for the Pfaffian state does not have a time-reversal (CT )/particle-hole symmetry [27], but Ref. [5] considered the eqn. (F.1)'s particle-hole conjugate wavefunction, dictated by the particle-hole transformation [64], and named it the anti-Pfaffian state: We can break down the Ψ APf state we are interested in [4,5] as a combination of two component pieces. The first piece is the ν = 1/2 Ψ APf with respect to the ν = 1 IQH state. The second piece can be viewed as a ν = 1 integer quantum Hall state (the IQH state with respect to the ν = 0 vacuum). The first part is nothing but the particle-hole conjugate of the ν = 1/2 Ψ Pf with respect to the ν = 0 vacuum. Indeed, the first line in eqn.  Particle-Hole Pfaffian (PH-Pfaffian, or PHPf) wavefunction [23] (see also [65] and other attempts [66,67]) can be written as The P LLL is the projection onto the lowest Landau level (LLL). From the first line in eqn. (F.4), we can see that the filling fraction is still ν = 1/2, as one can read off from the Laughlin factor using the same reasoning from the Pfaffian case. Moreover, the Pf The above discussion is consistent with the fact that the Ising TQFT contributes c − = −1/2 in eqn. (3.4), which can be induced from the (p x − ip y )-wave pairing of CF, with angular momentum ∝ (k x − ik y ) at L z = −1. For a Dirac CF picture, the pairing becomes the s-wave pairing with L z = 0. F.4 K = 8-state for κ xy = 3 K = 8-state wavefunction is a bosonic wavefunction but can be written as a fermionic wavefunction by dressing it with a fermionic tensor product state: The filling fraction is ν = 1/2 with an appropriate charge coupling, when the charge 2e quasi-excitations are coupled to the U(1) electromagnetic gauge field at level-1. The chiral central charge is c − = 1, as always for a Laughlin wavefunction. The is consistent with U(1) 8 TQFT with c − = 1. F.5 113-state for κ xy = 2 113-state wavefunction is a special case of the lmn wavefunction, known as the Halperin wavefunction (the multi-component generalization of Laughlin wavefunction) with l = 1, m = 1, n = 3 for some N + N electron system: The filling fraction is ν = 1/2 with an appropriate charge coupling. The chiral central charge is c − = 1 − 1 = 0 coming from two modes with opposite chiralities.
In general, we expect that quasiparticles and quasiholes of the above many-body wavefunctions in this Appendix agree with the anyons (and their quantum numbers) of TQFTs shown in Table 1, 2, 3, 4, and 5 in Sec. 3.
As for the Aχ piece, where the derivatives act on A but not χ, and the parity odd piece with a Levi-Civita symbol is present only if d = 3. At one derivative order, the parity odd piece vanishes upon integrating over p, and we are left with the same expression as the χA term (G.8) except now the derivative acts on A instead of χ. Thus, upon integration by parts, the Aχ term is an identical contribution to the effective action as the χA term. When d = 3, the coefficient of (∂ µ χ) A µ in the effective Lagrangian vanishes, as we found in (G.10).