From Dirac Semimetals to Topological Phases in Three Dimensions: a Coupled-Wire Construction

PHYSICAL REVIEW X 9, 011039 (2019)

From Dirac Semimetals to Topological Phases in Three Dimensions: A Coupled-Wire Construction

Syed Raza, Alexander Sirota, and Jeffrey C. Y. Teo* Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA

(Received 1 May 2018; revised manuscript received 11 January 2019; published 27 February 2019)

Weyl and Dirac (semi)metals in three dimensions have robust gapless electronic band structures. Their massless single-body energy spectra are protected by symmetries such as lattice translation, (screw) rotation, and time reversal. In this paper, we discuss many-body interactions in these systems. We focus on strong interactions that preserve symmetries and are outside the single-body mean-field regime. By mapping a Dirac (semi)metal to a model based on a three-dimensional array of coupled Dirac wires, we show (1) the Dirac (semi)metal can acquire a many-body excitation energy gap without breaking the relevant symmetries, and (2) interaction can enable an anomalous Weyl (semi)metallic phase that is otherwise forbidden by symmetries in the single-body setting and can only be present holographically on the boundary of a four-dimensional weak topological insulator. Both of these topological states support fractional gapped (gapless) bulk (respectively, boundary) quasiparticle excitations.

DOI: 10.1103/PhysRevX.9.011039 Subject Areas: Condensed Matter Physics, Strongly Correlated Materials, Topological Insulators

I. INTRODUCTION so that the net chirality, and, consequently, the anomaly, cancels. Or otherwise, a three-dimensional system of a Dirac and Weyl semimetals are nodal electronic phases single Weyl fermion must be holographically supported as of matter in three spatial dimensions. Their low-energy the boundary of a topological insulator in four dimensions emergent quasiparticle excitations are electronic Dirac [1] – and Weyl [2] fermions. (Contemporary reviews in con- [20 22]. On the other hand, a Dirac fermion in three densed electronic matter can be found in Refs. [3–10].) dimensions consists of a pair of Weyl fermions with They are three-dimensional generalizations of the Dirac opposite chiralities. Without symmetries, it is not stable fermions that appear in two-dimensional graphene [11] and and can turn massive upon inter-Weyl-species coupling. the surface boundary of a topological insulator [6,12–14]. With symmetries, a band crossing can be protected by the They follow massless quasirelativistic linear dispersions distinct symmetry quantum numbers the bands carry along near nodal points in the energy-momentum space close to a high symmetry axis. In this paper, we focus on the the Fermi level. Contrary to accidental degeneracies which fourfold degenerate Dirac nodal point protected by time- can be lifted by generic perturbations, these nodal points reversal (TR) and (screw) rotation symmetry. are protected by topologies or symmetries. In electronic systems, massless Dirac and Weyl fermions A Weyl fermion is chiral and has a nontrivial winding of appear in gap-closing phase transitions between spin-orbit a pseudospin texture near the singular nodal point in coupled topological insulators and normal insulators [23]. energy-momentum space. This would associate to a non- When inversion or time-reversal symmetry is broken, nodal conservative charge current under a parallel electric and Weyl points can be separated in energy-momentum space. magnetic field and is known as the Adler-Bell-Jackiw Such gapless electronic phases are contemporarily referred – (ABJ) anomaly [15,16]. Thus, in a true three-dimensional to as Weyl (semi)metals [24 27]. Their boundary surfaces lattice system, Weyl fermions must come in pairs [17–19] support open Fermi arcs [24] that connect surface-projected Weyl nodes. Weyl (semi)metals also exhibit exotic transport properties, such as negative magnetoresistance, non- *[email protected] local transport, chiral magnetic effect, and chiral vortical effect [28–33]. There have been numerous first-principle Published by the American Physical Society under the terms of calculations [34] on proposed materials such as the non- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to centrosymmetric ðLa=LuÞBi1−xSbxTe3 [35], the TlBiSe2 the author(s) and the published article’s title, journal citation, family [36], the TaAs family [37,38], trigonal Se=Te [39], and DOI. and the HgTe class [40], as well as the time-reversal

2160-3308=19=9(1)=011039(34) 011039-1 Published by the American Physical Society RAZA, SIROTA, and TEO PHYS. REV. X 9, 011039 (2019)

breaking pyrochlore iridates [24,41,42], magnetically Topological doped topological and trivial insulator multilayers [26], insulator spatial symmetry HgCr2Se4 [43] and Hg1−x−yCdxMnyTe [44]. At the same breaking charge U(1) charge U(1) and time, there have also been abundant experimental obser- preserving spatial symmetry preserving vations in bulk and surface energy spectra [45] as well as Dirac-Weyl Topological transport [46]. Angle-resolved photoemission spectroscopy (semi)metal many-body phase charge U(1) interaction (ARPES) showed bulk Weyl spectra and surface Fermi arcs breaking in TaAs [47–51] as well as similar materials such as NbAs, spatial symmetry Topological preserving NbP, and TaP [52,53]. Other materials such as Ag3BO3, superconductor TlTe2O6, and Ag2Se [54] were observed to host pinned Weyl nodes at high symmetry points. Negative magnetoresistance was reported in TaAs [55,56] as a suggestive FIG. 1. Symmetry-breaking single-body gapping versus sym- signature of the ABJ anomaly. Similar properties were also metry preserving many-body gapping of a Dirac-Weyl observed in TaP [57], NbP, and NbAs [58–60], although (semi)metal. not without controversies [61]. Weyl points with opposite chiralities cannot be separated is on symmetry-preserving many-body gapping inter- in energy-momentum space when both inversion and time- actions. The resulting insulating topological phase can reversal symmetries are present. Massless Dirac fermions carry long-range entanglement and a nontrivial topological appear between gap-closing phase transitions between order. Similar phenomena were theoretically studied on the topological and trivial (crystalline) insulators, such as Dirac surface state of a topological insulator [114–117] and Bi1−xSbx [62] and Pb1−xSnxTe [63]. Critical Dirac the Majorana surface state of a topological superconductor (semi)metals were investigated, for example, in the tunable [118,119], where symmetry-preserving many-body gap- – TlBiSe2−xSx [64 66],Bi2xInxSe3 [67,68], and Hg1−xCdxTe ping interactions are possible and lead to nontrivial surface [69], as well as the charge-balanced BaAgBi [70], PtBi2, topological orders that support anyonic quasiparticle SrSn2As2 [71], and ZrTe5 [72], whose natural states are excitations. believed to be close to a topological critical transition. Symmetry-preserving gapping interactions cannot be stud- A Dirac (semi)metallic phase can be stabilized when the ied using a single-body mean-field theory. This is because the Dirac band crossing is secured along a high symmetry axis Dirac-Weyl (semi)metallic phase is protected by symmetries and the two crossing bands carry distinct irreducible in the single-body setting and any mean-field model with an representations. Theoretical studies include the diamond- excitation energy gap must therefore break the symmetry structured β-crystobalite BiO2 family [73] [space group either explicitly or spontaneously. The coupled-wire con- (SG) No. 227, Fd3m], the orthorhombic body-centered struction can serve as a powerful tool in building an exactly BiZnSiO4 family [74] (SG No. 74, Imma), the tetragonal solvable interacting model and understanding many-body Cd3As2 [75] (SG No. 142, I41=acd), the hexagonal Na3Bi topological phases of this sort. The construction involves a family [76], as well as the filling-enforced nonsymmorphic highly anisotropic approximation where the electronic Dirac semimetals [77–82] such as the hexagonal TlMo3Te3 degrees of freedom (d.o.f.) are confined along an array of family [71] (SG No. 176, P63=m), the monoclinic continuous one-dimensional wires. Inspired by sliding Ca2Pt2Ga (SG No. 15, C2=c), AgF2,Ca2InOsO6 (SG Luttinger liquids [120–124], the coupled-wire construction No. 14, P21=n), and the orthorhombic CsHg2 (SG was pioneered by Kane, Mukhopadhyay, and Lubensky [125] No. 74, Imma) [83]. At the same time, there are numerous in the study of Laughlin [126] and Haldane-Halperin hier- experimental confirmations. They include angle-resolved archy [127,128] fractional quantum Hall states. Later, this photoemission spectroscopy (ARPES) observations on theoretical technique was applied in more general fractional Cd2As3 [84–86],Na3Bi [87,88], and ZrTe5 [72]; scanning quantum Hall states [129–133], anyon models [134,135], tunneling microscopy in Cd2As3 [89]; magnetotransport in spin liquids [136,137], (fractional) topological insulators Bi1−xSbx [90],Cd2As3 [91–98],Na3Bi [88,99], ZrTe5 [138–142], and superconductors [143,144],aswellasthe [72,100–102], HfTe5 [103], and PtBi2 [104]; magneto- exploration of symmetries and dualities [145,146].More- optics [105], and anomalous Nernst effect [106] in Cd2As3, over, coupled-wire construction has already been used to and many more. However, there are also contradicting investigate three-dimensional fractional topological phases pieces of evidence, especially in ZrTe5 and HfTe5 that [147–149] and Weyl (semi)metal [150] even in the strongly suggest a bulk band gap [107–113]. correlated fractional setting [151]. Dirac-Weyl (semi)metals are the origins of a wide variety The microscopic symmetry-preserving many-body inter- of topological phases in three dimensions (see Fig. 1). By actions in the Dirac surface state on a topological insulator introducing a spatial or charge Uð1Þ symmetry-breaking was discussed by Mross, Essin, and Alicea in Ref. [152]. single-body mass, they can be turned into a topological They mimicked the surface Dirac modes using a coupled- insulator or superconductor. The focus of this manuscript wire model and proposed explicit symmetric many-body

011039-2 FROM DIRAC SEMIMETALS TO TOPOLOGICAL PHASES … PHYS. REV. X 9, 011039 (2019) interactions that lead to a variation of gapped and gapless Dirac fermion model respects C2 and TR, no anomalies surface states. Motivated by this and also using a coupled- wire construction, the microscopic symmetry-preserving vortices of Dirac many-body gapping of the Majorana topological super- masses mx(r) + imy(r) conducting surface state was studied by one of us in 3D array of chiral Dirac strings Ref. [153]. violates TR, emergent AFTRs and C2 In this paper, we focus on (i) a coupled-wire realization of a Dirac-Weyl (semi)metallic phase protected by anti- single-body ferromagnetic time-reversal (AFTR) and screw twofold backscattering rotation symmetries, (ii) a set of exactly solvable interwire many-body interactions that introduces a finite excitation Coupled wire model energy gap while preserving the symmetries, and (iii) an interaction-enabled (semi)metallic electronic phase which is otherwise forbidden by symmetries in the single-body Dirac semimetal Surface states setting. preserves C2 and AFTRs AFTR breaking and preserving

sym-preserving bulk A. Summary of results many-body many-body interactions interactions We now highlight our results. By mapping a Dirac (semi) metal to a model based on a three-dimensional array of Pfaffian decomposition Pfaffian decomposition wires, we show that the Dirac semimetal can acquire a of nonchiral Dirac strings of chiral Dirac strings many-body excitation energy gap without breaking any relevant symmetries and leads to a three-dimensional Gapped topological state Fractional topological order. We also construct a new interaction- with 3D topological order surface states enabled Dirac semimetallic state which only has a single pair of Weyl nodes and preserves time-reversal symmetry. Single-body mass Such a state is forbidden in a single-body setting. A general Interaction-enabled anomalous outline of the construction of both these states is given in Dirac-Weyl semimetal Fig. 2 and also summarized below. The starting point of our model is a minimal Dirac FIG. 2. Logical outline of the paper. It shows the procedure of fermion model [Eq. (2.1) and Fig. 3] equipped with time- going from a Dirac fermion model to an interaction-enabled C reversal and (screw) C2 rotation symmetries. The model is gapped state with three-dimensional topological order. Here, 2 is anomaly free and so can be realized in a 3D lattice model. the twofold (screw) rotation symmetry, TR is time-reversal The first part of this paper addresses a mapping between the symmetry and AFTR is the antiferromagnetic time-reversal symmetry. isotropic massless Dirac fermion in the continuum limit and an anisotropic coupled-wire model where the effective low- fermions with opposite chiralities) emerges and captures energy d.o.f. are confined along a discrete array of 1D the low-energy long-length-scale electronic properties continuous wires. The mapping to a coupled-wire model is (Figs. 6 and 7). The coupled-wire Dirac model and its achieved by first introducing vortices (adding mass terms) massless energy spectrum are anomalous with respect to that break the symmetries microscopically, Eq. (2.2). These the AFTR and C2 symmetry. The three possible resolutions vortices are topological line defects that involve spatial of the anomaly are discussed in Sec. II A. The model with winding of symmetry-breaking Dirac mass parameters. an enlarged unit cell which leads to the two momentum- Consequently, these vortices host chiral Dirac electronic separated Weyl points to collapse to a single Dirac point channels, each of which corresponds to a gapless quasi-1D is also discussed (Fig. 9). The corresponding Fermi arc system where electronic quasiparticles can only propagate in a single direction along the channel and are localized along the perpendiculars Eq. (2.5). When assembled together onto a vortex lattice, the system recovers the screw C2 rotation symmetry as well as a set of emergent antiferromagnetic symmetries, which are combinations of the broken time-reversal and half- translations (Fig. 5). Upon nearest-wire single-body electron backscatterings, the electronic band structure in low energies disperses linearly and mirrors that of the continuous isotropic Dirac parent state. A symmetry-protected FIG. 3. The two pairs of counterpropagating Dirac bands along massless Dirac fermion (equivalently a pair of Weyl the kz axis distinguished by eigenvalues of C2 ¼i.

011039-3 RAZA, SIROTA, and TEO PHYS. REV. X 9, 011039 (2019) surface states are discussed in Sec. II B and shown in interactions. The coupled-wire construction suggests a new Figs. 12 and 13. interaction-enabled topological (semi)metal in which these The second part of this paper addresses nontrivial Weyl nodes can be realized (Fig. 19). symmetry-preserving many-body interacting effects The many-body interacting coupled-wire model can be beyond the single-body mean-field paradigm. We begin turned into a gapless system, where (1) all low-energy d.o.f. with the anisotropic array of chiral Dirac wires that are electronic and freely described in the single-body constitutes a Dirac (semi)metal protected by AFTR and noninteracting setting by two and only two separated (screw) C2 rotation symmetries (Fig. 6). We consider an Weyl nodes, (2) the high-energy gapped sector supports exactly solvable model of symmetry-preserving interwire fractionalization. Although the model is antiferromagnetic, many-body backscattering interactions. This model is we conjecture that similar anomalous Weyl (semi)metal inspired by and can be regarded as a layered version of can be enabled by interaction while preserving local time the symmetric massive interacting surface state of a reversal. topological insulator. It is based on a fractionalization The paper is organized as follows. In Sec. II,we scheme that divides a single chiral Dirac channel into a construct a single-body coupled-wire model of a Dirac- decoupled pair of identical chiral “Pfaffian” channels Weyl (semi)metal equipped with two emergent AFTR axes (Fig. 15). Each of these fractional channels carries half and a (screw) C2 rotation symmetry. In Sec. II A,we of the d.o.f. of the original Dirac wire. For instance, the establish the equivalence between the isotropic continuum fractionalization splits the electric and thermal currents limit and the anisotropic coupled-wire limit by a coarse- exactly in half. It leads to the appearance of fractional graining mapping. We also discuss the anomalous aspects quasiparticle excitations. For example, a chiral Pfaffian of the pair of Weyl fermions and different resolutions to the channel also runs along the 1D edge of the particle-hole anomaly. In Sec. II B, we describe the gapless surface states symmetric Pfaffian fractional quantum Hall state [154–156], of the coupled-wire model. AFTR breaking and preserving and supports charge e=4 Ising and e=2 semionic primary surfaces are considered separately in Secs. II B 1 and II B 2, fields. respectively. We consider an explicit combination of many-body In Sec. III, we move on to the effect of symmetry- interwire backscattering interactions that stabilize the preserving many-body interactions. The fractionalization of fractionalization. Similar coupled-wire constructions were a chiral Dirac channel is explained in Sec. III A, where we applied in the literature to describe the topological insula- establish the Pfaffian decomposition through bosonization tor’s surface state [152] and ν ¼ 1=2 fractional quantum techniques. The splitting of a Dirac channel is summarized Hall states [129,133]. They are higher-dimensional analogs in Fig. 16. In Sec. III B, we explicitly construct an exactly of the Affleck-Kennedy-Lieb-Tasaki spin chain model solvable interacting coupled-wire model that introduces a [157,158]. The pair of chiral Pfaffian channels along each finite excitation energy gap to the Dirac system while wire is backscattered in opposite directions to neighboring preserving relevant symmetries. The many-body interwire wires by the interaction (Fig. 17). As a result of this backscattering interactions are summarized in Fig. 17.In dimerization of fractional d.o.f., the model acquires a finite Sec. III C, we discuss a plausible stabilization mechanism excitation energy gap and at the same time preserves the of the desired interactions through an antiferromag- relevant symmetries. netic order. We speculate that such a symmetry-preserving gapping In Sec. IV, we discuss the other key result of the paper, a by many-body interactions leads to a three-dimensional variation of the model that enables an anomalous topo- topological order supporting exotic pointlike and linelike logical (semi)metal. We show how a single pair of Weyl quasiparticle excitations with fractional charge and statis- nodes in the presence of time-reversal symmetry can be tics. The complete characterization of the topological order realized through many-body interactions. Such a state is will be part of a future work [159]. Although there have forbidden in the single-body setting. In Sec. V, we elaborate been numerous field-theoretic discussions on possible on the gapless surface states of both new interacting phases properties of topologically ordered phases in 3D, this is discussed in Secs. III B and IV. the first work with a microscopic model that could possibly In Sec. VI, we conclude with discussions regarding the lead to its material realization. potential theoretical and high-energy impact of this work. In the single-body regime, an (antiferromagnetic) We also discuss the experimental realization and verifica- time-reversal symmetric Weyl (semi)metal realizable on a tion of the proposed states and some possible future three-dimensional lattice has a minimum of four momentum- directions. space-separated Weyl nodes. For a single pair of Weyl nodes with opposite chirality, time-reversal symmetry must be II. COUPLED-WIRE CONSTRUCTION broken. However, a key result of this paper is the realization OF A DIRAC SEMIMETAL of a single pair of momentum-space-separated Weyl nodes in We begin with a Dirac semimetal in three dimensions. It the presence of AFTR symmetry as enabled by many-body consists of a pair of massless Weyl fermions with opposite

011039-4 FROM DIRAC SEMIMETALS TO TOPOLOGICAL PHASES … PHYS. REV. X 9, 011039 (2019) chiralities. In this paper we do not distinguish between a symmetries. Symmetry-breaking intervalley scatterings Dirac and a Weyl semimetal. This is because the fermion introduce two coexisting mass terms doubling theorem [17–19] and the absence of the Adler- k r 0 k r μ r μ Bell-Jackiw anomaly [15,16] require Weyl fermions to HDiracð ; Þ¼HDiracð Þþmxð Þ x þ myð Þ y; ð2:2Þ always come in pairs in a three-dimensional lattice system. A Weyl semimetal therefore carries the same low energy where mx (or my) preserves (respectively, breaks) TR, and d.o.f. as a Dirac semimetal. We refer to the case when the both of them violate C2. We allow slow spatial modulation of pair of Weyl fermions are separated in momentum space as the mass parameters, which can be grouped into a single a translation symmetry protected Dirac semimetal. Here, complex parameter mðrÞ¼mxðrÞþimyðrÞ, and to be we assume the simplest case where the two Weyl fermions precise, momentum k should be taken as a differential overlap in energy-momentum space. Its low-energy band operator −i∇r when translation symmetry is broken. Hamiltonian takes the spin-orbit coupled form Nontrivial spatial windings of the symmetry-breaking mass parameters give rise to topological line defects or vortices 0 k ℏ k ⃗μ HDiracð Þ¼ v · s z; ð2:1Þ that host protected low-energy electronic d.o.f. Proli- feration of interacting vortices then provides a theoretical where s⃗¼ðs ;s ;s Þ are the spin-1=2 Pauli matrices, and path to multiple massive-massless topological phases while x y z restoring and modifying the original symmetries as they μz ¼1 indexes the two Weyl fermions. Normally, the masslessness of the Dirac system is emerge in the low-energy long-length-scale effective theory. protected by a set of symmetries. Here, we assume the A topological line defect is a vortex string of the mass TR T , which is represented in the single-body picture by parameter in three dimensions where the complex phase of m r m r eiφðrÞ winds nontrivially around the string. the spinful operator Tˆ ¼ is K where K is the complex ð Þ¼j ð Þj y The left diagram in Fig. 4 shows the spatial modulation of conjugation operator, and a twofold rotation C2 about the z φðrÞ along the xy cross-sectional plane normal to a axis. In the case when μ has a nonlocal origin such as z topological line defect, which runs along the z axis. In sublattice or orbital, it can enter the rotation operator. We this example, the complex phase φðrÞ winds by 6π around assume C2 is represented in the single-body picture by the line defect (represented by the red dot at the origin). The ˆ μ C2 ¼ isz z. It squares to minus one in agreement with the winding number of the complex phase in general can be fermionic statistics, and commutes with the local TR evaluated by the line integral operator. In momentum space, T flips k → −k while I I C2 rotates k ;k ;k → −k ; −k ;k . The band ð x y zÞ ð x y zÞ 1 1 ∇rmðrÞ Hamiltonian (2.1) shares simultaneous eigenstates with c ¼ dφðrÞ¼ · dr; ð2:3Þ 2π C 2πi C mðrÞ C2 along the kz axis. The two forward-moving bands have C2 eigenvalues þi while the two backward-moving ones where C is a (right-handed) closed path that runs once − have C2 eigenvalues i (see Fig. 3). Therefore, the band around the (oriented) line defect. Equation (2.3) is always crossing is C2 protected while the fourfold degeneracy is an integer given that the mass parameter mðrÞ is non- k 0 pinned at ¼ because of TR symmetry. Noticing that vanishing along C. each of the C2 ¼i sector along the kz axis is chiral (i.e., Massless chiral Dirac fermions run along these topo- consisting of a single propagating direction), it violates the logical line defects [160]. When focusing at kz ¼ 0, the fermion doubling theorem [17,18] and is anomalous. This differential operator (2.2) with a vortex along the z axis is can be resolved by assuming the C2 symmetry is actually a nonsymmorphic screw rotation in the microscopic lattice limit and squares to a primitive lattice translation in z. kz is now periodically defined (up to 2π=a) and the two C2 eigensectors wraps onto each other after each period. Focusing on the continuum limit where kz is small (when 2 ik a compared with 2π=a), C2 ¼ −e z ≈−1 and the C2 symmetry behaves asymptotically as a proper rotation. The primary focus of this paper is to explore symmetry- preserving-enabled interacting topological states that originate from the massless Dirac system. Contrary to its FIG. 4. Dirac string. (Left) Spatial winding of mass parameters robustness in the single-body noninteracting picture, we around a Dirac string going out of the paper represented by the show that the 3D Dirac fermion can acquire a many-body center red dot. Stream lines represent the vector field mass gap without violating the set of symmetries. To mðrÞ¼½mxðrÞ;myðrÞ. (Right) Energy spectrum of chiral Dirac illustrate this, we first make use of the fact that the fermions. Blue bands represent bulk continuum. Red bands Dirac system can be turned massive by breaking correspond to chiral Dirac fermions localized along the string.

011039-5 RAZA, SIROTA, and TEO PHYS. REV. X 9, 011039 (2019) identical to the 2D Jackiw-Rossi model [161] with chiral symmetry γ5 ¼ szμz. Each zero-energy mode corresponds to a massless chiral Dirac fermion with positive or negative group velocity in z depending on the sign of its γ5 eigenvalue. (For a concrete example, see Appendix A.) These quasi-one-dimensional low-energy electronic modes are similar to those that run along the edge of 2D Landau levels and Chern insulators, except they are now embedded in three dimensions. Their wave functions extend along the defect string direction but are localized and exponentially decay away from the defect line. Moreover, such an electronic channel is chiral in the sense that there is only FIG. 5. (Left) A 3D array of Dirac strings. (Right) Cross section a single propagating direction. The energy spectrum of the of the array. associates into-the-plane Dirac channel, represents out-of-plane ones. Stream lines represent the configuration topological line defect (for the example with winding m r r r 3 of the mass-parameter vector field ð Þ¼½mxð Þ;myð Þ of the number c ¼ ) is shown in the right diagram of Fig. 4, vortex lattice. in which, there are three chiral bands (red curves) inside the bulk energy gap representing the three chiral Dirac electrons. As a consequence of the chirality, the transport of a vortex lattice generated by the spatially varying Dirac charge and energy must also be unidirectional. The chiral mass electric and energy-thermal responses are, respectively, captured by the two conductances sdðx þ iyÞ mðrÞ¼m0 ; ð2:6Þ 2 2 2 jsdðx þ iyÞj δI e δI π k σ ¼ electric ¼ ν ; κ ¼ energy ¼ c B T; ð2:4Þ δV h δT 3h where sd is the (rescaled) Jacobian elliptic function [164] where ν is the filling fraction if the chiral channel is supported with simple zeros at p þ iq and poles at ðp þ 1=2Þþiðq þ by a 2D insulating bulk, and c is called the chiral central 1=2Þ for p, q integers. It consists of vortices with alter- charge. For the Dirac case, c ¼ ν is the number of chiral nating winding number c ¼1 at the zeros and poles in a Dirac channels. Here, c can be negative when the Dirac checkered board lattice configuration. On the cross-section fermions oppose the preferred orientation of the topological plot on the right side of Fig. 5, there is a Dirac string with line defect. In a more general situation, c ¼ cR − cL counts positive (or negative) winding at each (respectively, ). the difference between the number of forward propagating Each vortex string has a chiral Dirac fermion running and backward propagating Dirac fermions. There is a through it. Figure 6 shows the same two-dimensional slice mathematical index theorem [160,162,163] that identifies of the array, except suppressing the mass parameters which the topological winding number in (2.3) and the analytic correspond to irrelevant microscopic high-energy d.o.f. number of chiral Dirac fermions in (2.4). Hence, there is no We choose a unit cell labeled by ðp; qÞ, its x, y coordinates. ’ ⊙ need to distinguish the two c s. Each has both a forward-moving Dirac fermion ψ p;q The massless chiral Dirac channels, described by the ⊗ (shown as ) and a backward-moving one ψ p;q (shown low-energy effective theory as ).

XcR cXRþcL L ψ † ∂ ˜∂ ψ ψ † ∂ − ˜∂ ψ Dirac ¼ i að t þ v xÞ a þ i bð t v xÞ b; a¼1 b¼cRþ1 ð2:5Þ have an emergent conformal symmetry and the index c ¼ cR − cL is also the chiral central charge of the effective conformal field theory (CFT). We refer to the primitive topological line defect with c ¼1 that hosts one and only chiral Dirac fermion ψ as a Dirac string. (It should not be confused with the Dirac magnetic flux string that connects monopoles.) FIG. 6. Coupled Dirac wire model with tunneling amplitudes A three-dimensional array of Dirac strings (wires) can be t1, t2. Each unit cell (dashed box) consists a pair of counter- realized as a vortex lattice of the mass parameter m ¼ propagating Dirac strings, and . T 11; T 11¯ are the two mx þ imy in a Dirac semimetal. For example, Fig. 5 shows antiferromagnetic directions.

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2 F F This array configuration breaks TR as the symmetry C2 ¼ð−1Þ translationðaezÞ ≈ ð−1Þ : ð2:11Þ would have reversed the chirality (i.e., propagating direction) of each Dirac fermion. Instead, it has an emergent When adjacent vortex strings are near each other, their antiferromagnetic time-reversal (AFTR) symmetry, which Dirac fermion wave functions overlap and there are finite T T is generated by the operators 11 and 11¯ in the diagonal amplitudes of electron tunneling. We construct a coupled and off-diagonal directions. Each is composed of a time- Dirac wire model of nearest-wire single-body backscat- e e 2 reversal operation and a half-translation by ð x þ yÞ= or tering processes with π fluxes across each diamond −e e 2 ð x þ yÞ= . square (Fig. 6), where the tunneling amplitude t1 (or t2) in the (11) [respectively, ð11¯ Þ] direction is imaginary T ψ ⊗ T −1 ψ ⊙ T ψ ⊙ T −1 −ψ ⊗ 11 p;q 11 ¼ p;q; 11 p;q 11 ¼ pþ1;qþ1; (respectively, real). ⊗ −1 ⊙ ⊙ −1 ⊗ T ¯ ψ T ψ T ¯ ψ T −ψ 11 p;q 11¯ ¼ p−1;q; 11 p;q 11¯ ¼ p;qþ1: X H ¼ ℏv˜ðψ ⊙ †k ψ ⊙ − ψ ⊗ †k ψ ⊗ Þ ð2:7Þ p;q z p;q p;q z p;q p;q ψ ⊙ †ψ ⊗ − ψ ⊙ †ψ ⊗ These AFTR operators are nonlocal as they comewith lattice þ it1ð p;q p;q p−1;q−1 p;qÞþH:c: translation parts. They are antiunitary in the sense that ⊙ † ⊗ ⊙ † ⊗ þ t2ðψ −1 ψ p;q − ψ −1 ψ p;qÞþH:c:; ð2:12Þ T αψT −1 ¼ αT ψT −1 and hT ujT vi¼hujvi because the p ;q p;q local time-reversal symmetry is antiunitary. Similar to a where the first line is the kinetic Hamiltonian of individual spatial nonsymmorphic symmetry, the AFTR symmetries − ∂ ↔ square to the primitive translation operators Dirac channels under the Fourier transformation i z kz along the wire direction. This tight-binding Hamiltonian −1 F preserves the AFTR symmetry (2.7), THT H. T 11T 11¯ ¼ð−1Þ translationðe Þ; ¼ y ψ⃗ −1 RFourier transformation of the square lattice p;q ¼ T T e 2 − ⊙ ⊗ 11 11¯ ¼ translationð xÞ; ð2:8Þ 2π iðkxpþkyqÞψ⃗ ψ⃗ ψ ψ ½ðdkxdkyÞ=ð ÞRe k, ¼ð ; Þ turns H 2π 2 ψ⃗† ψ⃗ where ð−1ÞF is the fermion parity operator. Moreover, they (2.12) into ¼ ½ðdkxdkyÞ=ð Þ kHðkÞ k,where mutually commute ½T 11; T 11¯ ¼0. We notice in passing that ℏ˜ the AFTR symmetry is only an emergent symmetry in the vkz gðkx;kyÞ HðkÞ¼ ð2:13Þ low-energy effective theory. It is not preserved in the g ðkx;kyÞ −ℏvk˜ z microscopic Dirac model (2.2) and is broken by the mass parameter, mðrÞ ≠ m½r þðex eyÞ=2 . For instance, the is the Bloch band Hamiltonian, for gðkx;kyÞ¼ Jacobian elliptic Dirac mass function (2.6) actually has a −iðk þk Þ −ik −ik it1ð1 − e y x Þþt2ðe x − e y Þ. Here, momentum k periodic unit cell twice the size of that of the effective wire lives in the “liquid crystal” Brillouin zone (BZ) where −π ≤ model in Fig. 6. On the other hand, the Dirac mass (2.6) is k ;k ≤ π and −∞

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C2ð−kx; −ky;kzÞC2ðkx;ky;kzÞ¼−1, and the algebraic relations (2.10) with the AFTR operators

−1 C2ð−kÞT11ðkÞ¼−T11ðC2kÞ C2ðkÞ; −1 C2ð−kÞT11¯ ðkÞ¼−T11¯ ðC2kÞ C2ðkÞ; ð2:20Þ

for C2k ¼ð−kx; −ky;kzÞ.

A. The anomalous Dirac semimetal FIG. 7. Energy spectrum of the coupled Dirac wire model We notice that the coupled-wire Dirac model (2.12) and (2.12). its massless energy spectrum in Fig. 7 are anomalous with respect to the AFTR symmetries T 11 and T 11¯ as well as the C2 symmetry if it is proper symmorphic and not a screw T ψ⃗ T −1 k ψ⃗ T ψ⃗ T −1 k ψ⃗ rotation. This means that it cannot be realized in a single- 11 k 11 ¼ T11ð Þ −k; 11¯ k 11¯ ¼ T11¯ ð Þ −k; body three-dimensional lattice system with the AFTR or C2 0 −eiðkxþkyÞ symmetries. In a sense, it is not surprising at all since the T11ðkÞ¼ K; 10 chiral Dirac strings that constitute (2.12) are themselves violating fermion doubling [17,18]. Here, we further 0 −eiky ¯ k K elaborate on the anomalous Dirac spectrum (Fig. 7) where T11ð Þ¼ − ; ð2:15Þ e ikx 0 the pair of Weyl points are separately located at two TRIM K0 . We also comment on the nontrivial consequence of the where K is the complex conjugation operator. They satisfy anomaly and pave the path for later discussion on many- the appropriate algebraic relations (2.8) in momentum body interactions. space We begin with two 2D planes in momentum space parallel to kykz located at kx ¼π=2. They are represented −ik T11ð−kÞT11¯ ðkÞ¼T11¯ ð−kÞT11ðkÞ¼−e y ; by the two blue planes in Fig. 7. The AFTR or C2 −1 −1 −ik symmetries require the Chern invariants T11ð−kÞT11¯ ðkÞ ¼ T11¯ ð−kÞ T11ðkÞ¼e x ; ð2:16Þ Z i and the coupled-wire model (2.13) is AFTR symmetric Ch1 ¼ TrðP∂ P∂ PÞdk dk ð2:21Þ 2π ky kz y z

T11ðkÞHðkÞ¼Hð−kÞT11ðkÞ; at kx ¼π=2 to be opposite, where PðkÞ¼ð1 − T11¯ ðkÞHðkÞ¼Hð−kÞT11¯ ðkÞ: ð2:17Þ HðkÞ=jEðkÞjÞ=2 is the projection operator onto the negative energy band. This is because the AFTR symmetry is The Weyl points are at time-reversal invariant momenta antiunitary and preserves the orientation of the kykz plane, 2 (TRIM) K0 ≡ −K0 (modulo the reciprocal lattice 2πZ ), whereas C2 is unitary but reverses the orientation of the and the AFTR operators T11ðK0 Þ¼−iσyK and T11¯ ðK0 Þ¼ kykz plane. (See Appendix B for a detailed proof.) On the ∓ iσyK square to minus one. Hence, the Weyl points are not other hand, the two Chern invariants along the two planes only protected by the nonvanishing Fermi surface Chern must differ by 1 because they sandwich a single Weyl point invariant but also the Kramers’ theorem. In addition, the at Γ. This forces the Chern invariants to be a half integer 1 2 model is also C2 symmetric Ch1 ¼ = , which is anomalous. While the C2 anomaly can be resolved simply by C2ðkÞHðkÞ¼HðC2kÞC2ðkÞ; ð2:18Þ doubling the unit cell and assuming it originates from a microscopic nonsymmorphic screw axis, the AFTR where the twofold symmetry (2.9) is represented in the anomaly is stronger because the two antiferromagnetic single-body picture by a diagonal matrix combinations (2.8) generate lattice translations and fix the unit cell size. There are three resolutions. i 0 (1) The AFTR symmetries are broken by high energy C ψ⃗ C−1 k ψ⃗ k 2 k 2 ¼ C2ð Þ C2k;C2ð Þ¼ − d.o.f. when k is large. 0 −ie iðkxþkyÞ z (2) The spectrum in Fig. 7 is the holographic 3D ð2:19Þ boundary spectrum of an AFTR symmetric weak topological insulator in 4D. − 2 (suppressing the screw phase e ikza= in the continuum (3) The spectrum is generated by strong many-body limit a → 0). It agrees with the fermion statistics (2.11) interaction nonholographically in 3D.

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Below, we discuss the first two resolutions, and we leave (a) (b) the many-body interaction-enabled situation to Sec. IV.

1. Broken symmetries and coarse graining The mapping between the original Dirac fermion model and the emergent Dirac fermion model from a coupled-wire y y construction can be qualitatively understood as a coarse- Chern ribbons graining procedure. Here, the high-energy microscopic z x z x Chern ribbons electronic d.o.f. are integrated out. The procedure can be repeated indefinitely and resembles a real-space renorm- FIG. 8. Chiral Dirac channels ( and ) realized on the edge of alization. For example, the gapless Dirac electronic struc- Chern insulating ribbons (dark blue directed lines) stacked along 110¯ ture of the coupled-wire model can acquire a finite mass by the ð Þ normal direction. symmetry-breaking dimerizations. These dimerizations can be arranged in a topological manner that spatially wind symmetry is also broken and the coupled Dirac wire model nontrivially around a collective vortex. These second-stage now has an enlarged unit cell (light blue dashed boxes) that vortices can subsequently be assembled into an array consists of two pairs of counterpropagating chiral Dirac similar to the previous construction except now with a channels. All Chern ribbon patterns must break the C2 longer lattice constant. The system again recovers a symmetry about a Dirac wire because each wire is con- massless Dirac spectrum under intervortex electron tunnel- nected to one and only one Chern ribbon in a particular ing in low-energy and long length scale. The mapping direction. therefore establishes an equivalence between the continu- Now we go back to the vortex lattice generated by the ous isotropic massless Dirac fermion and the semidiscrete Jacobian elliptic Dirac mass function mðrÞ in (2.6) and anisotropic coupled Dirac wire model. consider its symmetries. For this purpose, we use the In the present case when the chiral Dirac channels symmetry properties of the (rescaled) Jacobian elliptic originate from vortex strings in an underlying microscopic function [164] Dirac insulator, the spatial modulation of mass parameters r mð Þ actually violates one of the AFTR symmetries, sd x iy −sd x 1 iy −sd x iy i ; ð þ Þ¼ ð þ þ Þ¼ ð þ þ Þ mðrÞ ≠ m½r þðex eyÞ=2, where stands for complex 1þi C conjugation. For instance, since all elliptic functions must sd xþiyþ ¼ −i ; contain at least two zeros and two poles in its periodic cell, 2 sdðxþiyÞ the Jacobian elliptic mass function (2.6) has longer periods sdð−x−iyÞ¼−sdðxþiyÞ; ð2:22Þ than ex and ey in Fig. 6, and thus must break T 11 or T 11¯ . The symmetry is broken only in the ultraviolet limit at large k z where C is some unimportant real constant that depends on where the chiral Dirac line nodes meet the microscopic bulk the modulus of sd and will never appear in the mass band (see Fig. 4) at high energy ∼jmðrÞj. In fact, the above function mðrÞ¼m0sdðx þ iyÞ=jsdðx þ iyÞj. We see from anomalous argument shows that all mass-parameter con- the minus sign in the first equation that the Jacobian elliptic figurations that produce the 3D vortex lattice array (Fig. 5) function, and consequently the mass function, have primi- must either (a) break both the AFTR symmetries T 11 and tive periods ex ey and therefore have a unit cell of size 2 T 11¯ , or (b) preserve one but violate translation so that the iπ=4 unit cell is enlarged and the two Weyl points collapse onto [see Fig. 9(a)]. Choosing m0 ¼jm0je , we see from the each other in momentum space. (See Figs. 8 and 9.) For instance, the microscopic system can be connected to (a) (b) a stack of Chern insulating ribbons (or lowest Landau levels) with alternating chiralities shown in Fig. 8. Instead of being supported by vortices of Dirac mass, the chiral Dirac wires are now realized as edge modes of Chern insulating strips. Each 2D ribbon (represented by thick dashed dark blue lines) is elongated in the out-of-paper z direction but is finite along the (110) direction and holds counterpropagating boundary chiral Dirac channels. The dark blue arrows represent the orientations of the Chern ribbons that accommodate the boundary Dirac channels FIG. 9. (a) The massive AFTR and C2 breaking coupled Dirac with the appropriate propagating directions. Here, the wire model. (b) The reduced Brillouin zone (BZ) after translation Chern ribbon pattern in Fig. 8(a) breaks both AFTR axes. symmetry breaking where the two Weyl points collapse to a The pattern in Fig. 8(b) preserves T 11¯ . However, translation single Dirac point at M.

011039-9 RAZA, SIROTA, and TEO PHYS. REV. X 9, 011039 (2019) second equation that T 11 (or T 11¯ ) is preserved (respec- j ¼ 1, 2. However, when the symmetries are broken, these tively, broken) hopping parameters do not have to agree. The Bloch band Hamiltonian after Fourier transforma- e e m r þ x y ¼mðrÞ; ð2:23Þ tion is 2 ℏvk˜ 1 hðk 0 ;k 0 Þ k z x y and thus the parent Dirac Hamiltonian (2.2) is T 11 Hð Þ¼ † ; hðkx0 ;ky0 Þ −ℏvk˜ z1 symmetric − −ik − ikx0 0 − 0 y0 e e iu1 it1e u2 t2e ˆ x þ y ˆ −1 hðkx0 ;ky0 Þ¼ ; ð2:27Þ TH −k; r þ T ¼ H ðk; rÞ; ð2:24Þ iky0 0 0 ikx0 Dirac 2 Dirac −u2 þ t2e −iu1 þ it1e

ˆ where the 2 × 2 identity matrix 1 and hðkx0 ;ky0 Þ acts on the for T ¼ isyK. Last, the third property of (2.22) entails the sublattice μ ¼ A, B d.o.f., and −π ≤ kx0 ;ky0 ≤ π are the mass function mðrÞ¼−mðC2rÞ is odd under C2, and consequently the parent Dirac Hamiltonian is (screw) rotated momenta. We perturb about the Dirac fixed point by Δ rotation symmetric introducing the dimerizations j ˆ k r ˆ −1 k r 0 − Δ 0 − Δ C2HDiracðC2 ;C2 ÞC2 ¼ HDiracð ; Þ; ð2:25Þ tj ¼ tj ¼ uj j ¼ uj j ð2:28Þ

ˆ −ikza=2 where C2 ¼ iszμz (or microscopically e iszμz) anti- for j ¼ 1, 2. About the Γ ¼ð0; 0; 0Þ point, μ μ commuting with the mass terms m1 x þ m2 y in HDirac [see Γ δk ℏ˜δ σ − δ σ − δ σ μ (2.2)], and C2k ¼ð−kx; −ky;kzÞ, C2r ¼ð−x; −y; zÞ. Hð þ Þ¼ v kz z t1 kx0 x t2 ky0 y x Remembering that the coupled-wire model (2.12) 2 − Δ1σ μ þ Δ2σ μ þ Oðδk Þ: ð2:29Þ (Fig. 6) descended from a vortex lattice of the microscopic y z y y r parent Dirac Hamiltonian (2.2), the Dirac mass mð Þ See Fig. 10 for its massive spectrum. actually allows the model to carry fewer symmetries than Here, the AFTR symmetry T 11 and the twofold rotation the low-energy effective Hamiltonian (2.12) suggests. Now C2 are represented in the single-body picture by that the translation symmetry is lowered, the BZ is reduced 0 1 [see Fig. 9(b)] so that the two Weyl points now coincide at 00−eikx 0 the origin Γ. This recovers an unanomalous Dirac semi- B C B 00 0 −1 C metallic model (2.1) around ðkx0 ;ky0 Þ¼ð0; 0Þ. The four- k B CK T11ð Þ¼@ A ; fold degenerate Dirac point is protected and pinned at Γ due 10 0 0 T — to the remaining AFTR symmetry 11 which takes the 0 eikx 00 ˆ 2 −1 0 1 role of a spinful time reversal (T ¼ ) in the continuum i 000 — limit and the C2 (screw) symmetry about the z axis. B − C B 0 ie iðkxþkyÞ 00C However, if any of these symmetries are further broken, the k B C C2ð Þ¼ − ð2:30Þ fourfold degeneracy of the Dirac point is not protected @ 00−ie ikx 0 A − [cf. the original continuum Dirac model (2.2)]. Figure 9(a) 00 0−ie iky shows a dimerized coupled Dirac wire model that introduces a finite mass for the Dirac fermion. We label the −ikza=2 μ;σ (again suppressing the C2 screw phase e in the Dirac fermion operators as ψ r;s , for σ ¼⊙; ⊗ the chirality, continuum limit a → 0). In the small kx, ky limit, T11ð0Þ¼ μ ¼ A, B the new sublattice label, and ðr; sÞ label the −iσ K and C2ð0Þ¼iσ . It is straightforward to check that coordinates of the unit cell according to the 45°-rotated x0, y z y0 axes. X X 0 μ;⊙† μ;⊙ μ;⊗† μ;⊗ H ¼ ℏv˜ðψ r;s kzψ r;s − ψ r;s kzψ r;s Þ r;s μ¼A;B A;⊙† A;⊗ 0 B;⊙† B;⊗ þ iu1ψ r;s ψ r;s − iu1ψ r;s ψ r;s þ H:c: B;⊙† A;⊗ 0 A;⊙† B;⊗ − u2ψ r;s ψ r;s þ u2ψ r;s ψ r;s þ H:c: − ψ A;⊙ †ψ A;⊗ 0 ψ B;⊙ †ψ B;⊗ it1 r−1;s r;s þ it1 rþ1;s r;s þ H:c: ψ B;⊙ †ψ A;⊗ − 0 ψ A;⊙ †ψ B;⊗ þ t2 r;sþ1 r;s t2 r;s−1 r;s þ H:c: ð2:26Þ

For instance, the model is identical to the AFTR and FIG. 10. Dirac mass gap 2jΔj introduced by AFTR and C2 0 0 Δ Δ Δ C2 symmetric one in (2.12) when tj ¼ tj ¼ uj ¼ uj for symmetry-breaking dimerization ¼ 1 þ i 2.

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(a) (b) about zw, and when cleaved along a 3D hypersurface normal to w, it generates the array of alternating chiral Dirac channels in Fig. 6. The 4D WTI model can also be regarded as a stack of 3D antiferromagnetic topological insulators (AFTI) [165]. Restricting to the 3D hyperplane normal to −ex þ ey, this model consists of alternating Chern insulating layers e e FIG. 11. (a) Dimerized model of a massive Dirac fermion. parallel to the wz plane stacked along the x þ y direction. Z (b) Vortex of dimerizations Δ ¼ Δ1 þ iΔ2 that leaves behind a This 3D model describes an AFTI with a nontrivial 2 massless localized chiral Dirac channel (blue dot). index. For instance, along the boundary surfaces normal to w or z that preserve the antiferromagnetic symmetry T 11, the model leaves behind a 2D array of alternating chiral the dimerization Δ2 preserves T 11 while both Δ1, Δ2 Dirac wires. The uniform nearest wire backscattering term break C2. t1 [see Eq. (2.12)] introduces a linear dispersion along the Since the coupled-wire model (2.29) and the parent 11 direction and gives rise to a single massless surface continuum Dirac model (2.2) have the same matrix and Dirac cone spectrum at a TRIM on the boundary of the symmetry structure, we can apply the same construction we 2 surface BZ where T 11 ¼ −1. The 4D WTI model is discussed before to the new coarse-grained model (2.29). 11¯ For instance, the noncompeting dimerizations ΔðrÞ¼ identical to stacking these 3D AFTI along the -off- diagonal direction −e þ e . A more detailed discussion on Δ1ðrÞþiΔ2ðrÞ can spatially modulate and form vortices x y in a longer length scale. Figure 11(b) shows a dimerization coupled-wire constructions of a 4D strong and weak pattern that corresponds to a single vortex in Δ. The solid topological insulator can also be found in Ref. [166]. (dashed) lines represent strong (respectively, weak) backscattering amplitudes. In the fully dimerized limit where the B. Surface Fermi arcs dashed bonds vanish, all Dirac channels are gapped except 1. AFTR breaking surfaces the one at the center (showed as a blue dot). In the weakly dimerized case, there is a collective chiral Dirac channel We discuss the surface states of the coupled Dirac wire whose wave function is a superposition of the original model (2.12). Similar to the boundary surface of a trans- channels and is exponentially localized at the Δ-vortex lation symmetry protected Dirac semimetal (or more core, but now with a length scale longer than that of the commonly called a Weyl semimetal), there are Fermi arcs original m-vortex lattice. These collective chiral Dirac Δ connecting the surface-projected Weyl points [3,6,24]. vortices can themselves form a coupled array, like (2.12), First, we consider the (100) surface normal to x axis and give a Dirac semimetal of even longer length scale. The (see Fig. 6). We assume the boundary cuts between unit ε 0 0 single-body coupled vortex construction is therefore a cells and set the Fermi energy at f ¼ .Atkz ¼ and ∈ −π π coarse-graining procedure that recovers equivalent emer- given a fixed ky ð ; Þ, the tight-binding model (2.13) gent symmetries at each step. is equivalent to the Su-Schriffer-Heeger model [167] or a 1D class AIII topological insulator [168,169] along the x massvortices direction protected by the chiral symmetry σzHðkxÞ¼ Dirac semimetal ⇄ chiral Dirac strings ð2:31Þ − σ coupled-wire model HðkxÞ z. It is characterized by the winding number Z π i 1 ∂gðkx;kyÞ wðkyÞ¼ dkx 2. Holographic projection from 4D 2π −π gðkx;kyÞ ∂kx The coupled-wire model (2.12) with two AFTR axes can ¼½1 þ sgnðkyt1=t2Þ=2: ð2:32Þ be supported by a weak topological insulator (WTI) in four dimensions. Instead of realizing the chiral Dirac channels When t1, t2 have the same (or opposite) sign, the quasi-1D using mass vortices of a 3D Dirac semimetal, they can be model is topological along the positive (respectively, generated as edge modes along the boundaries of 2D Chern negative) ky axis and thus carries a boundary zero mode. insulators (or lowest Landau levels). The 4D WTI is This corresponds to the Fermi line joining the two surface- constructed by stacking layers of Chern insulators parallel projected Weyl points at Γ¯ and M¯ [see Fig. 12(a)]. As the to the zw plane along the x and y directions. The Chern zero modes have a fixed chirality according to σ , they r z layers Lr, labeled by the checkerboard lattice vector ¼ propagate unidirectionally with the dispersion Eðk Þ¼ e e z rx x þ ry y on the xy plane, have alternating orientations ℏvk˜ σ . The cleaving surface breaks AFTR and C2 sym- 1 −1 z z so that Ch½Lr¼ if rx, ry are integers and Ch½Lr¼ if metries, and so does the Fermi arc in Fig. 12(a).For rx, ry are half-integers. The model therefore carries both instance, any one of the AFTR symmetries maps the AFTR symmetries T 11 and T 11¯ as well as the C2 rotation boundary surface to an inequivalent one that cuts through

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(a) (b) (a) (b)

FIG. 13. Fermi arcs (blue lines) on the (001) surface with alternative boundary conditions (a) gðkx;kyÞ¼−it1 and FIG. 12. Fermi arcs (blue lines) joining projected Weyl points − − iky 2 on the surface Brillouin zones along (a) the (100) surface and (b) gðkx;kyÞ¼ t2e in the insulating domain, for t2=t1 ¼ . (b) the (001) surface. ˆ fixed kx, ky, the differential operator Hðkx;kyÞ is identical unit cells instead of between them. As a result, the Fermi to the Jackiw-Rebbi model [170]. Deep in the insulator, arc will connect the Weyl points along the opposite side of gðkx;ky;z→ −∞Þ¼it1. There is an interface zero mode the ky axis for this surface. at the surface domain wall if g changes sign, i.e., if gðkx; iφ The (010) surface Fermi arc structure is qualitatively ky;z→ ∞Þ¼jgje has argument φ ¼ −signðt1Þπ=2. 110¯ equivalent to that of the (100) surface. The (110) and ð Þ When εf ¼ 0, the zero modes trace out a Fermi arc that surfaces that cleave along the diagonal and off-diagonal connects the two surface-projected Weyl points [see axes (see Fig. 6), respectively, preserve the AFTR sym- Fig. 12(b)]. metries T 11 and T 11¯ . There are no protected surface Fermi We notice that in the insulating phase (or on the arcs because the two bulk Weyl points project onto the boundary surface), Dirac wires can be backscattered with same point on the surface Brillouin zone. Last, we consider a different phase and dimerized out of the unit cell. These the (001) surface normal to the z axis, which is the direction different boundary conditions correspond to distinct sur- of the chiral Dirac strings that constitute the coupled-wire face Fermi arc patterns. Figure 13 shows two alternatives. model. A chiral Dirac channel cannot terminate on the (a) shows the zero-energy arcs when intracell backscatter- boundary surface. In a single-body theory, it must bend ing reverses sign t1 → −t1 in the insulating domain. and connect with an adjacent counterpropagating one. (b) shows a case when the dimerization is taken along Although the (001) plane is closed under the C2 as well the off-diagonal axis. These inequivalent boundary con- as both the AFTR symmetries, the surface bending of Dirac ditions differ by some three-dimensional integer quantum channels must violate at least one of them. Here, we Hall states, which correspond to additional chiral Fermi consider the simplest case where the counterpropagating arcs that wrap nontrivial cycles around the 2D toric surface pair of Dirac channels within a unit cell reconnects on the Brillouin zone. boundary surface. This boundary is equivalent to a domain wall interface separating the Dirac semimetal (2.12) from 2. AFTR preserving surfaces an insulator where Dirac channels backscatters to their We also notice that the Fermi arc structures in Figs. 12(b) counterpropagating partner within the same unit cell. and 13 are allowed because both the AFTR symmetries The domain wall Hamiltonian takes the form of a T 11, T 11¯ and the C2 symmetry are broken by the insulating differential operator domain. Any dimerization that preserves only one of T 11 X and T ¯ necessarily breaks translation symmetry, and Hˆ − ℏ˜ ψ ⊙ †∂ ψ ⊙ − ψ ⊗ †∂ ψ ⊗ 11 ¼ i vð m;j z m;j m;j z m;jÞ corresponds to an enlarged unit cell and a reduced m;j Brillouin zone (cf. Figs. 8 and 9). As a result, the two ψ ⊙ †ψ ⊗ θ ψ ⊙ †ψ ⊗ Γ¯ þ it1ð m;j m;j þ ðzÞ m−1;j−1 m;jÞþH:c: Weyl points would now collapse onto the same point. ⊙ † ⊗ ⊙ † ⊗ Any momentum plane that contains the kz direction and þ t2θðzÞðψ ψ þ ψ ψ ÞþH:c: ð2:33Þ m−1;j m;j m;j−1 m;j avoids the Γ point must have trivial Chern invariant, because it could always be deformed (while containing by replacing k ↔ −i∂ in (2.12). Here, θ z can be the z z ð Þ the k direction and avoiding the Γ point) to the reduced unit step function or any function that asymptotically z Brillouin zone boundary, where its Chern invariant would z → ∞ z → −∞ approaches 1 for or 0 for . The model be killed by the AFTR symmetry. therefore describes the Dirac semimetal (2.12) for positive However, the trivial bulk Chern invariant does not imply z, and an insulator for negative z where Dirac channels are the absence of surface state. This can be understood by pair annihilated within a unit cell by t1. After a Fourier ˆ looking at the surface boundary in real space. Here, we transformation, the Bloch Hamiltonian Hðkx;kyÞ is iden- assume the Dirac strings that constitute the coupled-wire ↔ − ∂ tical to (2.13) by replacing kz i z and gðkx;ky;zÞ¼ model (2.12) are supported by vortices of an underlying −iðk þk Þ −ik −ik it1½1 þ θðzÞe y x þt2θðzÞðe x þ e y Þ. Given any Dirac mass [see Fig. 5 and Eq. (2.2)]. The semimetallic

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anomalous. This means that there is no single-body energy gap opening mass term that preserves the symmetries, and there is no single-body fermionic lattice model in two dimensions that supports a massless Dirac fermion without breaking the symmetries. Neither of these statements hold true in the many-body setting. The surface Dirac fermion can acquire a TR and charge Uð1Þ preserving many-body interacting mass [114–117]. Consequently, this also enables a massless symmetry-preserving Dirac fermion in a pure 2D system without holographically relying on a semi- infinite 3D topological bulk. For instance, one can take a quasi-2D topological insulator slab with finite thickness and remove the Dirac fermion on one of the two surfaces by FIG. 14. Surface chiral Dirac channels of the coupled-wire introducing an interacting mass gap. This leaves a single model (2.12) terminated along the xy plane. massless Dirac fermion on the opposite surface without breaking symmetries. coupled-wire model terminates along the xy plane against A massless Dirac fermion in three-dimensional semi- vacuum, which is modeled by the Dirac insulator metallic materials can be protected in the single-body ℏ k ⃗μ μ 0 1 Hvacuum ¼ v · s z þ m0 x, say, with m0 > . Recall picture by screw rotation, time-reversal, and charge Uð Þ from (2.23) that the Dirac mass vortex configuration symmetries (see reviews, Refs. [3,6,9] and Sec. II). From a (2.6) is AFTR symmetric along the T 11 directions. The theory point of view, it can be supported on the 3D Dirac insulating vacuum is symmetric under local TR as boundary of a 4D weak topological insulator, where the well as continuous translation. It, however, breaks the two Weyl fermions are located at distinct time-reversal ˆ screw rotation symmetry C2 ¼ iszμz, but we here only invariant momenta (recall Fig. 7 and Sec. II A 2 for the focus on the AFTR symmetry. antiferromagnetic case). In this case, the massless fermions The surface boundary supports chiral Dirac channels that are protected by translation, time-reversal, and charge Uð1Þ connect the chiral Dirac strings in the semimetallic bulk symmetries. In this section, we address the following that are normal to the surface. The surface channels are issues. (1) We show by explicitly constructing an exactly shown in Fig. 14. The ( ) represent chiral vortices in the solvable coupled-wire model that the 3D Dirac fermion bulk that direct electrons away from (respectively, onto) the can acquire a many-body interacting mass while preserving surface. The vector field represents the Dirac mass mðrÞ¼ all symmetries. (2) We show in principle that an AFTR mxðrÞþimyðrÞ modulation in the semimetallic bulk near symmetric massless 3D Dirac system with two Weyl the surface. The surface Dirac line channels [160]—shown fermions separated in momentum space can be enabled by directed lines connecting the bulk Dirac strings , —are by many-body interactions without holographically relying on a higher-dimensional topological bulk. located where the TR symmetric Dirac mass mx changes sign across the surface boundary and the TR breaking Dirac We begin with the Dirac semimetallic coupled-wire model in Fig. 7 and Eq. (2.12). In particular, we focus mass my flips sign across the line channels along the surface. In other words, they are traced out of points on the on many-body interactions that facilitate the fractionalization of a ð1 þ 1ÞD chiral Dirac channel surface where mx < 0 and my ¼ 0. Each of these surface channels carries a chiral Dirac electronic mode that connects the bulk chiral Dirac vortices. They can couple Dirac ¼ Pfaffian ⊗ Pfaffian ð3:1Þ through interchannel electron tunneling, but the collective gapless surface state cannot be removed from low energy (see also Fig. 15). In a sense, each chiral Pfaffian channel by dimerization without breaking the AFTR symmetry T 11. carries half of the d.o.f. of the Dirac. For instance, it has half the electric and thermal conductances, which are charac- III. MANY-BODY INTERACTING VARIATIONS terized by the filling fraction ν ¼ 1=2 and the chiral central charge c ¼ 1=2 in (2.4). Throughout this paper, we refer to We discuss the effect of strong many-body interactions the low-energy effective CFT—that consists of an electri- in a Dirac semimetal in three dimensions. Before we do so, cally charged Uð1Þ4 bosonic component, say, moving in the it is worth stepping back and reviewing the two- R direction, and a neutral Majorana fermion component dimensional case in order to illustrate the issue and idea moving in the opposite L direction—simply as a Pfaffian that will be considered and generalized in three dimensions. CFT The massless Dirac fermion with H ¼ ℏvðkxsy − kysxÞ that appears on the surface of a topological insulator [6,12–14] 1 ⊗ is protected by TR and charge Uð1Þ symmetries and is Pfaffian ¼ Uð Þ4 Ising: ð3:2Þ

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Pfaffian governed by the chiral central charge c ¼ 1 − 1=2 ¼ Dirac Dirac 1=2. The chiral ð1 þ 1ÞD PHS Pfaffian CFT (3.2) is also Pfaffian present along the line interface separating a TR symmetric T -Pfaffian [116] domain and a TR breaking magnetic domain on the surface of a 3D topological FIG. 15. Gluing and splitting a pair of chiral Pfaffian 1D channels into and from a chiral Dirac channel. insulator. (Similar constructions can be applied to alternative TR symmetric topological insulator surface states [114,115,117], but they will not be considered in this paper.) Other than their thermal transport properties, the 1 [In this paper, we follow the level convention for Uð Þ in three Pfaffian FQH state can also be distinguished by the the CFT community [171]. The same theory may be more charge e=4 Ising anyon, which has spin h ¼ 1=8, −1=8,or 1 commonly referred to as Uð Þ8 in the fractional quantum 0 for the Moore-Read Pfaffian, anti-Pfaffian, or PHS Hall community. For clarification, see Lagrangian (3.3) Pfaffian states, respectively. and (3.4).] Since we will not be considering the Moore-Read While this is not the focus of this paper, here we clarify Pfaffian or its particle-hole conjugate anti-Pfaffian state, and disambiguate the three Pfaffian fractional quantum Hall we will simply refer to the PHS Pfaffian state as the Pfaffian (FQH) states that commonly appear in the literature. All state. The low-energy effective chiral ð1 þ 1ÞD CFT takes these ð2 þ 1ÞD states are theorized at filling fraction the decoupled form between the boson and fermion ν ¼ 1=2, although being applied to ν ¼ 5=2 in materials, L L L and have identical electric transport properties. However, Pfaffian ¼ charged þ neutral they have distinct thermal Hall transport behaviors. They 8 all have very similar anyonic quasiparticle (QP) structures. ∂ ϕ ∂ ϕ ∂ ϕ 2 γ ∂ − ˜∂ γ ¼ 2π t R x R þ vð x RÞ þ i Lð t v xÞ L; For instance, they all have four Abelian and two non- Abelian QP (up to the electron). On the other hand, the ð3:3Þ charge e=4 non-Abelian Ising anyons of the three states ℏ 1 ϕ 1 have different spin-exchange statistics. First, the gapless where we have set ¼ . Here, R is the free chiral Uð Þ4 1 1 L boundary of the Moore-Read Pfaffian FQH state [172–175] boson. It generates the ð þ ÞD theory charged, which is 2 1 can be described by the ð1 þ 1ÞD chiral CFT Uð1Þ4 ⊗ identical to the boundary edge theory of the ð þ ÞD Ising where the charged boson and neutral fermion sectors bosonic Laughlin ν ¼ 1=8 fractional quantum Hall state are copropagating. It therefore carries the chiral central described by the topological Chern-Simon theory charge c ¼ 1 þ 1=2 ¼ 3=2, which dictates the thermal Hall [178,179] response (2.4). Second, the “anti-Pfaffian” FQH state [176,177] is the particle-hole conjugate of the Moore- L K α ∧ α α ∧ 2þ1 ¼ d þ et dA ð3:4Þ Read Pfaffian state. Instead of half-filling the lowest 4π Landau level by electrons, one can begin with the com- with K ¼ 8 and t ¼ 2. The Uð1Þ4 CFT carries the electric pletely filled lowest Landau level, and half-fill it with holes. conductance σ ¼ tK−1t ¼ 1=2 in units of 2πe2 ¼ e2=h and In a sense the anti-Pfaffian state is obtained by subtracting a thermal conductance characterized by the chiral central the completely filled lowest Landau level by a Moore-Read charge c ¼ cR ¼ 1. Primary fields are of the form of Pfaffian state. Along the boundary, the ð1 þ 1ÞDCFT ϕ (normal ordered) chiral vertex operators ∶eim R∶, for m 1 ⊗ 1 ⊗ Uð Þ1=2 Uð Þ4 Ising consists of the forward propa- an integer, and carries charge q ¼ m=4 in units of e and 1 gating chiral Dirac Uð Þ1=2 sector that corresponds to the conformal scaling dimension (i.e., conformal spin) lowest Landau level, and the backward propagating Moore- h ¼ h ¼ m2=16. We summarize and abbreviate the oper- ¯ R Read Pfaffian Uð1Þ4 ⊗ Ising. Here, C can be interpreted as ator product expansion the time-reversal conjugate of the chiral CFT C. The ϕ ϕ ϕ 8 thermal transport is governed by the edge chiral central eim1 RðzÞeim2 RðwÞ ¼ eiðm1þm2Þ RðwÞðz−wÞm1m2= þ ð3:5Þ charge c ¼ 1–3=2 ¼ −1=2, which has an opposite sign from the filling fraction. Thus, unlike the Moore-Read by the Abelian fusion rule Pfaffian state, the net electric and thermal currents now ϕ ϕ ϕ travel in opposite directions along the edge. Last, the eim1 R × eim2 R ¼ eiðm1þm2Þ R ; ð3:6Þ recently proposed particle-hole symmetric (PHS) Pfaffian state [154–156], which is going to be the only Pfaffian FQH where z ∼ τ þ ix is the complex spacetime parameter and state considered in this paper (see Ref. [133] for a coupled- τ ¼ iπvt=2 is the Euclidean time. γ† γ wire construction), has the chiral edge CFT (3.2). As the L ¼ L is the free Majorana fermion. It generates the 1 1 L electrically charged boson and neutral fermion sectors ð þ ÞD theory neutral, which is equivalent to a chiral are counterpropagating, the net thermal edge transport is component of the critical Ising CFT or the boundary edge

011039-14 FROM DIRAC SEMIMETALS TO TOPOLOGICAL PHASES … PHYS. REV. X 9, 011039 (2019) theory of the ð2 þ 1ÞD Kitaev honeycomb model [180] in X Y XY its B phase with TR breaking (i.e., a chiral px þ ipy superconductor coupled with a Z2 gauge theory). It carries e2πihZ = = = MXY e2πi(hX +hY ) trivial electric conductance but contributes to a finite Z thermal conductance characterized by the chiral central Z charge c ¼ −cL ¼ −1=2. The Ising CFT has primary fields Z 1, γ , and σ , where the twist field (or Ising anyon) σ L L L ð3:11Þ carries the conformal spin h ¼ −hL ¼ −1=16. Again, we abbreviate the operator product expansions for hX;Y;Z the conformal spins for primary fields X, Y, Z. π XY 2π Unlike the gauge-dependent -exchange phase RZ , the 1 2π − − monodromy phase MXY ¼ e iðhZ hX hY Þ is gauge inde- γLðz¯ÞγLðw¯ Þ¼ þ; Z z¯ − w¯ pendent and physical. σ ¯ ψ LðwÞ The electronic quasiparticle is the composition el ¼ σ ðz¯Þγ ðw¯ Þ¼ þ; − 4ϕ L L 1=2 i R γ −1 ðz¯ − w¯ Þ e L so that it is fermionic and has electric charge in 1 units of e. Since electron is the fundamental building block σ ¯ σ ¯ ¯ − ¯ 3=8γ ¯ ψ LðzÞ LðwÞ¼ 1=8 þðz wÞ LðwÞ of the system, locality of el only allows primary fields X ðz¯ − w¯ Þ ψ that have trivial monodromy MX; el ¼ 1 with the electron. As a result, this restricts 1m, ψ m to even m and σm to odd m. by the fusion rule Last, the coupled-wire models constructed later will involve the Pfaffian channels that propagate in both forward and backward directions. We will denote the backward case by γ γ 1 σ γ σ L × L ¼ ; L × L ¼ L; Pfaffian, whose Lagrangian density is the time reversal of (3.3), i.e., replacing R ↔ L, i ↔ −i and ∂t ↔ −∂t. σL × σL ¼ 1 þ γL; ð3:7Þ

A. Gluing and splitting where z¯ ∼ τ − ix is the complex spacetime parameter and “ ” τ ¼ ivt˜ is the Euclidean time. A pair of copropagating Pfaffian CFT can be glued General primary fields of the Pfaffian CFT decompose together into a single chiral Dirac electronic channel. We A B first consider the decoupled pair L0 L L , into the Uð1Þ4 part and the Ising part. They take the form ¼ Pfaffian þ Pfaffian LA=B where Pfaffian is the Lagrangian density of one of the two ϕ ϕ ϕ Pfaffian CFT labeled by A, B. The pair of Majorana 1 im R ψ im R γ σ im R σ m ¼ e ; m ¼ e L; m ¼ e L: ð3:8Þ fermions can compose an electrically neutral Dirac fermion pffiffiffi γA γB 2 dL ¼ð L þ i LÞ= , which can then be bosonized ϕσ σ ∼ i L ϕ 1 The conformal spins and fusion rules also decompose so dL e , for L the chiral Uð Þ1=2 boson. The bare A that Lagrangian now becomes the multi-component Uð1Þ4 ⊗ 1 B ⊗ 1 Uð Þ4 Uð Þ1=2 boson CFT m2 m2 1 m2 −1 h1 ¼ ;hψ ¼ þ ;hσ ¼ ð3:9Þ 1 m 16 m 16 2 m 16 L ∂ ϕT ∂ ϕ ∂ ϕT ∂ ϕ 0 ¼ 2π t K x þ x V x ; ð3:12Þ

ϕ ϕA ϕB ϕσ 3 3 modulo 1, qm ¼ m=4 in units of e, and where ¼ð R; R; LÞ, K is the × diagonal matrix K ¼ diagð8; 8; −1Þ, and V is some nonuniversal velocity matrix. A primary field is a vertex operator eim·ϕ labeled by 1 × 1 ψ × ψ 1 A B m1 m2 ¼ m1 m2 ¼ m1þm2 an integral vector m ¼ðm ;m ; m˜ Þ. It carries conformal T −1 T −1 1 ψ ψ spin hm m K m=2 and electric charge qm t K m m1 × m2 ¼ m1þm2 ; ¼ ¼ in units of e, where t ¼ð2; 2; 0Þ is the charge vector. As 1 × σ ¼ ψ × σ ¼ σ ; m1 m2 m1 m2 m1þm2 n ¼ð1; −1; 4Þ is an electrically neutral null vector (i.e., σ σ 1 ψ nT n 0 t n 0 m1 × m2 ¼ m1þm2 þ m1þm2 : ð3:10Þ K ¼ and · ¼ ), it corresponds to the charge Uð1Þ preserving backscattering coupling

The 2π monodromy phase MXY ¼ RXYRYX between pri- δH − nT ϕ − 8ϕA − 8ϕB − 4ϕσ Z Z Z ¼ u cos ð K Þ¼ u cos ð R R LÞ mary fields X and Y with a fixed overall fusion channel Z can be deduced by the ribbon identity [180] ð3:13Þ

011039-15 RAZA, SIROTA, and TEO PHYS. REV. X 9, 011039 (2019) that gaps [181] and annihilates a pair of counterpropagating boson modes. The interacting Hamiltonian can also be expressed in terms of many-body backscattering of the Pfaffians’ primary fields

δH − ∶ † 4∶ ¼ u ðdLdRÞ þ H:c:; ð3:14Þ where d ¼ 1A1B is the electrically neutral Dirac fermion R 2 −2 FIG. 16. Schematics of splitting a chiral Dirac channel into a composed of the pair of oppositely charged semions in the pair of Pfaffian channels. two Pfaffian sectors. In strong coupling, the gapping Hamiltonian introduces an interacting mass and the ground state expectation value σ introducing nonlinear dispersion to the original chiral hΦi¼nπ=2, for n an integer and Φ ¼ 2ϕA − 2ϕB − ϕ .In R R L channel. In low energy, the three Dirac fermion modes low energy, it leaves behind the chiral boson combination 1 2 1;2 ϕ˜ ; − ϕ˜ ψ ∼ i R ψ ∼ i L ϕ˜ 2ϕA 2ϕB can be bosonized R e , L e and they are R ¼ R þ R, which has trivial operator product (i.e., commutes at equal time) with the order parameter Φ. The described by the multicomponent boson Lagrangian low-energy theory after projecting out the gapped sectors 1 becomes L˜ ∂ ϕ˜ T ˜ ∂ ϕ˜ ∂ ϕ˜ T ˜ ∂ ϕ˜ Dirac ¼ 2π t K x þ x V x ð3:18Þ 1 ˜ ˜ ˜ 2 ˜ ˜ 1 ˜ 2 ˜ L0 − δH → L ∂ ϕ ∂ ϕ v ∂ ϕ : ϕ ϕ ϕ ϕ ˜ ˜ Dirac ¼ 2π t R x R þ ð x RÞ ð3 15Þ for ¼ð R; R; LÞ, K is the diagonal matrix K ¼ diagð1; 1; −1Þ, and V˜ is some nonuniversal velocity matrix. which is identical to the bosonized Lagrangian density of a A general composite excitation can be expressed by a ˜ ψ el ∼ eim·ϕ m chiral Dirac fermion. For instance, the vertex operator R vertex operator , for an integral 3 vector, with spin ϕ˜ m 2 2 mT ˜ t˜ ei R ∼ 1A1B has the appropriate spin and electric charge of hm ¼j j = and electric charge qm ¼ K in units of e, 2 2 t˜ 1 1 1 an electronic Dirac fermion operator (h ¼ 1=2 and q ¼ 1 in where ¼ð ; ; Þ is the charge vector. ϕ˜ 2 Next we perform a fractional basis transformation units of e). Notice that the vertex operator ei R= has −1 ψ el monodromy with the local electronic R and therefore is ϕρ ϕ˜ 1 ϕ˜ 2 ϕ˜ not an allowed excitation in the fermionic theory. R ¼ R þ R þ L 1 1 We notice in passing that the gluing potential (3.13) σ 1 2 ϕ ¼ ϕ˜ − ϕ˜ þ ϕ˜ facilitates an anyon condensation process [182], where the R R 2 R 2 L maximal set of mutually local neutral bosonic anyon pairs 1 3 ϕσ ϕ˜ 1 ϕ˜ 2 ϕ˜ L ¼ R þ 2 R þ 2 L: ð3:19Þ A B A B 14 1−4 ; ψ 4 ψ −4 ; m m m m ˜ ψ A 1B 1A ψ B σA σB While the K matrix is invariant under the transformation, 4mþ2 −4m−2; 4mþ2 −4m−2; 4mþ1 −4m−1 ð3:16Þ ρ ϕρ t˜ → 1 0 0 ψ ∼ i R the charge vector changes to ð ; ; Þ. R e is the 1 2 is condensed, where m is an arbitrary integer. All primary local electronic Dirac fermion that carries spin = and iϕσ ∼ R=L fields that are nonlocal (i.e., with nontrivial monodromy) electric charge e, and dR=L e are counterpropagating with any of the condensed bosons in (3.16) are confined. electrically neutral Dirac fermions. As the K˜ matrix is still Any two primary fields that differ from each other by a diagonal, these fermions have trivial mutual 2π mono- condensed boson in (3.16) are now equivalent. The con- dromy and are local with respect to each other. However, it densation therefore leaves behind the electronic Dirac is important to notice that the neutral Dirac fermions dR=L fermion actually consist of fractional electronic components. Now, we focus on the two R-moving Dirac channels. By ψ el ψ A ≡ ψ B ≡ 1A1B R ¼ 4 4 2 2 ð3:17Þ pairing the Dirac fermions, they form two independent SUð2Þ1 Kac-Moody current operators [171] and its combinations. pffiffiffi On the other hand, a chiral Dirac channel can be A=B A=B J3 ðzÞ¼i2 2∂zϕ ðzÞ decomposed into a pair of chiral Pfaffian channels (see R A=B A=B Fig. 16 for a summary). First, perhaps from some channel A=B J1 ðzÞiJ2 ðzÞ i4ϕA=BðzÞ J ðzÞ¼ pffiffiffi ¼ e R ð3:20Þ reconstruction, we append to the chiral Dirac channel an 2 additional pair of counterpropagating Dirac modes. This 4ϕA ϕρ ϕσ 4ϕB ϕρ − ϕσ 2 can be realized by pulling a parabolic electronic-hole band where R ¼ R þ R and R ¼ R R. Both SUð Þ1 from the conduction-valence band to the Fermi level, or sectors are electrically charged so that the bosonic vertex

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A=B 2 separate the pair. This issue is addressed in the subsection operators J carries charge e. They obey the SUð Þ current algebra at level 1 below using many-body interaction in the coupled-wire model of a Dirac semimetal (or the PHS Pfaffian FQH state ffiffiffi λλ0 X3 p λλ0 in Ref. [133]). δ δij i 2δ εij JλðzÞJλ0 ðwÞ¼ þ k JλðwÞþ ð3:21Þ i j ðz−wÞ2 z−w k k¼1 B. Symmetry-preserving massive interacting model λ λ0 A ∼ ψ ρ We begin with the 3D array of chiral Dirac strings in for ; ¼ A, B. It is crucial to remember that J RdR ρ † Fig. 5. In Sec. II, we showed that the single-body coupled- and JB ∼ ψ d contains the fractional Dirac components R R wire model (2.12) described a Dirac semimetal with two d . Thus, the primitive local bosons are actually pairs of the R Weyl fermions (see Fig. 7). The system had emergent i8ϕA=B current operators, i.e., e R . Equivalently, this renorm- AFTR symmetries T 11 and T 11¯ along the diagonal and off- 4ϕA=B alizes the compactification radius of the boson R so diagonal axes [see Eq. (2.17)]. Together, they generate an that in a closed periodic spacetime geometry, we only emergent lattice translation symmetry with a two-wire unit require electronic Cooper pair combinations such as the cell, and separate the two Weyl points in the Brillouin zone. charge 2e local operators The symmetries are lowered beyond the effective model when the microscopic high-energy d.o.f. are included. For 8ϕA 4ϕ˜ 1 ϕ˜ 2 3ϕ˜ 1 4 2 † 3 i R ið Rþ Rþ LÞ ∼ ψ ψ ψ example, the mass function (2.6) that supports the Dirac e ¼ e ð RÞ Rð LÞ ; vortex string lattice has a four-wire periodic unit cell and 8ϕB 3ϕ˜ 2 ϕ˜ 2 3 † i R ið Rþ LÞ ∼ ψ ψ e ¼ e ð RÞ L ð3:22Þ only preserves one of the AFTR symmetries T 11 [see Eq. (2.23)]. With the lowered translation symmetry, the to be periodic. The incorporation of antiperiodic boundary two Weyl points now coincide at the same momentum. A=B 4ϕA=B i R Z condition for J ¼ e results in the 2-orbifold Interspecies (or intervalley) mixing is forbidden by the theory [183,184] Uð1Þ4 ¼ SUð2Þ1=Z2 for both A and B remaining AFTR symmetry and a (screw) twofold rotation sectors. For instance, the primitive twist fields are given by symmetry C2 about z [see Eqs. (2.18) and (2.25)]. ϕA=B A=B i R −1 Previously in Sec. II A 1, we introduced symmetry-break- e , which have monodromy phase with J . At this point, including the L-moving neutral Dirac ing wire dimerizations in (2.26) that led to a massive Dirac insulator. In this section, we construct many-body gapping sector, we have recovered the muticomponent boson ϕ ¼ ϕA ϕB ϕσ interactions that preserve the two AFTR symmetries ð R; R; LÞ described by the Lagrangian (3.12). Last, we T 11 and T 11¯ , the C2 symmetry, as well as charge Uð1Þ simply have to decompose the remaining neutral Dirac into pffiffiffi conservation. γA γB 2 Majorana components, dL ¼ð L þ i LÞ= . The A and B The many-body gapping scheme is summarized in Pfaffian sectors can then be independently generated by the Fig. 17. From the previous subsection, we saw that each 1 ϕA=B charged Uð Þ4 boson R and the neutral Majorana chiral Dirac channel can be decomposed into a pair of γA=B independent Pfaffian channels. They can then be back- fermion L . As a consistency check, the charge e fermionic (normal ordered) combinations defined in (3.8) scattered in opposite directions to neighboring wires. Figure 17(a) shows a particular dimerization pattern of A i4ϕA A iϕ˜ 1 ið3ϕ˜ 1 þϕ˜ 2 þ3ϕ˜ Þ the Pfaffian channels that preserves the symmetries. In this ψ 4 ∼ e R γ ∼ e R þ e R R L ; L case, the many-body backscattering interaction U is 4ϕB −ϕ˜ 1 ϕ˜ 2 −ϕ˜ ϕ˜ 1 2ϕ˜ 2 2ϕ˜ ψ B ∼ i R γB ∼ ið Rþ R LÞ − ið Rþ Rþ LÞ 4 e L e e ð3:23Þ directed along the diagonal axis. In the limit when U is much stronger than the single-body electron tunneling in are in fact local quasielectronic. the previous semimetallic model (2.12), the system decom- Unlike in the gluing case where there is a gapping poses into decoupled diagonal layers and it suffices to Hamiltonian (3.13) that pastes a pair of Pfaffians into a consider the interaction on a single layer. For convenience, Dirac, here in the splitting case we have simply performed we here change our spatial coordinates so that the diagonal some kind of fractional basis transformation that allows us axis is now labeled by y and the wires now propagate to express Dirac as a pair of Pfaffians. In fact, one can check along x. that the energy-momentum tensor of the Dirac theory Focusing on a single diagonal layer, the system in the (3.18) is identical to that of a pair of Pfaffians (3.3). noninteracting limit first consists of a 2D array of chiral However, this does not mean the Pfaffian primary fields are Dirac strings with alternating propagating directions [see natural stable excitations. In fact, as long as there is a pair of the left side of Fig. 17(b)]. We notice that this is identical to copropagating Pfaffian channels, all primary fields except the starting point of the coupled-wire construction of the the nonfractionalized electronic ones are unstable against topological insulator Dirac surface state considered by the gluing Hamiltonian δH in (3.13) and are generically Mross, Essin, and Alicea in Ref. [152]. For instance, gapped. In order for the Pfaffian CFT to be stabilized, one the alternating Dirac channels there were supported has to suppress δH. A possible way is to somehow spatially between magnetic strips with alternating orientations on

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1 2 ˜ 1;2 ˜ 3 ; iϕy ðxÞ 3 −iϕyðxÞ ψ y ðxÞ ∼ e ; ψ yðxÞ ∼ e : ð3:24Þ

The sliding Luttinger liquid [120–124] Lagrangian density is

∞ X ð−1ÞyK˜ L jk ∂ ϕ˜ j ∂ ϕ˜ k ˜ ∂ ϕ˜ j ∂ ϕ˜ k layer ¼ 2π t y x y þ Vjk x y x y; ð3:25Þ y¼−∞

˜ ˜ 1 1 −1 ˜ where K ¼ðKjkÞ3×3 ¼ diagð ; ; Þ, V is some nonuniversal velocity matrix, and repeating species indices j, k are (a) summed over. The boson operators obey the equal-time commutation relation (ETCR)

ϕ˜ j ϕ˜ j0 0 ½ yðxÞ; y0 ðx Þ jj0 − 0 ¼ cyy0 ðx x Þ y ˜ jj0 0 y jj0 ¼ iπð−1Þ δyy0 K sgnðx − xÞþiπð−1Þ δyy0 S maxfy;y0g 0 jj0 y−y0þ1 þ iπð−1Þ sgnðy − y ÞΣ σz ; ð3:26Þ

where sgnðsÞ¼s=jsj¼1 for s ≠ 0 and sgnð0Þ¼0, Dirac Pfaffian 0 1 0 1 01−1 11−1 B C B C Pfaffian @ −10 1A Σ @ 11−1 A splitting S ¼ ; ¼ ; 1 −10 −1 −11 (b) ð3:27Þ FIG. 17. Symmetry-preserving many-body gapping interaction. (a) Each represents a chiral Pfaffian channel into or out of and σz ¼1. The introduction of the specific Klein factors paper. The purple dashed line represents many-body gapping Sjj0 , Σjj0 and the undetermined sign σ are necessary for the interaction U in (3.47). (b) Coupled-wire model on a single layer z correct representations of the T 11 and C2 symmetries in the along the diagonal axis. bosonization setting, and these choices will be justified below. The first line of (3.26) is equivalent to the commu- the topological insulator surface, and a uniform nearest- tation relation between conjugate fields channel electron tunneling recovered the massless 2D ϕ˜ j ∂ ϕ˜ j0 0 2π −1 yδ ˜ jj0 δ − 0 Dirac spectrum protected by the AFTR symmetry. They ½ yðxÞ; x0 y0 ðx Þ ¼ ið Þ yy0 K ðx x Þð3:28Þ then proceeded to propose symmetry-preserving many- “ _” L body gapping interactions facilitated by adding 2D FQH which is set by the pq term in layer. The alternating strips between the channels. While this reconstruction signs ð−1Þy in (3.28) and (3.25) change the propagating trick can be applied on the 2D surface of a topological directions from wire to wire. The second and third lines insulator, it is not feasible in our 3D situation and would of (3.26) guarantee the correct anticommutation relations ˜ j0 require drastic modification of the bulk semimetal. ϕ˜ j iϕ i y y0 0 Instead, here we propose an alternative gapping scheme fe ;e g¼ between Dirac fermions along distinct 0 0 ˜ that does not involve additional topological phases. In channels j ≠ j or distinct wires y ≠ y .ThereasontheC2 other words, we are going to construct a 3D gapped and matrix is defined in this form will become clear in the layered topological phase solely from interacting elec- fractional basis discussed later in (3.43). tronic Dirac wires. The antiunitary AFTR symmetry along the diagonal T 11 First, in order to implement the splitting described in the direction transforms the bosons according to previous subsection, we assume each Dirac string consists 1 −1 y of two Dirac channels going in one direction and a third ˜ j −1 ˜ j þð Þ ˜ jj T 11ϕ T −ϕ K π: 3:29 Dirac channel going the opposite direction [see the left side y 11 ¼ yþ1 þ 2 ð Þ of Fig. 17(b)]. We denote the electronic Dirac fermions on 1 2 3 the yth wire by ψy ¼ðψ y; ψ y; ψ yÞ and bosonize The unitary C2 rotation takes

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π ˜ j −1 ˜ j ˜ j0 y j C2ϕ˜ j C−2 ϕ˜ j −1 y ˜ jjπ C2ϕ C C2 ϕ− −1 v ; 2 yðxÞ 2 ¼ y þð Þ K ; ð3:38Þ y 2 ¼ð Þj0 y þð Þ 2 0 1 0 1 0 1 122 1 3 2 F v which is consistent with C2 ¼ð−1Þ . Last, it is straightfor- ˜ B C B 2 C B C C2 ¼ @ 212A; v ¼ @v A ¼ @ −3 A: ward to check that the symmetry representations (3.29) and (3.30) are compatible with the algebraic relation (2.10), i.e., −2 −2 −3 v3 1 ˜ j −1 −1 ð3:30Þ C2T 11ϕyT 11 C2 F −1 ˜ j −1 −F ¼ð−1Þ T 11 C2ϕyC2 T 11ð−1Þ : ð3:39Þ Moreover, we choose the representation so that the sign σz in the ETCR (3.26) is preserved by the AFTR operator but Following the splitting scheme summarized in Fig. 16, is flipped by the C2 symmetry, we again define a fractional basis transformation −1 −1 [cf. Eq. (3.19)] T 11σzT 11 ¼ σz; C2σzC2 ¼ −σz: ð3:31Þ 0 1 0 10 1 ϕρ ϕ˜ 1 The ETCR (3.26) is consistent with the AFTR symmetry. y c11 1 c y B C B CB C T ϕ˜ j ϕ˜ j0 0 T −1 @ ϕσ1 A @ 1 −1=21=2 A@ ϕ˜ 2 A 3:40 This means that evaluating 11½ yðxÞ; y0 ðx Þ 11 by y ¼ y ð Þ taking the AFTR operator inside the commutator ϕσ2 11232 ˜ 3 y = = ϕy

0 ρ T ϕ˜ j T −1 T ϕ˜ j 0 T −1 ρ iϕ ½ 11 yðxÞ 11 ; 11 y0 ðx Þ 11 for each wire, so that ψ y ∼ e y is a Dirac fermion carrying σ1 ϕσ1 σ2 ϕσ2 0 0 ∼ i y ∼ i y ϕ˜ j ϕ˜ j 0 jj − 0 electric charge e, dy e (dy e ) is an electrically ¼½ 1ðxÞ; 0 1ðx Þ ¼ c 1 0 1ðx x Þð3:32Þ yþ y þ yþ ;y þ neutral Dirac fermion propagating in the same (respec- ψ ρ yields the same outcome as taking the TR of the purely tively, opposite) direction as y. For convenience, sometimes we combine the trans- imaginary scalar 1 2 3 formed bosonized variables into ϕy ¼ðϕy; ϕy; ϕyÞ¼ 0 0 ϕA ϕB ϕσ2 T jj − 0 T −1 − jj − 0 ð y ; y ; y Þ, which is related to the original local ones 11cyy0 ðx x Þ 11 ¼ cyy0 ðx x Þ: ð3:33Þ ϕJ Jϕ˜ j in (3.25) by y ¼ Gj y where The ETCR (3.26) is also consistent with the C2 0 1 symmetry 1=21=83=8 B C G ¼ @ 03=81=8 A: ð3:41Þ 0 0 ˜ j1 j1j2 − ˜ j2 C j1j2 − C−1 ðC2Þ 0 c−y1;−y2 ðx1 x2ÞðC2Þ 0 ¼ 2cy1y2 ðx1 x2Þ 2 : 11232 j1 j2 = = ð3:34Þ The AFTR symmetry operation (3.29) becomes

This is because the Klein factors (3.27) are C2 symmetric 1 −1 y T ϕI T −1 −ϕI þð Þ πκI 11 y 11 ¼ yþ1 þ ; ð3:42Þ ˜ ˜ T ˜ ˜ T 2 C2SC2 ¼ S; C2ΣC2 ¼ Σ: ð3:35Þ κI I ˜ jj 1 4 1 where ¼ GjK which is = for I ¼ , 2 and 0 for Notice that the undetermined sign σz, which is odd under I ¼ 3. The C2 transformation (3.30) becomes C2,in(3.26) is essential for the ETCR to be consistent with C2. π C ϕI C−1 I ϕJ −1 y I j The last term in the AFTR operation (3.29) makes sure 2 y 2 ¼ðC2ÞJ −y þð Þ Gjv 2 ; 0 1 0 1 01 0 3=2 T 2 ϕ˜ j T −2 ϕ˜ j −1 y ˜ jjπ 11 yðxÞ 11 ¼ yþ2 þð Þ K ; ð3:36Þ ˜ −1 B C B C C2 ¼ GC2G ¼ @10 0A;Gv ¼ @ −1 A: T 2 −1 F 2e 00−1 3 which is necessary for 11 ¼ð Þ translation Pð yÞ. iπ Nj Here, the fermion parity operator is ð−1ÞF ¼ e yj y , ð3:43Þ where The 3 × 3 C2 matrix takes a much simpler form here using Z ˜ j dx ˜ j the fractional basis than in (3.30). In fact, the original C2 Ny ¼ ∂xϕyðxÞð3:37Þ 2π matrix in the local basis in (3.30) was defined so that C2 ¼ ˜ −1 GC2G would act according to (3.43). Roughly speaking, is the number operator. The vector v in the C2 operation ignoring the constant phases Gv, the C2 symmetry switches δj ˜ j j0 2 ˜ jj ϕA ↔ ϕB ϕσ2 → −ϕσ2 (3.30) satisfies ½ j0 þðC2Þj0 v = ¼ K , and, consequently, y −y and sends y −y.

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2π 2 3 Next, we combine this copropagating pair of fermions although e iðNyþNyÞ ¼ 1, one cannot in general modify a to form two SU 2 current algebras [cf. Eqs. (3.20) and 2 3 ð Þ1 boson angle parameter simply by Θ → Θ þ 2πiðNy þ NyÞ (3.21)] because Θ and the number operators may not commute. ffiffiffi A=B p A=B For instance, using the Baker-Campbell-Hausdorff formula J ðy; wÞ¼i2 2∂ ϕ ðwÞ; i4ϕA=B i4ϕA=B 2πi N2 N3 3 w y and the ETCR (3.26), e and e þ ð yþ yÞ are off A=B 4ϕA=B J ðy; wÞ¼ei y ðwÞ; ð3:44Þ by a minus sign. The Pfaffian fractionalization is stabilized by the where w ∼ τ þð−1Þyx is the complex spacetime parameter. interwire many-body backscattering interaction [see A=B Fig. 17(b)] As a reminder, the charge e bosons J are nonelec- tronic fractional operators, although they carry nonfrac- X∞ tional statistics. U − ϕσ2 ϕσ2 4ϕA − 4ϕB ¼ u cos yþ1 sin y cos ð yþ1 y Þ The remaining counterpropagating neutral Dirac fermion y¼−∞ can be decomposed into real and imaginary components X∞ y A B ¼ −u ð−1Þ iγ 1γ cosðΘ 1 2Þ; ð3:47Þ σ σ2 σ2 yþ y yþ = dyðwÞ ∼ cos ϕy ðwÞþi sin ϕy ðwÞ: ð3:45Þ y¼−∞

Θ 4ϕA − 4ϕB π 2 3 Majorana fermions can be constructed by multiplying these for yþ1=2ðxÞ¼ yþ1ðxÞ y ðxÞþ ðNyþ1 þ Nyþ1Þ. components with “Jordan-Wigner” string A Previously in (3.23), we saw that the combinations ψ 4 ∼ 4ϕA 4ϕB Y 2 3 i γA ψ B ∼ i γB σ2 N þN e and 4 e can be decomposed into products γA ∼ ϕ −1 y0 y0 y cos y ð Þ ; of electron operators. Similarly, each interaction in the first 0 y >y line of (3.47) can be decomposed into products in the form Y 2 3 ϕσ2 4ϕA σ2 B B σ2 N 0 þN 0 ið 1 1Þ iðϕ 4ϕ Þ γy ∼ sin ϕy ð−1Þ y y ; ð3:46Þ of e yþ yþ e y y [with some scalar Uð1Þ coef- σ2 y0>y ficient], where the exponents ϕ 4ϕA=B are linear integral combinations of ϕ˜ j. Thus, the interaction can be j where Ny are the number operators defined in (3.37),so rewritten in terms of backscatterings of local electronic γλ γλ0 0 that they obey mutual fermionic statistics f yðxÞ; y0 ðx Þg ¼ operators. However, we will omit the electronic expression λλ0 0 0 δ δyy0 δðx − x Þ, for λ; λ ¼ A, B. Similar to the charge e as (3.47) is more useful in discussing the ground state and A=B σ symmetries. bosons J , the electrically neutral Dirac fermion d and, y U describes a symmetry-preserving exactly solvable γA=B consequently, the Majorana fermions y are also non- model. Using the ETCR (3.26) it is straightforward to electronic fractional operators. This AB decomposition check that the (normal ordered) order parameters splits each Dirac wire into a pair of decoupled Pfaffian sectors [see Fig. 17(b)]. F A B Θ iΘ 1 2ðxÞ O 1 2ðxÞ¼iγ 1ðxÞγ ðxÞ; O 1 2ðxÞ¼e yþ = Before we move on to the symmetric interaction, some yþ = yþ y yþ = j further elaborations are needed for the number operators Ny ð3:48Þ π j and their corresponding fermion parity operators ei Ny .In OF=Θ OF=Θ 0 0 our construction, the counterpropagating pair of channels mutually commute, i.e., ½ yþ1=2ðxÞ; y0þ1=2ðx Þ ¼ . with j ¼ 2, 3 are appended to the original one with j ¼ 1 to Therefore, the model is exactly solvable, and its ground make the Pfaffian fractionalization feasible. We choose the states are characterized by the ground state expectation Hilbert space so that the two additional fermion parity values (GEV) of the order parameters π 2 π 3 operators agree, ei Ny ¼ ei Ny . However, we allow fluctua- Θ iπðN2þN3Þ OF −1 y O 1 tions to the combined parity e y y and only require it l0h yþ1=2i¼ð Þ h yþ1=2i¼ ð3:49Þ 2π 2 3 squares to the identity, e iðNyþNyÞ ¼ 1. In other words, π 2 3 − π 2 3 so that the interacting energy hUi is minimized, where l0 is ei ðNyþNyÞ ¼ e i ðNyþNyÞ and it does not matter which one 2 3 some nonuniversal microscopic length scale. Pinning the we take as ð−1ÞNyþNy in the Jordan-Wigner string in (3.46). GEV hΘ 1 2i¼n 1 2π, for n 1 2 ∈ Z, gaps all d.o.f. This convention will also be useful later in seeing that the yþ = yþ = yþ = 1 A=B 2 A=B many-body interaction is exactly solvable and symmetry in the charged Uð Þ4 ¼ SUð Þ1 sector. The remaining preserving. Extra care is sometimes required. For example, neutral fermions are gapped by the decoupled Majorana unlike the original Dirac channel where the parity is simply backscattering 1 π 1 2π 1 −1 Ny i Ny iNy 1 ð Þ ¼ e because e ¼ , the individual parity X∞ 2;3 operators −1 Ny of these additional channels are not well δH −1 y OΘ γA γB ð Þ Majorana ¼ u ð Þ ih yþ1=2i yþ1 y : ð3:50Þ 2π 2;3 π 2;3 − π 2;3 defined because e iNy ≠ 1, i.e., ei Ny ≠ e i Ny . Also, y¼−∞

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Θ Θ π Θ 1 2 O i yþ1=2 It is worth noting that a -kink excitation of h yþ = i flips boson order parameter yþ1=2 ¼ e is conjugated the Majorana mass in (3.50) and therefore bounds a zero- and flipped under C2 energy Majorana bound state [185].Aπ kink at x0 can be ϕA ϕB Θ −1 Θ † i yþ1ðx0Þ i y ðx0Þ C O C −O created by the vertex operators e or e which 2 yþ1=2 2 ¼ −y−1=2 : ð3:56Þ carry 1=4 of an electric charge. (Recall the bosonic 4ϕA=B vertices ei y carry charge e.) This e=4 excitation there- When combined together, the minus signs in (3.55) and fore corresponds to the Ising anyon in the Pfaffian FQH (3.56) cancel and they show that the many-body interaction U state. in (3.47) preserves C2. From the AFTR symmetry action (3.42), one can Now that we have introduced symmetry-preserving show that the Majorana fermions (3.46) transform accord- gapping interactions on a single diagonal layer, we can ing to extend it to the entire 3D structure by transferring (3.47) to all layers using the off-diagonal AFTR operator T 11¯ [see A −1 A B −1 B Fig. 17(a)]. The resulting state belongs to a topological T 11γy T 11 ¼ γ 1; T 11γy T 11 ¼ −γ 1: ð3:51Þ yþ yþ phase in three dimensions with an excitation energy gap. T T OF γA γB It preserves both AFTR symmetries 11 and 11¯ as well as Therefore, the fermion order parameter yþ1=2 ¼ i yþ1 y the (screw) C2 symmetry. The choice of writing this paper (3.48) is translated under the antiunitary symmetry with a specific model with these symmetries was inten-

F −1 F tional; we wanted to work out the simplest example with T 11O T O : 3:52 yþ1=2 11 ¼ yþ3=2 ð Þ specific symmetries explicitly for illustrative reasons instead of doing a more general classification type argu- Θ The boson angle parameter yþ1=2 defined below (3.47) ment. We leave the SPT-SET correspondences for general −Θ − −1 yπ changes to yþ3=2 ð Þ under AFTR, and therefore symmetries as an open question, but we expect that the Θ Θ O i yþ1=2 the boson order parameter yþ1=2 ¼ e is flipped and methods presented in this work can be useful in explor- translated ing them.

T OΘ T −1 −OΘ 11 yþ1=2 11 ¼ yþ3=2: ð3:53Þ C. Antiferromagnetic stabilization The exactly solvable many-body interacting model Together, Eqs. (3.52) and (3.53) show that the many-body (3.47) (see also Fig. 17) shows that the Dirac semimetal interaction U in (3.47) is AFTR symmetric. (2.12) can acquire a many-body mass gap without breaking The C2 action (3.30) flips the number operator 2 3 −1 2 3 symmetries. However, it is not clear how dominant or stable C2 N N C −N− − N− , and therefore the parity ð y þ yÞ 2 ¼ y y the topological phase described by (3.47) is. There are operators appearing in the Jordan-Wigner string (3.46) are 2 3 −1 2 3 alternative interactions that lead to other metallic or C −1 NyþNy C −1 N−yþN−y C2 symmetric, 2ð Þ 2 ¼ð Þ . With the insulating phases that preserve or break symmetries. The help of the C2 action (3.43) in the fractional basis, one sees scaling dimensions and the relevance of the interaction C ϕσC−1 −1 yþ1 ϕσ C ϕσC−1 that 2 cos y 2 ¼ð Þ sin −y and 2 sin y 2 ¼ terms [186,187] can be tuned by the velocity matrix Vjk in yþ1 σ ð−1Þ cos ϕ−y and thus the Majorana fermions (3.46) (3.25) that is affected by forward-scattering interactions transform according to among copropagating channels. Instead of considering energetics, we focus on a topological deliberation— A −1 yþ1 B F2 3 C2γy C2 ¼ð−1Þ γ−yð−1Þ þ ; inspired by the coupled-wire construction of quantum — B −1 yþ1 A F2 3 Hall states [125,129] that can drastically reduce the C2γ C ¼ð−1Þ γ− ð−1Þ þ ; ð3:54Þ y 2 y number of possible interactions and may stabilize the Q ∞ 2 3 desired interactions when applied to materials. −1 F2þ3 −1 NyþNy where ð Þ ¼ y¼−∞ð Þ is the total fermion The coupled-wire model considered so far assumes all parity of channels 2 and 3. This shows the fermion order electronic Dirac modes at the Fermi level have zero parameter is odd under C2 momentum kx ¼ 0. This is convenient for the purpose of constructing an exactly solvable model because momentum C OF C−1 2 yþ1=2 2 is automatically conserved by the backscattering inter-

yþ2 B F2 3 yþ1 A F2 3 actions. However, this also allows a huge collection of ¼ ið−1Þ γ− −1ð−1Þ þ ð−1Þ γ− ð−1Þ þ y y competing interactions. We propose the application of a − γA γB −OF ¼ i −y −y−1 ¼ −y−1=2: ð3:55Þ commensurate modulation of magnetic field to restrict interactions that conserve momentum. There are multiple On the other hand, one can also show from the C2 action variations to the application, which depend on the details of (3.43) that the boson angle parameter changes as the Dirac material and the Dirac vortices. To illustrate the C Θ C−1 −Θ − −1 yπ 2 yþ1=2 2 ¼ −y−1=2 ð Þ and therefore the idea, we present one possible simple scenario.

011039-21 RAZA, SIROTA, and TEO PHYS. REV. X 9, 011039 (2019) pffiffiffi First, we go back to a single Dirac wire and consider a X∞ 2 2 1 r π ð m þ Þ e r nonlinear dispersion Bð Þ¼ Bm sin 11¯ · ; ð3:60Þ m¼−∞ a pffiffiffi pffiffiffi ℏv e e e 2 e −e e 2 0 − 1 − 2 − 3 11 ¼ð y þ zÞ= and 11¯ ¼ð y þ zÞ= , that pre- Ey¼2lðkxÞ¼ 2 ðkx kFÞðkx kFÞðkx kFÞ; b serves both the AFTR and C2 symmetries, ℏ 0 − v 1 2 3 Ey¼2lþ1ðkxÞ¼ 2 ðkx þ kFÞðkx þ kFÞðkx þ kFÞ; ð3:57Þ r e r e r − r b Bð þ a yÞ¼Bð þ a zÞ¼BðC2 Þ¼ Bð Þð3:61Þ where v and b are some nonuniversal velocity and wave- [see Fig. 18(b) for the 3D field configuration]. Moreover, number parameters. We assume k2

1;2 −1 y 1;2 2 ϕ˜ 1;2 ψ ∼ i½ð Þ ðkF xþbx= Þþ y ðxÞ The dashed band in Fig. 18(a) shows one commensurate y ðxÞ e ; 3 −1 y 3 2 −ϕ˜ 3 energy dispersion along an even wire. i½ð Þ ðk xþbx= Þ yðxÞ ψ yðxÞ ∼ e F ; ð3:63Þ Next, we consider a spatially modulating magnetic field BðrÞ¼BðrÞe11, where j → j ℏ where the momenta are shifted by kF kF þðe= cÞAx. The phase oscillation eikx is canceled in an interaction term (a) only when momentum is conserved, or otherwise the interaction would drop out after the integration over x. It is straightforward to check that the Majorana fermions σ (3.46), which contain the operators eiϕ , have zero momentum because of the Fermi wave-number commensurate condition (3.58). In addition, the boson backscatter- 4ϕA − 4ϕB ing cosð yþ1 y Þ in (3.47) preserves momentum because the magnetic field is also commensurate [see Eqs. (3.59) and (3.62)]. (b) B (c) B IV. INTERACTION-ENABLED DIRAC-WEYL a B SEMIMETAL Pfaffian a Dirac Pfaffian B So far in this section, we have been discussing the y z gapping of the Dirac semimetal while preserving the AFTR y B and C2 symmetries. Here, we focus on an opposite aspect x of the symmetric many-body interaction—the enabling of a x (semi)metallic phase that is otherwise forbidden by symmetries in the single-body setting. We noticed in Sec. II A FIG. 18. (a) The energy dispersion Ey¼2lðkxÞ with (solid curve) or without (dashed curve) the alternating magnetic field. (b) The that the pair of momentum-separated Weyl points in Fig. 7 alternating magnetic field configuration that preserves the AFTR is anomalous. In fact, it is well known already that Weyl and C2 symmetries. (c) The alternating magnetic field across a nodes [3,23–26], if separated in momentum space, must single layer along the xy plane. come in multiples of four in a lattice translation and

011039-22 FROM DIRAC SEMIMETALS TO TOPOLOGICAL PHASES … PHYS. REV. X 9, 011039 (2019) time-reversal symmetric three-dimensional noninteracting at opposite ends. The Fidkowski-Kitaev interaction, how- system. ever, allows one to construct a nontrivial ð1 þ 1ÞD topo- This no go theorem can be rephrased into a feature. logical model that preserves both TR and inversion but at (1) If the low energy excitations of a TR symmetric the same time supports four protected Majorana zero modes lattice (semi)metal in three dimensions consists of at each end [194]. Third, a single massless Dirac fermion in one pair of momentum-separated Weyl nodes, then ð1 þ 1ÞD is anomalous in a (spinful) TR and charge Uð1Þ the system must involve many-body interaction. preserving noninteracting lattice system. On the other hand, We refer to this TR and lattice translation sym- it can be enabled by many-body interactions. For instance, metric strongly correlated system as an interaction- when one of the two opposing surfaces of a topological enabled topological Dirac (semi)metal. We assume insulator slab is gapped by symmetry-preserving inter- the Weyl nodes are fixed at two TR invariant actions [114–117], a single massless Dirac fermion is left momenta, and therefore they are stable against behind on the opposite surface as the only gapless low symmetry-preserving deformations. Otherwise, if energy d.o.f. of the quasi-ð1 þ 1ÞD system. Similar slab the Weyl nodes are not located at high symmetry construction can be applied to the superconducting case, points, they can be moved and pair annihilated. and interactions can allow any copies of massless Majorana Also, as explained in the beginning of Sec. II fermions to manifest in ð2 þ 1ÞD with the presence of and contrary to the more common contemporary (spinful) TR symmetry. terminology, we prefer to call the (semi)metal On the contrary, there are anomalous gapless fermionic “Dirac” rather than “Weyl” because of the doubling. states that cannot be enabled even by strong interactions. Perhaps more importantly, we propose the following Chiral fermions that only propagate in a single direction conjecture. cannot be realized in a true ð1 þ 1ÞD lattice system. They (2) Beginning with the interaction-enabled Dirac (semi) can only be supported as edge modes of ð2 þ 1ÞD topo- metal, any single-body symmetry-breaking mass logical phases such as quantum Hall states [179] or chiral must lead to a 3D gapped topological phase that px þ ipy superconductors [195,196]. Otherwise, they cannot be adiabatically connected to a band insulator. would allow heat transfer [180,197,198] from a low We suspect this statement can be proven by a filling temperature reservoir to a high temperature one, thereby argument similar to that of Hasting-Oshikawa-Lieb- – violating the second law of thermodynamics. Similarly, a Schultz-Mattis [188 190], and may already be available single massless Weyl fermion can only be present as the in Ref. [191] by Watanabe, Po, and Vishwanath. This ð3 þ 1ÞD boundary state of a ð4 þ 1ÞD topological bulk conjecture applies to the coupled-wire situation where the [20–22,168,169]. It cannot exist in a true ð3 þ 1ÞD lattice gapped phase is long-range entangled and supports frac- system [17,18], or otherwise under a magnetic field there tional excitations. Its topological order is out of the scope of would be unbalanced chiral fermions propagating along the this paper, but will be presented in a future work [159].Ina field direction that constitute the ABJ anomaly [15,16,19]. broader perspective, this type of statement may provide In this section, we focus on the simplest anomalous connections between strongly interacting and noninteract- gapless fermionic states in ð3 þ 1ÞD that can be enabled by ing phases and help in understanding quantum phase interactions. As eluded in Sec. II A 2, a weak topological transitions of long-range entangled 3D phases from that insulator in ð4 þ 1ÞD can support the anomalous energy of single-body band insulating ones. spectrum in Fig. 7 on its boundary so that a pair of opposite Before discussing the three-dimensional case, we make Weyl points sit at two distinct TRIM on the boundary the connection to a few known interaction-enabled topo- Brillouin zone. A 4D WTI slab, where the fourth spatial logical phases with or without an energy gap in low dimension is open and the other three are periodic, has two dimensions. First, zero-energy Majorana fermions γ j ¼ ð3 þ 1ÞD boundaries and each carries a pair of Weyl γ† j in a true zero-dimensional noninteracting (spinless) TR fermions. The coupling between the two pairs of Weyl symmetric system must bipartite into an equal number of fermions is suppressed by the system thickness and bulk −1 positive chiral ones T γjT ¼þγj and negative chiral energy gap. By introducing symmetry-preserving gapping −1 ones T γlT ¼ −γl. Fidkowski and Kitaev showed in interactions on the bottom surface, the anomalous gapless Ref. [192] that under a combination of two-body inter- fermionic state is left behind on the top surface and is actions, eight Majoranas with the same chirality can acquire enabled in this quasi-ð3 þ 1ÞD setting [see Fig. 19(a)]. a TR preserving mass and be removed from low energy. Inspired by this construction, we propose a true This leaves behind a collection of zero-energy Majoranas ð3 þ 1ÞD coupled-wire model which has the anomalous that have a nontrivial net chirality of eight. Second, all energy spectrum in Fig. 7 and preserves the AFTR sym- 1D TR symmetric topological superconductors metries in both T 11 and T 11¯ directions as well as the C2 [6,12,13,168,169,193] in the Altland-Zirnbauer class (screw) rotation symmetry. The model is summarized in BDI must break inversion because the zero-energy Fig. 19(b). It consists of a checkerboard array of electronic Majorana boundary modes must have opposite chiralities wires, where each wire has two chiral Dirac channels

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anomalous (3+1)D (a) massless Weyl fermion pair backscattering terms that dimerize the remaining Dirac channels , and were described by (2.26) in Sec. II A 1. interaction-enabled (4+1)D weak topological insulator (3+1)D Dirac (semi)metal The resulting state is an insulating ð3 þ 1ÞD topological } phase with long-range entanglement. For instance, each many-body interacting U gapped (3+1)D boundary diagonal layer gapped by the many-body interaction has the identical topological order of the T -Pfaffian surface (b) into out of state of a topological insulator. plane plane

V. FRACTIONAL SURFACE STATES Chiral Dirac In Sec. II B, we discussed the surface states of the single- body coupled Dirac wire model (2.12) (see also Fig. 6). In Chiral Pfaffian particular, we show in Fig. 14 that an AFTR symmetry- fractionalize preserving surface hosts open chiral Dirac channels, which connect and leak into the 3D (semi)metallic bulk. Earlier in FIG. 19. (a) A quasi-ð3 þ 1ÞD interaction-enabled Dirac (semi) this section, we discussed the effects of many-body inter- metal constructed by a 4D slab of WTI. (b) Coupled-wire model action, which leads to two possible phases: (a) a gapped of an anomalous Dirac (semi)metal enabled by interaction with topological phase (see Sec. III B) that preserves one of the C2 rotation and both AFTR T 11; T 11¯ symmetries. two AFTR symmetries, say T 11, and (b) a gapless interaction-enabled Dirac semimetal (see Sec. IV) that preserves propagating into-paper and another two propagating out of the C2 rotation and both AFTR symmetries T 11 and T 11¯ . paper. Contrary to the model considered in Sec. II, here the Here, we describe the boundary states of the two interacting net chirality on each wire cancels and therefore the wires phases on a surface closed under the symmetries. are true ð1 þ 1ÞD systems without being supported by a First, we consider the coupled-wire model with the higher-dimensional bulk. Using the splitting scheme many-body interaction (3.47) (see also Fig. 17) and a described in Sec. III A, along each wire, one can fraction- boundary surface along the yz plane perpendicular to the alize a group of three Dirac channels ( ) into a wires. The surface network of fractional channels is shown pair of copropagating chiral Pfaffian channels (respec- in Fig. 20(a). We assume the bulk chiral Dirac wires ( ) tively, ). The two Pfaffian channels then can be back- are supported as vortices of Dirac mass in the bulk [recall scattered in opposite directions using the many-body (2.2)], where the texture of the mass parameters is repre- interaction U (dashed purple lines) described in Sec. III B. sented by the underlying vector field. The model is This introduces an excitation energy gap that removes juxtaposed along the yz boundary plane against the trivial ℏ k ⃗μ μ three Dirac channels per wire from low energy. Last, Dirac insulating state Hvacuum ¼ v · s z þ m0 x, which single-body backscatterings t1, t2 (solid directed blue models the vacuum. The line segments on the surface plane lines) among the remaining Dirac channels described where the Dirac mass m0μx changes sign host chiral Dirac in (2.12) and Fig. 6 give rise to the low-energy Weyl channels (cf. Sec. II B 2). spectrum in Fig. 7. Since the many-body interaction U and Unlike the single-body (semi)metallic case in Fig. 14 the single-body backscatterings t1, t2 preserve the C2 where the surface Dirac channels connect with the bulk rotation and both AFTR symmetries T 11 and T 11¯ ,the ones, now the many-body interacting bulk is insulating and model describes an interaction-enabled anomalous (semi) does not carry low-energy gapless excitations. Thus, the metal that is otherwise forbidden in a noninteracting surface Dirac channels here cannot leak into the bulk and nonholographic setting. must dissipate to other low-energy d.o.f. on the surface. The nonlocal antiferromagnetic nature of the time- The many-body interwire backscattering interaction in reversal symmetry is built-in in the present coupled-wire (3.47) (and Fig. 17) leaves behind chiral Pfaffian channels model. We speculate in passing that a local conventional on the surface. These fractional channels connect back to TR symmetric Dirac (semi)metallic phase consisting of a the surface Dirac channels in pairs. The surface network of single pair of momentum-space-separated Weyl nodes may chiral channels preserves the AFTR T 11 symmetry. also be enabled by interaction. On one hand, the AFTR However, the low-energy surface state is not protected. symmetry could be restored to a local TR symmetry by Electronic states can be localized by dimerizing the Pfaffian “melting” the checkerboard wire array. On the other hand, channels in the z (or 11¯ ) direction. there could also be an alternative wire configuration that Second, we consider the interaction-enabled Dirac semi- facilitates a coupled-wire model with a local conventional metallic model summarized in Fig. 19(b) in Sec. IV and TR symmetry. again let it terminate along the symmetry-preserving yz Last, we gap the interaction-enabled Dirac semimetallic plane perpendicular to the wires. The surface gapless model (Fig. 19) by a symmetry-breaking single-body channels are shown in Fig. 20(b). Here, the semimetallic mass. This can be achieved by introducing electronic bulk preserves C2 as well as the two AFTR symmetries T 11

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(a) (b) electronic Weyl nodes separated in momentum space. A brief conceptual summary was presented in Sec. IA and will not be repeated here. Instead, we include a short discussion on what are the broad implications of this work and then discuss possible future directions: Theoretical impact.—We believe this work is a first step towards a duality between quantum critical transitions of y z short-range entangled symmetry-protected topological phases and long-range entangled symmetry-enriched topo- x = Pfaffian channels = Dirac channels into bulk logical phases. The 3D topological order in these phases is = Dirac channels = Dirac channels out of bulk completely different from 2D topological order as it can have both point and linelike excitations and a much richer FIG. 20. Fractional surface states of (a) a 3D Dirac insulator structure [159]. There have been field-theoretical descrip- gapped by many-body interaction that preserves T 11, and (b) a tions, along the lines of BF and Chern-Simons theories, of 3D gapless interaction-enabled Dirac semimetal that preserves 3D topological phases that support these richer structures, T T 11, 11¯ , and C2. such as loop braiding. However, there have been only very few exact solvable examples and none of them patch the and T 11¯ . The bulk array of wires are true ð1 þ 1ÞD systems field-theoretical descriptions and microscopic electronic and are not supported as edge modes or vortices of a higher- systems. The construction presented in this work opens dimensional bulk. The pair of into-paper Dirac modes are a new direction towards making such a connection. The bent into the pair of out-of-paper ones along each wire at the models are exactly solvable, and they originate from a terminal. Similar to the previous case, the many-body bulk microscopic Dirac electronic system with local two-body interwire backscattering interaction leaves behind surface interactions. They also have potential impact on numerical chiral Pfaffian channels. Through the mode bending at the modeling. For example, the interacting coupled-wire model wire terminal, these Pfaffian channels join in pairs and can be approached by a lattice electronic model, which connect to the chiral Dirac channels in the bulk that forgoes exact solvability but potentially leads to new constitute the Dirac semimetal. In this case, the surface critical transitions between topological phases in 3D. — state is protected by C2, T 11, and T 11¯ , and is forced to carry High-energy impact. For a single pair of Weyl nodes fractional gapless excitations as a consequence and signa- with opposite chirality, time-reversal symmetry (TRS) must ture of the anomalous symmetries. For instance, the charge be broken. Hence, for time-reversal symmetric systems, at e=4 Ising-like quasiparticle and the charge e=2 semion can least four Weyl nodes are required. In this paper, we have in principle be detected by shot-noise tunneling experi- shown that, as enabled by many-body interactions, an ments. These gapless fractional excitations however are electronic system can support a single pair of Weyl nodes localized on the surface because the Dirac (semi)metallic in low energy without violating TRS (cf. Sec. IV). Such a bulk only supports gapless electronic quasiparticles. material, if it exists, can be verified experimentally by ARPES, and as a nontrivial consequence, our results assert that such a material must encode long-range entanglement. VI. CONCLUSION AND DISCUSSION The existence of a single pair of massless Weyl fermions Dirac and Weyl (semi)metals have generated immense without TRS breaking in 3+1D can potentially provide new theoretical and experimental interest. On the experimental theories beyond the standard model. front, this is fueled by an abundant variety of material Experimental realization.—There have been numerous classes and their detectable ARPES and transport signa- field-theoretical discussions on possible properties of tures. On the theoretical front, Dirac-Weyl (semi)metal is topologically ordered phases in 3D [199–206]. However, the parent state that, under appropriate perturbations, can unlike the 2D case there are no materials that exhibit give birth to a wide range of topological phases, such as topological order (quasiparticle excitations) in 3D. In this topological (crystalline) insulators and superconductors. In work, we show that an interacting Weyl or Dirac semimetal this work, we explored the consequences of a specific type is a good place to start for the following reasons. A of strong many-body interaction based on a coupled-wire symmetry-preserving gap must result in topological order description. In particular, we showed that (i) a 3D Dirac and fractionalization (cf. Sec. IV). While it is entirely likely fermion can acquire a finite excitation energy gap in the that interactions leads to a spontaneous symmetry-breaking many-body setting while preserving the symmetries that phase, we show that there is no obstruction to realizing forbid a single-body Dirac mass, and (ii) interaction can an interacting phase that preserves symmetries. Such a enable an anomalous antiferromagnetic time-reversal sym- gapping must support fractionalization, such as the e=4 metric topological (semi)metal whose low-energy gapless charged Ising-like and e=2 charged semionlike quasipar- d.o.f. are entirely described by a pair of noninteracting ticles in the bulk, as predicted by our work. These charged

011039-25 RAZA, SIROTA, and TEO PHYS. REV. X 9, 011039 (2019) particles can in principle be measured using a shot-noise studied in states with dx2 − y2 pairing [210],He3 in its experiment across a point contact. Moreover, the gapping superfluid A phase [211,212], and noncentrosymmetric procedure involves Pfaffian channels so there should be states [213,214]. Weyl and Dirac fermions were generalized excitations that mirror those in a Pfaffian state. There would in TR and mirror symmetric systems to carry Z2 topological also be linelike excitations in 3D for which the experimental charge [215]. General classification and characterization signature is not yet clear. Therefore, we believe an inter- of gapless nodal semimetals and superconductors were action-enabled, symmetry-preserving gapped Weyl-Dirac proposed [6,212,216–221]. It would be interesting to semimetal is a good candidate for realizing topologically investigate the effect of strong many-body interactions in ordered phases in 3D. As for the experimental verification general nodal systems. of the anomalous interaction-enabled Dirac semimetal, the Second, in Sec. II, we described a coarse-graining electronic energy spectrum of a single pair of momentum- procedure of the coupled-wire model that resembles a separated Weyl nodes in the presence of time-reversal real-space renormalization and allows one to integrate out symmetry can be measured using ARPES, assuming spon- high energy d.o.f. While this procedure was not required in taneous symmetry breaking is absent. Although the pro- the discussions that follow because the many-body inter- posed experimental signatures, if measured, will strongly acting model we considered was exactly solvable, it may be point towards the existence of such states, we cannot claim useful in the analysis of generic interactions and disorder. that such signatures provide a smoking gun evidence yet. The coarse-graining procedure relied on the formation of More work needs to be done for the complete characteri- vortices, which were introduced extrinsically. Like super- zation of the pointlike and linelike topological order of these conducting vortices, it would be interesting as a theory states and will be part of a future work. and essential in application to study the mechanism where Apart from the 3D topological order having a much the vortices of Dirac mass can be generated dynamically. richer structure than 2D topological order, the 3D case To this end, it may be helpful to explore the interplay presented in this work is qualitatively different from the between possible (anti)ferromagnetic orders and the spin- well-studied 2D case. In the 2D case, the massless Dirac momentum locked Dirac fermion through antisymmetric- surface state is anomalous and lives on the boundary of a exchange interactions like the Dzyaloshinskii-Moriya higher-dimensional bulk. This is qualitatively distinct from interaction [222,223]. the 3D Dirac-Weyl (semi)metal, which does not require Third, the symmetry-preserving many-body gapping holographic projection from a 4D bulk. In fact, a single 3D interactions considered in Sec. III have a ground state that Weyl fermion, which is supported on the boundary of a 4D exhibits long-range entanglement. This entails degenerate topological insulator, cannot be gapped while preserving ground states when the system is compactified on a closed charge Uð1Þ conservation even with many-body interaction three-dimensional manifold, and fractional quasiparticle due to chiral anomaly. This serves as a counterexample and quasistring excitations or defects. These topological which distinguishes the gappability of 2D versus 3D order properties were not elaborated in our current work but boundary state. Thus, it is not a priori an expected result will be crucial in understanding the topological phase [159] that a Dirac-Weyl (semi)metal can be gapped without as well as the future designs of detection and observation. It breaking symmetries. Moreover, the topological origin of would also be interesting to explore possible relationships 3D Dirac-Weyl (semi)metals relies on the addition of between the coupled-wire construction and alternative nonlocal spatial symmetry, in the current 3D case, the exotic states in three dimensions, such as the Haah’s code C2 screw rotation. This is distinct from the 2D Dirac [224,225]. surface case, where all symmetries are local. Fourth, the many-body interwire backscatterings pro- We conclude by discussing possible future directions. posed in Sec. III B were based on a fractionalization First, coupled-wire constructions can also be applied in scheme described in III A that decomposes a chiral superconducting settings and more general nodal electronic Dirac channel with ðc; νÞ¼ð1; 1Þ into a decoupled pair systems. For example, a Dirac-Weyl metal can be turned into of Pfaffian ones each with ðc; νÞ¼ð1=2; 1=2Þ. In theory, a topological superconductor [168,169,193] under appro- there are more exotic alternative partitions. For instance, if priate intraspecies (i.e., intravalley) s-wave pairing [207]. a Dirac channel can be split into three equal parts instead Pairing vortices host gapless chiral Majorana channels of two, an alternative coupled-wire model that put Dirac [207–209]. An array of these chiral vortices can form the channels on a honeycomb vortex lattice could be con- basis in modeling superconducting many-body topological structed by backscattering these fractionalized channels phases in three dimensions. On the other hand, instead of between adjacent pairs of wires. Such higher order decom- considering superconductivity in the continuous bulk, inter- positions may already be available as conformal embed- wire pairing can also be introduced in the coupled Dirac wire dings in the CFT context. For example, the affine SUð2Þ model and lead to new topological states [166]. Kac-Moody theory at level k ¼ 16 has the central charge Dirac-Weyl (semi)metals are a specific type of nodal c ¼ 8=3, and its variation may serve as the basis of a electronic matter. For example, nodal superconductors were “ternionic” model.

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ACKNOWLEDGMENTS Now, we separate the Hamiltonian This work is supported by the National Science 0 D Foundation under Grant No. DMR-1653535. We thank 0 ℏ Γ H ðkzÞ¼ vkz 5 þ † ; ðA4Þ Matthew Gilbert and Moon Jip Park for insightful D 0 discussions. where Γ5 ¼ diagð−12; 12Þ. We note that the zero momen- 0 tum sector H ðkz ¼ 0Þ has a chiral symmetry since it APPENDIX A: CHIRAL MODES ALONG anticommutes with with Γ5, and it reduces to the TOPOLOGICAL DEFECTS Jackiw-Rossi vortex problem in two dimensions [161]. The Dirac operator D† has only one normalizable zero In Sec. II, we begin with the Dirac Hamiltonian (2.2) −jmjr=ℏv iπ=4 −iπ=4 T mode u0ðrÞ ∝ e ðe ;e Þ , while its conjugate where the mass term winds around a vortex and as a 0 D has none. H ðkz ¼ 0Þ therefore has a zero eigenvector of consequence, it hosts a chiral Dirac channel along the T ψ 0ðrÞ¼½u0ðrÞ; 0 , which is also an eigenvector of Γ5.In vortex (also see Fig. 4). Here, we will demonstrate an the full Hamiltonian, the zero mode ψ 0ðrÞ has energy example of a simple vortex, and show that there is a chiral −ℏvk and corresponds a single midgap chiral Dirac Dirac zero mode. In general, the correspondence between z channel. the number of protected chiral Dirac channels and the vortex winding is a special case of the Atiyah-Singer Index theorem [162] and falls in the physical classification of APPENDIX B: SYMMETRY TRANSFORMATIONS topological defects [160]. OF CHERN INVARIANTS First, say we start with the Hamiltonian from (2.2). Then In Sec. II A, we discussed the Chern numbers on two- for simplicity we consider the particular Dirac mass mðrÞ¼ dimensional momentum planes of the anomalous Dirac iθ mxðrÞþimyðrÞ¼jmje that constitute a vortex along the (semi)metal. It was claimed that the Chern numbers (2.21) π 2 z axis, where θ is the polar angle on the xy plane. By on the two planes at kx ¼ = (see Fig. 7) are of opposite signs because of the AFTR and twofold C2 (screw) rotation replacing kx;y ↔ −i∂x;y, (2.2) becomes symmetries. In this appendix we will derive the symmetry flipping operations on the Chern invariants. H r ℏv −i∂ s − i∂ s k s μ ð Þ¼ ð x x y y þ z zÞ z We begin with a Bloch Hamiltonian HðkÞ that is k þjmj cos θμx þjmj sin θμy; ðA1Þ symmetric under the operation Gð Þ,

HðkÞ¼GðgkÞHðgkÞGðgkÞ−1 ðB1Þ where kz is still a good quantum number because translation in z is still preserved. The Hamiltonian can be transformed under a new basis into if G is unitary, or HðkÞ¼GðgkÞHðgkÞGðgkÞ−1 ðB2Þ −ℏvkz D H0 ¼ UHU−1 ¼ ; † ℏ D vkz if it is antiunitary. Let jumðkÞi be the occupied states of 0 1 0 0100 HðkÞ. We define jumðkÞi¼jGumðkÞi ¼ GðgkÞjumðgkÞi 0 k k k k B C (or jumð Þi ¼ jGumð Þi ¼ Gðg Þjumðg Þ i), which is B 0010C k U ¼ B C; ðA2Þ also an occupied state of Hð Þ, for unitary (respectively, @ 1000A antiunitary) symmetry. 0001 The Chern number (2.21) can equivalently be defined as Z i where the Dirac operator occupying the off-diagonal Ch1ðkxÞ¼ TrðF kÞ; ðB3Þ 2π N blocks is kx

where TrðF kÞ¼dTrðA Þ, N is the oriented k k plane −2 ℏ ∂ −iθ k kx y z † i v w jmje A D ¼ with fixed kx, and k is the Berry connection of the iθ mn jmje 2iℏv∂ ¯ occupied states A ¼hu ðkÞjdu ðkÞi. The Berry con- w k m n − ℏ ∂ − ∂ nection transforms according to − θσ i vð r i θ=rÞjmj ¼ e i z ; jmj iℏvð∂r þ i∂θ=rÞ 0mn 0 0 A k ≡ humðkÞjdunðkÞi ðA3Þ † ¼humðgkÞjGðgkÞ d½GðgkÞjunðgkÞi mn † iθ ¼ A k þhumðgkÞj½GðgkÞ dGðgkÞjunðgkÞi ðB4Þ where w ¼ x þ iy ¼ re and σz ¼ diagð1; −1Þ. g

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