From Dirac semimetals to topological phases in three dimensions: a coupled wire construction

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 manuscript, 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 (resp. boundary) quasiparticle excitations.


I. INTRODUCTION
Dirac and Weyl semimetals are nodal electronic phases of matter in three spatial dimensions. Their low-energy emergent quasiparticle excitations are electronic Dirac 1 and Weyl 2 fermions. (Contemporary reviews in condensed electronic matter can be found in Ref. 3-10.) They are three dimensional generalizations of the Dirac fermions that appear in two dimensional graphene 11 and the surface boundary of a topological insulator 6,[12][13][14] . They follow massless quasi-relativistic linear dispersions near nodal points in the energy-momentum space close to the Fermi level. Contrary to accidental degeneracies which can be lifted by generic perturbations, these nodal points are protected by topologies or symmetries.
A Weyl fermion is chiral and has a non-trivial winding of a pseudo-spin texture near the singular nodal point in energy-momentum space. This would associate to a non-conservative charge current under a parallel electric and magnetic field and is known as the Adler-Bell-Jackiw (ABJ) anomaly 15,16 . Thus, in a true three dimensional lattice system, Weyl fermions must come in pairs [17][18][19] so that the net chirality, and consequently the anomaly, cancels. Or otherwise, a three dimensional system of a single Weyl fermion must be holographically supported as the boundary of a topological insulator in four dimensions [20][21][22] . On the other hand, a Dirac fermion in three dimensions consists of a pair of Weyl fermions with opposite chiralities. Without symmetries, it is not stable and can turn massive upon inter-Weyl-species coupling. With symmetries, a band crossing can be protected by the distinct symmetry quantum numbers the bands carry along a high symmetry axis. In this article, we focus on the fourfold degenerate Dirac nodal point protected by time-reversal (TR) and (screw) rotation symmetry.
In electronic systems, massless Dirac and Weyl fermions appear in gap-closing phase transitions between spin-orbit coupled topological insulators and normal insulators 23 .
When inversion or time-reversal symmetry is broken, nodal Weyl points can be sepa-rated in energy-momentum space. Such gapless electronic phases are contemporarily referred to as Weyl (semi)metals [24][25][26][27] . Their boundary surfaces support open Fermi arcs 24 that connect surface-projected Weyl nodes. Weyl (semi)metals also exhibit exotic transport properties, such as negative magneto-resistance, nonlocal transport, chiral magnetic effect, and chiral vortical effect [28][29][30][31][32][33] . There have been numerous first principle calculations 34 on proposed materials such as the noncentrosymmetric (La/Lu)Bi 1−x Sb x Te 3 35 , the TlBiSe 2 family 36 , the TaAs family 37,38 , trigonal Se/Te 39 and the HgTe class 40 , as well as the time-reversal breaking pyrochlore iridates 24,41,42 , magnetically doped topological and trivial insulator multilayers 26 , HgCr 2 Se 4 43 and Hg 1−x−y Cd x Mn y Te 44 . At the same time, there have also been abundant experimental observations in bulk and surface energy spectra 45 as well as transport 46 . Angleresolved photoemission spectroscopy (ARPES) showed bulk Weyl spectra and surface Fermi arcs in TaAs [47][48][49][50][51] as well as similar materials such as NbAs, NbP and TaP 52,53 . Other materials such as Ag 3 BO 3 , TlTe 2 O 6 and Ag 2 Se 54 were observed to host pinned Weyl nodes at high symmetry points. Negative magneto-resistance was reported in TaAs 55,56 as a suggestive signature of the ABJ anomaly. Similar properties were also observed in TaP 57 , NbP and NbAs 58-60 , although not without controversies 61 .
Weyl points with opposite chiralities cannot be separated in energy-momentum space when both inversion and time reversal symmetries are present. Massless Dirac fermions appear between gap-closing phase transitions between topological and trivial (crystalline) insulators, such as Bi 1−x Sb x 62 and Pb 1−x Sn x Te 63 . Critical Dirac (semi)metals were investigated for example in the tunable TlBiSe 2−x S x 64-66 , Bi 2x In x Se 3 67, 68 and Hg 1−x Cd x Te 69 , as well as the charge balanced BaAgBi 70 , PtBi 2 , SrSn 2 As 2 71 and ZrTe 5 72 whose natural states are believed to be close to a topological critical transition. A Dirac (semi)metallic phase can be stablized when the Dirac band crossing is secured along a high symme-try axis and the two crossing bands carry distinct irreducible representations. Theoretical studies include the diamond-structured β-crystobalite BiO 2 family 73 (space group (SG) No. 227, Fd3m), the orthorhombic bodycentered BiZnSiO 4 family 74 (SG No. 74, Imma), the tetragonal Cd 3 As 2 75 (SG No. 142, I4 1 /acd), the hexagonal Na 3 Bi family 76 , as well as the filling-enforced nonsymmorphic Dirac semimetals 77-81 such as the hexagonal TlMo 3 Te 3 family 71 (SG No. 176, P6 3 /m), the monoclinic Ca 2 Pt 2 Ga (SG No. 15, C2/c), AgF 2 , Ca 2 InOsO 6 (SG No. 14, P2 1 /n), and the orthorhombic CsHg 2 (SG No. 74, Imma) 82 . At the same time, there are numerous experimental confirmations. They include ARPES observations on Cd 2 As 3 83-85 , Na 3 Bi 86,87 and ZrTe 5 72 ; scanning tunneling microscopy in Cd 2 As 3 88 ; magneto-transport in Bi 1−x Sb x 89 , Cd 2 As 3 90-97 , Na 3 Bi 87,98 , ZrTe 5 72,99-101 , HfTe 5 102 and PtBi 2 103 ; magneto-optics 104 and anomalous Nernst effect 105 in Cd 2 As 3 , and many more. However, there are also contradicting pieces of evidence, especially in ZrTe 5 and HfTe 5 that suggest a bulk band gap [106][107][108][109][110][111][112] . Dirac/Weyl (semi)metals are the origins of a wide variety of topological phases in three dimensions (see figure 1). By introducing a spatial or charge U (1) symmetry-breaking single-body mass, they can be turned into a topological insulator or superconductor. The focus of this manuscript is on symmetry-preserving many-body gapping interactions. The resulting insulating topological phase can carry long-range entanglement and a nontrivial topological order. Similar phenomena were theoretically studied on the Dirac surface state of a topological insulator [113][114][115][116] and the Majorana surface state of a topological superconductor 117,118 , where symmetrypreserving many-body gapping interactions are possible and lead to non-trivial surface topological orders that support anyonic quasiparticle excitations.

Dirac/Weyl (semi)metal
Symmetry-preserving gapping interactions cannot be studied using a single-body mean-field theory. This is because the Dirac/Weyl (semi)metallic phase is protected by symmetries in the single-body setting and any mean-field model with an excitation energy gap must therefore break the symmetry either explicitly or spontaneously. The coupled wire construction can serve as a powerful tool in building an exactly-solvable interacting model and understanding many-body topological phases of this sort. The construction involves a highly anisotropic approximation where the electronic degrees of freedom are confined along an array of continuous one-dimensional wires. Inspired by sliding Luttinger liquids [119][120][121][122][123] , the coupled wire construction was pioneered by Kane, Mukhopadhyay and Lubensky 124 in the study of Laughlin 125 and Haldane-Halperin hierarchy 126,127 fractional quantum Hall states. Later, this theoretical technique was applied in more general fractional quantum Hall states [128][129][130][131][132] , anyon models 133,134 , spin liquids 135,136 , (fractional) topological insulators [137][138][139][140][141] and superconductors 142,143 , as well as the exploration of symmetries and dualities 144,145 . Moreover, coupled wire construction has already been used to investigate three dimensional fractional topological phases 146 and Weyl (semi)metal 147 even in the strongly-correlated fractional setting 148 .
The microscopic symmetry-preserving many-body interactions in the Dirac surface state on a topological insulator was discussed by Mross, Essin and Alicea in Ref. 149. They mimicked the surface Dirac modes using a coupled wire model and proposed explicit symmetric many-body interactions that lead to a variation of gapped and gapless surface states. Motivated by this and also using a coupled wire construction, the microscopic symmetry-preserving many-body gapping of the Majorana topological superconducting surface state was studied by one of us in Ref. 150.
In this article, we focus on (i) a coupled wire realization of a Dirac/Weyl (semi)metallic phase protected by antiferromagnetic time-reversal (AFTR) and screw twofold rotation symmetries, (ii) a set of exactly-solvable inter-wire many-body interactions that introduces a finite excitation 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 setting.

A. Summary of results
We now highlight our results. The first part of this article addresses a mapping between the isotropic massless Dirac fermion in the continuum limit and an anisotropic coupled wire model where the effective low-energy degrees of freedom are confined along a discrete array of 1D continuous wires. We begin with a minimal Dirac (semi)metal equipped with time-reversal and (screw) C 2 rotation symmetries. The mapping to a coupled wire model is achieved by first introducing vortices that break the symmetries microscopically. These vortices are topological line defects that involve spatial winding of symmetry-breaking Dirac mass parameters. Consequently, these vortices host chiral Dirac electronic chan-nels, each of which corresponds to a gapless quasi-1D system where electronic quasiparticles can only propagate in a single direction along the channel and are localized along the perpendiculars.
When assembled together onto a vortex lattice, the system recovers the screw C 2 rotation symmetry as well as a set of emergent antiferromagnetic symmetries, which are combinations of the broken time-reversal and halftranslations. Upon nearest-wire single-body electron backscatterings, the electronic band structure disperses linearly and mirrors that of the continuous isotropic Dirac parent state. A symmetry-protected massless Dirac fermion (equivalently a pair of Weyl fermions with opposite chiralities) emerges and captures the low-energy long length scale electronic properties.
This mapping can be qualitatively understood as a coarse-graining procedure where high-energy microscopic electronic degrees of freedom are integrated out. The procedure can be repeated indefinitely and resembles a realspace renormalization. For example, the gapless Dirac electronic structure of the coupled wire model can acquire a finite mass by symmetry-breaking dimerizations. These dimerizations can be arranged in a topological manner that spatially wind non-trivially around a collective vortex. These second-stage vortices can subsequently be assembled into an array similar to the previous construction except now with a longer lattice constant. The system again recovers a massless Dirac spectrum under inter-vortex electron tunneling in low-energy and long length scale. The mapping therefore establishes an equivalence between the continuous isotropic massless Dirac fermion and the semi-discrete anisotropic coupled Dirac wire model.
The second part of this article addresses non-trivial symmetry-preserving many-body interacting effects beyond the single-body mean-field paradigm. We begin with the anisotropic array of chiral Dirac wires that constitutes a Dirac (semi)metal protected by antiferromagnetic time-reversal (AFTR ) and (screw) C 2 rotation symmetries. We consider an exactly-solvable model of symmetry-preserving inter-wire many-body backscattering interactions. This model is inspired by and can be regarded as a layered version of the symmetric massive interacting surface state of a topological insulator. It is based on a fractionalization scheme that divides a single chiral Dirac channel into a decoupled pair of identical chiral "Pfaffian" channels. Each of these fractional channels carries half of the degrees of freedom of the original Dirac wire. For instance, the fractionalization splits the electric and thermal currents exactly in half. It leads to the appearance of fractional quasiparticle excitations. For example, a chiral Pfaffian channel also runs along the 1D edge of the particle-hole symmetric Pfaffian fractional quantum Hall state [151][152][153] , and supports charge e/4 Ising and e/2 semionic primary fields.
We consider an explicit combination of many-body interwire backscattering interactions that stabilize the fractionalization. Similar coupled wire constructions were applied in the literature to describe topological insulator's surface state 149 and ν = 1/2 fractional quantum Hall states 128,132 . They are higher dimensional analogues of the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin chain model 154,155 . The pair of chiral Pfaffian channels along each wire is backscattered in opposite directions to neighboring wires by the interaction. As a result of this dimerization of fractional degrees of freedom, the model acquires a finite excitation energy gap and at the same time preserves the relevant symmetries.
The coupled wire construction also suggests new interaction-enabled topological (semi)metals.
In the single-body regime, an (antiferromagnetic) time-reversal symmetric Weyl (semi)metal realizable on a three dimensional lattice has a minimum of four momentum-spaceseparated Weyl nodes. The many-body interacting wire model can be turned into a gapless system where all lowenergy degrees of freedom are electronic and are freely described in the single-body non-interacting setting by two and only two separated Weyl nodes. Although the model is antiferromagnetic, we conjecture that similar anomalous Weyl (semi)metal can be enabled by interaction while preserving local time-reversal.
The paper is organized as follows. In section II, we construct a single-body coupled wire model of a Dirac/Weyl (semi)metal equipped with two emergent antiferromagnetic time-reversal (AFTR ) axes and a (screw) C 2 rotation symmetry. In section II A, we establish the equivalence between the isotropic continuum limit and the anisotropic coupled wire limit by a coarse-graining mapping. We also discuss the anomalous aspects of the pair of Weyl fermions and different resolutions to the anomaly. In section II B, we describe the gapless surface states of the coupled wire model. AFTR breaking and preserving surfaces are considered separately in section II B 1 and II B 2 respectively.
In section III, we move on to the effect of symmetrypreserving many-body interactions. The fractionalization of a chiral Dirac channel is explained in section III A, where we review the content of the fractional Pfaffian conformal field theory and establish the decomposition Dirac ≈ Pfaffian ⊗ Pfaffian through bosonization techniques. The splitting of a Dirac channel is summarized in figure 15. In section III B, we explicitly construct an exactly-solvable interacting coupled wire model that introduces a finite excitation energy gap to the Dirac system while preserving the relevant symmetries. The many-body interwire backscattering interactions are summarized in figure 16. In section III C, we discuss a plausible stabilization mechanism of the desired interactions through an antiferromagnetic order. In section III D, we discuss a variation of the model that enables an anomalous topological (semi)metal with two Weyl nodes through interaction. In section III E, we elaborate on the gapless surface states of these new interacting phases.

II. COUPLED WIRE CONSTRUCTION OF A DIRAC SEMIMETAL
We begin with a Dirac semimetal in three dimensions. It consists of a pair of massless Weyl fermions with opposite chiralities. In this article we do not distinguish between a Dirac and a Weyl semimetal. This is because the fermion doubling theorem [17][18][19] and the absence of the Adler-Bell-Jackiw anomaly 15,16 require Weyl fermions to always come in pairs in a three dimensional lattice system. A Weyl semimetal therefore carries the same low energy degrees of freedom as a Dirac semimetal. We refer to the case when the pair of Weyl fermions are separated in momentum space as a translation symmetry protected Dirac semimetal. Here, we assume the simplest case where the two Weyl fermions overlap in energy-momentum space. Its low-energy band Hamiltonian takes the spin-orbit coupled form where s = (s x , s y , s z ) are the spin-1/2 Pauli matrices, and µ z = ±1 indexes the two Weyl fermions. Normally the masslessness of the Dirac system is protected by a set of symmetries. Here, we assume the time reversal (TR) T , which is represented in the single-body picture by the spinful operatorT = is y K where K is the complex conjugation operator, and a twofold rotation C 2 about the z-axis. In the case when µ z has a non-local origin such as sublattice or orbital, it can enter the rotation operator. We assume C 2 is represented in the singlebody picture byĈ 2 = is z µ z . It squares to minus one in agreement with the fermionic statistics, and commutes with the local TR operator. In momentum space, T flips The band Hamiltonian (2.1) shares simultaneous eigenstates with C 2 along the k z -axis. The two forward moving bands have C 2 eigenvalues +i while the two backward moving ones have C 2 eigenvalues −i (see figure 2). Therefore the band crossing is C 2 -protected while the fourfold degeneracy is pinned at k = 0 because of TR symmetry. Noticing that each of the C 2 = ±i sector along the k z -axis is chiral (i.e. consisting of a single propagating direction), it violates the fermion doubling theorem 17,18 and is anomalous. This can be resolved by assuming the C 2 symmetry is actually a non-symmorphic screw rotation in the microscopic lattice limit and squares to a primitive lattice translation in z. k z is now periodically defined (up to 2π/a) and the two C 2 eigen-sectors wraps onto each other after each period. Focusing on the continuum limit where k z is small (when compared with 2π/a), C 2 2 = −e ikza ≈ −1 and the C 2 symmetry behaves asymptotically as a proper rotation.
The primary focus of this article is to explore symmetry preserving/enabled interacting topological states that originate from the massless Dirac system. Contrary to its robustness in the single-body non-interacting picture, we show that the 3D Dirac fermion can acquire a many-body mass gap without violating the set of symmetries. To illustrate this, we first make use of the fact that the Dirac system can be turned massive by breaking symmetries. Symmetry breaking inter-valley scatterings introduce two coexisting mass terms where m x (or m y ) preserves (resp. breaks) TR , and both of them violate C 2 . We allow slow spatial modulation of the mass parameters, which can be grouped into a single complex parameter m(r) = m x (r) + im y (r), and to be precise, momentum k should be taken as a differential operator −i∇ r when translation symmetry is broken. Nontrivial spatial windings of the symmetry breaking mass parameters give rise to topological line defects or vortices that host protected low-energy electronic degrees of freedom. Proliferation of interacting vortices then provides a theoretical path to multiple massive/massless topological phases while restoring and modifying the original symmetries as they emerge in the low-energy long-length scale effective theory. A topological line defect is a vortex string of the mass parameter in three dimensions where the complex phase of m(r) = |m(r)|e iϕ(r) winds non-trivially around the string. The left diagram in figure 3 shows the spatial modulation of ϕ(r) along the xy cross-sectional plane normal to a topological line defect, which runs along the z axis. In this example, the complex phase ϕ(r) winds by 6π around the line defect (represented by the red dot at the origin). The winding number of the complex phase in general can be evaluated by the line integral where C is a (righthanded) closed path that runs once around the (oriented) line defect. Eq.(2.3) is always an integer given that the mass parameter m(r) is nonvanishing along C. Massless chiral Dirac fermions run along these topological line defects 156 . When focusing at k z = 0, the differential operator (2.2) with a vortex along the z-axis is identical to the 2D Jackiw-Rossi model 157 with chiral symmetry γ 5 = s z µ z . 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 γ 5 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 a single propagating direction. The energy spectrum of the topological line defect (for the example with winding number c = 3) is shown in the right diagram of figure 3, in which, there are three chiral bands (red curves) inside the bulk energy gap representing the 3 chiral Dirac electrons. As a consequence of the chirality, the transport of charge and energy must also be uni-directional. The chiral electric and energy-thermal responses are respectively captured by the two conductances where ν is the filling fraction if the chiral channel is supported by a 2D insulating bulk, and c is called the chiral central charge. For the Dirac case, c = ν is the number of chiral Dirac channels. Here c can be negative when the Dirac fermions oppose the preferred orientation of the topological line defect. In a more general situation, c = c R − c L counts the difference between the number of forward propagating and backward propagating Dirac fermions. There is a mathematical index theorem 156,158,159 that identifies the topological winding number in (2.3) and the analytic number of chiral Dirac fermions in (2.4). Hence, there is no need to distinguish the two c's. The massless chiral Dirac channels, described by the low-energy effective theory have an emergent conformal symmetry and the index c = c R − c L is also the chiral central charge of the effective conformal field theory (CFT). We refer to the primitive topological line defect with c = ±1 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.)  A three-dimensional array of Dirac strings (wires) can be realized as a vortex lattice of the mass parameter m = m x + im y in a Dirac semimetal. For example, figure 4 shows a vortex lattice generated by the spatially-varying Dirac mass where sd is the (rescaled) Jacobian elliptic function 160 with simple zeros at p+iq and poles at (p+1/2)+i(q+1/2) for p, q integers. It consists of vortices with alternating winding number c = ±1 at the zeros and poles in a checkered board lattice configuration. On the cross section plot on the right side of figure 4, there is a Dirac string with positive (or negative) winding at each • (resp. ×).
Each vortex string has a chiral Dirac fermion running through it. Figure 5 shows the same two-dimensional slice of the array, except suppressing the mass parameters which correspond to irrelevant microscopic highenergy degrees of freedom. We choose a unit cell labeled by (p, q), its x, y coordinates. Each has both a forward moving Dirac fermion ψ p,q (shown as •) and a backward moving one ψ ⊗ p,q (shown as ×). This array configuration breaks TR as the symmetry would have reversed the chirality (i.e. propagating direction) of each Dirac fermion. Instead, it has an emergent anti-ferromagnetic time reversal (AFTR) symmetry, which is generated by the operators T 11 and T1 1 in the diagonal and off-diagonal directions. Each is composed of a time reversal operation and a half-translation by (e x + e y )/2 or (−e x + e y )/2.
These AFTR operators are non-local as they come with lattice translation parts. They are anti-unitary in the sense that T αψT −1 = α * T ψT −1 and T u|T v = u|v * because the local time reversal symmetry is anti-unitary. Similar to a spatial non-symmorphic symmetry, the AFTR symmetries square to the primitive translation operators where (−1) F is the fermion parity operator. Moreover they mutually commute [T 11 , T1 1 ] = 0. We notice in passing that the AFTR symmetry is only an emergent symmetry in the low-energy effective theory. It is not preserved in the microscopic Dirac model (2.2) and is broken by the mass parameter, m(r) = m(r + (e x ± e y )/2) * . For instance, the Jacobian elliptic Dirac mass function (2.6) actually has a periodic unit cell twice the size of that of the effective wire model in figure 5. On the other hand, the Dirac mass (2.6) is odd under C 2 , m(C 2 r) = −m(r). This sign is canceled by the C 2 rotations of the Dirac matrices,Ĉ 2 µ x,yĈ −1 2 = −µ x,y , that couple with the Dirac mass in the Hamiltonian (2.2). Therefore the Dirac wire model in figure 5 has a twofold axis along one of the Dirac string, say ψ 0,0 . The Dirac channel fermions transform unitarily according to where the factor of i ensures the fermionic −1 twist phase for a 2π rotation, and the second eqaulity in (2.9) is determined by the first one together with (2.7) and the symmetry relations Again, in order for the rotation symmetric wire model to be free of anomalies, C 2 should really be a screw rotation with respect to some microscopic lattice that has become irrelevant in the low-energy continuum picture.
When adjacent vortex strings are near each other, their Dirac fermion wave functions overlap and there are finite amplitudes of electron tunneling. We construct a coupled Dirac wire model of nearest-wire single-body backscattering processes with ±π fluxes across each diamond square (figure 5), where the tunneling amplitude t 1 (or t 2 ) in the (11) (resp.(11)) direction is imaginary (resp. real).
where the first line is the kinetic Hamiltonian of individual Dirac channels under the Fourier transformation −i∂ z ↔ k z along the wire direction. This tightbinding Hamiltonian preserves the AFTR symmetry (2.7), T HT −1 = H. Fourier transformation of the square lattice is the Bloch band Hamiltonian, for g(k x , k y ) = it 1 (1 − e −i(ky+kx) ) + t 2 (e −ikx − e −iky ). Here momentum k lives in the "liquid crystal" Brillouin zone (BZ) where −π ≤ k x , k y ≤ π and −∞ < k z < ∞ (in the continuum limit a → 0 and π/a → ∞). The energy spectrum of the two-band model is given by E ± (k) = ± |g(k x , k y )| 2 + 2ṽ2 k 2 z (see figure 6). It gives two linearly dispersing Weyl cones of opposite chiralities in the BZ centered at K + 0 = Γ = (0, 0, 0) and K − 0 = M = (π, π, 0). Near these points, the Hamiltonians are of the linear form H( where σ = (σ x , σ y , σ z ) are Pauli matrices acting on the (ψ , ψ ⊗ ) degrees of freedom. The velocity matrices are whose determinant's sign decides the ± chirality of the Weyl fermion at Γ and M , i.e. the ±1 Fermi surface Chern invariants 3,6,24 . The AFTR symmetries (2.7) in the single-body picture are expressed under Fourier transformation as where K is the complex conjugation operator. They satisfy the appropriate algebraic relations (2.8) in momentum space (2.16) and the coupled wire model (2.13) is AFTR symmetric The Weyl points are at time reversal invariant momenta (TRIM) K ± 0 ≡ −K ± 0 (modulo the reciprocal lattice 2πZ 2 ), and the AFTR operators T 11 (K ± 0 ) = −iσ y K and T1 1 (K ± 0 ) = ∓iσ y K square to minus one. Hence the Weyl points are not only protected by the non-vanishing Fermi surface Chern invariant but also the Kramers' theorem.
In addition, the model is also C 2 symmetric where the twofold symmetry (2.9) is represented in the single-body picture by a diagonal matrix (supressing the screw phase e −ikza/2 in the continuum limit a → 0). It agrees with the fermion statistics (2.11) C 2 (−k x , −k y , k z )C 2 (k x , k y , k z ) = −1, and the algebraic relations (2.10) with the AFTR operators

A. The anomalous Dirac semimetal
We notice that the coupled wire Dirac model (2.12) and its massless energy spectrum in figure 6 are anomalous with respect to the AFTR symmetries T 11 and T1 1 as well as the C 2 symmetry if it is proper symmorphic and not a screw rotation. This means that it cannot be realized in a single-body three dimensional lattice system with the AFTR or C 2 symmetries. In a sense, it is not surprising at all since the chiral Dirac strings that constitute (2.12) are themselves violating fermion doubling 17,18 . Here we further elaborate on the anomalous Dirac spectrum (figure 6) where the pair of Weyl points are separately located at two TRIM K ± 0 . We also comment on the non-trivial consequence of the anomaly and pave the path for later discussion on many-body interactions.
We begin with two 2D planes in momentum space parallel to k y k z located at k x = ±π/2. They are represented by the two blue planes in figure 6. The AFTR or C 2 symmetries require the Chern invariants is the projection operator onto the negative energy band. This is because the AFTR symmetry is anti-unitary and preserves the orientation of the k y k z plane, whereas C 2 is unitary but reverses the orientation of the k y k z plane. (See appendix B for a detailed proof.) On the other hand, the two Chern invariants along the two planes must differ by 1 because they sandwich a single Weyl point at Γ. This forces the Chern invariants to be a half integer Ch 1 = ±1/2, which is anomalous. While the C 2 anomaly can be resolved simply by doubling the unit cell and assuming it originates from a microscopic non-symmorphic screw axis, the AFTR anomaly is stronger because the two antiferromagnetic combinations (2.8) generate lattice translations and fix the unit cell size. There are three resolutions.
1. The AFTR symmetries are broken by high energy degrees of freedom when k z is large.
2. The spectrum in figure 6 is the holographic 3D boundary spectrum of an AFTR symmetric weak topological insulator in 4D.
3. The spectrum is generated by strong many-body interaction non-holographically in 3D.
Below we discuss the first two resolutions, and we leave the many-body interaction-enabled situation to section III D.

Broken symmetries and coarse-graining
In the present case when the chiral Dirac channels originate from vortex strings in an underlying microscopic Dirac insulator, the spatial modulation of mass parameters m(r) actually violate one of the AFTR symmetries, m(r) * = m(r + (e x ± e y )/2), where * stands for complex conjugation. For instance, since all elliptic functions must contain at least two zeros and two poles in its periodic cell, the Jacobian elliptic mass function (2.6) has longer periods than e x and e y in figure 5, and thus must break T 11 or T1 1 . The symmetry is broken only in the ultra-violet limit at large k z where the chiral Dirac line nodes meet the microscopic bulk band (see figure 3) at high energy ∼ |m(r)|. In fact, the above anomalous argument shows that all mass parameter configurations that produce the 3D vortex lattice array (figure 4) must either (a) break both the AFTR symmetries T 11 and T1 1 , or (b) preserve one but violate translation so that the unit cell is enlarged and the two Weyl points collapse onto each other in momentum space. (See figure 7 and 8.) For instance, the microscopic system can be connected to a stack of Chern insulating ribbons (or lowest Landau levels) with alternating chiralities shown in figure 7. 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 counter-propagating boundary chiral Dirac channels. The dark blue arrows represent the orientations of the Chern ribbons that accommodate the boundary Dirac channels with the appropriate propagating directions. Here the Chern ribbon pattern in figure 7(a) breaks both AFTR axes. The pattern in figure 7(b) preserves T1 1 . However, translation symmetry is also broken and the coupled Dirac wire model now has an enlarged unit cell (light blue dashed boxes) that consists of two pairs of counter-propagating chiral Dirac channels. All Chern ribbon patterns must break the C 2 symmetry about a Dirac wire because each wire is connected to one and only one Chern ribbon in a particular direction. Now we go back to the vortex lattice generated by the Jacobian elliptic Dirac mass function m(r) in (2.6) and consider its symmetries. For this purpose, we use the symmetry properties of the (rescaled) Jacobian elliptic function 160 where C is some unimportant real constant that depends on the modulus of sd and will never appear in the mass function m(r) = m 0 sd(x + iy)/|sd(x + iy)|. We see from the minus sign in the first equation that the Jacobian elliptic function, and consequently the mass function, have primitive periods e x ± e y and therefore have a unit cell of size 2 (see figure 8(a)). Choosing m 0 = |m 0 |e iπ/4 , we see from the second equation that T 11 (or T1 1 ) is preserved (resp. broken) and thus the parent Dirac Hamiltonian Lastly, the third property of (2.22) entails the mass function m(r) = −m(C 2 r) is odd under C 2 , and consequently the parent Dirac Hamiltonian is (screw) rotation symmetriĉ anticommuting with the mass terms m 1 µ x + m 2 µ y in H Dirac (see (2.2)), and Remembering that the coupled wire model (2.12) (figure 5) descended from a vortex lattice of the microscopic parent Dirac Hamiltonian (2.2), the Dirac mass m(r) actually allows the model to carry fewer symmetries than the low-energy effective Hamiltonian (2.12) suggests. Now that the translation symmetry is lowered, the BZ is reduced (see figure 8(b)) so that the two Weyl points now coincide at the origin Γ. This recovers an unanomalous Dirac semimetallic model (2.1) around (k x , k y ) = (0, 0). The fourfold degenerate Dirac point is protected and pinned at Γ due to the remaining AFTR symmetry T 11 -which takes the role of a spinful time reversal (T 2 = −1) in the continuum limit -and the C 2 (screw) symmetry about the z-axis. However, if any of these symmetries is further broken, the fourfold degeneracy of the Dirac point is not protected (c.f. the original continuum Dirac model (2.2)). Figure 8(a) shows a dimerized coupled Dirac wire model that introduces a finite mass for the Dirac fermion. We label the Dirac fermion operators as ψ µ,σ r,s , for σ = , ⊗ the chirality, µ = A, B the new sublattice label, and (r, s) label the coordinates of the unit cell according to the 45 • -rotated x , y -axes.
For instance, the model is identical to the AFTR and C 2 symmetric one in (2.12) when t j = t j = u j = u j for j = 1, 2. However, when the symmetries are broken, these hopping parameters do not have to agree. The Bloch band Hamiltonian after Fourier transformation is where the 2 × 2 identity matrix 1 1 and h(k x , k y ) acts on the sublattice µ = A, B degrees of freedom, and −π ≤ k x , k y ≤ π are the rotated momenta. We perturb about the Dirac fixed point by introducing the dimerizations for j = 1, 2. About the Γ = (0, 0, 0) point, See figure 9 for its massive spectrum. Here the AFTR symmetry T 11 and the twofold rotation C 2 are represented in the single-body picture by (again suppressing the C 2 screw phase e −ikza/2 in the continuum limit a → 0). In the small k x , k y -limit, T 11 (0) = −iσ y K and C 2 (0) = iσ z . It is straightforward to check that the dimerization ∆ 2 preserves T 11 while both ∆ 1 , ∆ 2 breaks C 2 .
Since the coupled wire model (2.29) and the parent continuum Dirac model (2.2) have the same matrix and symmetry structure, we can apply the same construction we discussed before to the new coarse-grained model (2.29). For instance, the non-competing dimerizations ∆(r) = ∆ 1 (r) + i∆ 2 (r) can spatially modulate and form vortices in a longer length scale. Figure 10(b) shows a dimerization pattern that corresponds to a single vortex in ∆. The solid (dashed) lines represent strong (resp. weak) backscattering amplitudes. In the fully dimerized limit where the dashed bonds vanish, all Dirac channels are gapped except the one at the center (showed as a blue dot). In the weakly dimerized case, there is a collective chiral Dirac channel whose wave function is a superposition of the original channels and is exponentially localized at the ∆-vortex core, but now with a length scale longer than that of the original m-vortex lattice. These collective chiral Dirac ∆-vortices can themselves form a coupled array, like (2.12), and give a Dirac semimetal of even longer length scale. The single-body coupled vortex construction is therefore a coarse-graining procedure that recovers equivalent emergent symmetries at each step.

Holographic projection from 4D
The coupled wire model (2.12) with two AFTR axes can be supported by a weak topological insulator (WTI) in four dimensions. Instead of realizing the chiral Dirac channels using mass vortices of a 3D Dirac semimetal, they can be generated as edge modes along the boundaries of 2D Chern insulators (or lowest Landau levels). The 4D WTI is constructed by stacking layers of Chern insulators parallel to the zw-plane along the x and y directions. The Chern layers L r , labeled by the checkerboard lattice vector r = r x e x + r y e y on the xy-plane, have alternating orientations so that Ch[L r ] = 1 if r x , r y are integers and Ch[L r ] = −1 if r x , r y are half-integers. The model therefore carries both AFTR symmetries T 11 and T1 1 as well as the C 2 rotation about zw, and when cleaved along a 3D hyper-surface normal to w, it generates the array of alternating chiral Dirac channels in figure 5.
The 4D WTI model can also be regarded as a stack of 3D antiferromagnetic topological insulators (AFTI) 161 . Restricting to the 3D hyperplane normal to −e x +e y , this model consists of alternating Chern insulating layers parallel to the wz-plane stacked along the e x + e y direction. This 3D model describes an AFTI with a non-trivial Z 2 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 Dirac wires. The uniform nearest wire backscattering term t 1 (see (2.12)) introduces a linear dispersion along the 11-direction and gives rise to a single massless surface Dirac cone spectrum at a TRIM on the boundary of the surface BZ where T 2 11 = −1. The 4D WTI model is identical to stacking these 3D AFTI along the11-offdiagonal direction −e x + e y . A more detailed discussion on coupled wire constructions of a 4D strong and weak topological insulator can also be found in Ref. 162.

AFTR breaking surfaces
We discuss the surface states of the coupled Dirac wire model (2.12). Similar to the boundary surface of a translation symmetry protected Dirac semimetal (or more commonly called a Weyl semimetal), there are Fermi arcs connecting the surface-projected Weyl points 3,6,24 . First we consider the (100) surface normal to x-axis (see figure 5). We assume the boundary cuts between unit cells and set the Fermi energy at ε f = 0. At k z = 0 and given a fixed k y ∈ (−π, π), the tight-binding model (2.13) is equivalent to the Su-Schriffer-Heeger model 163 or a 1D class AIII topological insulator 164,165 along the x-direction protected by the chiral symmetry When t 1 , t 2 have the same (or opposite) sign, the quasi-1D model is topological along the positive (resp. negative) k y -axis and thus carries a boundary zero mode. This corresponds to the Fermi line joining the two surface projected Weyl points at Γ and M (see figure 11(a)).
As the zero modes have a fixed chirality according to σ z , they propagate uni-directionally with the dispersion E(k z ) = ṽk z σ z . The cleaving surface breaks AFTR and C 2 symmetries, and so does the Fermi arc in figure 11(a). For instance, any one of the AFTR symmetries maps the boundary surface to an inequivalent one that cuts through unit cells instead of between them. As a result, the Fermi arc will connect the Weyl points along the opposite side of the k y -axis for this surface. The (010) surface Fermi arc structure is qualitatively equivalent to that of the (100) surface. The (110) and (110) surfaces that cleave along the diagonal and offdiagonal axes (see figure 5) respectively preserve the AFTR symmetries T 11 and T1 1 . There are no protected surface Fermi arcs because the two bulk Weyl points project onto the same point on the surface Brillouin zone. Lastly, we consider the (001) surface normal to the zaxis, which is the direction of the chiral Dirac strings that constitute the coupled wire model. A chiral Dirac channel cannot terminate on the boundary surface. In a single-body theory, it must bend and connect with an adjacent counter-propagating one. Although the (001) plane is closed under the C 2 as well as both the AFTR symmetries, the surface bending of Dirac channels must violate at least one of them. Here we consider the simplest case where the counter-propagating pair of Dirac channels within a unit cell re-connects on the boundary surface. This boundary is equivalent to a domain wall interface separating the Dirac semimetal (2.12) from an insulator where Dirac channels backscatters to their counter-propagating partner within the same unit cell.
The domain wall Hamiltonian takes the form of a differential operator by replacing k z ↔ −i∂ z in (2.12). Here θ(z) can be the unit step function or any function that asymptotically approaches 1 for z → ∞ or 0 for z → −∞. The model therefore describes the Dirac semimetal (2.12) for positive z, and an insulator for negative z where Dirac channels are pair annihilated within a unit-cell by t 1 . After a Fourier transformation, the Bloch HamiltonianĤ(k x , k y ) is identical to (2.13) by replacing k z ↔ −i∂ z and g(k . Given any fixed k x , k y , the differential operatorĤ(k x , k y ) is identical to the Jackiw-Rebbi model 166 . When ε f = 0, the zero modes trace out a Fermi arc that connects the two surface projected Weyl points (see figure 11(b)). We notice that in the insulating phase (or on the boundary surface), Dirac wires can be backscattered with a different phase and dimerized out of the unit cell. These different boundary conditions correspond to distinct surface Fermi arc patterns. Figure 12 shows two alternatives. (a) shows the the zero energy arcs when intra-cell backscattering reverses sign t 1 → −t 1 in the insulating domain. (b) shows a case when the dimerization is taken along the off-diagonal axis. These inequivalent boundary conditions differ by some three dimensional integer quantum Hall states, which correspond to additional chiral Fermi arcs that wrap non-trivial cycles around the 2D toric surface Brillouin zone.

AFTR preserving surfaces
We also notice that the Fermi arc structures in figures 11(b) and 12 are allowed because both the AFTR symmetries T 11 , T1 1 and the C 2 symmetry are broken by the insulating domain. Any dimerization that preserves only one of T 11 and T1 1 necessarily breaks translation symmetry, and corresponds to an enlarged unit cell and a reduced Brillouin zone (c.f. figure 7 and 8). As a result, the two Weyl points would now collapse onto the same Γ point. Any momentum plane that contains the k zdirection and avoids the Γ point must have trivial Chern invariant, because it could always be deformed (while containing the k z -direction and avoiding the Γ point) to the reduced Brillouin zone boundary, where its Chern invariant would be killed by the AFTR symmetry.
However, the trivial bulk Chern invariant does not imply the absence of surface state. This can be understood by looking at the surface boundary in real space. Here, we assume the Dirac strings that constitute the coupled wire model (2.12) are supported by vortices of an underlying Dirac mass (see figure 4 and eq.(2.2)). The semimetallic coupled wire model terminates along the xy-plane against vacuum, which is modeled by the Dirac insulator H vacuum = vk · sµ z + m 0 µ x , say with m 0 > 0. Recall from (2.23) that the Dirac mass vortex configuration (2.6) is AFTR symmetric along the T 11 -directions. The Dirac insulating vacuum is symmetric under local TR as well as continuous translation. It however breaks the screw rotation symmetryĈ 2 = is z µ z , but we here only focus on the AFTR symmetry. The surface boundary supports chiral Dirac channels that connect the chiral Dirac strings in the semimetallic bulk that are normal to the surface. The surface channels are shown in figure 13. The × (•) represent chiral vortices in the bulk that direct electrons away from (resp. onto) the surface. The vector field represents the Dirac mass m(r) = m x (r) + im y (r) modulation in the semimetallic bulk near the surface. The surface Dirac line channels 156 -shown by directed lines connecting the bulk Dirac strings ×, • -are located where the TR symmetric Dirac mass m x changes sign across the surface boundary and the TR breaking Dirac mass m y flips sign across the line channels along the surface. In other words, they are traced out of points on the surface where m x < 0 and m y = 0. Each of these surface channels carries a chiral Dirac electronic mode that connects the bulk chiral Dirac vortices. They can couple through inter-channel electron tunneling, but the collective gapless surface state cannot be removed from low-energy by dimerization without breaking the AFTR symmetry T 11 .

III. MANY-BODY INTERACTING VARIATIONS
We discuss the effect of strong many-body interactions in a Dirac semimetal in three dimensions. Before we do so, it is worth stepping back and reviewing the two dimensional case in order to illustrate the issue and idea that will be considered and generalized in three dimensions. The massless Dirac fermion with H = v(k x s y − k y s x ) that appears on the surface of a topological insulator 6,12-14 is protected by time reversal (TR) and charge U (1) symmetries and is 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 U (1) preserving many-body interacting mass. [113][114][115][116] 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 introducing an interacting mass gap. This leaves a single massless Dirac fermion on the opposite surface without breaking symmetries.
A massless Dirac fermion in three dimensional semimetallic materials can be protected in the singlebody picture by screw rotation, time reversal and charge U (1) symmetries (see reviews Ref. 3, 6, and 9 and section II). From a theory point of view, it can be supported on the 3D boundary of a 4D weak topological insulator, where the two Weyl fermions are located at distinct time reversal invariant momenta (recall figure 6 and section II A 2 for the antiferromagnetic case). In this case, the massless fermions are protected by translation, time reversal and charge U (1) symmetries. In this section, we address the following issues. (1) We show by explicitly constructing an exactly solvable coupled wire model that the 3D Dirac fermion can acquire a many-body interacting mass while preserving all symmetries. (2) We show in principle that an antiferromagnetic time reversal (AFTR) symmetric massless 3D Dirac system with two Weyl fermions separated in momentum space can be enabled by many-body interactions without holographically relying on a higher dimensional topological bulk.
We begin with the Dirac semimetallic coupled wire model in figure 6 and (2.12). In particular, we focus on many-body interactions that facilitate the fractionalization of a (1 + 1)D chiral Dirac channel Dirac = Pfaffian ⊗ Pfaffian

4).)
While this is not the focus of this article, here we clarify and disambiguate the three "Pfaffian" fractional quantum Hall (FQH) states that commonly appear in the literature. All these (2 + 1)D states are theorized at filling fraction ν = 1/2, although being applied to ν = 5/2 in materials, and have identical electric transport properties. However, they have distinct thermal Hall transport behaviors. They all have very similar anyonic quasiparticle (QP) structures. For instance, they all have four Abelian and two non-Abelian QP (up to the electron). On the other hand, the charge e/4 non-Abelian Ising anyons of the three states have different spin-exchange statistics. First, the gapless boundary of the Moore-Read Pfaffian FQH state [168][169][170][171] can be described by the (1 + 1)D chiral CFT U (1) 4 ⊗ Ising where the charged boson and neutral fermion sectors are co-propagating. It therefore carries the chiral central charge c = 1 + 1/2 = 3/2, which dictates the thermal Hall response (2.4). Second, the "anti-Pfaffian" FQH state 172,173 is the particle-hole conjugate of the Moore-Read Pfaffian state. Instead of half-filling the lowest Landau level by electrons, one can begin with the completely filled lowest Landau level, and half-fill it with holes. In a sense the anti-Pfaffian state is obtained by subtracting the completely filled lowest Landau level by a Moore-Read Pfaffian state. Along the boundary, the (1 + 1)D CFT U (1) 1/2 ⊗U (1) 4 ⊗ Ising consists of the forward propagating chiral Dirac U (1) 1/2 sector that corresponds to the lowest Landau level, and the backward propagating Moore-Read Pfaffian U (1) 4 ⊗ Ising. Here C can be interpreted as the time-reversal conjugate of the chiral CFT C. The thermal transport is governed by the edge chiral central charge c = 1 − 3/2 = −1/2, which has an opposite sign from the filling fraction. Thus, unlike the Moore-Read Pfaffian state, the net electric and thermal currents now travel in opposite directions along the edge. Lastly, the recently proposed particle-hole symmetric (PHS) Pfaffian state [151][152][153] , which is going to be the only Pfaffian FQH state considered in this article (see Ref. 132 for a coupled wire construction), has the chiral edge CFT (3.2). As the electrically charged boson and neutral fermion sectors are counter-propagating, the net thermal edge transport is governed by the chiral central charge c = 1 − 1/2 = 1/2. The chiral (1 + 1)D PHS Pfaffian CFT (3.2) is also present along the line interface separating a TR symmetric T -Pfaffian 115 domain and a TR breaking magnetic domain on the surface of a 3D topological insulator. (Similar constructions can be applied to alternative TR symmetric topological insulator surface states 113,114,116 , but they will not be considered in this article.) Other than their thermal transport properties, the three Pfaffian FQH state can also be distinguished by the charge e/4 Ising anyon, which has spin h = 1/8, −1/8 or 0 for the Moore-Read Pfaffian, anti-Pfaffian or PHS Pfaffian states respectively.
Since we will not be considering the Moore-Read Pfaffian or its particle-hole conjugate anti-Pfaffian state, we will simply refer to the PHS Pfaffian state as the Pfaffian state. The low-energy effective chiral (1+1)D CFT takes the decoupled form between the boson and fermion where we have set = 1. Here φ R is the free chiral U (1) 4 boson. It generates the (1 + 1)D theory L charged , which is identical to the boundary edge theory of the (2 + 1)D bosonic Laughlin ν = 1/8 fractional quantum Hall state described by the topological Chern-Simon theory 174,175 where z ∼ τ + ix is the complex space-time parameter and τ = iπvt/2 is the Euclidean time. γ † L = γ L is the free Majorana fermion. It generates the (1 + 1)D theory L neutral , which is equivalent to a chiral component of the critical Ising CFT or the boundary edge theory of the (2 + 1)D Kitaev honeycomb model 176 in its B-phase with TR breaking (i.e. a chiral p x + ip y superconductor coupled with a Z 2 gauge theory). It carries trivial electric conductance but contributes to a finite thermal conductance characterized by the chiral central charge c = −c L = −1/2. The Ising CFT has primary fields 1, γ L and σ L , where the twist field (or Ising anyon) σ L carries the conformal spin h = −h L = −1/16. Again, we abbreviate the operator product expansions wherez ∼ τ − ix is the complex space-time parameter and τ = iṽt is the Euclidean time.
General primary fields of the Pfaffian CFT decompose into the U (1) 4 part and the Ising part. They take the form The conformal spins and fusion rules also decompose so that modulo 1, q m = m/4 in units of e, and (3.10) The 2π monodromy phase M XY Z = R XY Z R Y X Z between primary fields X and Y with a fixed overall fusion channel Z can be deduced by the ribbon identity 176 for h X,Y,Z the conformal spins for primary fields X, Y, Z. Unlike the gauge dependent π-exchange phase R XY Z , the 2π-monodromy phase M XY The electronic quasiparticle is the composition ψ el = e −i4φ R γ L so that it is fermionic and has electric charge −1 in units of e. Since electron is the fundamental building block of the system, locality of ψ el only allows primary fields X that have trivial monodromy M X,ψ el = 1 with the electron. As a result, this restricts 1 m , ψ m to even m and σ m to odd m. Lastly, 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 Pfaffian, whose Lagrangian density is the time reversal of (3.3), i.e. replacing R ↔ L, i ↔ −i and ∂ t ↔ −∂ t .  can also be expressed in terms of many-body backscattering of the Pfaffians' primary fields : +h.c. (3.14) where d R = 1 A 2 1 B −2 is the electrically neutral Dirac fermion composed of the pair of oppositely charged semions in the two Pfaffian sectors.
In strong coupling, the gapping Hamiltonian introduces an interacting mass and the ground state expectation value Φ = nπ/2, for n an integer and Φ = 2φ A R − 2φ B R − φ σ L . In low energy, it leaves behind the chiral boson combinationφ R = 2φ A R + 2φ B R , which has trivial operator product (i.e. commutes at equal time) with the order parameter Φ. The low-energy theory after projecting out the gapped sectors becomes which is identical to the bosonized Lagrangian density of a chiral Dirac fermion. For instance, the vertex operator ψ el R ∼ e iφ R ∼ 1 A 2 1 B 2 has the appropriate spin and electric charge of an electronic Dirac fermion operator (h = 1/2 and q = 1 in units of e). Notice that the vertex operator e iφ R /2 has −1 monodromy with the local electronic ψ el R and therefore is not an allowed excitation in the fermionic theory.
We notice in passing that the gluing potential (3.13) facilitates an anyon condensation process 178 , where the maximal set of mutually local neutral bosonic anyon pairs is condensed, where m is an arbitrary integer. All primary fields that are non-local (i.e. with non-trivial monodromy) with any of the condensed bosons in (3.16) are confined. Any two primary fields that differ from each other by a condensed boson in (3.16) are now equivalent. The condensation therefore leaves behind the electronic Dirac fermion and its combinations.  On the other hand, a chiral Dirac channel can be decomposed into a pair of chiral Pfaffian channels (see figure 15 for a summary). First, perhaps from some channel re-construction, we append to the chiral Dirac channel an additional pair of counter-propagating Dirac modes. This can be realized by pulling a parabolic electronic/hole band from the conduction/valence band to the Fermi level, or introducing non-linear dispersion to the original chiral channel. In low-energy, the three Dirac fermion modes can be bosonized ψ 1,2 R ∼ e iφ 1,2 R , ψ L ∼ e −iφ L and they are described by the multicomponent boson Lagrangian 1, −1), andṼ is some non-universal velocity matrix. A general composite excitation can be expressed by a vertex operator e im· φ , for m an integral 3-vector, with spin h m = |m| 2 /2 and electric charge q m = m TKt in units of e, wheret = (1, 1, 1) is the charge vector.
Next we perform a fractional basis transformation While theK matrix is invariant under the transformation, the charge vector changes tot → (1, 0, 0). ψ ρ R ∼ e iφ ρ R is the local electronic Dirac fermion that carries spin 1/2 and electric charge e, and d R/L ∼ e iφ σ R/L are counter-propagating electrically neutral Dirac fermions. As theK matrix is still diagonal, these fermions have trivial mutual 2π-monodromy and are local with respect to each other. However, it is important to notice that the neutral Dirac fermions d R/L actually consist of fractional electronic components. Now we focus on the two R-moving Dirac channels. By pairing the Dirac fermions, they form two independent SU (2) 1 Kac-Moody current operators 167 Both SU (2) 1 sectors are electrically charged so that the bosonic vertex operators J A/B ± carries charge ±e. They obey the SU (2) current algebra at level 1 to be periodic. The incorporation of anti-periodic boundary condition for J are in fact local quasi-electronic. Unlike in the gluing case where there is a gapping Hamiltonian (3.13) that pastes a pair of Pfaffians into a Dirac, here in the splitting case we have simply performed some kind of fractional basis transformation that allows us to express Dirac as a pair of Pfaffians. In fact, one can check that the energy-momentum tensor of the Dirac theory (3.18) is identical to that of a pair of Pfaffians (3.3). However, this does not mean the Pfaffian primary fields are natural stable excitations. In fact, as long as there is a pair of co-propagating Pfaffian channels, all primary fields except the non-fractionalized electronic ones are unstable against the gluing Hamiltonian δH in (3.13) and are generically gapped. In order for the Pfaffian CFT to be stablized, one has to suppress δH. A possible way is to somehow spatially separate the pair. This issue is addressed in the subsection below using many-body interaction in the coupled wire model of a Dirac semimetal (or the PHS Pfaffian FQH state in Ref. 132).

B. Symmetry preserving massive interacting model
We begin with the 3D array of chiral Dirac strings in figure 4. In section II, we showed that the single-body coupled wire model (2.12) described a Dirac semimetal with two Weyl fermions (see figure 6). The system had emergent antiferromagnetic time reversal (AFTR) symmetries T 11 and T1 1 along the diagonal and off-diagonal axes (see (2.17)). Together they generate an emergent lattice translation symmetry with a 2-wire unit cell, and separate the two Weyl points in the Brillouin zone. The symmetries are lowered beyond the effective model when the microscopic high-energy degrees of freedom are included. For example, the mass function (2.6) that supports the Dirac vortex string lattice has a 4-wire periodic unit cell and only preserves one of the AFTR symmetries T 11 (see (2.23)). With the lowered translation symmetry, the two Weyl points now coincide at the same momentum. Inter-species (or inter-valley) mixing is forbidden by the remaining AFTR symmetry and a (screw) twofold rotation symmetry C 2 about z (see (2.18) and (2.25)). Previously in section II A 1, we introduced symmetry breaking wire dimerizations in (2.26) that led to a massive Dirac insulator. In this section, we construct many-body gapping interactions that preserves the two AFTR symmetries T 11 and T1 1 , the C 2 symmetry, as well as charge U (1) conservation. The many-body gapping scheme is summarized in figure 16. From the previous subsection, we saw that each chiral Dirac channel can be decomposed into a pair of independent Pfaffian channels. They can then be backscattered in opposite directions to neighboring wires. Figure 16(a) shows a particular dimerization pattern of the Pfaffian channels that preserves the symmetries. In this case, the many-body backscattering interaction U is directed along the diagonal axis. In the limit when U is much stronger than the single-body electron tunneling in the previous semimetallic model (2.12), the system decomposes into decoupled diagonal layers and it suffices to consider the interaction on a single layer. For convenience, we here change our spatial coordinates so that the diagonal axis is now labeled by y and the wires now propagate along x.
Focusing on a single diagonal layer, the system in the non-interacting limit first consists of a 2D array of chiral Dirac strings with alternating propagating directions (see the left side of figure 16(b)). We notice that this is identical to the starting point of the coupled wire construction of the topological insulator Dirac surface state considered by Mross, Essin and Alicea in Ref. 149. For instance, the alternating Dirac channels there were supported between magnetic strips with alternating orientations on the topological insulator surface, and an uniform nearest-channel electron tunneling recovered the massless 2D Dirac spectrum protected by the AFTR symmetry. They then proceeded to propose symmetry preserving many-body gapping interactions facilitated by adding 2D FQH strips between the channels. While this reconstruction trick can be applied on the 2D surface of a topological insulator, it is not feasible in our 3D situation and would require drastic modification of the bulk semimetal. Instead, here we propose an alternative gapping scheme that does not involve additional topological phases. In other words, we are going to construct a 3D gapped and layered topological phase solely from interacting electronic Dirac wires.
First, in order to implement the splitting described in the previous subsection, we assume each Dirac string consists of two Dirac channels going in one direction and a third Dirac channel going the opposite direction (see the left side of figure 16(b)). We denote the electronic Dirac fermions on the y th wire by ψ y = (ψ 1 y , ψ 2 y , ψ 3 y ) and bosonize The sliding Luttinger liquid [119][120][121][122][123] Lagrangian density is where sgn(s) = s/|s| = ±1 for s = 0 and sgn(0) = 0, 27) and σ z = ±1. The introduction of the specific Klein factors S jj , Σ jj and the undetermined sign σ z are necessary for the correct representations of the T 11 and C 2 symmetries in the bosonization setting, and these choices will be justified below. The first line of (3.26) is equivalent to the commutation relation between conjugate fields which is set by the "pq" term in L layer . The alternating signs (−1) y in (3.28) and (3.25) changes the propagating directions from wire to wire. The second and third line of (3.26) guarantee the correct anticommutation relations {e ±iφ j y , e ±iφ j y } = 0 between Dirac fermions along distinct channels j = j or distinct wires y = y . The reason thẽ C 2 matrix is defined in this form will become clear in the fractional basis discussed later in (3.43).
The anti-unitary AFTR symmetry along the diagonal T 11 direction transforms the bosons according to The unitary C 2 rotation takes Moreover, we choose the representation so that the sign σ z in the ETCR (3.26) is preserved by the AFTR operator but is flipped by the C 2 symmetry, The ETCR (3.26) is consistent with the AFTR symmetry. This means that evaluating T 11 φj y (x),φ j y (x ) T −1 11 by taking the AFTR operator inside the commutator yields the same outcome as taking the TR of the purely imaginary scalar The ETCR (3.26) is also consistent with the C 2 symmetry This is because the Klein factors (3.27) are C 2 symmetric Notice that the undetermined sign σ z , which is odd under C 2 , in (3.26) is essential for the ETCR to be consistent with C 2 . The last term in the AFTR operation (3.29) makes sure which is necessary for T 2 11 = (−1) F translation(2e y ). Here the fermion parity operator is (−1) F = e iπ yj N j y , where is the number operator. The vector v in the C 2 operation (3.30) satisfies (δ j j + (C 2 ) j j )v j /2 =K jj , and consequently which is consistent with C 2 2 = (−1) F . Lastly, it is straightforward to check that the symmetry representations (3.29) and (3.30) are compatible with the algebraic relation (2.10), i.e. for each wire, so that ψ ρ y ∼ e iφ ρ y is a Dirac fermion carrying electric charge e, d σ1 y ∼ e iφ σ1 y (d σ2 y ∼ e iφ σ2 y ) is an electrically neutral Dirac fermion propagating in the same (resp. opposite) direction as ψ ρ y . For convenience, sometimes we combine the transformed bosonized variables into φ y = (φ 1 y , φ 2 y , φ 3 y ) = (φ A y , φ B y , φ σ2 y ), which is related to the original local ones in (3.25)  The AFTR symmetry operation (3.29) becomes where κ I = G I jK jj which is 1/4 for I = 1, 2 and 0 for I = 3. The C 2 transformation (3.30) becomes .
The 3×3 C 2 matrix takes a much simpler form here using the fractional basis than in (3.30). In fact, the original C 2 matrix in the local basis in (3.30) was defined so that C 2 = GC 2 G −1 would act according to (3.43). Roughly speaking, ignoring the constant phases Gv, the C 2 symmetry switches φ A y ↔ φ B −y and sends φ σ2 y → −φ σ2 −y . Next, we combine these co-propagating pair of fermions to form two SU (2) 1 current algebras (c.f. (3.20) and (3.21)) where w ∼ τ + (−1) y x is the complex spacetime parameter. As a reminder, the charge ±e bosons J A/B ± are non-electronic fractional operators, although they carry non-fractional statistics.
The remaining counter-propagating neutral Dirac fermion can be decomposed into real and imaginary components d σ y (w) ∼ cos φ σ2 y (w) + i sin φ σ2 y (w). (3.45) Majorana fermions can be constructed by multiplying these components with "Jordan-Wigner" string 46) where N j y are the number operators defined in (3.37), so that they obey mutual fermionic statistics {γ λ y (x), γ λ y (x )} = δ λλ δ yy δ(x − x ), for λ, λ = A, B. Similar to the charge ±e bosons J A/B ± , the electrically neutral Dirac fermion d σ y and consequently the Majorana fermions γ A/B y are also non-electronic fractional operators. This AB-decomposition splits each Dirac wire into a pair of decoupled Pfaffian sectors (see figure 16(b)).
Before we move on to the symmetric interaction, some further elaborations are needed for the number operators N j y and their corresponding fermion parity operators e iπN j y . In our construction, the counter-propagating pair of channels with j = 2, 3 are appended to the original one with j = 1 to make the Pfaffian fractionalization feasible. We choose the Hilbert space so that the two additional fermion parity operators agree, e iπN 2 y = e iπN 3 y . However, we allow fluctuations to the combined parity e iπ(N 2 y +N 3 y ) and only require it squares to the identity, e 2πi(N 2 y +N 3 y ) = 1. In other words, e iπ(N 2 y +N 3 y ) = e −iπ(N 2 y +N 3 y ) and it does not matter which one we take as (−1) N 2 y +N 3 y in the "Jordan-Wigner" string in (3.46). This convention will also be useful later in seeing that the many-body interaction is exactly solvable and symmetry preserving. Extra care is sometimes required.
). Previously in (3.23), we saw that the combinations ψ A 4 ∼ e i4φ A γ A and ψ B 4 ∼ e i4φ B γ B can be decomposed into products of electron operators. Similarly, each interaction in the first line of (3.47) can be decomposed into products in the form of e ±i(φ σ2 (with some scalar U (1) coefficient), where the exponents φ σ2 ±4φ A/B are linear integral combinations ofφ j . Thus, the interaction can be re-written in terms of backscatterings of local electronic operators. However, we will omit the electronic expression as (3.47) is more useful in discussing ground state and symmetries. U describes a symmetry-preserving exactly solvable model. Using the ETCR (3.26) it is straightforward to check that the (normal ordered) order parameters y +1/2 (x ) = 0. Therefore, the model is exactly solvable, and its ground states are characterized by the ground state expectation values (GEV) of the order parameters so that the interacting energy U is minimized, where l 0 is some non-universal microscopic length scale. Pinning the GEV Θ y+1/2 = n y+1/2 π, for n y+1/2 ∈ Z, gaps all degrees of freedom in the charged U (1) sector. The remaining neutral fermions are gapped by the decoupled Majorana backscattering It is worth noting that a π-kink excitation of Θ y+1/2 flips the Majorana mass in (3.50) and therefore bounds a zero energy Majorana bound state 181 . A π-kink at x 0 can be created by the vertex operators e ±iφ A y+1 (x0) or e ±iφ B y (x0) which carry ±1/4 of an electric charge. (Recall the bosonic vertices e i4φ A/B y carry charge e.) This e/4 excitation therefore corresponds to the Ising anyon in the Pfaffian FQH state.
From the AFTR symmetry action (3.42), one can show that the Majorana fermions (3.46) transform according to Therefore the fermion order parameter O F y+1/2 = iγ A y+1 γ B y (3.48) is translated under the antiunitary symmetry The boson angle parameter Θ y+1/2 defined below (3.47) changes to −Θ y+3/2 − (−1) y π under AFTR , and therefore the boson order parameter O Θ y+1/2 = e iΘ y+1/2 is flipped and translated Together, (3.52) and (3.53) show that the many-body interaction U in (3.47) is AFTR symmetric. The C 2 action (3.30) flips the number operator C 2 (N 2 −y , and therefore the parity operators appear in the "Jordan-Wigner" string (3.46) are C 2 symmetric, C 2 (−1) N 2 With the help of the C 2 action (3.43) in the fractional basis, one sees that C 2 cos φ σ y C −1 2 = (−1) y+1 sin φ σ −y and C 2 sin φ σ y C −1 2 = (−1) y+1 cos φ σ −y and thus the Majorana fermions (3.46) transform according to y is the total fermion parity of channel 2 and 3. This shows the fermion order parameter is odd under C 2 On the other hand, one can also show from the C 2 action (3.43) that the boson angle parameter changes as C 2 Θ y+1/2 C −1 2 = −Θ −y−1/2 − (−1) y π and therefore the boson order parameter O Θ y+1/2 = e iΘ y+1/2 is conjugated and flipped under C 2 When combined together, the minus signs in (3.55) and (3.56) cancel and they show that the many-body interaction U in (3.47) preserves C 2 . Now that we have introduced symmetry preserving gapping interactions on a single diagonal layer, we can extend it to the entire 3D structure by transferring (3.47) to all layers using the off-diagonal AFTR operator T1 1 (see figure 16(a)). The resulting state belongs to a topological phase in three dimensions with an excitation energy gap. It preserves both AFTR symmetries T 11 and T1 1 as well as the (screw) C 2 symmetry.

C. Antiferromagnetic stablization
The exactly-solvable many-body interacting model (3.47) (see also figure 16) shows that the Dirac semimetal (2.12) can acquire a many-body mass gap without breaking symmetries. However, it is not clear how dominant or stable the topological phase described by (3.47) is. There are alternative interactions that lead to other metallic or insulating phases that preserve or break symmetries. The scaling dimensions and the relevance of the interaction terms 182,183 can be tuned by the velocity matrix V jk in (3.25) that is affected by forward scattering interactions among co-propagating channels. Instead of considering energetics, we focus on a topological deliberation -inspired by the coupled wire construction of quantum Hall states 124,128 -that can drastically reduce the number of possible interactions and may stablize the desired interactions when applied to materials. The coupled wire model considered so far assumes all electronic Dirac modes at the Fermi level have zero momentum k x = 0. This is convenient for the purpose of constructing an exactly solvable model because momentum is automatically conserved by the backscattering interactions. However, this also allows a huge collection of competing interactions. We propose the application of a commensurate modulation of magnetic field to restrict interactions that conserve momentum. There are multiple variations to the application, which depend on the details of the Dirac material and the Dirac vortices. To illustrate the idea, we present one possible simple scenario. First we go back to a single Dirac wire and consider a non-linear dispersion where v and b are some non-universal velocity and wave number parameters. We assume k 2 F < k 3 F < k 1 F so that when the Fermi energy is at ε F = 0, there are two right (left) moving modes at k x = k 1 F , k 2 F and one left (resp. right) moving one at k X = k 3 F along an even (resp. odd) wire. This matches the three-channel Dirac wire (3.18) used in the splitting scheme in section III A. We assume the three Fermi wave numbers satisfy a commensurate condition and we set The dashed band in figure 17(a) shows one commensurate energy dispersion along an even wire. Next, we consider a spatially modulating magnetic field B(r) = B(r)e 11 , where B(r) = ∞ m=−∞ B m sin π √ 2(2m + 1) a e1 1 · r , (3.60) e 11 = (e y + e z )/ √ 2 and e1 1 = (−e y + e z )/ √ 2, that preserves both the AFTR and C 2 symmetries, B(r + ae y ) = B(r + ae z ) = B(C 2 r) = −B(r) (3.61) (see figure 17(b) for the 3D field configuration). Moreover, we assume the field is commensurate with the Fermi wave numbers so that the magnetic flux per unit length across the xy layer between adjacent wires (see figure 17(c)) is where L is the wire length, φ 0 = hc/e is the magnetic flux quantum. Equivalently, the average magnetic field strength in the normal z-direction between adjacent wires is where a is the displacement between adjacent counter-propagating wires. We choose the vector potential A x (y, z) = [(−1) y +(−1) z −1]|B z |a/2 and A y = A z = 0 along the (y, z) th wire.
Along a wire on the xy plane where z = 0, the three electronic Dirac channels are now bosonized by where the momenta are shifted by k j F → k j F + (e/ c)A x . The phase oscillation e ikx is canceled in an interaction term 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 (3.46), which contain the operators e ±iφ σ , have zero momentum because of the Fermi wave number commensurate condition (3.58). In addition, the boson backscattering cos(4φ A y+1 − 4φ B y ) in (3.47) preserves momentum because the magnetic field is also commensurate (see (3.59) and (3.62)).

D. Interaction-enabled antiferromagnetic Dirac semimetal
So far in this section, we have been discussing the gapping of the Dirac semimetal while preserving the AFTR and C 2 symmetries. In this subsection, we focus on an opposite aspect of the symmetric many-body interaction -the enabling of a (semi)metallic phase that is otherwise forbidden by symmetries in the single-body setting. We noticed in subsection II A that the pair of momentumseparated Weyl points in figure 6 is anomalous. In fact, it is well-known already that Weyl nodes 3,23-26 , if separated in momentum space, must come in multiples of four in a lattice translation and time reversal symmetric three dimensional non-interacting system.
This no go theorem can be rephrased into a feature.
1. If the low energy excitations of a TR symmetric lattice (semi)metal in three dimensions consists of one pair of momentum-separated Weyl nodes, then the system must involve many-body interaction.
We refer to this TR and lattice translation symmetric strongly-correlated system as an interaction-enabled topological Dirac (semi)metal. We assume the Weyl nodes are fixed at two TR invariant momenta, and therefore they are stable against symmetry-preserving deformations. Otherwise, if the Weyl nodes are not located at high symmetry points, they can be moved and pair annihilated. Also, as explained in the beginning of section II and contrary to the more common contemporary terminology, we prefer to call the (semi)metal "Dirac" rather than "Weyl" because of the doubling. Perhaps more importantly, we propose the following conjecture.
2. Beginning with the interaction-enabled Dirac (semi)metal, any single-body symmetry-breaking mass must lead to a 3D gapped topological phase that cannot be adiabatically connected to a band insulator.
We suspect this statement can be proven by a filling argument similar to that of Hasting-Oshikawa-Lieb-Schultz-Mattis [184][185][186] , and may already be available in Ref. 187 by Watanabe, Po and Vishwanath. This conjecture applies to the coupled wire situation where the gapped phase is long-range entangled and supports fractional excitations. Its topological order is out of the scope of this article, but will be presented in a future work 188 . In a broader perspective, this type of statements may provide connections between strongly-interacting and non-interacting phases and help understanding quantum phase transitions of long-range entangled 3D phases from that of single-body band insulating ones. Before discussing the three dimensional case, we make the connection to a few known interaction-enabled topological phases with or without an energy gap in low dimensions. First, zero energy Majorana fermions γ j = γ † j in a true zero dimensional non-interacting (spinless) TR symmetric system must bipartite into an equal number of positive chiral ones T γ j T −1 = +γ j and negative chiral ones T γ l T −1 = −γ l . Fidkowski and Kitaev showed in Ref. 189 that under a combination of two-body interactions, eight Majoranas with the same chirality can acquire a TR preserving mass and be removed from low energy. This leaves behind a collection of zero energy Majoranas that have a non-trivial net chirality of eight. Second, all (1+1)D TR symmetric topological BDI superconductors 6,12,13,164,165,190 must break inversion because the zero energy Majorana boundary modes must have opposite chiralities at opposite ends. The Fidkowski-Kitaev interaction however allows one to construct a non-trivial (1 + 1)D topological model that preserves both TR and inversion but at the same time supports four protected Majorana zero modes at each end 191 . Third, a single massless Dirac fermion in (2 + 1)D is anomalous in a (spinful) TR and charge U (1) preserving non-interacting lattice system. On the other hand, it can be enabled by many-body interactions. For instance, when one of the two opposing surfaces of a topological insulator slab is gapped by symmetry-preserving interactions [113][114][115][116] , a single massless Dirac fermion is left behind on the opposite surface as the only gapless low energy degrees of freedom of the quasi-(2 + 1)D system. Similar slab construction can be applied to the superconducting case, and interactions can allow any copies of massless Majorana fermions to manifest in (2 + 1)D with the presence of (spinful) TR symmetry.
On the contrary, there are anomalous gapless fermionic states that cannot be enabled even by strong interactions. Chiral fermions that only propagate in a single direction cannot be realized in a true (1 + 1)D lattice system. They can only be supported as edge modes of (2 + 1)D topological phases such as quantum Hall states 175 or chiral p x + ip y superconductors 192,193 . Otherwise, they would allow heat transfer 176,194,195 from a low temperature reservoir to a high temperature one, thereby violating the second law of thermodynamics. Similarly, a single massless Weyl fermion can only be present as the (3+1)D boundary state of a (4 + 1)D topological bulk [20][21][22]164,165 . It cannot exist in a true (3 + 1)D lattice system 17,18 , or otherwise under a magnetic field there would be unbalanced chiral fermions propagating along the field direction that constitute the ABJ-anomaly 15,16,19 . In this section, we focus on the simplest anomalous gapless fermionic states in (3 + 1)D that can be enabled by interactions. As eluded in section II A 2, a weak topological insulator in (4 + 1)D can support the anomalous energy spectrum in figure 6 on its boundary so that a pair of opposite Weyl points sit at two distinct TRIM on the boundary Brillouin zone. A 4D WTI slab, where the fourth spatial dimension is open and the other three are periodic, has two (3 + 1)D boundaries and each carries a pair of Weyl fermions. The coupling between the two pairs of Weyl fermions are suppressed by the system thickness and bulk energy gap. By introducing symmetry-preserving gapping interactions on the bottom surface, the anomalous gapless fermionic state is left behind on the top surface and is enabled in this quasi-(3 + 1)D setting (see figure 18(a)).
Inspired by this construction, we propose a true (3 + 1)D coupled wire model which has the the anomalous energy spectrum in figure 6 and preserves the AFTR symmetries in both T 11 and T1 1 directions as well as the C 2 (screw) rotation symmetry. The model is summarized in figure 18(b). It consists of a checkerboard array of electronic wires, where each wire has two chiral Dirac channels propagating into-paper and another two propagating out-of-paper. Contrary to the model considered in section II, here the net chirality on each wire cancels and therefore the wires are true (1 + 1)D systems without being supported by a higher dimensional bulk. Using the splitting scheme described in section III A, along each wire, one can fractionalize a group of three Dirac channels • • × (× × •) into a pair of co-propagating chiral Pfaffian channels (resp. ++). The two Pfaffian channels then can be backscattered in opposite directions using the many-body interaction U (dashed purple lines) described in section III B. This introduces an excitation energy gap that removes three Dirac channels per wire from low energy. Lastly, single-body backscatterings t 1 , t 2 (solid directed blue lines) among the remaining Dirac channels •× described in (2.12) and figure 5 give rise to the lowenergy Weyl spectrum in figure 6. Since the many-body interaction U and the single-body backscatterings t 1 , t 2 preserve the C 2 rotation and both AFTR symmetries T 11 and T1 1 , the model describes an interaction-enabled anomalous (semi)metal that is otherwise forbidden in a non-interacting non-holographic setting.
The non-local anti-ferromagnetic nature of the time reversal symmetry is built-in in the present coupled wire model. We speculate in passing that a local conventional TR symmetric Dirac (semi)metallic phase consisting of a single pair of momentum-space-separated Weyl nodes may also be enabled by interaction. On one hand, the AFTR symmetry could be restored to a local TR symmetry by "melting" the checkerboard wire array. On the other hand, there could also be an alternative wire configuration that facilitates a coupled wire model with a local conventional TR symmetry.
Lastly, we gap the interaction-enabled Dirac semimetallic model ( figure 18) by a symmetry-breaking single-body mass. This can be achieved by introducing electronic backscattering terms that dimerize the remaining Dirac channels •×, and were described by (2.26) in section II A 1. The resulting state is an insulating (3 + 1)D topological phase with long-range entanglement. For instance, each diagonal layer gapped by the many-body interaction U has the identical topological order of the T -Pfaffian surface state of a topological insulator.

E. Fractional surface states
In section II B, we discussed the surface states of the single-body coupled Dirac wire model (2.12) (see also figure 5). In particular, we showed in figure 13 that an AFTR symmetry preserving surface hosts open chiral Dirac channels, which connect and leak into the 3D (semi)metallic bulk. Earlier in this section, we discussed the effects of many-body interaction, which leads to two possible phases: (a) a gapped topological phase (see section III B) that preserves one of the two AFTR symmetries, say T 11 , and (b) a gapless interaction-enabled Dirac semimetal (see section III D) that preserves the C 2 rotation and both AFTR symmetries T 11 and T1 1 . Here, we describe the boundary states of the two interacting phases on a surface closed under the symmetries. First, we consider the coupled wire model with the many-body interaction (3.47) (see also figure 16) and a boundary surface along the yz-plane perpendicular to the wires. The surface network of fractional channels is shown in figure 19(a). We assume the bulk chiral Dirac wires (×•) are supported as vortices of Dirac mass in the bulk (recall (2.2)), where the texture of the mass parameters is represented by the underlying vector field. The model is juxtaposed along the yzboundary plane against the trivial Dirac insulating state H vacuum = vk · sµ z + m 0 µ x , which models the vacuum. The line segments on the surface plane where the Dirac mass m 0 µ x changes sign host chiral Dirac channels (c.f. subsection II B 2).
Unlike the single-body (semi)metallic case in figure 13 where the surface Dirac channels connects with the bulk ones, now the many-body interacting bulk is insulating and does not carry low-energy gapless excitations. Thus, the surface Dirac channels here cannot leak into the bulk and must dissipate to other low-energy degrees of freedom on the surface. The many-body interwire backscattering interaction in (3.47) (and figure 16) leaves behind chiral Pfaffian channels on the surface. These fractional channels connect back to the surface Dirac channels in pairs. The surface network of chiral channels preserves the AFTR T 11 symmetry. However, the low-energy surface state is not protected. Electronic states can be localized by dimerizing the Pfaffian channels in the z (or 11) direction.
Second, we consider the interaction-enabled Dirac semimetallic model summarized in figure 18(b) in section III D and again let it terminate along the symmetry preserving yz-plane perpendicular to the wires. The surface gapless channels are shown in figure 19(b). Here, the semimetallic bulk preserves C 2 as well as the two AFTR symmetries T 11 and T1 1 . The bulk array of wires are true (1 + 1)D systems and are not supported as edge modes or vortices of a higher dimensional bulk. The pair of intopaper Dirac modes are bent into the pair of out-of-paper ones along each wire at the terminal. Similar to the previous case, the many-body bulk interwire backscattering interaction leaves behind surface chiral Pfaffian channels. Through the mode bending at the wire terminal, these Pfaffian channels join in pairs and connect to the chiral Dirac channels in the bulk that constitute the Dirac semimetal. In this case, the surface state is protected by C 2 , T 11 and T1 1 , and is forced to carry fractional gapless excitations as a consequence and signature of the anomalous symmetries. For instance, the charge e/4 Ising-like quasiparticle and the charge e/2 semion can in principle be detected by shot noise tunneling experiments. These gapless fractional excitations however are localized on the surface because the Dirac (semi)metallic bulk only supports gapless electronic quasiparticles.

IV. CONCLUSION AND DISCUSSION
Dirac and Weyl (semi)metals have generated immense theoretical and experimental interest. On the experimental front, this is fueled by an abundant variety of material classes and their detectable ARPES and transport signatures. On the theoretical front, Dirac/Weyl (semi)metal is the parent state that, under appropriate perturbations, can give birth to a wide range of topological phases, such as topological (crystalline) insulators and superconductors. In this work, we explored the consequences of a specific type of strong many-body interaction based on a coupled-wire description. In particular, we showed that (i) a 3D Dirac fermion can acquire a finite excitation energy gap in the many-body setting while preserving the symmetries that forbid a single-body Dirac mass, and (ii) interaction can enable an anomalous antiferromagnetic time-reversal symmetric topological (semi)metal whose low-energy gapless degrees of freedom are entirely described by a pair of non-interacting electronic Weyl nodes separated in momentum space. A brief conceptual summary was presented in section I A and will not be repeated here. Instead, we conclude by discussing possible future directions.
First, coupled wire constructions can also be applied in superconducting settings and more general nodal electronic systems. For example, a Dirac/Weyl metal can be turned into a topological superconductor 164,165,190 under appropriate intra-species (i.e. intra-valley) s-wave pairing 196 . Pairing vortices host gapless chiral Majorana channels [196][197][198] . An array of these chiral vortices can form the basis in modeling superconducting many-body topological phases in three dimensions. On the other hand, instead of considering superconductivity in the continuous bulk, inter-wire pairing can also be introduced in the coupled Dirac wire model and lead to new topological states 162 . Dirac/Weyl (semi)metals are a specific type of nodal electronic matter. For example, nodal superconductors were studied in states with dx 2 -y 2 pairing 199 , He 3 in its superfluid A-phase 200,201 , and non-centrosymmetric states 202,203 . Weyl and Dirac fermions were generalized in TR and mirror symmetric systems to carry Z 2 topological charge 204 . General classification and characterization of gapless nodal semimetals and superconductors were proposed 6,201,[205][206][207][208][209][210] . It would be interesting to investigate the effect of strong many-body interactions in general nodal systems.
Second, in section II, we described a coarse-graining procedure of the coupled wire model that resembles a real-space renormalization and allows one to integrate out high energy degrees of freedom. While this procedure was not required in the discussions that follow because the many-body interacting model we considered was exactly solvable, it may be useful in the analysis of generic interactions and disorder. The coarsegraining procedure relied on the formation of vortices, which were introduced extrinsically. Like superconducting vortices, it would be interesting as a theory and essential in application to study the mechanism where the vortices of Dirac mass can be generated dynamically. To this end, it may be helpful to explore the interplay between possible (anti)ferromagnetic orders and the spinmomentum locked Dirac fermion through antisymmetric exchange interactions like the Dzyaloshinskii-Moriya interaction 211,212 .
Third, the symmetry-preserving many-body gapping interactions considered in section III have a ground state that exhibits long-range entanglement. This entails degenerate ground states when the system is compactified on a closed three dimensional manifold, and fractional quasi-particle and quasi-string excitations or defects. These topological order properties were not elaborated in our current work but will be crucial in understanding the topological phase 188 as well as the future designs of detection and observation. It would also be interesting to explore possible relationships between the coupled wire construction and alternative exotic states in three dimensions, such as the Haah's code 213,214 .
Fourth, the many-body inter-wire backscatterings proposed in section III B were based on a fractionalization scheme described in III A that decomposes a chiral Dirac channel with (c, ν) = (1, 1) into a decoupled pair of Pfaf-fian ones each with (c, ν) = (1/2, 1/2). In theory, there are more exotic alternative partitions. For instance, if a Dirac channel can be split into three equal parts instead of two, an alternative coupled wire model that put Dirac channels on a honeycomb vortex lattice could be constructed by backscattering these fractionalized channels between adjacent pairs of wires. Such higher order decompositions may already be available as conformal embeddings in the CFT context. For example, the affine SU (2) Kac-Moody theory at level k = 16 has the central charge c = 8/3, and its variation may serve as the basis of a "ternionic" model. In section II, we begin with the Dirac Hamiltonian (2.2) where the mass term winds around a vortex and as a consequence, it hosts a chiral Dirac channel along the vortex (also see figure 3). Here we will demonstrate an example of a simple vortex, and show that there is a chiral Dirac zero mode. In general, the correspondence between the number of protected chiral Dirac channels and the vortex winding is a special case of the Atiyah-Singer Index theorem 158 and falls in the physical classification of topological defects 156 .
First, say we start with the Hamiltonian from (2.2). Then for simplicity we consider the particular Dirac mass m(r) = m x (r) + im y (r) = |m|e iθ that constitute a vortex along the z-axis, where θ is the polar angle on the xyplane. By replacing k x,y ↔ −i∂ x,y , (2.2) becomes H(r) = v(−i∂ x s x − i∂ y s y + k z s z )µ z + |m| cos θµ x + |m| sin θµ y (A1) where k z is still a good quantum number because translation in z is still preserved.

Now we separate the Hamiltonian
where Γ 5 = diag(−1 1 2 , 1 1 2 ). We note that the zero momentum sector H (k z = 0) has a chiral symmetry since it anticommutes with with Γ 5 , and it reduces to the Jackiw-Rossi vortex problem in two-dimensions 157 . The Dirac operator D † has only one normalizable zero mode u 0 (r) ∝ e −|m|r/ v (e iπ/4 , e −iπ/4 ) T , while its conjugate D has none. H (k z = 0) therefore has a zero eigenvector of ψ 0 (r) = (u 0 (r), 0) T , which is also an eigenvector of Γ 5 . In the full Hamiltonian, the zero mode ψ 0 (r) has energy − vk z and corresponds a single mid-gap chiral Dirac channel.

Appendix B: Symmetry transformations of Chern invariants
In section II A, we discussed the Chern numbers on two-dimensional momentum planes of the anomalous Dirac (semi)metal. It was claimed that the Chern numbers (2.21) on the two planes at k x = ±π/2 (see figure 6) are of opposite signs because of the AFTR and twofold C 2 (screw) rotation symmetries. In this appendix we will derive the symmetry flipping operations on the Chern invariants.
We begin with a Bloch Hamiltonian H(k) that is symmetric under the operation G(k), if it is antiunitary. Let |u m (k) be the occupied states of H(k). We define |u m (k) = |Gu m (k) = G(gk)|u m (gk) (or |u m (k) = |Gu m (k) = G(gk)|u m (gk) * ), which is also an occupied state of H(k), for unitary (resp. antiunitary) symmetry. The Chern number (2.21) can equivalently be defined as where Tr (F k ) = dTr (A k ), N kx is the oriented k y k z -plane with fixed k x , and A k is the Berry connection of the occupied states A mn k = u m (k)|du n (k) . The Berry connection transforms according to if G is antiunitary, because the connection is skewhermitian A = −A † . Therefore F k = F gk + dTr P gk ∧ G(gk) † dG(gk) (B5) for an unitary symmetry, or for an antiunitary one. Here P (k) = n |u n (k) u n (k)| is the projection operator on to the occupied energy states at momentum k. Since the trace of Berry curvature Tr(F) does not depend on the gauge choice of occupied states, Tr(F k ) = Tr(F k ). We notice the final terms in both (B5) and (B6) integrate to zero over the closed periodic momentum plane N kx . This is because they are total derivatives, and unlike A k , P k and G(k) are defined non-singularly on the entire Brillouin zone (see (2.15) and (2.19)). Now we derive the relation between the Chern number (B3) between k x and −k x using the antiunitary AFTR and the unitary C 2 symmetries. The AFTR symmetries flip all momentum axes T 11 , T1 1 : (k x , k y , k z ) → (−k x , −k y , −k z ), while the C 2 symmetry flips only two C 2 : (k x , k y , k z ) → (−k x , −k y , k z ). Thus, T 11 , T1 1 : N kx → N −kx maps between opposite planes while preserving their orientations, but C 2 : N kx → −N −kx is orientation reversing. Lastly, we substitute (B5) and (B6) into (B3), and apply a change of integration variable k ↔ gk. The AFTR and C 2 requires the Chern number to flip under k x ↔ −k x Ch 1 (k x ) = −Ch 1 (−k x ). (B7)