Theory of a 3+1D fractional chiral metal: interacting variant of the Weyl semimetal

Theory of a 3+1D fractional chiral metal: interacting variant of the Weyl semimetal

Tobias Meng Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany    Adolfo G. Grushin Department of Physics, University of California, Berkeley, CA 94720, USA Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany    Kirill Shtengel Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany Department of Physics and Astronomy, University of California at Riverside, Riverside CA 92511, USA    Jens H. Bardarson Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany

Formulating consistent theories describing strongly correlated metallic topological phases is an outstanding problem in condensed matter physics. In this work we derive a theory defining a fractionalized analogue of the Weyl semimetal state: the fractional chiral metal. Our approach is to construct a 4+1D quantum Hall insulator by stacking 3+1D Weyl semimetals in a magnetic field. In a strong enough field the low-energy physics is determined by the lowest Landau level of each Weyl semimetal, which is highly degenerate and chiral, motivating us to use a coupled-wire approach. The one-dimensional dispersion of the lowest Landau level allows us to model the system as a set of degenerate 1+1D quantum wires that can be bosonized in the presence of electron-electron interactions and coupled such that a gapped phase is obtained, whose response to an electromagnetic field is given in terms of a Chern–Simons field theory. At the boundary of this phase we obtain the field theory of a 3+1D gapless fractional chiral state, which we show is consistent with a previous theory for the surface of a 4+1D Chern–Simons theory. The boundary’s response to an external electromagnetic field is determined by a chiral anomaly with a fractionalized coefficient. We suggest that such anomalous response can be taken as a working definition of a fractionalized strongly correlated analogue of the Weyl semimetal state.

I Introduction

Recent exciting developments in condensed matter physics concern a variety of topological phases. These are phases that are not classified by broken symmetries and local order parameters Qi and Zhang (2011); Hasan and Kane (2010). While the term “topological phases” was originally used to refer only to topologically ordered gapped phases with long-range entanglement, it is now understood to encompass a broader class of phases. This includes symmetry-protected topological phases Senthil (2015)—gapped states with no long-range entanglement that are distinct from the trivial phase only in the presence of certain symmetries—and now frequently even gapless states Turner and Vishwanath (); Hosur and Qi (2013).

The prototypical gapless topological phase of matter is the nodal semimetal in which nodal points act as sources and sinks of Berry’s phase making them topologically stable. Such states are realized in certain phases of liquid helium and have been discussed extensively in that context Volovik (2003), while their condensed matter realization in the 3+1D Weyl semimetals is a very recent experimental achievement Weng et al. (2015); Huang et al. (2015); Xu et al. (2015a); Lv et al. (2015a); Xu et al. (2015b); Lv et al. (2015b); Yang et al. (2015). The non-trivial topological structure is induced by spin-orbit coupling. The surface states corresponding to a pair of Weyl nodes disperse chiraly and result in open Fermi surfaces—the Fermi arcs. The presence of these exotic surface states is closely related to quantum anomalies which also govern the unusual response of the system to external fields Nielsen and Ninomiya (1983a). Whether strong electronic correlations can stabilize exotic cousins of nodal phases is the open question that motivates our work.

One fruitful strategy to describe and analyze unconventional 3+1D gapless states of matter is inspired by the fact that a boundary between gapped phases, one topological and another one not, is expected to be gapless 111Since topological invariants are quantized, the adiabatic theorem prohibits changing them without closing a gap; one just needs to interpret the coordinate perpendicular to the boundary as a parameter being varied adiabatically. One possible exception to this rule are so-called quantum doubles describing a topological state which possesses a pair of invariants of opposite sign; in this case these invariants can “annihilate” one another without closing a gap. A toric code provides a simple example of such behavior.. Accordingly, the first key idea of this manuscript is to approach the putative fractional Weyl semimetal as a surface state of a higher-dimensional gapped topological state, namely that of a 4+1D fractional quantum Hall state Zhang and Hu (2001). Previous studies have described such a state in terms of Landau levels for Dirac fermions in higher dimensions Li et al. (2012, 2013), quaternions Li and Wu (2013), and ground state wave functions Wang and Zhang (2014). Our understanding of these phases, however, is still much less advanced in comparison to their 2+1D counterparts. While some general results, such as the connection between charge fractionalization and a non-trivial ground state degeneracy in a gapped 2+1D system on a torus Oshikawa and Senthil (2006), should carry through in higher dimensions, other aspects of topological order, including for example exotic quasiparticle statistics, are much less transparent. Describing the 4+1D fractional quantum Hall state, although not the main goal of this study, will therefore be a useful spin-off that adds to the existing body of knowledge of this state.

In this work, we combine the above strategy—approaching a putative fractional 3+1D metallic state as the surface state of a fractional 4+1D quantum Hall state—with a second powerful approach, the so-called coupled-wire construction. The advantage of the combined strategy is that, unlike previous parton constructions Witczak-Krempa et al. (2014); Wang (2015), it does not postulate fractionalization from the start, allows us to address both the gapped bulk and the gapless surfaces of a 4+1D quantum Hall state, and remains analytically tractable. The main idea is to trade the isotropy usually inherent to low-energy topological quantum field theories for the analytical control over electron-electron interactions provided by Luttinger liquid theories describing (coupled) one-dimensional systems. Starting with the seminal studies Kane et al. (2002); Teo and Kane (2014), coupled-wire constructions have been successfully employed to describe a large variety of chiral topological phases in 2+1D Lu and Vishwanath (2012); Vaezi and Barkeshli (2014); Klinovaja and Loss (2014); Meng et al. (2014a); Sagi and Oreg (2014); Klinovaja and Tserkovnyak (2014); Neupert et al. (2014); Klinovaja et al. (2015); Santos et al. (2015); Meng and Sela (2014); Sagi et al. (2015); Cano et al. (2015); Szumniak et al. (); Klinovaja et al. (); Oreg et al. (2014); Meng et al. (2014b); Cornfeld et al. (2015); Cornfeld and Sela (2015), including surfaces of topological 3+1D states Mross et al. (2015); Sahoo et al. (), topological superconductors Seroussi et al. (2014); Mong et al. (2014); Vaezi (2014), and spin liquids Meng et al. (2015); Gorohovsky et al. (2015); Huang et al. (). First generalizations of the coupled-wire approach to higher dimensions have been discussed in Vazifeh (2013); Meng (2015); Sagi and Oreg (2015); Iadecola et al. (). In the following, we adapt the coupled-wire construction for the description of topological 4+1D phases, and apply it to a specific class of 4+1D fractional quantum Hall states 222Different generalizations of the quantum Hall effect to 4+1D exist, which can for instance involve Abelian and non-Abelian gauge fields Zhang and Hu (2001); Karabali and Nair (2002) – our constructions is part of the former category.

In the next section we summarize our main results and discuss the main ideas. This section is aimed at those readers that are not experts in coupled-wire constructions but are interested in the main ideas behind the calculation. All the technical details are given in section III and require some background knowledge; those not interested in these details can safely skip this section and go directly to the discussion in section IV, where we also discuss the connection to current experimental prospects.

Ii Summary of main results

In this section we discuss the general philosophy and the key ideas of our work; the technical details of our calculation are given in the next section.

The central result of our work is a coupled-wire construction of a 4+1D fractional quantum Hall insulator that has 3+1D fractional chiral metals at its surfaces, and the conjecture of a fractionalized gapless 3+1D phase composed of two fractional chiral metals of opposite chiralities.

The 4+1D quantum Hall insulator we construct has a current density response to an external electromagnetic field given by


where with integer , and is the totally antisymmetric tensor (here and henceforth we use units where ). The field theory underlying this response is the 4+1D Chern–Simons theory


According to Eq. (1) a combination of a three dimensional magnetic field and an electric field both perpendicular to the -direction generates a current


parallel to the -direction 333Here and in the remainder of the paper, we define the three-dimensional vector fields and by their components and (). While not representing the full electromagnetic field strength tensor in (4+1)D, the definition of these fields allows us to explicitly make contact to (3+1)D Weyl semimetals, and underlines the invariance of our construction with respect to rotations around the axis.. This result we interpret as the chiral anomaly induced response of a surface 3+1D fractional chiral metal.

This fractional chiral metal interpretation is motivated by an analogy with the edge states in the 2+1D fractional quantum Hall effect. There, the 2+1D Hall current can be understood as arising from a chiral charge that is pumped from one edge to the other. The chiral charge is therefore not separately conserved on each edge and the theory describing a given edge is anomalous. The change in chiral charge is proportional to the electric field inducing the Hall current, with a coefficient that is a fraction of that obtained in the noninteracting integer case. Analogously, the response (3) represents the pumping of chiral charge from one anomalous 3+1D metallic surface to another. For , the coefficient of this anomaly is fractionalized with respect to the well-known coefficient of the chiral anomaly of noninteracting Weyl fermions, which is obtained for  Nielsen and Ninomiya (1983a). We thus define the obtained surface state as a fractional chiral metal.

The fundamental idea of our construction is as follows: A 4+1D quantum Hall state is constructed by regularly stacking 3+1D Weyl semimetals along a fourth spatial direction at , with and the lattice spacing. By suitably coupling nodes of opposite chiralities in neighboring Weyl semimetals the bulk is gapped out, as shown schematically in Fig. 1.

To describe each semimetal, we restrict ourselves to the minimal two-band model of an inversion-symmetric time-reversal-broken Weyl semimetal, which has two Weyl nodes of opposite chirality at the same energy separated in momentum space. The Hamiltonian describing the Weyl semimetal at is


where is a three dimensional momentum and is a spinor of creation operators for electrons of spin } and momentum at . The spin label more generally denotes the two bands, but for simplicity we always refer to it as spin. Close to the two Weyl nodes at with chirality , that is for , the Hamiltonian matrix is given to lowest order in by


Here is a vector of the three Pauli matrices and is the Fermi velocity. The detailed form of away from the Weyl nodes is not important for our construction as long as the separation is large enough; we comment on the precise conditions where appropriate.

Figure 1: Construction of a 4+1D integer quantum Hall effect from coupled Weyl semimetals. A tunnel coupling between the left-handed node of each Weyl semimetal with the right-handed node of its neighbor at smaller , depicted by red arrows, induces a gap for all bulk nodes. In a slab of finite extent , single gapless nodes remain at the 3+1D surfaces at and .

The coupling of the 3+1D Weyl semimetals resulting in a gapped 4+1D quantum Hall state is most transparent in the noninteracting case. In this case, the right-handed Weyl nodes at are coupled to the left-handed nodes at by tunneling, as depicted by red arrows in Fig. 1. Since the two nodes that are so coupled have opposite chirality they can annihilate and gap each other out, resulting in a gapped state. In a finite slab with , however, the left-handed node at and the right-handed node at do not have partner nodes to pair up with. Instead, they form 3+1D chiral gapless surface states that are higher dimensional analogues of the chiral 1+1D edge modes of a 2+1D quantum Hall state. Like these modes, they escape the fermion doubling theorem Nielsen and Ninomiya (1981a, b, c) by the fact that they live on the 3+1D surfaces of a topological 4+1D state. We therefore identify the gapped state just constructed as an integer 4+1D quantum Hall state.

In the presence of interactions the construction of a gapped state is more involved and we rely on a coupled-wire construction related to that of Kane and collaborators for 2+1D fractional quantum Hall states Kane et al. (2002). As in the noninteracting case, we tunnel-couple the left and right handed nodes in neighboring Weyl semimetals, but now the combination of interactions and tunneling leads to several different gapped states, just as in the 2+1D fractional quantum Hall case. The way this essentially works is that one of the dimension in the Weyl semimetal, say , is made into effective one dimensional quantum wires by quenching the kinetic energy along the other and directions. These effective wires are then coupled through the fourth dimension. In order to make this calculation controlled, we need the coupling of the wires to be the dominant coupling, such that it leads to a nontrivial gapped phase. To this end we apply a strong magnetic field , with the unit vector in the direction, to each of the Weyl semimetals. This results in the formation of Landau levels that disperse only in and thereby naturally form a basis of quantum wires that can be coupled, see Fig. 2. At low energies , with the magnetic length defined as , each Weyl node can be approximated by its gapless zeroth Landau level, which is is composed of highly degenerate chiral modes with a degeneracy factor . Dispersing linearly, these chiral modes are readily bosonized. The right-handed chiral modes in one Weyl semimetal are tunnel-coupled to the left-handed chiral modes in the neighboring Weyl semimetal, see Fig. 2. The inclusion of interaction allows for correlated tunneling in which one particle tunnels between two Weyl semimetals while at the same time particles in each Weyl semimetal change their chirality from right-handed to left-handed or vice versa, see Fig. 3. This is the same multi-particle process as the one of Kane and collaborators leading to the fractional quantum Hall states, and like in that case a current along occurs as a result of applying an electric field along the wire direction . Here, however, since each of the modes is degenerate the current is also proportional to the degeneracy factor and hence to the magnetic field . This is the origin of the chiral anomaly form of the current density in Eq. (3).

Up to this point we have described how the chiral fractional metal emerges as the boundary state of a 4+1D quantum Hall insulator. It remains to obtain the field theory description of this boundary state. To that end we take a manifold finite in the direction and impose gauge invariance, to obtain the field theory describing a single surface:


where is a scalar field defined at that surface, is the external field strength and is a constant whose sign defines the chirality of the surface. This we recognize as the action of a 2+1D quantum Hall effect edge described by with an extra functional dependence that, as above, has its origin in the Landau level degeneracy. Remarkably this result as derived from the coupled-wire construction is consistent with earlier attempts to describe the edge theory of the 4+1D Chern Simons theory (2) based on the current algebra perspective Gupta and Stern (1995).

Before going into the details of the calculations, a brief note on coupling scales and renormalization. The application of a strong magnetic field has the advantage of allowing us to identify the dominant couplings at the microscopic level, leading to a well defined phase. We further assume that these couplings remain the leading ones under renormalization, and do not attempt a systematic renormalization group analysis of all interaction terms here, as this would take us way beyond the scope of this work. Instead, in analogy with the canonical discussion of 2+1D fractional quantum Hall states, we simply assume that, if necessary, one can always adjust the values of the microscopic parameters such that the renormalization flow is towards the gapped phases we have identified.

Iii A coupled-wire construction of 4+1D fractional quantum Hall states

iii.1 A single Weyl node in a magnetic field

As discussed in the previous section our coupled-wire construction approach relies on coupling 3+1D Weyl nodes subject to a magnetic field. Therefore we first recount the physics of a single isotropic 3+1D Weyl node in a magnetic field , where denotes the unit vector in -direction (with ) Nielsen and Ninomiya (1983a); Ashby and Carbotte (2013). In the Landau gauge, this magnetic field is associated with the vector potential .

A Weyl node of chirality is described by the first-quantized Hamiltonian


where is the three-dimensional momentum measured with respect to the Weyl node, and is the vector of Pauli matrices. Choosing the magnetic field such that , with the electron charge, we define the magnetic length and introduce the dimensionless creation and annihilation operators


These obey the bosonic commutator relation . In the eigenstates of the Hamiltonian takes the form


Denoting the Landau level quantum number—the integer eigenvalues of —by the spectrum of (9) comprises particle-hole symmetric bands with dispersion and a chiral linearly dispersing lowest Landau level , as illustrated in Fig. 2. Since the bands are independent of the momentum eigenvalue it labels the degenerate states inside each Landau level whose number is


where is the length of the system in -direction.

When the magnetic field is sufficiently large for all energy scales of interest to be smaller than , the low-energy physics is determined by the gapless lowest Landau level only. Thus a single Weyl node can be approximated by a macroscopically degenerate set of right- or left-moving chiral electrons with a one-dimensional dispersion . The eigenvectors of this lowest Landau level take the form and are therefore spin-polarized.

Figure 2: Stacking of Weyl semimetals along the fourth spatial dimension subject to a magnetic field . Each Weyl semimetal contains the Landau levels of a pair of Weyl nodes. The left panels depicts the energy of a Weyl node of chirality in a magnetic field described by the Hamiltonian as a function of the momentum measured with respect to the Weyl node. The Landau levels form quadratically dispersing, particle-hole symmetric bands. Each Landau level is macroscopically degenerate with respect to . The right panel illustrates the low-energy description of the stacked 4+1D system at finite , which reduces to the gapless lowest Landau levels. As discussed in Sec. III.2, a complex hopping connects neighboring Weyl semimetals.

iii.2 The coupled-wire Hamiltonian and correlated tunneling processes

We now construct the full 4+1D system by stacking individual Weyl semimetals in the limit of strong magnetic field where we can restrict our model to the gapless lowest Landau levels. We choose the stacking to be along the additional discrete spatial dimension with , see Fig. 2. The associated Hamiltonian reads


where describes the individual Weyl semimetals, and and encode, respectively, tunneling terms between them and the electron–electron interactions.

Let us start by detailing . We take the Fermi energy in each of the semimetals to reside at the Weyl nodes, which are located at momentum . Sec. III.1 then implies that the low-energy form of is captured by linearly dispersing right- and left-moving modes dispersing only with . Adapting the standard low-energy treatment of one-dimensional systems Giamarchi (2003), we approximate by a model with unbounded linear dispersions,


Here, the chiral modes and precisely correspond to the linearly dispersing low-energy excitations in the lowest Landau levels shown in Fig. 2. As customary in bosonization, the translation from the initial operators and , creating and annihilating an electron of momenta in the lowest Landau level at position , to the chiral low-energy modes is via


Next, we address the explicit form of the tunneling term . At vanishing magnetic field, we require the tunneling to preserve the momenta and . In the presence of a strong magnetic field, this translates into a conservation of the Landau level index and the momentum . The electron momentum , however, is allowed to be shifted by the tunneling, by an amount that depends on the precise state to be generated. Physically, a finite momentum shift demands that the electrons couple to a vector potential with an -component of . This can be achieved with a complex hopping whose phase relates to the vector potential via the Peierls substitution. Such a Peierls phase of the complex hopping is indeed equivalent to a momentum shift for an electron tunneling from to . Denoting the tunneling strength by , we thus find that the projection of the tunneling Hamiltonian to the lowest Landau levels reads


where denotes the Fourier transform of with respect to the third coordinate.

The Hamiltonian , finally, describes electron-electron interactions, whose presence is a crucial ingredient to fractional quantum Hall states. The screening of long range interactions by the large density of states of the gapless lowest Landau level motivates us to neglect non-local interactions. Since in addition the wave function of an electron with degeneracy index is proportional to a Gaussian centered at , the largest contribution to the local interaction involves electrons with the same . We thus specialize to couplings between the densities of electrons with identical at the closest possible coordinates and , where is the lattice constant along . Using , we obtain the interaction Hamiltonian as


The effects of further interaction processes not relevant to our discussion are briefly addressed in Sec. III.3 below.

In 2+1D coupled-wire constructions, topologically ordered states are generated by correlated tunnelings of electrons between wires; these are processes in which an electron tunnels from one wire to a neighboring wire, while simultaneously a number of additional electrons in both wires are backscattered Kane et al. (2002). We generalize this class of processes to 4+1D by analyzing the correlated tunnelings depicted in Fig. 3, in which an electron tunnels from the Weyl semimetal at to the neighboring semimetal at , while at the same time electrons are backscattered between the Weyl nodes of both semimetals.

Microscopically, the correlated tunnelings in Fig. 3 are obtained from the Hamiltonian in Eq. (11) by treating and as perturbations to the decoupled Weyl semimetals described by . Let us start by illustrating the derivation in the simplest case . This process is generated by the combined perturbative expansion of and to first order in the tunneling and second order in , see Fig. 4. As shown by the dotted arrow, the local interaction first causes two electrons at the left-handed node in the Weyl semimetal at to scatter off each other with a momentum transfer of ; one electron thus ends up at the right-handed node, while the other electron occupies an intermediate high-energy state at momentum . The high-energy electron then hops, as indicated by the solid arrow, to the Weyl semimetal at by virtue of the complex tunneling . It thereby acquires a momentum shift and ends up at . Finally, the interaction mediates a scattering, depicted by the dash-dotted arrow, between the tunneling electron and an electron that is initially close to the left-handed node at . Because the total correlated tunneling process has to conserve energy (and momentum), this second electron should be scattered to the right-handed Weyl node at (or any other Weyl node for that matter, but these other processes do not generate the fractional quantum Hall states we are interested in; see also Sec. III.3). The second scattering process is thus also associated with a momentum transfer of . The tunneling electron has thereafter acquired a total -momentum shift of . The process shown in Fig. 4, which stabilizes a fractional quantum Hall state, is a correlated tunneling that transfers the hopping electron from the left-handed Weyl node at to the right-handed node at . This requires the total momentum shift for the tunneling electron to be , and thus fixes for this particular process to conserve momentum.

To obtain a low-energy description of this process, we integrate out the high-energy intermediate states of the hopping electron. We then obtain an effective three-particle interaction that annihilates two left movers with identical quantum number at the left-handed node at , and creates two right movers at the right-handed node at . Due to their fermionic character, these right and left movers cannot be at the same position, but need to be slightly displaced. Since we generate the process using the interaction in Eq. (15) involving derivatives , this is indeed the case. For , we find the low-energy Hamiltonian of the correlated tunneling shown in Fig. 4 to read


For , when the process depicted in Fig. 4 does not conserve momentum, Eq. (16) acquires additional oscillating factors that suppress the scattering. This is analogous to the observation that a 2+1D Laughlin state only exists at specific filling fractions (i.e., specific strengths of the applied magnetic field), and that the 2+1D coupled-wire construction of these states involves a momentum shift proportional to the applied field Kane et al. (2002).

The analogy to the 2+1D case, where Laughlin states exist for many filling factors, immediately suggests that there should be a number of momentum shifts at which other correlated tunnelings conserve momentum in our 4+1D system. This is indeed the case for with , when processes depicted in Fig. 3 with a backscattering of electrons between the Weyl nodes become resonant. These higher-order correlated tunnelings are generated in first order in and -th order in , and are thus . The first interaction processes now scatter electrons in the Weyl semimetals from the left-handed the the right-handed node, thereby transferring a momentum of to one other electron that is initially in the vicinity of the left-handed node. This latter electron is thus pushed to a momentum . It then tunnels to , and thereby acquires a momentum shift of , which puts it at a momentum . In the final interaction processes, electrons are scattered from the left-handed node to the right-handed node at while transferring a momentum of to the electron that has tunneled. This latter electron consequently end up at momentum , i.e., at the right-handed node. As a subtlety, we remark that all intermediate states of the tunneling electron should be at high energies (within the lowest Landau level approximation), and thus far away from the Weyl nodes. As a result, the possible values of for which our construction is valid are constrained by the periodicity of the Brillouin zone to satisfy and by the lowest Landau level approximation to .

Figure 3: Correlated tunnelings between neighboring Weyl semimetals leading to fractional quantum Hall states. While an electron hops from the left-handed Weyl-node at to the right-handed node at , electrons are scattered from the left-handed node to the right-handed node in both Weyl semimetals connected by the tunneling. This correlated process is indicated by the arrows, whose labels indicate the number of electrons transported along the respective arrow.
Figure 4: Generation of the correlated scattering corresponding to in Fig. 3. Empty circles depict initial states of the electrons, filled circles correspond to their final states, and the dotted circles indicate the intermediate virtual states of the hopping electron. The dotted horizontal lines depict the Fermi level (and thus the energy of the Weyl nodes).

The low-energy Hamiltonian describing these higher-order correlated tunnelings can be obtained in analogy to Eq. (16) by integrating out the intermediate high-energy states of the tunneling electron. We then obtain a scattering process involving low-energy electrons that includes the annihilation of left moving electrons with identical at , as well as the creation of right moving electrons at . Correspondingly, the low-energy Hamiltonian again contains derivatives with respect to that account for the small spatial displacements of the individual electrons. We obtain


Just as Eq. (16), this Hamiltonian is only valid if the resonance condition is met. All correlated tunnelings with violate momentum conservation, and are suppressed by oscillating factors.

iii.3 Other interaction processes

The interaction in Eq. (15) is not of the most generic form, but is rather optimized to explore a specific set of states in our 4+1D system. Namely, we are interested in the states generated by the couplings shown in Fig. 3. Those are the 4+1D analogues of the correlated scatterings generating Laughlin states in a 2+1D coupled-wire system Kane et al. (2002). We disregard possible competing states generated by Eq. (15), or by a more general interaction. This includes locked charge density waves in neighboring Weyl semimetals generated by backscattering interactions, or a 1/3-Laughlin-crystal type of order Kane et al. (2002), see Fig. 5.

In general, any of the couplings present in a given system may determine the low-energy physics. Technically, this happens if the respective term has a large coupling constant, while all other terms have small coupling constants. The hierarchy of coupling constants is either due to a fine-tuning of their bare values, or can be generated by their RG flow. The structure of the flow is ultimately determined by the interaction itself, which can again be tuned to favor a particular coupling.

Since the remainder of this paper aims at characterizing the phases generated by the correlated tunnelings depicted in Fig. 3, we neglect all other interaction processes by assuming that the system parameters have been be adjusted accordingly. Note, however, that we analyze a topological state of matter that is not symmetry-protected. The topological response given in Eq. (3) is thus insensitive to the addition of other interactions provided these remain sufficiently small and do not alter the RG scaling of the correlated tunneling process depicted in Fig. 3 by preventing it from being the most relevant term. This philosophy has already proven very useful for the exploration of topological phases in 2+1D coupled-wire constructions. A full RG analysis of all sine-Gordon terms that could possibly be present in a coupled-wire system is, however, to date lacking even for these much simpler systems. It constitutes an important open problem for the field in general, and is beyond the scope of the present work.

Figure 5: Two of the possible competing interaction processes that would lead to a charge-density-wave order (top panel), and a 1/3-Laughlin-crystal-type of order (bottom panel) Kane et al. (2002).

iii.4 Bosonization

We bosonize the chiral modes using the standard prescription Giamarchi (2003)


where is a Klein factor, denotes a high momentum cutoff, and is a compact notation for the operators introduced in Eq. (12). The chiral bosonized fields obey


where the scalar takes the value () for (). In the Luttinger liquid Hamiltonian describing the low-energy physics of the bosonized modes , we retain the bosonized version of , the correlated tunnelings , and interactions between the densities of the chiral modes. In bosonization, the latter are given by . We neglect all further interactions, including in particular the ones shown in Fig. 5 and other interaction processes that turn right movers into left movers. Upon bosonization, these scatterings give rise to sine-Gordon terms that compete with the correlated tunnelings ; as for the terms depicted in Fig. 5, we consider system parameters such that all of these sine-Gordon terms are irrelevant for our system.

The gapless motion described by and the chiral density-density interactions can be combined into a quadratic bosonized Hamiltonian reading




depends on the Fermi velocity and the density-density interactions between the different bosonized modes Giamarchi (2003). In order to bosonize the tunnelings in Eq. (17), it is useful to introduce new fields


which obey


This definition, together with Eq. (18), allows us to cast the leading terms in the operator product expansion of Eq. (17) into the form


Because the argument of each sine-Gordon term in Eq. (24) commutes with itself at different positions, each term can order individually by pinning its argument to one of its minima. Since the arguments also commute between different sine-Gordon terms, all of them can order simultaneously. This fully gaps the bulk. If the system is finite along and has 3+1D surfaces at and , two sets of surface modes remain gapless. These are the modes and which simply do not have a partner mode to pair up with and thus do not appear in any of the sine-Gordon terms.

iii.5 Field theory

In order to show that the bulk of the gapped state obtained above behaves as a fractional 4+1D quantum Hall state, we calculate its response to an external electromagnetic field in a quantum field theory representation using the action formalism. This further allows us to relate the boundary modes to a 3+1D chiral anomaly with a fractional coefficient as compared with the noninteracting case.

In previous sections we established that a stack of 3+1D Weyl semimetals is described by a Hamiltonian of the form , where is the bosonized free theory of Eq. (III.4) and encodes the relevant correlated tunneling terms of Eq. (24). In order to explicitly derive the response of this system to an external electromagnetic field we closely follow Ref. Santos et al., 2015, in which a Chern-Simon theory of 2+1D fractional topological insulators was obtained, while highlighting the differences.

The starting point of the derivation is to implement, within the coupled-wire construction, minimal coupling of the electromagnetic field to the fermionic current through where the summation over is implied. It is technically convenient to treat the different components separately. The components were already included in the construction of the Landau levels, and the theory respects gauge invariance in these coordinates through the Landau level degeneracy prefactor , which is proportional to the (gauge invariant) magnetic field . The rest of the components are obtained by demanding that our theory is invariant under the gauge transformation:


with an analogous relation for . Here and for the remainder of this section indices imply summation over only while imply summation over all indices. We suppress in the notation the explicit dependence of all fields on ; the dependence on and is encoded in the Landau level quantum numbers and . Within the realm of bosonized fields the gauge transformation (25) translates, via Eq. (18), into


The gauge invariant generalization of the tunneling term Eq. (24), which contains the electromagnetic response of the 4+1D state along the direction, then takes the action form


We have assumed is a smooth function of on scales , where is the lattice constant in the discrete fourth dimension.

The remaining components involve with the particle density and the current density along the direction. To write the corresponding coupling in terms of the fields, we note that the one-dimensional particle density of electrons at is given by


This implies that a kink in one of the sine-Gordon terms of Eq. (24) carries a charge of . The (one-dimensional) particle current density , on the other hand, is related to the particle density by the continuity equation


Using this continuity equation and the density expression (29) we obtain the electromagnetic response


with the totally antisymmetric tensor and summation over is implied. By combining the above action components, Eqs. (28) and (31), with the independent kinetic term obtained from Eq. (III.4), the full electromagnetic response of the system finally takes the form


In the next step towards showing that the response (32) reduces to that of a 4+1D fractional quantum Hall effect, we use the fact that the cosines in Eq. (28) pin their arguments to one of their minima when these terms become relevant. This in turn fixes the difference between left and right chiral fields to


With this strong-coupling condition we write the action in Eq. (31) in terms of and only


Importantly, the summands are now independent of which allows us to replace the sum with the Landau level degeneracy . This action yields the current density


One immediately identifies this as the response (3) of a 4+1D quantum Hall effect to an external electromagnetic field that satisfies and .

To finally connect this response to a 4+1D Chern–Simons term, we express the Landau level degeneracy as


where . With this, the action in Eq. (34) becomes


where and . This action can be interpreted as a part of the isotropic fractionalized Chern–Simons action


This action exactly generates the response given by (1) with a fractional , with the current density along taking the form


reproduced in Eq. (3).

Due to its intrinsic anisotropy, the coupled-wire construction only recovers the part of Eq. (39) given by Eq. (35). Note, however, that we are free to redefine the direction of the external magnetic field as well as the direction along and perpendicular to the wires. Had we chosen other directions, we would have separately obtained all the parts that compose the Chern–Simons action (38). Another argument for the correctness of the isotropic Chern–Simons functional form of (38) is that the complete isotropy is required from gauge invariance in the bulk. A way to understand this is by analogy with emergent 2+1D Chern–Simons terms in non-relativistic field theories (dimensionality is irrelevant for the present argument). If Lorentz invariance is preserved, one expects the functional form for a generic bosonic field , with all terms having the same coefficient. The absence of Lorentz invariance naively would lead one to expect different prefactors for and but this statement is incorrect; a Chern–Simons field theory is only gauge invariant if all of its terms are present with the same coefficient Zee (2010).

Considering all the above, we conclude that Eq. (39) represents the isotropic current response of the gapped state found in our coupled-wire construction. This is the response of a fractional 4+1 D quantum Hall insulator to an external electromagnetic field, as advertised by Eq. (3).

iii.6 Surface theory: Fractional chiral metal

Having obtained a bulk description of the 4+1D fractional quantum Hall state, we now characterize its boundary modes—the chiral fractional metal.

Our object of interest is the 3+1D theory at the surface of the 4+1D quantum Hall insulator. From our findings in the previous section, see Eqs. (34) and (36), the 3+1D surface state should be constructed from copies of the edge of a 2+1D Chern–Simons theory, each copy labelled by . A useful and general way to access such surface theories is to describe the quantum Hall states with an effective theory of conserved current operators in terms of bosonic fields (the number of indices is determined by dimensionality) so that satisfies by construction Wen (2007); Zee (2010). In our case, the effective action is a sum of 2+1D actions for each copy labelled by


This theory recovers the characteristic 2+1D Chern–Simons term after integrating out the gauge fields


resulting in the fractional Hall conductivity for each . The Chern–Simons theory Eq. (34) in its isotropic form is obtained by summing over which results in an overall prefactor of in front of the 2+1D Chern–Simons theory (42) provided . For notational simplicity and until otherwise stated, we now drop the label .

To write down the edge theory of each 2+1D dimensional ‘slice’ defined in the coordinate space, consider an infinite strip of length in the direction. Firstly, it is possible to check Santos et al. (2015) that gauge invariance at the boundary requires that where is a scalar field, which we will interpret physically shortly. By using this constraint and partial integration of the action (40) we can rewrite its first term as (see Appendix A for details)

where we have identified the two chiral bosonized fields defined in (18) with the scalar field at the edges such that and . In other words we have associated a physical meaning to : it is the bosonic scalar field that represents the two chiral modes at the boundary.

The surface theory is completed by the non-universal Hamiltonian defined by the first term in Eq. (21). Adding it to Eq. (LABEL:eq:Ssurf), the full surface theory is