From an array of quantum wires to three-dimensional fractional topological insulators

The coupled-wires approach has been shown to be useful in describing two-dimensional strongly interacting topological phases. In this manuscript, we extend this approach to three-dimensions, and construct a model for a fractional strong topological insulator. This topologically ordered phase has an exotic gapless state on the surface, called a fractional Dirac liquid, which cannot be described by the Dirac theory of free fermions. Like in noninteracting strong topological insulators, the surface is protected by the presence of time-reversal symmetry and charge conservation. We show that upon breaking these symmetries, the gapped fractional Dirac liquid presents unique features. In particular, the gapped phase that results from breaking time-reversal symmetry has a halved fractional Hall conductance of the form ${\sigma }_{xy}=\frac{1}{2}\frac{e^2}{mh}$ if the filling is $\nu =1/m$. On the other hand, if the surface is gapped by proximity coupling to an $s$-wave superconductor, we end up with an exotic topological superconductor. To reveal the topological nature of this superconducting phase, we partition the surface into two regions: one with broken time-reversal symmetry and another coupled to a superconductor. We find a fractional Majorana mode, which cannot be described by a free Majorana theory, on the boundary between the two regions. The density of states associated with tunneling into this one-dimensional channel is proportional to ${\omega }^{m-1}$, in analogy to the edge of the corresponding Laughlin state.

Figure 1

A schematic depiction of the array of wires studied in this manuscript. The coupled-wires approach allows us to treat interacting terms using the Abelian bosonization framework, making it very useful in constructing strongly interacting topological phases. Throughout this work, we focus on the construction of a fractional strong topological insulator.

Figure 2

The fractional strong topological insulator phase we construct has an exotic fractional Dirac liquid on the surface. As in the noninteracting case, this gapless surface is protected by time-reversal symmetry and charge conservation. Upon breaking any of these symmetries, we end up with a gapped surface displaying unique topological properties. For example, by breaking time-reversal symmetry, we get a halved fractional quantum Hall effect, characterized by a surface Hall conductance of the form ${\sigma }_{xy}=\frac{1}{2}\frac{e^2}{mh}$, where $m$ is an odd integer (in a state associated with the 2D Laughlin state at filling $\nu =1/m$). On the magnetic domain wall shown in (a), one therefore finds a chiral Luttinger liquid, similar to the edge mode of the corresponding 2D Laughlin-state. If the system is gapped by proximity to an $s$-wave superconductor (i.e., by breaking charge conservation), we get an exotic time-reversal invariant topological superconductor. It is interesting to study the boundary between a superconducting region and a magnetic region, as depicted in Fig (b). A fractional Majorana mode, which cannot be described by a free Majorana theory, is found on the interface between the two regions. The tunneling density of states associated with this 1D channel is proportional to ${\omega }^{m-1}$, in analogy to the edge of the corresponding Laughlin state.

Figure 3

(a) The simplified model we use to describe a weak topological insulator. Each plane forms an $S_z$-conserving 2D topological insulator. The edge of each layer therefore contains counter-propagating spin-up and spin-down modes, represented by red and blue arrows. (b) A diagrammatic depiction of the surface of the simplified model. The vertical direction represents the layer index, and the horizontal direction represents spin. The symbol $⨂$ ($⨀$) corresponds to a right (left) mover. We introduce nearest-layer coupling terms, represented by arrows connecting different chiral modes. Time-reversal symmetry relates the amplitudes of the terms represented by the full arrows to the amplitudes of the terms represented by the dashed arrows. The above model can be decomposed into two decoupled gapless Hamiltonians, represented by the green and purple arrows.

Figure 4

(a) The simplified model we use to describe an antiferromagnetic topological insulator and its fractional analog. Each layer can be thought of as a single-component QHE system at filling $\pm \nu$ (where $\nu =1$ in the integer case, and in the fractional case we consider $\nu =1/m$, where $m$ is an odd integer). Due to the existence of a modified time-reversal symmetry, given by the product of the original time-reversal operator and a translation by a unit cell, the $xz$ surface remains gapless even in the presence of interlayer coupling terms. The surface can be described in terms of coupled 1D channels, which enables a simple derivation of universal surface properties. We argue that these properties remain true for a strong topological insulator and its fractional analogue, where the surface is protected by the local time-reversal symmetry. (b) A diagrammatic representation of $xz$ surface of the above model in terms of the coupled 1D chiral edge modes. (c) The diagram that corresponds to the situation where the modified time-reversal symmetry is explicitly broken by introducing a term of the form (11) with ${\tilde{t}}^{\prime }=\tilde{t}$.

Figure 5

A diagrammatic representation, in terms of the $\tilde{\psi }$ (or $\chi$) modes defined at the beginning of Sec. 2b, of the situation where the coefficient ${\tilde{t}}^{\prime }$ of the time-reversal breaking term changes abruptly from $-\tilde{t}$ to $\tilde{t}$ around $n=1$. A decoupled chiral $\tilde{\psi }$ mode is seen to be localized near the boundary at $n=1$, leading to the conclusion that the gapped surface has a halved fractional quantum Hall effect.

Figure 6

A diagrammatic representation of the surface model in terms of the fractional Majorana modes ${\gamma }_n^i$ [defined after Eq. (14)] in three situations: (a) The modified time-reversal symmetry is broken by the perturbation (11) with ${\tilde{t}}^{\prime }=\tilde{t}$. (b) Charge conservation is broken by a superconducting term of the form (13) with $\mathrm{\Delta }=\tilde{t}$. (c) The region with $n<1$ of the surface is gapped by a superconducting term, and the region with $n>1$ is gapped by a modified time-reversal symmetry breaking term. A chiral fractional Majorana mode is attached to the boundary at $n=1$. Here the vertical direction represents the index $i$, and the horizontal direction represents the layer index $n$. Dotted symbols represent the chiral fractional Majorana modes.

Figure 7

The wire construction we use as a realization of a 2D fractional topological insulator. This model is the starting point in our construction of a 3D fractional strong topological insulator (see Secs. 4b and 4c).

Figure 8

The spectrum of the system depicted in Fig. 7, when the interwire terms are switched off for (a) $\nu =1$ and (b) $\nu =1/3$. We note that only the spin-up sector is presented here. The energies in blue, cyan, red, and green correspond to wires number 1, 2, 3, and 4, respectively.

Figure 9

The diagrams which present the Fermi-momenta of the 2D model as a function of the wire index (labeled by $(i,\alpha )$, where $i$ and $\alpha$ are indices enumerating the unit cells and the position within the unit cell, respectively) for (a) $\nu =1$, and (b) $\nu =1/3$. Notice that only the modes with spin-up are presented here.

Figure 10

A diagrammatic representation of the reduced 2D tight binding model, describing the interwire part of the full 3D model. Here, each “lattice point” corresponds to a linearly dispersing mode in the $x$ direction, with a well defined $y$ and $z$ coordinates. The horizontal axis describes the layer and spin indices ($n,s$), and the vertical axis describes the position within each layer ($i,\alpha$). Again, the symbol $⨂$ ($⨀)$ represents a right (left) mover. The green arrows represent the terms in the intralayer Hamiltonian $H_y$, and the brown arrows represent interlayer terms which couple the spin-up modes, and take the form $-2t_{z}cos\left(k_{z}a\right){\tau }_x$ in $k$-space. We emphasize that the reduced model defined in Eqs. (42) and (43) contains additional terms which are not depicted here. If all the terms are considered, the full 3D model is topologically nontrivial if the reduced model forms a 2D topological insulator. Thus the analysis of some aspects of the strongly interacting 3D phase is reduced to the analysis of the 2D noninteracting topological phase.