Topological Nematic States and Non-Abelian Lattice Dislocations

An exciting new prospect in condensed matter physics is the possibility of realizing fractional quantum Hall states in simple lattice models without a large external magnetic field. A fundamental question is whether qualitatively new states can be realized on the lattice as compared with ordinary fractional quantum Hall states. Here we propose new symmetry-enriched topological states, topological nematic states, which are a dramatic consequence of the interplay between the lattice translational symmetry and topological properties of these fractional Chern insulators. The topological nematic states are realized in a partially filled flat band with a Chern number $N$, which can be mapped to an $N$-layer quantum Hall system on a regular lattice. However, in the topological nematic states the lattice dislocations can act as wormholes connecting the different layers and effectively change the topology of the space. Consequently, lattice dislocations become defects with a nontrivial quantum dimension, even when the fractional quantum Hall state being realized is, by itself, Abelian. Our proposal leads to the possibility of realizing the physics of topologically ordered states on high-genus surfaces in the lab even though the sample has only the disk geometry.

Popular Summary

Amazing things can happen when electrons get together. In the right circumstances, new collective states emerge whose properties differ dramatically from those of independent electrons. A classic experimental example of such a state, the fractional quantum Hall (FQH) state, was discovered in the early 1980s in perfect fluid-like sheets of electrons at ultralow temperatures and very high magnetic fields. Strong interactions between the electrons combined with the magnetic field make them coalesce and the resulting state features what seems like fragments of the original electrons—stable particles with charges that are rational fractions of the elementary electron charge. A new type of long-range quantum-mechanical correlation, called “topological order,” can be used to characterize these FQH states. More recently, researchers have begun to explore related states that may emerge in very different physical settings: where a magnetic field is absent and the motion of electrons is confined to a lattice as opposed to the continuous, fluid-like motion in the traditional setting. In this paper, we examine theoretical models that describe such new settings, and find entirely new topologically ordered states. For example, we find that the presence of the lattice opens a new door to states that could be used to robustly encode quantum information in future computing schemes.

Each energy band in the models we have investigated is characterized by an integer Chern number $N$, a topologically invariant quantum number related to how the state of the electrons changes with their momenta. Unlike traditional FQH systems, the lattice systems allow $N$ to be greater than 1. The key observation that leads to our new results is this: An $N>1$ lattice system can be mapped onto $N$ parallel layers of continuous FQH systems. Electronic states of the former evolve in a complex way as the electrons move from one lattice point to another, which is mathematically equivalent to cyclically permuting the $N$ parallel layers. One consequence of this complexity is the emergence of a new type of topologically ordered states. These states are analogs of conventional FQH states of $N$ correlated layers, and may explicitly break rotational symmetry. Another, even more interesting consequence of the mapping is that any lattice dislocations, which disrupt the regular atomic alignment, effectively act as “wormholes” that connect the different layers together. The wormholes change the topology of the space of the $N$-layer system, introducing degeneracy to the ground state of the lattice system. Since this degeneracy depends only on the topology, the degenerate ground state can be used to encode quantum information in a fault-tolerant way.

Our proposal points out a new path toward topological quantum computation that could be realized in a wide class of systems. This should motivate researchers to find suitable physical realizations of these states and to develop the mathematical framework to describe these new topological phenomena.

Figure 1

(a) Each state shifts over two lattice spacings: $⟨\widehat{x}⟩\to ⟨\widehat{x}⟩+2$, as $k_y\to k_y+2\pi$. Thus, there are two families of states (red and blue lines). (b) The states can be mapped to two families of states, each parametrized by a single parameter $K_y$. (c) Illustration of the fact that the Chern-number-2 lattice system is mapped to a bilayer quantum Hall system, with the two layers corresponding to the two families of Wannier states shown in panel (b).

Figure 2

(a) Illustration of an $x$ dislocation. (b) Left panel: Illustration that an $x$ dislocation leads to a branch cut around which the two effective layers are exchanged. Right panel: A reflection of the top layer maps the branch cut between a pair of dislocations into a wormhole connecting the two layers. (c) A torus with two pairs of $x$ dislocations is equivalent to two tori connected by two wormholes, which is a genus-3 surface. This picture illustrates the fact that dislocations carry nontrivial topological degeneracy.

Figure 3

Topological degeneracy seen from the edge-state perspective. (a) The dislocations are oriented along a single line, and then the system is cut along the line, yielding gapless counterpropagating edge states along the line. The original FQH state is obtained from gluing the system back together by turning on appropriate interedge-tunneling terms. (b) Depiction of the two branches (red and blue) of counterpropagating edge excitations. The arrows between the edge states indicate the kinds of electron-tunneling terms that are added. Away from the dislocations, in the $A$ regions, the usual electron-tunneling terms involving tunneling between the same layers, ${\Psi }_{eRI}^{†}{\Psi }_{eLI}+\mathrm{H}.\mathrm{c}.$, are added. In the regions that include the branch cuts separating the dislocations, twisted-tunneling terms are added: ${\Psi }_{eR1}^{†}{\Psi }_{eL2}+{\Psi }_{eR2}^{†}{\Psi }_{eL1}+\mathrm{H}.\mathrm{c}.$ ${\alpha }_i$ indicate the midpoints of the $A$ or $B$ regions.

Figure 4

Illustration of different definitions of Wannier states, which lead to different types of topological nematic states. ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ are two reciprocal vectors defining a Brillouin zone. The Wannier-state basis can be constructed by taking the Fourier transform of Bloch states along one periodic direction of the Brillouin zone, which are marked by the red, blue, and purple lines with arrows. The red, blue, and purple lines correspond to topological nematic states of the types (1,0), (0,1), and (1,1), respectively (see text).

Figure 5

Illustration of two lattice configurations to detect the topological nematic states. (a) A torus with a kink along the dashed line such that all points above the dashed line are translated by one lattice spacing in the $x$ direction relative to a regular lattice model, and with an even number of sites in the $x$ direction. (b) A regular lattice with an odd number of sites in the $x$ direction. In both cases, a periodic boundary condition is imposed on both directions. For both configurations, the ($mml$) topological nematic state of type (1,0) has a reduced ground-state degeneracy of $|m+l|$ instead of the degeneracy of $|m^2-l^2|$ on a torus with an even number of sites in the $x$ direction.

Figure 6

(a) Illustration of the domain wall between two topological nematic states (1,0) (upper half) and (0,1) (lower half). (b) Illustration that the left- and right-moving edge states along the domain wall transform differently under lattice translation $T_x$. The right movers from the (1,0) state are exchanged by $T_x$ while the left movers from the (0,1) state are separately translation invariant.