Topological Nematic States and Non-Abelian Lattice Dislocations

TL;DR

Topological nematic states arise from the fusion of topology and lattice symmetry in partially filled Chern bands. The authors map a Chern number $C=N$ band to $N$ coupled Landau levels, show that lattice dislocations act as branch cuts or wormholes that permute layers and alter topology, and derive nontrivial ground-state degeneracies even for Abelian FQH states. They develop edge-state and effective field theory pictures, classify generic nematic types by lattice vectors, and propose numerical diagnostics (even/odd effects) and domain-wall protected gapless modes. The work opens a route to realize high-genus topological physics in planar samples and suggests broader implications for dislocation braiding and topological liquid-crystal phases.

Abstract

An exciting new prospect in condensed matter physics is the possibility of realizing fractional quantum Hall (FQH) 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 translation symmetry and topological properties of these fractional Chern insulators. When a partially filled flat band has a Chern number N, it can be mapped to an N-layer quantum Hall system. We find that lattice dislocations can act as wormholes connecting the different layers and effectively change the topology of the space. Lattice dislocations become defects with non-trivial quantum dimension, even when the FQH 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.

Figures (6)

• Figure 1: A. Each state shifts over two lattice spacings: $\langle \hat{x} \rangle \rightarrow \langle \hat{x} \rangle + 2$, as $k_y \rightarrow k_y + 2\pi$. 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: (Upper pannel) Illustration that an $x$-dislocation leads to a branch cut around which the two effective layers are exchanged. (Lower pannel) A reflection of the top layer maps the branch cut between a pair of dislocations into a "worm hole" connecting the two layers. C: A torus with two pairs of $x$-dislocations is equivalent to two tori connected by two "worm holes", which is a genus $3$ surface. This picture illustrates the fact that dislocations carry nontrivial topological degeneracy.
• Figure 3: The edge state understanding of the topological degeneracy. 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 the system back together by turning on appropriate inter-edge tunneling terms. B. Depiction of the two branches (red and blue) of counter-propagating 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, $\Psi_{eRI}^\dagger \Psi_{eLI} + H.c.$, are added. In the regions including the branch cuts separating the dislocations, twisted tunneling terms are added: $\Psi_{eR1}^\dagger \Psi_{eL2} + \Psi_{eR2}^\dagger \Psi_{eL1} + H.c.$. $\alpha_i$ indicate the mid-points of the $A$ or $B$ regions.
• Figure 4: Illustration of different definitions of Wannier states, which lead to different types of topological nematic states. ${\bf e}_1$ and ${\bf e}_2$ 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, marked by the red, blue and purple lines with arrows. The red, blue and purple lines correspond to topological nematic states of the type $(1,0)$, $(0,1)$ 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 in hopping along the dash line, and with even number of sites in $x$ direction. (b) A torus with no kink in hopping, but with odd number of sites in $x$ direction. In both cases, periodic boundary condition is imposed to 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 even number of sites in $x$ direction.