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.
Highlights
Among the most important discoveries in condensed matter physics are the integer quantum Hall and fractional quantum Hall (FQH) states [1,2,3,4], which provided the first examples of electron fractionalization in more than one dimension and paved the way for our current understanding of topological order [5,6,7]
The topological order of the FQH states is characterized by ground state degeneracies that depend on spatial topology [8], and fractionalized quasiparticles [4], while the quantized Hall conductance is determined by a topological invariant—the Chern number—which for a band insulator can be determined by the momentum-space flux of the Berry’s phase gauge field. [9]
Since Chern numbers are generic properties of any band structure, it is natural to expect that quantum Hall states can be realized in lattice systems without an applied magnetic field [10,11], and such ‘‘quantum anomalous Hall’’ (QAH) states may be realizable experimentally [12,13]
Summary
Chern number cannot be written as a superposition of Wannier functions that are localized in both directions in real space. The many-body lattice FQH states constructed using these Wannier states break rotational symmetry, which is why we will refer to them as topological nematic states. The Wannier states satisfy the twistedboundary condition jWðky þ 2; nÞi 1⁄4 jWðky; n þ C1Þi and correspondingly xnðky þ 2Þ 1⁄4 xnðkyÞ þ C1 Because of such a twisted-boundary condition, the Wannier states for C1 1⁄4 1 can be labeled by one parameter, K 1⁄4 ky þ 2n, and the Wannier-state basis jWKi is in one-to-one correspondence with the Landau-level wave functions in the Landau gauge in the ordinary quantum Hall problem [22]. (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). Since the (mml) states are the most stable and simplest two-component FQH states, we assume that they are stabilized in the ground state of some physically reasonable Hamiltonians
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