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arXiv:1501.06917 · Physical Review B

Two-dimensional topological order of kinetically constrained quantum particles

Stefanos Kourtis, Claudio Castelnovo

Published Apr 20, 2015• 8 citations
arXivHTMLPDF

Abstract

We investigate how imposing kinetic restrictions on quantum particles that would otherwise hop freely on a two-dimensional lattice can lead to topologically ordered states. The kinetically constrained models introduced here are derived as a generalization of strongly interacting particle systems in which hoppings are given by flux-lattice Hamiltonians and may be relevant to optically driven cold-atom systems. After introducing a broad class of models, we focus on particular realizations and show numerically that they exhibit topological order, as witnessed by topological ground-state degeneracies and the quantization of corresponding invariants. These results demonstrate that the correlations responsible for fractional quantum Hall states in lattices can arise in models involving terms other than density-density interactions.

DOI
10.1103/physrevb.91.155134

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We investigate how imposing kinetic restrictions on quantum particles that would otherwise hop freely on a two-dimensional lattice can lead to topologically ordered states. The kinetically constrained models introduced here are derived as a generalization of strongly interacting particle systems in which hoppings are given by flux-lattice Hamiltonians and may be relevant to optically driven cold-atom systems. After introducing a broad class of models, we focus on particular realizations and show numerically that they exhibit topological order, as witnessed by topological ground-state degeneracies and the quantization of corresponding invariants. These results demonstrate that the correlations responsible for fractional quantum Hall states in lattices can arise in models involving terms other than density-density interactions.
Motivated by experimental progress in the growth of heavy transition metal oxides, we theoretically study a class of lattice models of interacting fermions with strong spin-orbit coupling. Focusing on interactions of intermediate strength, we derive a low-energy effective field theory for a fully gapped, topologically ordered, fractionalized state with an eightfold ground-state degeneracy. This state is a fermionic symmetry-enriched topological phase with particle-number conservation and time-reversal symmetry. The topological terms in the effective field theory describe a quantized magnetoelectric response and nontrivial mutual braiding statistics of dynamical extended vortex loops with emergent fermions in the bulk. We explicitly compute the expected mutual statistics in a specific model on the pyrochlore lattice within a slave-particle mean-field theory. We argue that our model also provides a possible condensed-matter realization of oblique confinement.
We develop a method to characterize topological phase transitions for strongly correlated Hamiltonians defined on two-dimensional lattices based on the many-body Berry curvature. Our goal is to identify a class of quantum critical points between topologically nontrivial phases with fractionally quantized Hall (FQH) conductivity and topologically trivial gapped phases through the discontinuities of the many-body Berry curvature in the so-called flux Brillouin zone (fBZ), the latter being defined by imposing all possible twisted boundary conditions. For this purpose, we study the finite-size signatures of several quantum phase transitions between fractional Chern insulators and charge-ordered phases for two-dimensional lattices by evaluating the many-body Berry curvature numerically using exact diagonalization. We observe degeneracy points (nodes) of many-body energy levels at high-symmetry points in the fBZ, accompanied by diverging Berry curvature. We find a correspondence between the number and order of these nodal points, and the change of the topological invariants of the many-body ground states across the transition, in close analogy with Weyl nodes in noninteracting band structures. This motivates us to apply a scaling procedure, originally developed for noninteracting systems, for the Berry curvature at the nodal points. This procedure offers a useful tool for the classification of topological phase transitions in interacting systems harboring FQH like topological order.
We present a detailed microscopic investigation of fractional quantum Hall states with gapped boundaries in a coupled bilayer lattice model featuring holes whose counterpropagating chiral edge states are hybridized and gapped out. We focus on an lattice limit for cold-atom experiments, in which each hole just consists of a single removed site. Although the holes distort the original band structure and lead to in-gap remnants of the continuum edge modes, we find that the lowest nearly flat band representing a higher-genus system may naturally form by controlling the local hopping terms that gap out the boundaries. Remarkably, local interactions in this new flat band lead to various Abelian and non-Abelian fractional quantum Hall states with gapped boundaries residing on emergent higher-genus surfaces, which we identify by extracting the nontrivial topological ground-state degeneracies and the fractional statistics of quasiparticles. These results demonstrate the feasibility of realizing novel fractional quantum Hall states with gapped boundaries even in the extreme lattice limit, thus enabling a possible new route towards universal topological quantum computation.
Topologically-ordered matter is a novel quantum state of matter observed only in a small number of physical systems, notably two-dimensional electron systems exhibiting fractional quantum Hall effects. It was recently proposed that a simple form of topological matter may be created in interacting systems of rotating ultra-cold atoms. We describe ensemble measurements on small, rotating clusters of interacting bosonic atoms, demonstrating that they can be induced into quantum ground states closely analogous to topological states of electronic systems. We report measurements of inter-particle correlations and momentum distributions of Bose gases in the fractional quantum Hall limit, making comparison to a full numerical simulation. The novel experimental apparatus necessary to produce and measure properties of these deeply entangled quantum states is described.
The fractional quantum Hall (FQH) effect illustrates the range of novel phenomena which can arise in a topologically ordered state in the presence of strong interactions. The possibility of realizing FQH-like phases in models with strong lattice effects has attracted intense interest as a more experimentally accessible venue for FQH phenomena which calls for more theoretical attention. Here we investigate the physical relevance of previously derived geometric conditions which quantify deviations from the Landau level physics of the FQHE. We conduct extensive numerical many-body simulations on several lattice models, obtaining new theoretical results in the process, and find remarkable correlation between these conditions and the many-body gap. These results indicate which physical factors are most relevant for the stability of FQH-like phases, a paradigm we refer to as the geometric stability hypothesis, and provide easily implementable guidelines for obtaining robust FQH-like phases in numerical or real-world experiments.
Abstract We demonstrate the emergence of a topological ordered phase for non-Hermitian systems. Specifically, we elucidate that systems with non-Hermitian two-body interactions show a fractional quantum Hall (FQH) state. The non-Hermitian Hamiltonian is considered to be relevant to cold atoms with dissipation. We conclude the emergence of the non-Hermitian FQH state by the presence of the topological degeneracy and by the many-body Chern number for the ground state multiplet showing Ctot = 1. The robust topological degeneracy against non-Hermiticity arises from the manybody translational symmetry. Furthermore, we discover that the FQH state emerges without any repulsive interactions, which is attributed to a phenomenon reminiscent of the continuous quantum Zeno effect.