Truncation of lattice fractional quantum Hall Hamiltonians derived from conformal field theory

Abstract

Conformal field theory has recently been applied to derive few-body Hamiltonians whose ground states are lattice versions of fractional quantum Hall states. The exact lattice models involve interactions over long distances, which is difficult to realize in experiments. It seems, however, that such long-range interactions should not be necessary, as the correlations decay exponentially in the bulk. This poses the question, whether the Hamiltonians can be truncated to contain only local interactions without changing the physics of the ground state. Previous studies have in a couple of cases with particularly much symmetry obtained such local Hamiltonians by keeping only a few local terms and numerically optimizing the coefficients. Here, we investigate a different strategy to construct truncated Hamiltonians, which does not rely on optimization, and which can be applied independent of the choice of lattice. We test the approach on two models with bosonic Laughlin-like ground states with filling factor 1/2 and 1/4, respectively. We first investigate how the coupling strengths in the exact Hamiltonians depend on distance, and then we study the truncated models. For the case of 1/2 filling, we find that the truncated model with truncation radius √ 2 lattice constants on the square lattice and 1 lattice constant on the triangular lattice has an approximate twofold ground state degeneracy on the torus, and the overlap per site between these states and the states constructed from conformal field theory is higher than 0.99 for the lattices considered. For the model at 1/4 filling, our results give some hints that a truncation radius of √ 5 on the square lattice and √ 7 on the triangular lattice might be enough, but the finite size effects are too large to judge whether the topology is, indeed, present in the thermodynamic limit. The states with high overlap also have the expected topological entanglement entropies.

Abelian topological order of ν=2/5 and 3/7 fractional quantum Hall states in lattice models

“…Continuum FQH states in the Jain hierarchy (ν = 1/3, 2/5, 3/7,...) have been shown numerically [1,2,23], and for the Laughlin state also experimentally [24,25], to possess Abelian topological order for the Coulomb interaction. The analysis of corresponding lattice FQH states, on the other hand, is complicated by several factors, including the limited number of viable experimental systems [15] and the difficulty of engineering long-range interactions [26]. Coupled with this, it has been shown that the lattice can host fundamentally different phases of matter [27], as well as states with non-Abelian statistics at equivalent filling factors [13,28].…”

Few-particle dynamics of fractional quantum Hall lattice models

“…We would like to split this Hamiltonian into a noninteracting part plus interactions. By multiplying out the terms in the defining Hamiltonian, one obtains [38]…”

“…Following Ref. [38], we present and use the Hamiltonians in a form that distinguishes one-body, two-body, and three-body terms. In Sec.…”

Non-Abelian chiral spin liquid on spin-1 kagome lattice: truncation of an exact Hamiltonian and numerical optimization

“…[44], and studied further in [48]. Moreover, a general procedure of truncating the long-range terms was formulated for Hamiltonians constructed from conformal field theory [49]. While so far it was applied only to the parent Hamiltonians of Abelian states, the longrange Hamiltonian from Ref.…”