Fractional Chern insulators on cylinders: Tao-Thouless states and beyond

TL;DR

This paper investigates finite-circumference effects in topological phases on cylinders, revealing how different parameter scalings influence the emergence of charge density waves or topological degeneracies, thus clarifying lattice effects in topological state studies.

Contribution

It introduces a parameter scaling approach for studying topological phases on cylinders, distinguishing between Tao-Thouless states and minimally entangled states, and analyzes their relation to continuum and lattice effects.

Findings

Different scaling schemes yield distinct topological signatures.

Tao-Thouless states emerge on thin cylinders with charge density wave order.

Uniform topological states with degeneracy are identified as minimally entangled states.

Abstract

Topological phases in two-dimensional quantum lattice models are often studied on cylinders for revealing different topological properties and making the problem numerically tractable. This makes a proper understanding of finite-circumference effects crucial for reliably extrapolating the results to the thermodynamic limit. Using matrix product states, we investigate these effects for the Laughlin-1/2 phase in the Hofstadter-Bose-Hubbard model, which can be viewed as the lattice discretization of the bosonic quantum Hall problem in the continuum. We propose a scaling of the model's parameters with the cylinder circumference that simultaneously approaches the continuum and thermodynamic limits. We find that different scaling schemes yield distinct topological signatures: we either retrieve a spontaneous formation of charge density wave ordering reminiscent of the Tao-Thouless states, known from the continuum problem on thin cylinders, or we find uniform states with a topological degeneracy that can be identified as minimally entangled states known from studies of chiral spin liquids on cylinders. Finally, we carry out a similar analysis of the non-Abelian Moore-Read phase in the same model. Our results clarify the role of symmetries in numerical studies of topologically ordered states on cylinders and highlight the role of lattice effects.