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

TL;DR

This work analyzes fractional Chern insulators on infinite cylinders to understand how finite circumference and lattice effects shape topological signatures. Using matrix product states, it introduces two scaling schemes, $\alpha \propto 1/N_y$ and $\alpha \propto 2/N_y$, to connect lattice models to the continuum and thermodynamic limits. For the Laughlin-1/2 phase, the first scaling yields Tao-Thouless CDW states with SU(2)$_1$-type entanglement structure and a Chern number $\mathcal{C}=1/2$, while the second scaling reveals two nearly degenerate uniform states forming minimally entangled states with an emergent $\mathrm{O}(2)$ symmetry and $\mathcal{C}=1/2$; a similar analysis of the Moore-Read phase shows lattice-induced sector structure and flux-quantized responses, though some Majorana-related sectors remain challenging to access variationally. The results illuminate how symmetries constrain cylinder diagnostics of topological order and highlight the interplay between continuum-like physics and lattice-induced effects in FCIs. Overall, the study provides practical scaling strategies for lattice FQH numerics and clarifies finite-circumference signatures linked to Tao-Thouless precursors and 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.

Figures (17)

• Figure 1: Sketch of the Tao-Thouless state with filling $\nu=1/2$ on the cylinder with circumference $L_y$. The one-particle orbitals $\varphi_n$ are separated by a distance $\Delta_x$; in the Tao-Thouless state, the shaded orbitals are filled and the white ones are empty.
• Figure 2: The single-particle band structure of the Hofstadter-Bose-Hubbard model in Eq. \ref{['eq:ham']} with $\alpha=1/N_y$ (top) and $\alpha=2/N_y$ (bottom) for $N_y=6$. The inset shows the single-particle orbitals [Eq. \ref{['eq:orbitals']}] on the cylinder in the continuum, with the associated lattice discretization.
• Figure 3: Generic MPS representation of the ground state of the Hofstadter-Bose-Hubbard model [Eq. \ref{['eq:ham']}] on the cylinder. We snake a 1D path through the cylinder, giving rise to an effective 1D model with $N_y$ unit cell. For analyzing the symmetry structure of the MPS, we can group each unit cell or rung in a super-site with $N_y$ physical legs (in the actual numerical optimization we always decompose into different MPS tensors). Depending on the situation, we can consider a multi-rung unit cell, where each MPS tensor $A_i$ carries a corresponding $\mathrm{U}(1)$ charge $q_i$. Note also that the translation operator $\mathcal{T}_y$ and the reflection operator $\mathcal{R}_y$ act as the product of operators acting locally as a permutation within the super-site space.
• Figure 4: Top: the equivalence between two different ways of structuring the U(1) charges in the MPS, where we have used the super-site representation as in Fig. \ref{['fig:mps']}. We start from the MPS representation where we put integer charge on one of the tensors (a), we split this charge into half (b), we drag this half-integer charge unto the next tensor (c), and fuse the double-line leg by shifting all the virtual charges by an amount of $1/2$ (d). Bottom: the two quasiparticle configurations. We start from the MPS representation in (d), and make a defect between tensors $A_1$ and $A_2$ (e) or vice-versa (f). The corresponding tensors $B_{12}$ and $B_{21}$ carry an extra charge label $q$, which denotes the global charge of the excitation relative to the ground state charge and has to be half-integer. In the excitation ansatz, we make another momentum superposition of such a configuration, and variationally optimize over the tensors $B_{12}$ or $B_{21}$.
• Figure 5: Scaling of the charge density wave order parameter (top) and the entanglement entropy (bottom) for $(\alpha,n_b)=(1/N_y,1/(2N_y))$ as a function of $N_y$. For the former, we fit to the expected exponential decay as a function of $N_y$. For the latter, we fit to a linear behavior as a function of $\sqrt{N_y}$ giving us a value for the topological entanglement entropy $\gamma\approx0.415$.