Dimensional crossover and cold-atom realization of topological Mott insulators

Abstract

Interacting cold-atomic gases in optical lattices offer an experimental approach to outstanding problems of many body physics. One important example is the interplay of interaction and topology which promises to generate a variety of exotic phases such as the fractionalized Chern insulator or the topological Mott insulator. Both theoretically understanding these states of matter and finding suitable systems that host them have proven to be challenging problems. Here we propose a cold-atom setup where Hubbard on-site interactions give rise to spin liquid-like phases: weak and strong topological Mott insulators. They represent the celebrated paradigm of an interacting and topological quantum state with fractionalized spinon excitations that inherit the topology of the non-interacting system. Our proposal shall help to pave the way for a controlled experimental investigation of this exotic state of matter in optical lattices. Furthermore, it allows for the investigation of a dimensional crossover from a two-dimensional quantum spin Hall insulating phase to a three-dimensional strong topological insulator by tuning the hopping between the layers.

Figure 1. Phase diagram and spectra of the lattice model.

(a) 2D–3D crossover phase diagram as a function of layer coupling t_z and staggered lattice potential λ. (b) Bulk gap as a function of t_z and λ. (c) Surface state spectrum (blue) of the isotropic 3D system t_z = t, λ = λ_c at the x = 0 surface as a function of momenta k_y and k_z. Bulk states are shown as gray dots. We use open (periodic) boundary conditions along x (y, z). (d–e) One-dimensional cuts of the spectrum for fixed values of k_z = 0 and k_z = π. Gapless edge states at x = 0 (x = L) edge are shown in blue (red) yielding z₀ = 1 and z_π = 0 and thus ν₀ = z₀ + z_π = 1.

Figure 2. Twisting the Hubbard interaction.

Part (a) shows the relabeling of the spin σ and spatial even/odd site μ degrees of freedom. To obtain an identical Hamiltonian H_2D + H_z one must realize different hopping elements between nearest and next-nearest neighbor sites as illustrated in (b) for the x-direction. The required hoppings along the y- and z-direction are described in detail in Supplementary Discussion S2.

Figure 3. Properties of the interacting system according to slave-rotor theory.

(a) Interacting phase diagram as a function of interaction U and hopping t_z for fixed λ/t = 0.25. Upon increasing U the phases found at U = 0 remain mostly intact with renormalized parameters. For an extended correlated semi-metallic (SM) phase appears at . At a critical value U_c/t (green line) the system enters a Mott insulating (MI) state with weak and strong topological MI (WTMI and STMI) as well as gapless MI (GMI) phases. The GMI phase exhibits a semimetallic spinon spectrum. Dashed lines are a guide to the eye showing that Mott phases persist for U > U_c. (b) Interacting phase diagram and (c) bulk gap at U_c as a function of staggering λ and hopping t_z. The STMI phase occupies a large part of the phase diagram and features a significant bulk gap. (d–e) One-dimensional cut through the spinon bandstructure in the STMI phase for the isotropic system t_z = t and λ/t = 0.15. We use open (periodic) boundary conditions along x (y, z). Bulk states are shown in yellow, gapless spinon edge states at the x = 0 (x = L) edge are shown in blue (red).