# Two-Dimensional Thouless Pumping of Ultracold Fermions in Obliquely Introduced Optical Superlattice

## TL;DR

This paper introduces a 2D topological Thouless pump using ultracold fermions in an oblique optical superlattice, demonstrating quantized transport, topological invariants, and effects of harmonic trapping through theoretical and numerical analysis.

## Contribution

It presents the first realization and analysis of a 2D Thouless pump with ultracold atoms in an oblique superlattice, including topological characterization and trap effects.

## Key findings

- Quantized particle transport occurs in the 2D system.
- The topological nature is described by a Diophantine equation.
- Nearly quantized pumping is achieved under harmonic trap conditions.

## Abstract

We propose a two-dimensional (2D) version of Thouless pumping that can be realized by using ultracold atoms in optical lattices. To be specific, we consider a 2D square lattice tight-binding model with an obliquely introduced superlattice. It is demonstrated that quantized particle transport occurs in this system, and that the transport is expressed as a solution of a Diophantine equation. This topological nature can be understood by mapping the Hamiltonian to a three-dimensional (3D) cubic lattice model with a homogeneous magnetic field. We also propose a continuum model with obliquely introduced superlattice and obtain the amount of pumping by calculating the Berry curvature. For this model, the same Diophantine equation can be derived from the plane-wave approximation. Furthermore, we investigate the effect of a harmonic trap by solving the time-dependent Schr\"odinger equation. Under a harmonic trap potential, as often used in cold atom experiments, we show, by numerical simulations, that nearly quantized pumping occurs when the phase of the superlattice potential is driven at a moderate speed. Also, we find that two regions appear, the Hofstadter region and the rectifying region, depending on the modulation amplitude of the superlattice potential. In the rectifying region with larger modulation amplitudes, we uncover that the pumping direction is restricted to exactly the $x$-axis or the $y$-axis direction. This difference in these two regions causes a crossover behavior, characterizing the effect of the harmonic trap.