Measuring Berry curvature with quantum Monte Carlo

Abstract

The Berry curvature and its descendant, the Berry phase, play an important role in quantum mechanics. They can be used to understand the Aharonov-Bohm effect, define topological Chern numbers, and generally to investigate the geometric properties of a quantum ground state manifold. While Berry curvature has been well-studied in the regimes of few-body physics and non-interacting particles, its use in the regime of strong interactions is hindered by the lack of numerical methods to solve it. In this paper we fill this gap by implementing a quantum Monte Carlo method to solve for the Berry curvature, based on interpreting Berry curvature as a leading correction to imaginary time ramps. We demonstrate our algorithm using the transverse-field Ising model in one and two dimensions, the latter of which is non-integrable. Despite the fact that the Berry curvature gives information about the phase of the wave function, we show that our algorithm has no sign or phase problem for standard sign-problem-free Hamiltonians. Our algorithm scales similarly to conventional methods as a function of system size and energy gap, and therefore should prove a valuable tool in investigating the quantum geometry of many-body systems.

4 pages + 2 page appendix, 2 figures

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• Dynamic scaling of the restoration of rotational symmetry in Heisenberg quantum antiferromagnets
• Simulation of the many-body dynamical quantum Hall effect in an optical lattice
• Efficient Berry Phase Calculation via Adaptive Variational Quantum Computing Approach