Large-$N$ Chern insulators: lattice field theory and quantum simulation approaches to correlation effects in the quantum anomalous Hall effect

TL;DR

This paper explores the complex phase structure of four-Fermi lattice models relevant to the quantum anomalous Hall effect, using a combination of analytical, numerical, and quantum simulation techniques to understand strong correlation effects.

Contribution

It introduces a multidisciplinary approach combining lattice field theory, tensor networks, and quantum simulation to study correlation effects in large-N Chern insulators.

Findings

Rich phase diagram with Chern insulators and fermion condensates

Tensor-network algorithms improve understanding of strong-coupling regimes

Quantum simulation schemes with ultra-cold atoms enable experimental exploration

Abstract

Four-Fermi quantum field theories in (2+1) dimensions lie among the simplest models in high-energy physics, the understanding of which requires a non-perturbative lattice formulation addressing their strongly-coupled fixed points. These lattice models are also relevant in condensed matter, as they offer a neat playground to explore strong correlations in the quantum anomalous Hall (QAH) effect. We give a detailed description of our multidisciplinary approach to understand the fate of the QAH phases as the four-Fermi interactions are increased, which combines strong-coupling and effective-potential techniques, unveiling a rich phase diagram with large-$N$ Chern insulators and Lorentz-breaking fermion condensates. Moreover, this toolbox can be enlarged with recent advances in quantum information science, as we show that tensor-network algorithms based on projected entangled pairs can be used to improve our understanding of the strong-coupling limit. We also present a detailed scheme that uses ultra-cold atoms in optical lattices with synthetic spin-orbit coupling to build quantum simulators of these four-Fermi models. This yields a promising alternative to characterise the strongly-coupled fixed points and, moreover, could also explore real-time dynamics and finite-fermion densities.