Efficient and Flexible Approach to Simulate Low-Dimensional Quantum Lattice Models with Large Local Hilbert Spaces

TL;DR

This paper introduces a novel mapping technique that creates artificial symmetries in quantum lattice models, significantly reducing computational costs and enabling efficient simulation of systems with large local Hilbert spaces, demonstrated on the Holstein model.

Contribution

The authors develop a new mapping that constructs artificial $U(1)$ symmetries, improving the efficiency of tensor network methods for large local Hilbert space systems in quantum lattice models.

Findings

Achieved simulations with up to 501 lattice sites and 63 phonons per site.

Established a connection between Schmidt values and single-site reduced density matrices.

Provided bounds on numerical complexity compared to standard methods.

Abstract

Quantum lattice models with large local Hilbert spaces emerge across various fields in quantum many-body physics. Problems such as the interplay between fermions and phonons, the BCS-BEC crossover of interacting bosons, or decoherence in quantum simulators have been extensively studied both theoretically and experimentally. In recent years, tensor network methods have become one of the most successful tools to treat lattice systems numerically. Nevertheless, systems with large local Hilbert spaces remain challenging. Here, we introduce a mapping that allows to construct artificial $U(1)$ symmetries for any type of lattice model. Exploiting the generated symmetries, numerical expenses that are related to the local degrees of freedom decrease significantly. This allows for an efficient treatment of systems with large local dimensions. Further exploring this mapping, we reveal an intimate connection between the Schmidt values of the corresponding matrix-product-state representation and the single-site reduced density matrix. Our findings motivate an intuitive physical picture of the truncations occurring in typical algorithms and we give bounds on the numerical complexity in comparison to standard methods that do not exploit such artificial symmetries. We demonstrate this new mapping, provide an implementation recipe for an existing code, and perform example calculations for the Holstein model at half filling. We studied systems with a very large number of lattice sites up to $L=501$ while accounting for $N_{\rm ph}=63$ phonons per site with high precision in the CDW phase.