Comparative Study of State-of-the-Art Matrix-Product-State Methods for Lattice Models with Large Local Hilbert Spaces

TL;DR

This paper compares three advanced matrix-product-state methods for simulating lattice models with large local Hilbert spaces, focusing on their efficiency and accuracy in analyzing the Holstein model.

Contribution

It provides a systematic comparison of three state-of-the-art MPS methods tailored for high-dimensional local spaces in lattice models.

Findings

All three methods effectively analyze the Holstein model.

The methods vary in computational efficiency and accuracy.

Finite-size scaling reveals strengths and limitations of each approach.

Abstract

Lattice models consisting of high-dimensional local degrees of freedom without global particle-number conservation constitute an important problem class in the field of strongly correlated quantum many-body systems. For instance, they are realized in electron-phonon models, cavities, atom-molecule resonance models, or superconductors. In general, these systems elude a complete analytical treatment and need to be studied using numerical methods where matrix-product states (MPS) provide a flexible and generic ansatz class. Typically, MPS algorithms scale at least quadratic in the dimension of the local Hilbert spaces. Hence, tailored methods, which truncate this dimension, are required to allow for efficient simulations. Here, we describe and compare three state-of-the-art MPS methods each of which exploits a different approach to tackle the computational complexity. We analyze the properties of these methods for the example of the Holstein model, performing high-precision calculations as well as a finite-size-scaling analysis of relevant ground-state obervables. The calculations are performed at different points in the phase diagram yielding a comprehensive picture of the different approaches.