Matrix Product States Algorithms and Continuous Systems

TL;DR

This paper introduces a versatile matrix product states (MPS) based method for analyzing continuous-variable many-body systems, demonstrating high accuracy in quantum models and extending to solving complex partial differential equations.

Contribution

It presents a new MPS algorithm applicable to continuous systems, achieving high precision in quantum and PDE problems, and offers a practical iterative technique for multi-variable equations.

Findings

Accurately computed properties of quantum harmonic oscillator chains.

Determined the central charge at the quantum critical point of the rotor model.

Solved Poisson-like equations with high precision using the new iterative method.

Abstract

A generic method to investigate many-body continuous-variable systems is pedagogically presented. It is based on the notion of matrix product states (so-called MPS) and the algorithms thereof. The method is quite versatile and can be applied to a wide variety of situations. As a first test, we show how it provides reliable results in the computation of fundamental properties of a chain of quantum harmonic oscillators achieving off-critical and critical relative errors of the order of 10^(-8) and 10^(-4) respectively. Next, we use it to study the ground state properties of the quantum rotor model in one spatial dimension, a model that can be mapped to the Mott insulator limit of the 1-dimensional Bose-Hubbard model. At the quantum critical point, the central charge associated to the underlying conformal field theory can be computed with good accuracy by measuring the finite-size corrections of the ground state energy. Examples of MPS-computations both in the finite-size regime and in the thermodynamic limit are given. The precision of our results are found to be comparable to those previously encountered in the MPS studies of, for instance, quantum spin chains. Finally, we present a spin-off application: an iterative technique to efficiently get numerical solutions of partial differential equations of many variables. We illustrate this technique by solving Poisson-like equations with precisions of the order of 10^(-7).