Exploiting translational invariance in Matrix Product State simulations of spin chains with periodic boundary conditions

TL;DR

This paper introduces an efficient matrix product state algorithm for simulating translationally invariant spin chains with periodic boundary conditions, significantly reducing computational costs especially for large systems.

Contribution

The authors develop a novel MPS algorithm that minimizes computational cost for ground state approximations, achieving N times faster performance than previous methods for certain conditions.

Findings

Algorithm reduces computational cost to D^3 for   N, independent of system size.

Successfully applied to critical and noncritical spin models, accurately capturing correlation decay.

Identifies a bond dimension D(N) for faithful correlation representation in critical systems.

Abstract

We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension D of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain of N sites and correlation length \xi, the computational cost formally scales as g(D,\xi /N)D^3, where g(D,\xi /N) is a nontrivial function. For \xi << N, this scaling reduces to D^3, independent of the system size N, making our algorithm N times faster than previous proposals. We apply the method to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-1/2 models as well as for the noncritical Heisenberg spin-1 model. In the critical case, for any chain length N, we find a model-dependent bond dimension D(N) above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.