Finite Density Matrix Renormalisation Group Algorithm for Anyonic Systems

TL;DR

This paper introduces a finite density matrix renormalisation group algorithm tailored for anyonic systems, enabling efficient ground state calculations and demonstrating superior performance on the Golden Chain benchmark compared to existing methods.

Contribution

The paper develops and implements a DMRG algorithm specifically for finite anyonic systems, optimizing computational scaling and demonstrating its effectiveness on a key benchmark system.

Findings

Accurately computes ground state energy of the Golden Chain

Achieves results comparable to or better than existing algorithms

Reduces computational cost significantly

Abstract

The numerical study of anyonic systems is known to be highly challenging due to their non-bosonic, non-fermionic particle exchange statistics, and with the exception of certain models for which analytical solutions exist, very little is known about their collective behaviour as a result. Meanwhile, the density matrix renormalisation group (DMRG) algorithm is an exceptionally powerful numerical technique for calculating the ground state of a low-dimensional lattice Hamiltonian, and has been applied to the study of bosonic, fermionic, and group-symmetric systems. The recent development of a tensor network formulation for anyonic systems opened up the possibility of studying these systems using algorithms such as DMRG, though this has proved challenging both in terms of programming complexity and computational cost. This paper presents the implementation of DMRG for finite anyonic systems, including a detailed scheme for the implementation of anyonic tensors with optimal scaling of computational cost. The anyonic DMRG algorithm is demonstrated by calculating the ground state energy of the Golden Chain, which has become the benchmark system for the numerical study of anyons, and is shown to produce results comparable to those of the anyonic TEBD algorithm and superior to the variationally optimised anyonic MERA, at far lesser computational cost.