Finite density matrix renormalization group algorithm for anyonic systems

Abstract

The numerical study of anyonic systems is known to be highly challenging due to their nonbosonic, nonfermionic particle-exchange statistics, and with the exception of certain models for which analytical solutions exist, very little is known about their collective behavior as a result. Meanwhile, the density matrix renormalization 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 in terms of both 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 time-evolving block decimation algorithm and superior to the variationally optimized anyonic multiscale renormalization ansatz, at far lesser computational cost.