Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

Thorsten B. Wahl,Arijeet Pal,Steven H. Simon

doi:10.1103/physrevx.7.021018

Abstract

We propose a tensor network encoding the set of all eigenstates of a fully many-body localized system in one dimension. Our construction, conceptually based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built from two layers of unitary matrices which act on blocks of $\ell$ contiguous sites. We argue this yields an exponential reduction in computational time and memory requirement as compared to all previous approaches for finding a representation of the complete eigenspectrum of large many-body localized systems with a given accuracy. Concretely, we optimize the unitaries by minimizing the magnitude of the commutator of the approximate integrals of motion and the Hamiltonian, which can be done in a local fashion. This further reduces the computational complexity of the tensor networks arising in the minimization process compared to previous work. We test the accuracy of our method by comparing the approximate energy spectrum to exact diagonalization results for the random field Heisenberg model on 16 sites. We find that the technique is highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the predicted dynamical phase transition. To demonstrate the power of our technique, we study a system of 72 sites and we are able to see clear signatures of the phase transition. Our work opens a new avenue to study properties of the many-body localization transition in large systems.

Highlights

Many-body localization (MBL), a phenomenon conjectured by Anderson in 1958 for disordered, interacting quantum particles [1], occurs in an isolated quantum system when it fails to reach thermal equilibrium
In conventional tensor network states, symmetries of the model can be imposed on the individual tensors [54,55]: Any tensor network state that is invariant under a symmetry can be written as a tensor network state, where all its individual tensors form a projective representation of the corresponding symmetry group; that is, they are invariant up to a phase under the action of the symmetry
Besides improving upon the tensor network ansatz proposed in Ref. [48], we optimize the network by minimizing a different figure of merit given by the magnitude of the commutator of the Hamiltonian and the approximate qLIOMs produced by the tensor network ansatz

Summary

Introduction

Many-body localization (MBL), a phenomenon conjectured by Anderson in 1958 for disordered, interacting quantum particles [1], occurs in an isolated quantum system when it fails to reach thermal equilibrium. It was shown to exist within perturbation theory for short-ranged interacting models with sufficiently strong disorder for states even at a finite energy density [2,3]. As opposed to a thermalizing system where the eigenstates exhibit volume law entanglement and satisfy the eigenstate-thermalization hypothesis (ETH) [6,7], for a one-dimensional system exhibiting FMBL, all the eigenstates of the Hamiltonian are expected to obey an area law [8,9]. Topological and symmetry-breaking orders, which are destroyed by thermal fluctuations at equilibrium, can be extended to highly excited states at a finite energy density because of MBL [11,12,13,14,15].

Objectives

Results

Conclusion