Displacement Transformations as a Tool to Study Many-Body Localization

Pablo Serna,Miguel Ortuño,Andrés M Somoza

doi:10.3389/fphy.2021.767001

Abstract

We obtain eigenstates of interacting disorder Hamiltonians using unitary displacement transformations that rotate the state of the system. The method generates excited states if the strength of these transformations is chosen to optimize the energy, while decreasing the energy variance. We apply the method to analyse the localization properties of one-dimensional spinless fermions with short range interactions, reaching relatively large system sizes. We quantify the degree of localization through the size and disorder dependence of the inverse participation ratio.

Highlights

In strongly disordered interacting systems, particles are localized, a phenomenon known as manybody localization (MBL) [1,2,3,4,5,6]
In a previous publication [14], two of us together with Louk Rademaker improved the numerical efficiency of the displacement transformations technique by focusing on a specific reference state and considering only those transformations that affect the energy of this state
We focus on the effect of the most influential displacement transformations to gain information on the localization properties of the state

Summary

INTRODUCTION

In strongly disordered interacting systems, particles are localized, a phenomenon known as manybody localization (MBL) [1,2,3,4,5,6]. A practical computational approach to obtain the IOM and diagonalize interacting fermionic Hamiltonians was developed with the help of displacement transformations [11, 12], a type of unitary transformation (see Eq 2). The flow equation method [15,16,17,18,19,20,21] is a continuous versions of our model, in which a set of differential equations is solved to implement the displacement transformations. Discrete transformations can handle better the storage of the states, which can be done in a sparse representation. This opens the possibility of reaching larger system sizes.

MODEL HAMILTONIAN AND DISPLACEMENT TRANSFORMATIONS

ENERGY MINIMIZATION

EXCITED STATES

LOCALIZATION

DISCUSSION AND OUTLOOK

DATA AVAILABILITY STATEMENT