Stable Unitary Integrators for the Numerical Implementation of Continuous Unitary Transformations

TL;DR

This paper develops specialized numerical integrators for continuous unitary transformations in quantum systems, addressing inefficiencies and unitarity loss in standard methods, and demonstrates their effectiveness through a many-body localization example.

Contribution

The paper introduces new integrators that efficiently implement continuous unitary transformations while preserving unitarity, improving upon standard methods.

Findings

New integrators improve efficiency in numerical flow equations.

The methods maintain unitarity during the transformation process.

Application to many-body localization demonstrates practical effectiveness.

Abstract

The technique of continuous unitary transformations has recently been used to provide physical insight into a diverse array of quantum mechanical systems. However, the question of how to best numerically implement the flow equations has received little attention. The most immediately apparent approach, using standard Runge-Kutta numerical integration algorithms, suffers from both severe inefficiency due to stiffness and the loss of unitarity. After reviewing the formalism of continuous unitary transformations and Wegner's original choice for the infinitesimal generator of the flow, we present a number of approaches to resolving these issues including a choice of generator which induces what we call the "uniform tangent decay flow" and three numerical integrators specifically designed to perform continuous unitary transformations efficiently while preserving the unitarity of flow. We conclude by applying one of the flow algorithms to a simple calculation that visually demonstrates the many-body localization transition.