Stable Unitary Integrators for the Numerical Implementation of Continuous Unitary Transformations

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.

13 pages, 4 figures, Comments welcome

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Cited by in corpus (12)

• Time Evolution of Many-Body Localized Systems with the Flow Equation Approach
• Strong Disorder RG approach - a short review of recent developments
• Anderson Localization on the Bethe Lattice using Cages and the Wegner Flow
• Quasi-Many-Body Localization of Interacting Fermions with Long-Range Couplings
• Local Integrals of Motion in Quasiperiodic Many-Body Localized Systems
• Dynamics of Disordered Quantum Systems Using Flow Equations
• Dissipative flow equations
• Scaling theory for Mott-Hubbard transitions
• Holographic unitary renormalization group for correlated electrons -- I: a tensor network approach
• Pymablock: an algorithm and a package for quasi-degenerate perturbation theory
• Construction of Many-Body Eigenstates with Displacement Transformations
• Holographic entanglement renormalisation for fermionic quantum matter