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.

Anderson localization on the Bethe lattice using cages and the Wegner flow

“…We use flows to (almost) diagonalize a potentially dense matrix while keeping track of how much level repulsion has occured between the eigenstates associated with each pair of graph sites. As discussed in [29], one can monitor the level repulsion between two eigenvalues over the course of the flow using the Ξ metric defined elementwise, for a real Hamiltonian, as…”

“…We utilized a high-efficiency stabilized third-order integrator to numerically implement the Wegner flow as detailed in [29]. By "third-order", we mean the accumulated error should scale approximately like the inverse cube of the number of unitary steps taken for a given τ max and error tolerance.…”

“…The flows were performed out to a hypothetical time of τ max = 2000, which, according to the heuristics in [29], should in general be sufficient to resolve at least 99% of a coupling with a lever spacing of at least 0.024. At this point, the Hamiltonian is almost entirely diagonalized.…”

“…For more details, see section IV. [26][27][28][29] Of course, the infinite Bethe lattice can not be simulated on a finite computer. As discussed in section III, many finite analogs of the Bethe lattice have been used in previous research.…”

Dynamics of disordered quantum systems using flow equations

“…Thirdly, the computational aspects of flow equations as applied to many-body systems have not historically received a great deal of attention (with notable exceptions, e.g. [32]): further developments and optimisations in coding and applying continuous unitary transforms could pave the way to a more widespread adoption of this method.…”