Flow equations for Hamiltonians: Contrasting different approaches by using a numerically solvable model

TL;DR

This paper compares different flow equation approaches for Hamiltonians using a numerically solvable model, highlighting the importance of optimization and the challenges in operator flow truncation.

Contribution

It introduces a general truncation scheme for Hamiltonian and operator flow and provides the first explicit analysis of operator flow behavior near resonances.

Findings

A general truncation scheme yields good Hamiltonian flow results.

Operator flow truncation after linear or bilinear terms is insufficient near resonances.

High-order series expansion terms are significant close to resonances.

Abstract

To contrast different generators for flow equations for Hamiltonians and to discuss the dependence of physical quantities on unitarily equivalent, but effectively different initial Hamiltonians, a numerically solvable model is considered which is structurally similar to impurity models. By this we discuss the question of optimization for the first time. A general truncation scheme is established that produces good results for the Hamiltonian flow as well as for the operator flow. Nevertheless, it is also pointed out that a systematic and feasible scheme for the operator flow on the operator level is missing. For this, an explicit analysis of the operator flow is given for the first time. We observe that truncation of the series of the observable flow after the linear or bilinear terms does not yield satisfactory results for the entire parameter regime as - especially close to resonances - even high orders of the exact series expansion carry considerable weight.