Non-perturbative flow equations from continuous unitary transformations

TL;DR

This paper introduces a non-perturbative method using continuous unitary transformations to accurately map spectra and compute expectation values in many-body quantum systems, demonstrated on the Lipkin model.

Contribution

It develops a novel nonlinear PDE approach for flow equations that is non-perturbative and system-size controlled, enabling high-accuracy spectral and eigenstate analysis.

Findings

High-accuracy spectrum mapping for the Lipkin model

Universal function for large particle systems

Efficient computation of expectation values

Abstract

We use a novel parameterization of the flowing Hamiltonian to show that the flow equations based on continuous unitary transformations, as proposed by Wegner, can be implemented through a nonlinear partial differential equation involving one flow parameter and two system specific auxiliary variables. The implementation is non-perturbative as the partial differential equation involves a systematic expansion in fluctuations, controlled by the size of the system, rather than the coupling constant. The method is applied to the Lipkin model to construct a mapping which maps the non-interacting spectrum onto the interacting spectrum to a very high accuracy. This function is universal in the sense that the full spectrum for any (large) number of particles can be obtained from it. In a similar way expectation values for a large class of operators can be obtained, which also makes it possible to probe the stucture of the eigenstates.