Universal asymptotic behavior in flow equations of dissipative systems

TL;DR

This paper reveals a universal asymptotic behavior in flow equations of dissipative systems, independent of initial parameters, and introduces a stable, analytical approach for evaluating correlation functions in such systems.

Contribution

It demonstrates a universal attractor in flow equations for dissipative Hamiltonians, enabling more stable numerical evaluation and analytical insights into low-energy behavior.

Findings

Universal asymptotic behavior depends only on bath properties.

Flow equations possess a universal attractor.

Analytical low-energy correlation functions derived.

Abstract

Based on two dissipative models, universal asymptotic behavior of flow equations for Hamiltonians is found and discussed. Universal asymptotic behavior only depends on fundamental bath properties but not on initial system parameters, and the integro-differential equations possess an universal attractor. The asymptotic flow of the Hamiltonian can be characterized by a non-local differential equation which only depends on one parameter - independent of the dissipative system or truncation scheme. Since the fixed point Hamiltonian is trivial, the physical information is completely transferred to the transformation of the observables. This yields a more stable flow which is crucial for the numerical evaluation of correlation functions. Furthermore, the low energy behavior of correlation functions is determined analytically. The presented procedure can also be applied if relevant perturbations are present as is demonstrated by evaluating dynamical correlation functions for sub-Ohmic environments. It can further be generalized to other dissipative systems.