Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems

We describe a numerical algorithm for approximating the equilibrium-reduced density matrix and the effective (mean force) Hamiltonian for a set of system spins coupled strongly to a set of bath spins when the total system (system + bath) is held in canonical thermal equilibrium by weak coupling with a super-bath. Our approach is a generalization of now standard typicality algorithms for computing the quantum expectation value of observables of bare quantum systems via trace estimators and Krylov subspace methods. In particular, our algorithm makes use of the fact that the reduced system density, when the bath is measured in a given random state, tends to concentrate about the corresponding thermodynamic averaged reduced system density. Theoretical error analysis and numerical experiments are given to validate the accuracy of our algorithm. Further numerical experiments demonstrate the potential of our approach for applications including the study of quantum phase transitions and entanglement entropy for long range interaction systems. Published under an exclusive license by AIP Publishing.

Automated summary

In this paper, a numerical algorithm is introduced for approximating the equilibrium-reduced density matrix and the effective Hamiltonian for a system of strongly coupled system spins and bath spins. The algorithm is a generalization of typicality algorithms and uses the tendency of the reduced system density to concentrate about the corresponding thermodynamic averaged density. The accuracy of the algorithm is validated through theoretical error analysis and numerical experiments, and its potential for applications such as studying quantum phase transitions and entanglement entropy for long range interaction systems is demonstrated.