Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials

TL;DR

This paper introduces an exact mathematical transformation that maps complex quantum system-reservoir models to simpler one-dimensional chains, enabling efficient simulation and analysis of open quantum systems.

Contribution

It provides an exact unitary transformation using orthogonal polynomials to map system-reservoir Hamiltonians to nearest-neighbour chain models with computable parameters.

Findings

Derived relations between chain parameters and orthogonal polynomial coefficients.

Proved properties of the chain system for various spectral functions.

Demonstrated applications to physical systems with analytic chain expressions.

Abstract

By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbour interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation, and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions, and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short range interactions of the effective chain system permits these open quantum systems to be efficiently simulated by the density matrix renormalization group methods.