Numerical analysis of the dissipative two-state system with the density-matrix Hilbert-space-reduction algorithm

TL;DR

This paper uses a Hilbert-space-reduction algorithm combined with Lanczos diagonalization to analyze the ground state and dynamical properties of a dissipative two-state quantum system, providing detailed computational methodology and results.

Contribution

It applies a Hilbert-space-reduction scheme to efficiently study the dissipative two-state system's ground state and dynamical susceptibility, aligning with recent quantum Monte-Carlo findings.

Findings

Damping rate and oscillation frequency are obtained over a relevant frequency range.

Results agree with recent quantum Monte-Carlo studies.

Critical dissipation decreases with increased tunneling amplitude.

Abstract

Ground state of the dissipative two-state system is investigated by means of the Lanczos diagonalization method. We adopted the Hilbert-space-reduction scheme proposed by Zhang, Jeckelmann and White so as to reduce the overwhelming reservoir Hilbert space to being tractable in computers. Both the implementation of the algorithm and the precision applied for the present system are reported in detail. We evaluate the dynamical susceptibility (resolvent) with the continued-fraction-expansion formula. Through analysing the resolvent over a frequency range, whose range is often called `interesting' frequency, we obtain the damping rate and the oscillation frequency. Our results agree with those of a recent quantum Monte-Carlo study, which concludes that the critical dissipation from oscillatory to over-damped behavior decreases as the tunneling amplitude is strengthened.