Critical properties of dissipative quantum spin systems in finite dimensions

TL;DR

This paper analytically investigates the critical properties of finite-dimensional dissipative quantum spin systems, revealing phase transition characteristics and critical exponents through a spherical model approach, supported by simulations and RG calculations.

Contribution

It introduces a spherical model approach to analyze dissipative quantum spin systems, connecting static critical exponents to classical models in higher dimensions.

Findings

Critical exponents match those of classical spherical models in d+2 dimensions.

Dynamical exponent z=2 for dissipative quantum spin systems.

Results align with Monte Carlo and renormalization group studies.

Abstract

We study the critical properties of finite-dimensional dissipative quantum spin systems with uniform ferromagnetic interactions. Starting from the transverse-field Ising model coupled to a bath of harmonic oscillators with Ohmic spectral density, we generalize its classical representation to classical spin systems with $O(n)$ symmetry and then take the large-$n$ limit to reduce the system to the spherical model. The exact solution to the resulting spherical model with long-range interactions along the imaginary-time axis shows a phase transition with static critical exponents coinciding with those of the conventional short-range spherical model in $d+2$ dimensions, where $d$ is the spatial dimensionality of the original quantum system. This implies the dynamical exponent to be $z=2$. These conclusions are consistent with the results of Monte Carlo simulations and renormalization group calculations for dissipative transverse-field Ising and $O(n)$ models in one and two dimensions. The present approach therefore serves as a useful tool to analytically investigate the properties of quantum phase transitions of the dissipative transverse-field Ising and related models. Our method may also offer a platform to study more complex phase transitions in dissipative finite-dimensional quantum spin systems, which recently receive renewed interest under the context of quantum annealing in a noisy environment.