Quantum phase transitions in the sub-ohmic spin-boson model: Failure of the quantum-classical mapping

TL;DR

This paper demonstrates that the quantum-classical mapping fails for the sub-ohmic spin-boson model, revealing a non-trivial quantum phase transition not captured by classical analogs due to long-range interactions in imaginary time.

Contribution

It proves the failure of the quantum-classical mapping in the sub-ohmic spin-boson model and characterizes its quantum phase transition using epsilon expansion and numerical methods.

Findings

Quantum transition controlled by an interacting fixed point at small s

Classical long-range Ising model exhibits mean-field behavior for 0 < s < 1/2

Failure of quantum-classical mapping due to long-range imaginary time interactions

Abstract

The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that such a mapping fails for the sub-ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) ~ omega^s. Using an epsilon expansion we prove that this model has a quantum transition controlled by an interacting fixed point at small s, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to have an upper-critical dimension at s = 1/2, with mean-field transition behavior controlled by a non-interacting fixed point for 0 < s < 1/2. The failure of the quantum-classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum model.