Non-degenerate Ground State of the Spin-Boson Model under Abelian Diagonalization

TL;DR

This paper rigorously establishes that the ground state of the spin-boson model is non-degenerate and has a definite parity, providing analytical conditions for parity breaking and aligning with numerical phase diagrams.

Contribution

The authors derive necessary and sufficient conditions for ground state parity in the spin-boson model using unitary transformations and matrix algebra, offering analytical insights into parity breaking.

Findings

Ground state energy is lower than degenerate energies and has definite parity.

Analytical expression for parity-breaking critical value matches numerical phase diagrams.

The phase diagram does not fully characterize the ground state of the system.

Abstract

By utilizing a unitary transformation, we derive the necessary and sufficient conditions for the degeneracy between the even- and odd-parity energy states of the spin-boson model (SBM). Employing the Rayleigh quotient of matrix algebra, we rigorously prove that the ground state energy of the SBM is lower than the systems lowest possible degenerate energy and possesses a definite parity. Based on the necessary and sufficient conditions for parity breaking, we provide an analytical expression for the parity-breaking critical value, which is closely related to the expansion order and computational accuracy. This expression reproduces the SBM phase diagram obtained by quantum Monte Carlo (QMC) and logarithmic discretization numerical renormalization group (NRG) methods. However, this phase diagram does not characterize the ground state of the system.