Ground state of a two-level system with phonon coupling

Abstract

The ground state of a two-level system coupled to a dispersionless phonon bath is studied using both a connected moments expansion and a truncated Lanczos tridiagonal scheme. We consider the spin-boson Hamiltonian, $\ifmmode \hat{H}\else \^{H}\fi{}=\ensuremath{-}{\ensuremath{\delta}}_{o}{\ensuremath{\sigma}}_{x}+{\ensuremath{\sum}}_{\mathbf{k}}\ensuremath{\Elzxh}{\ensuremath{\omega}}_{\mathbf{k}}{a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\mathbf{k}}+{\ensuremath{\sum}}_{\mathbf{k}}{g}_{\mathbf{k}}{(a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{+a}_{\mathbf{k}}){\ensuremath{\sigma}}_{z},$ where ${\ensuremath{\delta}}_{0}$ is the bare tunneling matrix element and ${g}_{\mathbf{k}}$ represents the coupling to the $\mathbf{k}$ phonon modes. Such systems have found relevance in applications to molecular polaron formation, exciton motion, and attenuation of sound in glasses. Our results are then compared to those of variational methods as well as an exact numerical diagonalization.