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Physical review. B, Condensed matter

Two-state system coupled to phonons: A renormalization-group analysis of the transition

Vincent Hakim, A. Muramatsu, F. Guinea

Published Jul 1, 1984• 54 citations
Publisher

Abstract

We present a systematic renormalization-group analysis of the transition occurring in a symmetric two-state system with a linear dissipative coupling to a heat bath. We define the theory as a double expansion in $\ensuremath{\Delta}$ (tunneling frequency) and $\ensuremath{\delta}$ (distance from the critical coupling to the heat bath at $\ensuremath{\Delta}=0$), following a similar strategy as in the sine-Gordon theory. We reproduce previous results and obtain higher-order terms and a new universal correction in the flow equations.

DOI
10.1103/physrevb.30.464

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We present a systematic renormalization-group analysis of the transition occurring in a symmetric two-state system with a linear dissipative coupling to a heat bath. We define the theory as a double expansion in \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ (tunneling frequency) and \ensuremathδ\ensuremath{\delta}\ensuremathδ (distance from the critical coupling to the heat bath at \ensuremathΔ=0\ensuremath{\Delta}=0\ensuremathΔ=0), following a similar strategy as in the sine-Gordon theory. We reproduce previous results and obtain higher-order terms and a new universal correction in the flow equations.
Analysis of a dissipative two-state system at zero temperature shows that the model Hamiltonian may be exactly reduced to a modified quantum sine-Gordon model, which describes the effective interactions between the low-frequency phonons under coupling with a tunneling system. Directly considering the infrared divergence encountered in the conventional perturbation treatment, we have developed a new perturbation approach for the effective Hamiltonian, and derived the exact critical conditions for the localization transition. In the critical regime, a gap will be opened near zero momentum in the elementary excitation spectrum of the low-frequency phonons, and the corresponding ground state wave function is found to be a pairing quasi-particle state, analogous to the BCS superconducting state.
The static properties of a dissipative two-state system are studied by perturbation, mapping to the quantum sine-Gordon model, and a variational approach. We find that its properties at zero temperature depend not only on the spectral density J(\ensuremath{\omega})\ensuremath{\sim}tsuml{\mathit{tsum}}_{\mathit{l}}tsuml​gl2{\mathit{g}}_{\mathit{l}}^{2}gl2​\ensuremath{\delta}(\ensuremath{\omega}-${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}$), but also on the explicit form of the coupling strength gl{\mathit{g}}_{\mathit{l}}gl​\ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}^{\ensuremath{\lambda}}$ (${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}$ is the phonon frequency). For weak coupling (\ensuremath{\lambda}\ensuremath{\ge}1), the effect of displacement of the phonon state is dominant; while for strong couplings (\ensuremath{\lambda}1), one must take account of both displacement and deformation. Displaced squeezed states are proposed as ground states of the bath under strong coupling,and we show that the localization transition of the Ohmic dissipation occurs at ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$=3-2\ensuremath{\lambda} instead of ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$=1 as for the weak-coupling case.
The decoherence of a two-state system coupled with a sub-Ohmic bath is investigated theoretically by means of the perturbation approach based on a unitary transformation. It is shown that the decoherence depends strongly and sensitively on the structure of environment. Nonadiabatic effect is treated through the introduction of a function ${\ensuremath{\xi}}_{k}$ which depends on the boson frequency and renormalized tunneling. The results are as follows: (i) the nonequilibrium correlation function P(t)P(t)P(t), the dynamical susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$, and the equilibrium correlation function C(t)C(t)C(t) are analytically obtained for $s\ensuremath{\leqslant}1$; (ii) the phase diagram of thermodynamic transition shows the delocalized-localized transition point ${\ensuremath{\alpha}}_{l}$ which agrees with exact results and numerical data from the numerical renormalization group; (iii) the dynamical transition point ${\ensuremath{\alpha}}_{c}$ between coherent and incoherent phase is explicitly given. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the coherent dynamics disappear); (iv) the Shiba's relation and sum rule are exactly satisfied when $\ensuremath{\alpha}\ensuremath{\leqslant}{\ensuremath{\alpha}}_{c}$; (v) an underdamping-overdamping transition point ${\ensuremath{\alpha}}_{c}^{*}$ exists in the function $S(\ensuremath{\omega})$. Consequently, the dynamical phase diagrams in both ohmic and sub-Ohmic case are mapped out. For $\ensuremath{\Delta}⪡{\ensuremath{\omega}}_{c}$, the critical couplings (${\ensuremath{\alpha}}_{l},{\ensuremath{\alpha}}_{c}$, and ${\ensuremath{\alpha}}_{c}^{*}$) are proportional to ${\ensuremath{\Delta}}^{1\ensuremath{-}s}$.
We present a systematic analysis of the ground state properties of a tunneling system coupled to an Ohmic bath. The nonadiabatic effects induced by coupling with tunneling particle, on the ground state properties of phonons are taken into account. For this we have considered both displacement and deformation of phonons via a variational treatment. The symmetry breaking transition, or the delocalization-localization transition, of the tunneling particle is presented and is compared with earlier known results. We also show that in the critical regime a gap will be opened near the zero momentum in the spectrum of the phonons. The modified phonon spectrum along with the phonon displacement field correlation functions have been calculated. It is shown that in the critical regime the tunneling system stabilizes the lattice against long wavelength fluctuations.