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Physical Review Letters

Universality in the dissipative two-state system

Maura Sassetti, Ulrich Weiß

Published Oct 29, 1990• 75 citations
Publisher

Abstract

The thermodynamics and dynamics of the spin-boson model are formulated in terms of the exact formal solutions. The low-temperature features of the free energy, the static susceptibility, and the equilibrium correlation function are presented for Ohmic and super-Ohmic dissipation, and various universal properties are deduced. In the Ohmic case the Wilson ratio and the spectral properties at low frequencies (Shiba's relation) are calculation in the range 01 of the coupling strength. The corresponding universal relations are determined also for super-Ohmic dissipation.

DOI
10.1103/physrevlett.65.2262

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The thermodynamics and dynamics of the spin-boson model are formulated in terms of the exact formal solutions. The low-temperature features of the free energy, the static susceptibility, and the equilibrium correlation function are presented for Ohmic and super-Ohmic dissipation, and various universal properties are deduced. In the Ohmic case the Wilson ratio and the spectral properties at low frequencies (Shiba's relation) are calculation in the range 01 of the coupling strength. The corresponding universal relations are determined also for super-Ohmic dissipation.
The thermodynamics of the dissipative two-state system is calculated exactly for all temperatures and level asymmetries for the case of Ohmic dissipation. We exploit the equivalence of the two-state system to the anisotropic Kondo model and extract the thermodynamics of the former by solving the thermodynamic Bethe-ansatz equations of the latter. The universal scaling functions for the specific heat ${C}_{\ensuremath{\alpha}}(T)$ and static dielectric susceptibility ${\ensuremath{\chi}}_{\ensuremath{\alpha}}(T)$ are extracted for all dissipation strengths $0<\ensuremath{\alpha}<1.$ The logarithmic corrections to these quantities at high temperatures are found in the Kondo limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}{1}^{\ensuremath{-}},$ whereas for $\ensuremath{\alpha}<1$ we find the expected power law temperature dependences with the powers being functions of the dissipative coupling \ensuremath{\alpha}. The low-temperature behavior is always that of a Fermi liquid.
Using a real-time renormalization-group method we determine the complete dynamics of the spin-boson model with Ohmic dissipation for coupling strengths $\ensuremath{\alpha}\ensuremath{\lesssim}0.1--0.2.$ We calculate the relaxation and dephasing time, static susceptibility, and correlation functions. Our results are consistent with quantum Monte Carlo simulations and the Shiba relation. We present reliable results for finite cutoff and finite bias in a regime where perturbation theory in $\ensuremath{\alpha}$ or in tunneling breaks down. Furthermore, an unambiguous comparison to results from the Kondo model is achieved.
Scaling and universality in the Ohmic two-state system is investigated by exploiting the equivalence of this model to the anisotropic Kondo model. For the Ohmic two-state system, we find universal scaling functions for the specific heat ${C}_{\ensuremath{\alpha}}(T)$, static susceptibility ${\ensuremath{\chi}}_{\ensuremath{\alpha}}(T)$, and spin relaxation function ${S}_{\ensuremath{\alpha}}(\ensuremath{\omega})$, depending on the reduced temperature $T/{\ensuremath{\Delta}}_{r}$ (frequency $\ensuremath{\omega}/{\ensuremath{\Delta}}_{r}$), with ${\ensuremath{\Delta}}_{r}$ the renormalized tunneling frequency, and uniquely specified by the dissipation strength $\ensuremath{\alpha}$ ( $0<\ensuremath{\alpha}<1$). The scaling functions can be used to extract $\ensuremath{\alpha}$ and ${\ensuremath{\Delta}}_{r}$ in experimental realizations.
Abstract We introduce a new theoretical approach to dissipative quantum systems. By means of a continuous sequence of infinitesimal unitary transformations, we decouple the small quantum system that one is interested in from its thermodynamically large environment. This yields a trivial final transformed Hamiltonian. Dissipation enters through the observation that generically observables “decay” completely under these unitary transformations, i.e. are completely transformed into other terms. As a nontrivial example the spin‐boson model is discussed in some detail. For the super‐Ohmic bath we obtain a very satisfactory description of short, intermediate and long time scales at small temperatures. This can be tested from the generalized Shiba‐relation that is fulfilled within numerical errors.
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.
We present results for the dynamics of an impurity spin coupled to a magnetic field and to two ohmic baths that are out of equilibrium due to the application of a bias voltage. Both the nonequilibrium steady state and the rate constants describing the approach to the steady state are found to depend sensitively on the relative strengths of a magnetic field and a voltage dependent decoherence rate. Computation of physical quantities, including the frequency dependent ratio of the response to the correlation functions and the probabilities of the two spin states, allows the extraction of voltage dependent effective temperatures. The temperatures extracted from different quantities differ from one another in magnitude and in their dependence on parameters, and, in general, are nonmonotonic.