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arXiv:cond-mat/9903104 · Physical review. B, Condensed matter

Thermodynamics of the dissipative two-state system: A Bethe-ansatz study

Theo A. Costi, Gergely Zaránd

Published May 15, 1999• 55 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.59.12398

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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\ensuremathα(T){C}_{\ensuremath{\alpha}}(T)C\ensuremathα​(T) and static dielectric susceptibility \ensuremathχ\ensuremathα(T){\ensuremath{\chi}}_{\ensuremath{\alpha}}(T)\ensuremathχ\ensuremathα​(T) are extracted for all dissipation strengths 0<\ensuremathα<1.0<\ensuremath{\alpha}<1.0<\ensuremathα<1. The logarithmic corrections to these quantities at high temperatures are found in the Kondo limit \ensuremathα\ensuremath→1\ensuremath−,\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}{1}^{\ensuremath{-}},\ensuremathα\ensuremath→​1\ensuremath−, whereas for \ensuremathα<1\ensuremath{\alpha}<1\ensuremathα<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.
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.
We construct and solve numerically the thermodynamic Bethe-ansatz equations for the spin-anisotropic two-channel Kondo model in arbitrary external field h. At high temperatures the specific heat and the susceptibility show power-law dependence. For $\stackrel{\ensuremath{\rightarrow}}{h}0$ and at temperatures below the Kondo temperature TK{T}_{K}TK​ a two-channel Kondo effect develops characterized by a Wilson ratio 8/3,8/3,8/3, and a logarithmic divergence of the susceptibility and the linear specific-heat coefficient. A finite magnetic field, h>0,h>0,h>0, drives the system to a Fermi-liquid fixed point with an unusual Wilson ratio which depends sensitively on h.
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 thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic S=12S=\frac{1}{2}S=21​ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on TTT for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.