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arXiv:cond-mat/9712040 · Physical Review Letters

Scaling and Universality in the Anisotropic Kondo Model and the Dissipative Two-State System

Theo A. Costi

Published Feb 2, 1998• 70 citations
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Abstract

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.

DOI
10.1103/physrevlett.80.1038

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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\ensuremathα(T){C}_{\ensuremath{\alpha}}(T)C\ensuremathα​(T), static susceptibility \ensuremathχ\ensuremathα(T){\ensuremath{\chi}}_{\ensuremath{\alpha}}(T)\ensuremathχ\ensuremathα​(T), and spin relaxation function S\ensuremathα(\ensuremathω){S}_{\ensuremath{\alpha}}(\ensuremath{\omega})S\ensuremathα​(\ensuremathω), depending on the reduced temperature T/\ensuremathΔrT/{\ensuremath{\Delta}}_{r}T/\ensuremathΔr​ (frequency \ensuremathω/\ensuremathΔr\ensuremath{\omega}/{\ensuremath{\Delta}}_{r}\ensuremathω/\ensuremathΔr​), with \ensuremathΔr{\ensuremath{\Delta}}_{r}\ensuremathΔr​ the renormalized tunneling frequency, and uniquely specified by the dissipation strength \ensuremathα\ensuremath{\alpha}\ensuremathα ( 0<\ensuremathα<10<\ensuremath{\alpha}<10<\ensuremathα<1). The scaling functions can be used to extract \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathΔr{\ensuremath{\Delta}}_{r}\ensuremathΔr​ in experimental realizations.
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.
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.
We study the entanglement between a qubit and its environment from the spin-boson model with Ohmic dissipation. Through a mapping to the anisotropic Kondo model, we derive the entropy of entanglement of the spin $E(\ensuremath{\alpha},\ensuremath{\Delta},h)$, where $\ensuremath{\alpha}$ is the dissipation strength, $\ensuremath{\Delta}$ is the tunneling amplitude between qubit states, and hhh is the level asymmetry. For $1\ensuremath{-}\ensuremath{\alpha}\ensuremath{\gg}\ensuremath{\Delta}/{\ensuremath{\omega}}_{c}$ and $(\ensuremath{\Delta},h)\ensuremath{\ll}{\ensuremath{\omega}}_{c}$, we show that the Kondo energy scale TK{T}_{K}TK​ controls the entanglement between the qubit and the bosonic environment (${\ensuremath{\omega}}_{c}$ is a high-energy cutoff). For $h\ensuremath{\ll}{T}_{K}$, the disentanglement proceeds as (h/TK)2(h/{T}_{K}{)}^{2}(h/TK​)2; for $h\ensuremath{\gg}{T}_{K}$, EEE vanishes as $({T}_{K}/h{)}^{2\ensuremath{-}2\ensuremath{\alpha}}$, up to a logarithmic correction. For a given hhh, the maximum entanglement occurs at a value of $\ensuremath{\alpha}$ which lies in the crossover regime $h\ensuremath{\sim}{T}_{K}$. We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise.
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.
Wilson's momentum shell renormalization group method is used to solve for the dynamics of the dissipative two-state system. We utilize the mapping of the spin-boson model onto the anisotropic Kondo model (AKM) and solve for the dynamics of the latter. We find that the AKM captures the physics of the dissipative two-state system for dissipation strength $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}4$ corresponding to $\ensuremath{\infty}\ensuremath{\ge}{J}_{\ensuremath{\Vert}}\ensuremath{\ge}\ensuremath{-}\ensuremath{\infty}$ in the AKM. The dynamics of the AKM shows a smooth crossover between two strong coupling regimes corresponding to weak and strong dissipation in the spin-boson model.