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Physical Review Letters

Exact, asymptotic, three-dimensional, space- and time-dependent, Green’s functions in the multichannel Kondo effect

Andreas W. W. Ludwig, Ian K. Affleck

Published Nov 25, 1991• 197 citations
Publisher

Abstract

Exact 3D space- and time-dependent Green's functions are computed for the multichannel Kondo effect in the universal large-distance, long-time regime, using conformal field theory. They exhibit crossover from Fermi-liquid form far away, to singular non-Fermi-liquid behavior, near the impurity, where charge, spin, and ``flavor'' degrees of freedom become ``deconfined.'' Explicit expressions are given for the resistivity, \ensuremath{\rho}, for all k and for the spin-density and pair-density Green's functions, for k=2. For k=2, \ensuremath{\rho}(T)-\ensuremath{\rho}(0)\ensuremath{\propto}${\mathit{T}}^{1/2}$ and the pair susceptibility \ensuremath{\propto}lnT.

DOI
10.1103/physrevlett.67.3160

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Exact 3D space- and time-dependent Green's functions are computed for the multichannel Kondo effect in the universal large-distance, long-time regime, using conformal field theory. They exhibit crossover from Fermi-liquid form far away, to singular non-Fermi-liquid behavior, near the impurity, where charge, spin, and flavor'' degrees of freedom become deconfined.'' Explicit expressions are given for the resistivity, \ensuremath{\rho}, for all k and for the spin-density and pair-density Green's functions, for k=2. For k=2, \ensuremath{\rho}(T)-\ensuremath{\rho}(0)\ensuremath{\propto}T1/2{\mathit{T}}^{1/2}T1/2 and the pair susceptibility \ensuremath{\propto}lnT.
We present an analytical solution of a SU(N)\ifmmode\times\else\texttimes\fi{}SU(M) generalization of the multichannel single-impurity Kondo model in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, M\ensuremath{\rightarrow}\ensuremath{\infty}, with \ensuremath{\gamma}=M/N fixed. Non-Fermi-liquid behavior of the single electron Green function and of the local spin and flavor susceptibilities occurs in both regimes, N\ensuremath{\le}M and N>M, with leading critical exponents identical to those found in the conformal field theory solution for all N and M (with M\ensuremath{\ge}2). We explain this remarkable agreement and connect it to ``spin-flavor separation,'' the essential feature of the non-Fermi-liquid, fixed point.
The multichannel Kondo model with SU(N)\mathrm{SU}(N)SU(N) spin symmetry and SU(K)\mathrm{SU}(K)SU(K) channel symmetry is considered. The impurity spin is chosen to transform as an antisymmetric representation of SU(N),\mathrm{SU}(N),SU(N), corresponding to a fixed number of Abrikosov fermions ${\ensuremath{\sum}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}^{\ifmmode\dagger\else\textdagger\fi{}}{f}_{\ensuremath{\alpha}}=Q.$ For more than one channel (K>1),(K>1),(K>1), and all values of NNN and Q,Q,Q, the model displays non-Fermi behavior associated with the overscreening of the impurity spin. Universal low-temperature thermodynamic and transport properties of this non-Fermi-liquid state are computed using conformal field theory methods. A large-NNN limit of the model is then considered, in which $K/N\ensuremath{\equiv}\ensuremath{\gamma}$ and $Q/N\ensuremath{\equiv}{q}_{0}$ are held fixed. Spectral densities satisfy coupled integral equations in this limit, corresponding to a (time-dependent) saddle point. A low-frequency, low-temperature analysis of these equations reveals universal scaling properties in the variable $\ensuremath{\omega}/T,$ in agreement with conformal invariance. The universal scaling form is obtained analytically and used to compute the low-temperature universal properties of the model in the large-NNN limit, such as the T=0T=0T=0 residual entropy and residual resistivity, and the critical exponents associated with the specific heat and susceptibility. The connections with the ``noncrossing approximation'' and the previous work of Cox and Ruckenstein are discussed.
A conformal-field-theory approach has recently been developed for quantum-impurity problems including the ``overscreened'' multichannel Kondo model. We present the details of our calculation of the single-fermion Green's function. The universal zero-temperature resistivity and leading temperature-dependent term are derived. Effects of particle-hole symmetry breaking are included. We also give our perturbative proof of the ``g theorem'' governing the change in residual entropy under a renormalization-group flow between two boundary fixed points.
A new, general approach to finding exact solutions of Quantum Impurity Problems is reviewed, in which a local quantum mechanical degree of freedom is coupled to an extended variable, having gapless excitations. The principle ideas, using Conformal Field Theory techniques, are demonstrated in the example of the Multi-Channel Kondo effect, which exhibits unusual non-Fermi-liquid behavior at low temperatures and which appears to be relevant for Heavy Fermion Materials. In particular, static and dynamic Green's functions as well as transport and pairing properties are calculated exactly at low temperatures. Quantum Impurity Problems provide realizations of non-trivial, fully interacting Conformal Field Theories.
Symmetry-breaking perturbations destabilize the critical points of the two-channel and two-impurity Kondo models, thereby leading to a crossover from non-Fermi liquid behavior to standard Fermi liquid physics. Here we use an analogy between this crossover and one occurring in the boundary Ising model to calculate the full crossover Green function analytically. In remarkable agreement with our numerical renormalization group calculations, the single exact function applies for an arbitrary mixture of the relevant perturbations in each model. This rich behavior resulting from finite channel asymmetry, interlead charge transfer, and/or magnetic field should be observable in quantum dot or tunneling experiments.