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arXiv:cond-mat/9409048 · Physical review. B, Condensed matter

Solution of the two-impurity Kondo model: Critical point, Fermi-liquid phase, and crossover

Junwu Gan

Published Apr 1, 1995• 52 citations
arXivHTMLPDF

Abstract

An asymptotically exact solution is presented for the two-impurity Kondo model for a finite region of the parameter space surrounding the critical point. This region is located in the most interesting intermediate regime where the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is comparable to the Kondo temperature. After several exact simplifications involving reduction to one dimension and Abelian bosonization, the critical point is explicitly identified, making clear its physical origin. By using controlled low-energy projection, an effective Hamiltonian is derived for the finite region in the phase diagram around the critical point. The completeness of the effective Hamiltonian is rigorously proved from general symmetry considerations. The effective Hamiltonian is solved exactly not only at the critical point but also for the surrounding Fermi-liquid phase. Analytic crossover functions from the critical to Fermi-liquid behavior are derived for the specific heat and staggered susceptibility. It is shown that applying a uniform magnetic field has negligible effect on the physical behavior inside our solution region. A detailed comparison is made with the numerical renormalization-group and conformal-field-theory results. The excellent agreement is exploited to argue for the universality of the critical point, which in turn implies universal behavior everywhere inside our solution region.

DOI
10.1103/physrevb.51.8287

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An asymptotically exact solution is presented for the two-impurity Kondo model for a finite region of the parameter space surrounding the critical point. This region is located in the most interesting intermediate regime where the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is comparable to the Kondo temperature. After several exact simplifications involving reduction to one dimension and Abelian bosonization, the critical point is explicitly identified, making clear its physical origin. By using controlled low-energy projection, an effective Hamiltonian is derived for the finite region in the phase diagram around the critical point. The completeness of the effective Hamiltonian is rigorously proved from general symmetry considerations. The effective Hamiltonian is solved exactly not only at the critical point but also for the surrounding Fermi-liquid phase. Analytic crossover functions from the critical to Fermi-liquid behavior are derived for the specific heat and staggered susceptibility. It is shown that applying a uniform magnetic field has negligible effect on the physical behavior inside our solution region. A detailed comparison is made with the numerical renormalization-group and conformal-field-theory results. The excellent agreement is exploited to argue for the universality of the critical point, which in turn implies universal behavior everywhere inside our solution region.
The two-impurity Kondo model with particle-hole symmetry is known to have a critical point at comparable RKKY interaction and Kondo temperature. Using bosonization and controlled low energy projection, we identify the critical point and map it to a solvable resonant level model. The physics of the critical point is shown to be the same as the two-channel Kondo model. An effective Hamiltonian is derived from which the solutions for the critical point and surrounding Fermi-liquid phase follow immediately. The universality of the critical point is strongly supported by the overall excellent agreement with the numerical renormalization group and conformal field theory results.
The problem of two magnetic impurities in a normal metal exposes the two opposite tendencies in the formation of a singlet ground state, driven respectively by the single-ion Kondo effect with conduction electrons to screen impurity spins or the Ruderman-Kittel-Kasuya-Yosida interaction between the two impurities to directly form impurity spin singlet. However, whether the competition between these two tendencies can lead to a quantum critical point has been debated over more than two decades. Here, we study this problem by applying the newly proposed natural orbitals renormalization group method to a lattice version of the two-impurity Kondo model with a direct exchange KKK between the two impurity spins. The method allows for unbiased access to the ground state wave functions and low-lying excitations for sufficiently large system sizes. We demonstrate the existence of a quantum critical point, characterized by the power-law divergence of impurity staggered susceptibility with critical exponent $\ensuremath{\gamma}=0.60(1)$, on the antiferromagnetic side of KKK when the interimpurity distance RRR is even lattice spacing, while a crossover behavior is recovered when RRR is odd lattice spacing. These results have ultimately resolved the long-standing discrepancy between the numerical renormalization group and quantum Monte Carlo studies, confirming a link of this two-impurity Kondo critical point to a hidden particle-hole symmetry predicted by the local Fermi liquid theory.
Numerical renormalization-group and conformal-field-theory work indicate that the two-impurity Kondo Hamiltonian has a non-Fermi-liquid critical point separating the Kondo-screening phase from the interimpurity singlet phase when particle-hole (P-H) symmetry is maintained. We clarify the circumstances under which this critical point occurs, pointing out that there are two types of P-H symmetry. Only one of them guarantees the occurrence of the critical point. Much of the previous numerical work was done on models with the other type of P-H symmetry. We analyze this critical point using the boundary conformal-field-theory technique. The finite-size spectrum is presented in detail and compared with about 50 energy levels obtained using the numerical renormalization group. Various Green's functions, general renormalization group behavior, and a hidden SO(7) are analyzed.
The two-impurity Kondo Hamiltonian is known to possess a multicritical point separating stable Fermi-liquid phases in which the two impurities either form a singlet or else are screened by the conduction electrons. We propose an exact critical theory for this non-Fermi-liquid multicritical point, using conformal field theory methods, which predicts all critical properties including the finite-size spectrum, specific heat, susceptibilities, renormalization-group behavior, residual entropy, scattering matrix, and local pair Green's function.
We propose that real-space properties of the two-impurity Kondo model can be obtained from an effective spin model where two single-impurity Kondo spin chains are joined via an Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the two impurity spins. We then use a density matrix renormalization group approach, valid in all ranges of parameters, to study its features using two complementary quantum-entanglement measures, the negativity and the von Neumann entropy. This nonperturbative approach enables us to uncover the precise dependence of the spatial extent ξ(K) of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an exponential suppression of the Kondo temperature T(K)~1/ξ(K) with the size of the effective impurity spin in the limit of large ferromagnetic RKKY coupling, a striking display of "Kondo resonance narrowing" in the two-impurity Kondo model. We also show how the antiferromagnetic RKKY interaction produces an effective decoupling of the impurities from the bulk already for intermediate strengths of this interaction, and, furthermore, exhibit how the non-Fermi liquid quantum critical point is signaled in the quantum entanglement between various parts of the system.
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.