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Physical Review Letters

Ground-State Properties in the Kondo Problem

Victor J. Emery, Alan H. Luther

Published Jun 21, 1971• 27 citations
Publisher

Abstract

The equilibrium Kondo problem is solved using a spin-boson representation of the partition function. For antiferromagnetic coupling, a canonical transformation removes infrared singularities in the energy and magnetization series. At $T=0$ the moment vanishes, the energy is nonanalytic in ${J}_{\ensuremath{\perp}}$, the magnetization saturates at a field ${H}_{S}$ of order ${(\frac{{{J}_{\ensuremath{\perp}}}^{2}}{{J}_{\ensuremath{\parallel}}})}^{\frac{1}{2{J}_{\ensuremath{\parallel}}}}$, and the susceptibility $\ensuremath{\chi}$ is proportional to ${{H}_{S}}^{\ensuremath{-}1}$. ${J}_{\ensuremath{\parallel}}$ and ${J}_{\ensuremath{\perp}}$ are longitudinal and transverse exchange constants.) For ferromagnetic coupling the free moment is only slightly renormalized.

DOI
10.1103/physrevlett.26.1547

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The equilibrium Kondo problem is solved using a spin-boson representation of the partition function. For antiferromagnetic coupling, a canonical transformation removes infrared singularities in the energy and magnetization series. At T=0T=0T=0 the moment vanishes, the energy is nonanalytic in J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​, the magnetization saturates at a field HS{H}_{S}HS​ of order (J\ensuremath⊥2J\ensuremath∥)12J\ensuremath∥{(\frac{{{J}_{\ensuremath{\perp}}}^{2}}{{J}_{\ensuremath{\parallel}}})}^{\frac{1}{2{J}_{\ensuremath{\parallel}}}}(J\ensuremath∥​J\ensuremath⊥​2​)2J\ensuremath∥​1​, and the susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ is proportional to HS\ensuremath−1{{H}_{S}}^{\ensuremath{-}1}HS​\ensuremath−1. J\ensuremath∥{J}_{\ensuremath{\parallel}}J\ensuremath∥​ and J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ are longitudinal and transverse exchange constants.) For ferromagnetic coupling the free moment is only slightly renormalized.
By nonperturbatively solving the nonequilibrium Anderson two-impurity model with the hierarchical equations of motion approach, we report a robust ferromagnetic (FM) phase in series-coupled double quantum dots, which can suppress the antiferromagnetic (AFM) phase and dominate the phase diagram at finite bias and detuning energy in the strongly correlated limit. The FM exchange interaction origins from the passive parallel spin arrangement caused by the Pauli exclusion principle during the electrons transport. At very low temperature, the Kondo screening of the magnetic moment in the FM phase induces some nonequilibrium Kondo effects in magnetic susceptibility, spectral functions and current. In the weakly correlated limit, the AFM phase is found still stable, therefore, a magnetic-field-free internal control of spin states can be expected through the continuous FM-AFM phase transition.
We investigate the magnetic order and related strongly correlated effects in a one-dimensional Ising-Kondo lattice with transverse field. This model is the anisotropic limit of the conventional isotropic Kondo lattice model, in the sense that the itinerant electrons interact with the localized magnetic moments via only longitudinal Kondo exchange. Adopting the numerical density-matrix renormalization group method, we map out the ground-state phase diagram in various parameter spaces. Depending on the Kondo coupling and filling number, three distinct phases, including a metallic paramagnetic, a metallic ferromagnetic, and a gapped spin-density wave phase, are obtained. The spin-density wave is characterized by an ordering wave vector, which coincides with the nesting wave vector of the Fermi surface. This makes the corresponding magnetic transition a spin analog of the Peierls transition occurring in the one-dimensional metal. Moreover, by analyzing the momentum distribution function and charge correlation function, the conduction electrons are shown to behave like free spinless fermions in the ferromagnetic phase. We finally discuss the effect of the repulsive Hubbard interaction between conduction electrons. Our paper enriches the Kondo physics and deepens the current understanding of the heavy fermion compounds.
The bosonized Kondo Hamiltonian is obtained by comparing the long-time limit of the reduced grand partition function for the impurity spin. We analyze this bosonized Kondo Hamiltonian at zero temperature and derive an effective Hamiltonian describing the effective interaction between the conduction electrons via an impurity-spin scattering. The infrared divergences encountered in the conventional perturbation theory are caused by this growth effective coupling at the low-energy limit. With the Bogoliubov transformation, we have developed a perturbation approach for this effective Hamiltonian and investigated the nontrivial ferromagnetic-antiferromagnetic crossover of the impurity spin. The critical condition derived here is in agreement with the renormalization-group numerical result. In particular, the ground-state wave function and its excitation spectrum of the conduction electrons are also obtained. Moreover, this effective Hamiltonian may be mapped onto a modified quantum sine-Gordon model. Paralleling the renormalization-group theory of the quantum sine-Gordon model, we straightforwardly reproduce previous results and derive the higher-order terms and a new universal correction in the flow equations.