Particle-Hole Asymmetry in the Two-Impurity Kondo Model

Abstract

A modified numerical renormalization-group procedure, preserving the particle-hole asymmetry of the two-impurity Kondo model, calculates the susceptibility, over 13 decades of temperature, and lowtemperature specific heat. For large ferromagnetic RKKY couplings, a two-stage Kondo effect screens the triplet-correlated moments. For large antiferromagnetic couplings, the impurities develop singlet correlations and contribute Van Vleck terms to the low-temperature properties. For vanishing particlehole asymmetry, previously found critical behavior emerges at low temperatures. PACS numbers: 75.20.Hr, 75.30.Hx, 75.30.Mb, 75.40.Cx The two-impurity Kondo problem stands out in the theory of strongly correlated electrons. Its Hamiltonian has two energy scales: the conduction-electron mediated (RKKY [1]) interaction I, which tends to align the impurity spins, and the Kondo temperature T K , below which the conduction band tends to screen their magnetic moments [2,3]. These competing energies generate Fermiand non-Fermi-liquid behaviors [4-6], ferromagnetic and antiferromagnetic correlations, Curie and Van Vleck susceptibilities, one-and two-stage Kondo effects [7], and a strange sensitivity to particle-hole asymmetry [8][9][10][11]. All this richness surfaces in our susceptibility and lowtemperature specific-heat curves. Our unified picture accommodates every breakthrough in the long history of the model.A scaling analysis of the susceptibility predicted, among other features, a two-stage Kondo effect and a smooth transition from the Kondo regime (k B T K ¿ jIj) to the RKKY regime (jIj ¿ k B T K ) [7]. Years later, a numerical renormalization-group (NRG) diagonalization [12] of a somewhat different, particle-hole symmetric Hamiltonian discovered an unstable, non-Fermi-liquid fixed point separating the antiferromagnetic RKKY regime from the Kondo regime [5,9]. By the time the low-energy spectrum and its physical properties were calculated analytically [6,11], it had become apparent that only special models, which we call symmetrical, encompass this fixed point: The model Hamiltonian must remain invariant under a special particle-hole transformation [8][9][10][11]. Moreover, since this symmetry generally generates only ferromagnetic RKKY interactions, to allow tuning to the critical antiferromagnetic coupling the model must have an additional, adjustable impurity-impurity interaction [11]. Generally asymmetric [13] and lacking the added interaction, the conventional model displays only Fermi-liquid behavior [4].The critical properties understood, interest reverts to more general features. Our NRG diagonalization [14] of the (conventional) two-impurity Hamiltonian yields quantitatively reliable temperature-dependent uniform susceptibilities. The results fit the image of temperature-dependent screening in competition with impurity alignment [4,7], and, in particular, substantiate the conclusions of Jayaprakash, and Wilkins [7], from the antiferromagnetic formation of an impurity singlet to the ferromagnetic two-stage Kondo...

Numerical renormalization group study of two-channel three-impurity triangular clusters

“…(1), is particle-hole symmetric; in fact, it has a larger SU(2) isospin symmetry of which the particle-hole transformation symmetry is merely a subgroup 25,67,70 . We performed all calculations taking explicitly into account spin SU(2), isospin SU(2) and mirror Z 2 symmetry groups 7,12,20,25,97,98,99 . We have used the discretization scheme described in Ref.…”

Quantum phase transition of Ising-coupled Kondo impurities

“…Twoimpurity Kondo models with K ⊥ = K z have been widely studied. 12,13,14,15,16,17,18,19 As argued in Refs. 16,17 the resulting phase diagram depends on the presence or absence of particle-hole symmetry (which does, however, not modify the phase diagram for K ⊥ = 0 as pointed out above).…”

“…11 We will discuss different formulations and applications of our model in the body of the paper. Coupled impurities or two-level systems have been investigated in a number of papers, 12,13,14,15,16,17,18,19 where most attention has been focussed on the case of SU(2)-symmetric direct exchange coupling between the impurity spins, K S 1 · S 2 . Here, two different regimes are possible as function of the inter-impurity exchange K: The model has two phases: At small Jz and large Kz, the ground state is doubly degenerate and the two impurity spins are locked into a "frozen mini-domain".…”

Kondo versus indirect exchange: Role of lattice and actual range of RKKY interactions in real materials

“…Further QMC studies on two and three dimensional systems, with both a quadratic dispersion and a lattice, did not find any evidence of such a state. 51,54,55 Later analysis revealed that the existence of such critical point required the presence of a very particular kind of particle-hole symmetry, [56][57][58] which is realized in our problem when R is even. Our simulations have confirmed the QMC results, with a fast decay of the correlations, and the absence of anomalous behavior.…”