Emery-Kivelson Line and Universality of Wilson Ratio of Spin Anisotropic Kondo Model.

Yuval-Anderson’s scaling analysis and Affleck-Ludwig’s Conformal Field Theory approach are applied to the k channel spin anisotropic Kondo model. Detailed comparisons with the available Emery-Kivelson’s Abelian Bosonization approaches are made. It is shown that the EK line exists for any k, although it can be mapped to free fermions only when k = 1 or 2. The Wilson ratio is universal if k = 1 or 2, but not universal if k > 2. The leading low temperature correction to the electron resistivity is not affected by the spin anisotropy for any k. A new universal ratio for k > 2 is proposed to compare with experiments. Typeset using REVTEX 1 In the general multichannel Kondo model, a magnetic impurity with spin s couples to k degenerate bands of spin 1/2 conduction electrons by Heisenberg exchange interaction. When k ≤ 2s (underscreening and complete-screening), the electron-impurity system can be described by a local Fermi liquid at very low temperatures. However, when k > 2s (overscreening ), Noziéres and Blandin [1] showed that the low temperature physics, being controlled by a non-trivial zero temperature fixed point, is not described by Fermi liquid behavior. Bethe ansatz [2], Boundary conformal field theory (BCFT) [3], Numerical renormalization group (NRG) [4], Bosonization [5] etc. have been utilized to investigate the nature of this non-fermi liquid fixed point. By using BCFT approach, Affleck and Ludwig (AL) identified the non-trivial fixed point symmetry as Affine Kac-Moody algebra ŜUk(2)× ŜU 2(k)×U(1). Near the fixed point, they classified all the possible perturbations according to the representation theory of the underlying KM algebra at the fixed point. AL only considered isotropic case, although they showed [4] that spin anisotropy is always irrelevant for s = 1/2 in the sense that no relevant operators will appear by allowing spin anisotropy . BCFT is very elegant and applicable for any channel. The symmetry is also explicitly demonstrated in this approach. But the relation between the boundary operators of BCFT and the original scaling variables is not transparent and it is hard to apply CFT near the weak coupling fixed line. Emery and Kivelson (EK) [5,6], using Abelian Bosonization, found an alternative solution to k = 2, s = 1/2 anisotropic Kondo model. By using a canonical transformation U = ezs and refermionization, they located a exactly solvable line (EK line) which is analogous to the Toulouse line [7] for the ordinary single channel Kondo model. They also found one leading irrelevant operator Sz∂xΦs away from the EK line. By doing perturbative calculations around the EK line with this operator, they recovered the generic low temperature behaviors of the impurity specific heat and susceptibility. Sengupta and Georges [10], using EK’s method, calculated the Wilson ratio independently. Later EK’s approach was extended to the 4 channel case [9] and applied to the electron assisted tunneling of a heavy particle between two sites in a metal [11]. EK’s approach can be applied easily, but it is lim2 ited to special values of k. The symmetry is hidden and there is no systematic classification of all the possible operators in this approach. The two channel Kondo model has been argued to describe uranium based heavy fermion systems and tunneling of a non-magnetic impurity between two sites in a metal [12]. k = 1, 2 and k = 3 magnetic Kondo models have also been proposed to interpret Ce based alloys [13]. It is believed that the general spin anisotropy of the form in Eq.1 is ubiquitous in all the experimental systems mentioned above, therefore it is important to investigate its effects in detail. In this paper, fixing the impurity spin s = 1/2, we extend the scaling analysis of YA near the weak coupling fixed line to the multi-channel Kondo model and AL’s approach near the intermediate coupling fixed point to the spin anisotropic Kondo model. We also make detailed comparisons among YA’s, AL’s and EK’s approaches. We find 1-1 mapping between the boundary operators in CFT and those in EK’s solution, therefore establish the relation between the scaling parameters in the two approaches. We define the EK line as a special line in parameter space where the impurity part of Kondo system completely decouples from the uniform external magnetic field, hence χimp ≡ 0 [14]. The EK line defined in this way coincides with the conventional EK line if k = 2 and the decoupling line [6] if k = 1. We show that the EK line is a natural property for any k channel Kondo model, although this line can be mapped to free theory only when k = 1, 2 [15]. The reason that only one leading irrelevant operator was found in EK’s approach to 2 (1) channel Kondo model is due to first order (second order) KM null states [17], therefore the Wilson ratio is universal for the general spin anisotropic 2 (1) channel-Kondo model Eq.1. However, the Wilson ratio is not universal for k > 2 due to two or more leading irrelevant operators with the same scaling dimension. It is shown that the general spin anisotropy does not affect the leading low temperature correction to the electron resistivity for any k. A new universal ratio for k > 2 is proposed to compare with experiments. The three-dimensional Kondo problem can be mapped to a one dimensional problem [3]. Following closely the notation of AL, we write the following k channel general spin 3 anisotropic Kondo Hamiltonian in a uniform magnetic field