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Journal of Physics C Solid State Physics

Some numerical results on the Kondo problem and the inverse square one-dimensional Ising model

Philip Warren Anderson, G. Yuval

Published Apr 15, 1971• 198 citations
Publisher

Abstract

The authors present some numerical results for the one dimensional Ising model with inverse-square long range interaction obtained by the use of the scaling technique developed in connection with the Kondo problem. Certain new remarks on the Kondo problem are also included.
DOI
10.1088/0022-3719/4/5/011

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We report some new results obtained for the thermodynamic properties and correlation functions of the one-dimensional Ising model with an inverse-square interaction between the spins. Using the Anderson and Yuval renormalization-group equations, we evaluate the correlation function and magnetic susceptibility for T<TcT<{T}_{c}T<Tc​. Monte Carlo data for the magnetization, specific heat, and spin-spin correlation function have also been obtained.
Abstract The previously developed n-vicinity method [3] allows us to calculate accurately critical values of inverse temperatures for Ising models with short-range interaction. We generalize the method to the case of long-range interactions in spin systems and obtain theoretical formulas for the inverse temperatures in terms of the spin interaction constants. The comparison of our theoretical estimates with computer simulations for the two- and three-dimensional Ising models shows that the larger the dimension of the problem the better their agreement.
To specify the difference between the one-dimensional Kondo problem and the three-dimensional Kondo problem, we consider a model which describes the interaction between one-dimensional free electrons and a magnetic impurity. Bethe ansatz exact results and scaling analysis show that if J F > -| J B |, the Kondo screening occurs. There is no ferromagnetic Kondo screening in such a system but the long range antiferromagnetic Kondo coupling is relevant. As soon as the Coulomb interaction is introduced, the crossovers between the scaling trajectories in the parameter region J F < -| J B | and the trajectories in other parameter regions arise and ferromagnetic Kondo screening may occur.
The critical behaviour of the one-dimensional Ising model with long-range ferromagnetic interactions decaying with distance r as 1/r 1+ sigma has been studied by scaling the range of interactions. Exact calculations have been done for a system with finite ranges up to the tenth neighbour for 0< sigma <or=1. Applying the range scaling, the critical temperature, critical exponent nu and the anomalous dimension of the order parameter have been calculated. Additional analysis of the convergence of the method has been performed by applying the Vanden Broeck and Schwartz extrapolation procedure in addition to the simple least-squares approximation and by evaluating the convergence exponents.
The authors present the results of the coherent anomaly method when applied to Ising models in one dimension with long-range interactions. This class of systems acts as an interesting and challenging test for the method because the critical exponents as well as the critical temperature values of which are given by the method, depend on the rate of fall-off of the interactions. Thus one can see how accurately the method gives results which correctly reflect this dependency. The results obtained from this method are compared with results obtained by a variety of other methods.