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Journal of Physics A Mathematical and General

Tunnelling of a large spin: mapping onto a particle problem

G. Scharf, Walter Felipe Wreszinski, van, J. Leo van Hemmen

Published Sep 11, 1987• 77 citations
Publisher

Abstract

The tunnelling of a single quantum spin is studied in the limit of large spin quantum number S. The problem is mapped onto a particle problem on the positive half-line with a Hamiltonian which is invariant under inversion x to 1/x. Not only the ground-state energy but also all the other energy levels and corresponding level splittings (if any) are computed by using the conventional WKB methods for the particle problem and an excellent agreement with numerical data is found.

DOI
10.1088/0305-4470/20/13/032

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The tunnelling of a single quantum spin is studied in the limit of large spin quantum number S. The problem is mapped onto a particle problem on the positive half-line with a Hamiltonian which is invariant under inversion x to 1/x. Not only the ground-state energy but also all the other energy levels and corresponding level splittings (if any) are computed by using the conventional WKB methods for the particle problem and an excellent agreement with numerical data is found.
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We present the calculation and comparison of tunnel splitting at excited levels of biaxial spin models by various methods, including the generalized instanton method, the generalized path integral method for coherent spin states, the perturbation method and the exact method by numerical diagonalization of the Hamiltonian. It is found that, for integer spin with spin number around 10, tunnel splitting predicted by the generalized path integral for coherent spin states is about 10 − n times of the exact numerical result for the n th excited level, while the ratio of the results of the perturbation method and the exact numerical method diverges in the large spin limit. We thus conclude that the generalized instanton method is the best approximate way for calculating tunnel splitting in spin models.
We analyze the ground-state energy, magnetization, magnetic susceptibility, and Kondo screening cloud of the symmetric single-impurity Anderson model (SIAM) that is characterized by the bandwidth WWW, the impurity interaction strength UUU, and the local hybridization VVV. We compare Gutzwiller variational and magnetic Hartree-Fock results in the thermodynamic limit with numerically exact data from the density-matrix renormalization group (DMRG) method on large rings. To improve the DMRG performance, we use a canonical transformation to map the SIAM onto a chain with half the system size and open boundary conditions. We compare to Bethe ansatz results for the ground-state energy, magnetization, and spin susceptibility that become exact in the wide-band limit. Our detailed comparison shows that the field-theoretical description is applicable to the SIAM on a ring for a broad parameter range. Hartree-Fock theory gives an excellent ground-state energy and local moment for intermediate and strong interactions. However, it lacks spin fluctuations and thus cannot screen the impurity spin. The Gutzwiller variational energy bound becomes very poor for large interactions because it does not describe properly the charge fluctuations. Nevertheless, the Gutzwiller approach provides a qualitatively correct description of the zero-field susceptibility and the Kondo screening cloud. The DMRG provides excellent data for the ground-state energy and the magnetization for finite external fields. At strong interactions, finite-size effects make it extremely difficult to recover the exponentially large zero-field susceptibility and the mesoscopically large Kondo screening cloud.
We consider a small and fixed number of fermions in an isolated one-dimensional trap (microcanonical ensemble). The ground state of the system is defined at T = 0, with the lowest single-particle levels occupied. The number of particles in this ground state fluctuates as a function of excitation energy. By breaking up the energy spectrum into particle and hole sectors, and mapping the problem onto the classic number partitioning theory, we formulate a new method of calculating the exact particle number fluctuation more efficiently than the direct combinatorics method. The exact ground-state number fluctuation for particles interacting via an inverse-square pairwise interaction is also calculated.