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arXiv:0801.2060 · Journal of Mathematical Physics

The transmission property of the discrete Heisenberg ferromagnetic spin chain

Qing Ding, Wei Lin

Published Sep 1, 2008• 2 citations
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Abstract

We present a mechanism for displaying the transmission property of the discrete Heisenberg ferromagnetic (DHF) spin chain via a geometric approach. With the aid of a discrete nonlinear Schrödinger-like equation which is the discrete gauge equivalent to the DHF, we show that the determination of transmitting coefficients in the transmission problem is always bistable. Thus, a definite algorithm and general stochastic algorithms are presented. A new invariant periodic phenomenon of the nontransmitting behavior for the DHF, with a large probability, is revealed by an adoption of various stochastic algorithms.

DOI
10.1063/1.2976036

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We present a mechanism for displaying the transmission property of the discrete Heisenberg ferromagnetic (DHF) spin chain via a geometric approach. With the aid of a discrete nonlinear Schrödinger-like equation which is the discrete gauge equivalent to the DHF, we show that the determination of transmitting coefficients in the transmission problem is always bistable. Thus, a definite algorithm and general stochastic algorithms are presented. A new invariant periodic phenomenon of the nontransmitting behavior for the DHF, with a large probability, is revealed by an adoption of various stochastic algorithms.
Abstract The quantum mechanical underpinnings of magnetism are explored via the Heisenberg model of antiferromagnetism. The Lanczos algorithm is developed and applied to obtain ground state properties of the anisotropic antiferromagnetic Heisenberg spin chain. In particular, the phase diagram for the system magnetization is determined. A quantum Monte Carlo method that is appropriate for discrete systems is also presented. The method leverages the similarity between the Schrödinger equation and the diffusion equation to compute energy levels. The formalism necessary to compute ground state matrix elements is also developed. Finally, the method is tested with an application to the spin chain.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.