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Journal of Mathematical Sciences

Time-Dependent Correlators of Local Spins of the One-Dimensional XY Heisenberg Chain

V. S. Kapitonov, A. G. Pronko

Published Apr 4, 2003• 14 citations
Publisher
DOI
10.1023/a:1022655914303

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We analyze the convergence of the irreversible event-chain Monte Carlo algorithm for continuous spin models in the presence of topological excitations. In the two-dimensional XY model, we show that the local nature of the Markov-chain dynamics leads to slow decay of vortex-antivortex correlations while spin waves decorrelate very quickly. Using a Fréchet description of the maximum vortex-antivortex distance, we quantify the contributions of topological excitations to the equilibrium correlations, and show that they vary from a dynamical critical exponent at the critical temperature to in the limit of zero temperature. We confirm the event-chain algorithm's fast relaxation (corresponding to z = 0) of spin waves in the harmonic approximation to the XY model. Mixing times (describing the approach towards equilibrium from the least favorable initial state) however remain much larger than equilibrium correlation times at low temperatures. We also describe the respective influence of topological monopole-antimonopole excitations and of spin waves on the event-chain dynamics in the three-dimensional Heisenberg model.
This paper is concerned with the nonmonotonic frequency dependence of $S(k, \ensuremath{\omega})$, the wave-number-frequency transform of the time-dependent spin-spin correlation functions for a one-dimensional Heisenberg magnet at infinite temperature, as calculated by Carboni and Richards. It is argued that this nonmonotonic feature of $S(k, \ensuremath{\omega})$ is a one-dimensional effect, and supporting evidence is presented via a phenomenological calculation of $S(k, \ensuremath{\omega})$, utilizing the equivalence of the Heisenberg chain with a fermion system.