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arXiv:1206.2827 · Physical Review B

Thermally driven classical Heisenberg model in one dimension

Debarshee Bagchi, P. K. Mohanty

Published Dec 14, 2012• 8 citations
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Abstract

We study thermal transport in a classical one-dimensional Heisenberg model employing a discrete-time odd-even precessional update scheme. This dynamics equilibrates a spin chain for any arbitrary temperature and finite value of the integration time step $\ensuremath{\Delta}t$. We rigorously show that in presence of driving, the system attains local thermal equilibrium, which is a strict requirement of Fourier law. In the thermodynamic limit, heat current for such a system obeys Fourier law for all temperatures, as has been recently shown [A. V. Savin, G. P. Tsironis, and X. Zotos, Phys. Rev. B 72, 140402(R) (2005)]. Finite systems, however, show an apparent ballistic transport which crosses over to a diffusive one as the system size is increased. We provide exact results for current and energy profiles in zero- and infinite-temperature limits.

DOI
10.1103/physrevb.86.214302

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We numerically investigate thermal conduction in quantum dynamics. A one-dimensional quantum spin system is used as a model, which we can make integrable or nonintegrable by controlling parameter values. A different method to simulate heat reservoirs is introduced. Using this method, we perform numerical simulations of the spin chain in contact with two heat reservoirs at different temperatures. As a consequence, a flat temperature profile is observed in the integrable case, while a finite temperature gradient is found in a nonintegrable system. This result suggests that the Fourier heat law may be realized in nonintegrable systems and is consistent with the classical case reported in the literature. \textcopyright{} 1996 The American Physical Society.
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We investigate the open dynamics of a chain of interacting spins using the quantized version of the GENERIC equation from classical out-of-equilibrium thermodynamics. We focus on both equilibrium and nonequilibrium scenarios for chains of different sizes. While in the equilibrium case we demonstrate thermal equilibration to the correct many-body Gibbs density matrix, in the nonequilibrium dynamics we show a ballistic-to-diffusive transition in the steady-state energy current and a scaling that is consistent with Fourier's law of heat transfer.
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We study the real-time dynamics of spin chains driven out of thermal equilibrium by an initial temperature gradient ${T}_{L}\ensuremath{\ne}{T}_{R}$ using density matrix renormalization group methods. We demonstrate that the nonequilibrium energy current saturates fast to a finite value if the linear-response thermal conductivity is infinite, i.e., if the Drude weight DDD is nonzero. Our data suggest that a nonintegrable dimerized chain might support such dissipationless transport (D>0D>0D>0). We show that the steady-state value JE{J}_{E}JE​ of the current for arbitrary ${T}_{L}\ensuremath{\ne}{T}_{R}$ is of the functional form ${J}_{E}=f({T}_{L})\ensuremath{-}f({T}_{R})$, i.e., it is completely determined by the linear conductance. We argue for this functional form, which is essentially a Stefan-Boltzmann law in this integrable model; for the XXX ferromagnet, fff can be computed via the thermodynamic Bethe ansatz in good agreement with the numerics. Inhomogeneous systems exhibiting different bulk parameters as well as Luttinger liquid boundary physics induced by single impurities are discussed briefly.
The problem of spin diffusion is studied numerically in the one-dimensional classical Heisenberg model using a deterministic odd-even spin precession dynamics. We demonstrate that spin diffusion in this model is normal in the infinite-temperature limit and one obtains a long-time diffusive tail in the decay of the autocorrelation function. Some variations of the model with different coupling schemes and anisotropy are also studied and we find normal diffusion in all of them. A systematic finite-size analysis also suggests normal diffusive spreading of spin fluctuations, contrary to previous claims of anomalous diffusion. We also briefly discuss spin diffusion in this model at finite temperatures.