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Physical review. B, Condensed matter

Exact solution of the Heisenberg equation of motion for the surface spin in a semi-infiniteS=1/2XYchain at infinite temperatures

Surajit Sen

Published Oct 1, 1991• 37 citations
Publisher

Abstract

The Heisenberg equation of motion for the surface spin operator in a semi-infinite S=1/2 XY chain is exactly solved at infinite temperatures via the recurrence-relations method of Lee. It is found that the time evolution of the surface spin shows Bessel-like on-site relaxation that is drastically different from the corresponding Gaussian relaxation of a bulk spin. This difference is shown to be related to broken translational symmetry in the former case. The memory function and the random force for the surface spin is exactly determined. It is found that the surface spin dynamics is, in some sense, ``harmonic'' in nature.

DOI
10.1103/physrevb.44.7444

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The Heisenberg equation of motion for the surface spin operator in a semi-infinite S=1/2 XY chain is exactly solved at infinite temperatures via the recurrence-relations method of Lee. It is found that the time evolution of the surface spin shows Bessel-like on-site relaxation that is drastically different from the corresponding Gaussian relaxation of a bulk spin. This difference is shown to be related to broken translational symmetry in the former case. The memory function and the random force for the surface spin is exactly determined. It is found that the surface spin dynamics is, in some sense, ``harmonic'' in nature.
The time evolution of a spin operator Sjx{\mathit{S}}_{\mathit{j}}^{\mathit{x}}Sjx​(t) in a nearest-neighbor S=1/2 semi-infinite XY chain is considered at T=\ensuremath{\infty}. It is shown that the surface spin relaxes very differently compared to the subsurface spin, which already behaves like a bulk spin. The transition from surface to bulk dynamics is discussed using the recurrence-relations method of solving the Heisenberg equation of motion.
We present a study on the nature of the dynamical on-site spin-pair correlation in an s=1/2 isotropic Heisenberg chain at T=\ensuremath{\infty} using the Mori-Lee formalism for solving the Heisenberg equation of motion and the direct summation method of evaluating infinite continued fractions. Our results are consistent with existing short-time analyses and support a power-law decay of the spin-pair correlations. Our study reveals that the frequency spectra corresponding to the on-site dynamical correlations can be seen as consisting of a Gaussian central peak presumably arising from the XY part of the Hamiltonian and a second broad finite frequency peak, which probably arises from the Ising part of the Hamiltonian and is slightly shifted toward a higher frequency than the Ising frequency.
Results are derived for the dynamic correlation functions between a pair of spin operators at any two sites within a semi-infinite Heisenberg ferromagnet at low temperatures, T<<Tc. The correlation functions are found to contain terms describing the effects of the bulk spin-wave modes and the surface spin-wave modes, together with appropriate weighting factors. The results are employed to evaluate the spin deviation in the ferromagnet as a function of temperature and distance from the surface.
We present an exact solution of the long-time relaxational behavior of the magnetization in the Ising and XY\mathrm{XY}XY chains in a quenched random field. The random field is assumed to be of infinite strength but present at only a fraction of the spin sites. We use Glauber dynamics for the Ising case, and a suitably generalized master equation for the XY\mathrm{XY}XY case. In both models we find that for T=0T=0T=0, as time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ the magnetization decays as exp (${\ensuremath{-}Ct}^{\frac{1}{3}}$), where CCC is a constant. At finite temperatures the ultimate asymptotic behavior is purely exponential.
Relaxation phenomena in a class of nondissipative systems with two highly disparate time scales (i.e., Brownian systems) have unique commonalities quantifiable via two scalars. It is shown that the exactly solvable problem of the dynamics of a weakly linked impurity spin in a s=1/2 XY chain belongs to this dynamical universality class and so does that of a heavy mass in an infinite harmonic oscillator chain and a spinless quasi two-dimensional attractive Fermi gas in the long-wavelength limit. The case of the strongly linked impurity in the XY chain is also discussed along with the corresponding limits in the harmonic oscillator chain and the electron-gas problems. \textcopyright{} 1996 The American Physical Society.