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Physical review. B, Condensed matter

Transition from surface-spin to bulk-spin relaxation in anS=1/2 nearest-neighborXYchain at infinite temperature

Surajit Sen, S. D. Mahanti, Zhi-Xiong Cai

Published May 1, 1991• 27 citations
Publisher

Abstract

The time evolution of a spin operator ${\mathit{S}}_{\mathit{j}}^{\mathit{x}}$(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.

DOI
10.1103/physrevb.43.10990

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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.
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.
A Mori-Lee theoretic study of spin relaxation in the neighborhood of a magnetic impurity in the s=1/2 nearest-neighbor (NN) Heisenberg chain at high temperatures is presented. Our calculations suggest that in the limit of zero impurity coupling, the spectral function for the ``surface spin'' is consistent with a set of discrete frequencies given approximately by wn{\mathit{w}}_{\mathit{n}}wn​\ensuremath{\approxeq}[n(w0{\mathit{w}}_{0}w0​+1/4)-1/4] against a shallow continuum background, where w0{\mathit{w}}_{0}w0​\ensuremath{\approxeq}3J/4 with 2J being the coupling between NN spins. The frequencies broaden as the impurity coupling is increased to the NN coupling.
We study the time evolution of classical spin systems with purely relaxational dynamics, quenched from $T\ensuremath{\gg}{T}_{c}$ to the critical point, in the semi-infinite geometry. Shortly after the quench, as in the bulk, a nonequilibrium regime governed by universal power laws is also found near the surface. We show for "ordinary" and "special" transitions that the corresponding critical exponents differ from their bulk values, but can be expressed via scaling relations in terms of known bulk and surface exponents. To corroborate our scaling analysis, we present perturbative ($\ensuremath{\epsilon}$-expansion) and Monte Carlo results.
Using the spin-wave approximation elementary excitations of a semi-infinite two-dimensional S=12S=\frac12S=21​ Heisenberg antiferromagnet are considered. The spectrum consists of bulk modes -- standing spin waves and a quasi-one-dimensional mode of boundary spin waves. These latter excitations eject bulk modes from two boundary rows of sites, thereby dividing the antiferromagnet into two regions with different dominant excitations. As a result absolute values of nearest-neighbor spin correlations on the edge exceed the bulk value.
Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.