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Physical Review B

Sound pumping in an alternating quantum spin chain

Andrei A. Zvyagin, G. A. Zvyagina

Published Mar 16, 2006• 0 citations
Publisher

Abstract

The pumping of the energy of a longitudinal sound by a quantum spin chain with the alternation of coupling constants is studied theoretically. It is shown that the power of the sound wave, absorbed by the dimerized spin chain, is zero in the dynamical regime, i.e., at times, smaller than relaxation times. In the steady-state regime, for times, larger than the relaxation time, the dimerized quantum spin chain can absorb the power of the sound. We have shown that the absorption of the sound exists when the frequency of the wave is between two threshold values, related to two critical fields, at which quantum phase transitions take place. Contrary, the calculated magnetic field dependence of the absorbed power of the sound does not clearly manifest two critical fields. Nevertheless, we have shown that it is possible to extract the information about the value of the field, at which the spin gap of low-lying excitations is closed, from the temperature dependence of the absorption of the longitudinal sound.

DOI
10.1103/physrevb.73.104422

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The pumping of the energy of a longitudinal sound by a quantum spin chain with the alternation of coupling constants is studied theoretically. It is shown that the power of the sound wave, absorbed by the dimerized spin chain, is zero in the dynamical regime, i.e., at times, smaller than relaxation times. In the steady-state regime, for times, larger than the relaxation time, the dimerized quantum spin chain can absorb the power of the sound. We have shown that the absorption of the sound exists when the frequency of the wave is between two threshold values, related to two critical fields, at which quantum phase transitions take place. Contrary, the calculated magnetic field dependence of the absorbed power of the sound does not clearly manifest two critical fields. Nevertheless, we have shown that it is possible to extract the information about the value of the field, at which the spin gap of low-lying excitations is closed, from the temperature dependence of the absorption of the longitudinal sound.
The dynamical mean-field-like theory of the electron spin resonance (ESR) in spin-1/2 chains with nearest and next-nearest neighbor interactions (or zigzag spin ladders) and alternation is developed. It is explained that either the magnetic anisotropy of spin-spin couplings or the Dzyaloshinskii-Moriya interaction causes shifts of the resonance position in spin chains, comparing to only magnetically isotropic interactions. The power of the ac ESR field, absorbed by the spin chain, is calculated. It is shown that the temperature dependence of the shift of the resonance position is related to the temperature dependence of the ESR absorption. Depending on the strength of next-nearest neighbor interactions, the shift of the ESR position can become either exponentially small or can be increased near quantum critical points. The absorbed by the alternating spin chain power of the ac magnetic field is calculated, manifesting two resonances. The influence of mentioned spin-spin interactions on the linewidth of the ESR is analyzed.
Critical behaviors of the sound attenuation in a spin-1 Ising system with bilinear (J) and biquadratic (K) interactions are investigated within the framework of cluster variation method and Onsager theory of irreversible thermodynamics. The sound wave is assumed to couple mainly to the order parameter fluctuations which decay via order parameter relaxation processes. Two relaxation times are obtained and an expression is found for the sound attenuation coefficient (α) in terms of these relaxation times. The temperature behavior of the sound attenuation near the phase transition temperatures (Tc) is analyzed according to various values of Onsager coefficients (γij) and sound frequency (ω). For T<Tc it is found that the maxima of the attenuation shifted to lower temperatures with increasing ω and γij (i≠j) values. For T>Tc the data give evidence that there is no relaxational contribution to sound attenuation coming from order parameter fluctuations. On the other hand, a convergence is found in attenuation just below the critical and the tricritical points as (Tc−T), while a jump-discontinuity is observed for the first-order behavior. The frequency variation of the sound attenuation is also investigated and in addition to ω2-attenuation dependence observed in the hydrodynamic regime it is observed that in the high frequency region the attenuation is independent of ω and the ratio of two interaction parameters (J/K).
Recently several compounds were observed to reveal properties of a one-dimensional spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain with gapless low-lying spin excitations in an absence of an external magnetic field. However even a very weak external magnetic field yields an onset of a spin gap for those low-lying excitations. Usually this onset of a field-induced spin gap is related to the presence of a staggered Dzyaloshinskii-Moriya interaction, or to the staggered effective g factors of magnetic ions. Experiments on thermodynamic properties cannot distinguish between those two origins for the spin-gap formation. Here we propose the setup for an experiment, in which one can clearly distinguish between the staggered Dzyaloshinskii-Moriya coupling and staggered external magnetic field: an electronic paramagnetic resonance in the parallel pumping geometry. The characteristics of the system under conditions of such an experiment, i.e., the renormalization of the homogeneous and staggered magnetizations, power of the microwave magnetic field, absorbed by the system, are calculated. Several important features of the solution are discussed.
Exact analytic calculations in spin-1/2 XY\mathrm{XY}XY chains show the presence of long-time tails in the asymptotic dynamics of spatially inhomogeneous excitations. The decay of inhomogeneities for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty},$ is given in the form of a power law $(t/{\ensuremath{\tau}}_{Q}{)}^{\ensuremath{-}{\ensuremath{\nu}}_{Q}},$ where the relaxation time ${\ensuremath{\tau}}_{Q}$ and the exponent ${\ensuremath{\nu}}_{Q}$ depend on the wave vector Q, characterizing the spatial modulation of the initial excitation. We consider several variants of the XY\mathrm{XY}XY model (dimerized, with staggered magnetic field, with bond alternation, and with isotropic and uniform interactions), that are grouped into two families, whether the energy spectrum has a gap or not. Once the initial condition is given, the nonequilibrium problem for the magnetization is solved in closed form, without any other assumption. The long-time behavior for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ can be obtained systematically in a form of an asymptotic series through the stationary phase method. We found that gapped models show critical behavior with respect to Q, in the sense that there exist critical values Qc{Q}_{c}Qc​ where the relaxation time ${\ensuremath{\tau}}_{Q}$ diverges and the exponent ${\ensuremath{\nu}}_{Q}$ changes discontinuously. At those points, a slowing down of the relaxation process is induced, similarly to phenomena occurring near phase transitions. Long-lived excitations are identified as incommensurate spin density waves that emerge in gapped systems, as a consequence of both approximate nesting of the spectrum and the degeneracy of some stationary points. In contrast, gapless models do not present the above anomalies as a function of the wave vector Q.
A theoretical study of the low-temperature dynamics of the antiferromagnetic Heisenberg chain in an external field is presented. The time- and frequency-dependent transverse and longitudinal spin fluctuations and energy fluctuations are calculated by means of a continued fraction representation. The frequency moments are evaluated analytically by a computer program. The transfer-operator technique is used to calculate all static correlation functions. For modest magnetic fields, a two-peak structure in the dynamic form factor of the transverse spin fluctuations is observed. Apart from the usual spin-wave excitations, the longitudinal-spin-fluctuation spectra show a broad second resonance for some temperatures and fields. The energy fluctuations display a dominant oscillatory time dependence.