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Physical review. B, Condensed matter

Magnons in a one-dimensional spin glass: The high-field and zero-field limits

Ibrahim Avgin, David L. Huber

Published Nov 1, 1993• 11 citations
Publisher

Abstract

The high-field and zero-field behavior of the linearized magnetic excitations (harmonic magons) in a one-dimensional \ifmmode\pm\else\textpm\fi{}J Heisenberg spin glass are studied. In the high-field limit---a field strong enough to ensure complete alignment of the ground state---the density of states (DOS), the inverse localization length (ILL), and the dynamic structure factor are calculated over the interval -4JE-H4J (H is applied field) by employing the coherent-exchange approximation (CEA), negative-eigenvalue counting, and matrix diagonalization. In the low-energy regime (0.001E-H0.1J), the CEA closely approximates the exact results reproducing, in particular, the anomalous power-law behavior of the DOS \ensuremath{\rho}(E)\ensuremath{\sim}(E-H${)}^{1/3}$ and the ILL 1/L(E)\ensuremath{\sim}(E-H${)}^{2/3}$ for the symmetric distribution of the exchange interactions (concentration c=0.5). In zero field, the DOS and the ILL are calculated using negative-eigenvalue counting for 0E4J and 0.0001E0.01J. For c=0.5, a connection is established between the zero-field and the high-field limits, and for other concentrations, a phenomenological approach is developed in the low-energy regime where it is found that \ensuremath{\rho}(E)=f(c)${\mathit{E}}^{\mathrm{\ensuremath{-}}1/3}$ and 1/L(E)=g(c)${\mathit{E}}^{2/3}$.

DOI
10.1103/physrevb.48.13625

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The high-field and zero-field behavior of the linearized magnetic excitations (harmonic magons) in a one-dimensional \ifmmode\pm\else\textpm\fi{}J Heisenberg spin glass are studied. In the high-field limit---a field strong enough to ensure complete alignment of the ground state---the density of states (DOS), the inverse localization length (ILL), and the dynamic structure factor are calculated over the interval -4JE-H4J (H is applied field) by employing the coherent-exchange approximation (CEA), negative-eigenvalue counting, and matrix diagonalization. In the low-energy regime (0.001E-H0.1J), the CEA closely approximates the exact results reproducing, in particular, the anomalous power-law behavior of the DOS \ensuremath{\rho}(E)\ensuremath{\sim}(E-H)1/3{)}^{1/3})1/3 and the ILL 1/L(E)\ensuremath{\sim}(E-H)2/3{)}^{2/3})2/3 for the symmetric distribution of the exchange interactions (concentration c=0.5). In zero field, the DOS and the ILL are calculated using negative-eigenvalue counting for 0E4J and 0.0001E0.01J. For c=0.5, a connection is established between the zero-field and the high-field limits, and for other concentrations, a phenomenological approach is developed in the low-energy regime where it is found that \ensuremath{\rho}(E)=f(c)E\ensuremath−1/3{\mathit{E}}^{\mathrm{\ensuremath{-}}1/3}E\ensuremath−1/3 and 1/L(E)=g(c)E2/3{\mathit{E}}^{2/3}E2/3.
The authors present a calculation of the Lyapunov exponent for transverse magnon modes in a Heisenberg spin-glass chain with positive and negative exchange interactions, both analytically as a microcanonical average and also numerically in the usual canonical statistical ensemble. In the positive- omega regime they find that the localization length obeys xi ( omega ) varies as omega -1 at omega to 0 if p>q, where p, q=1-p are the +1 and -1 spin concentrations, respectively. It is shown that the previously reported xi ( omega ) varies as omega -2/3 singularity, related to the anomalous dispersion law omega varies as k 3/2 , holds only in the limit of equal concentrations p=q=0.5. The corresponding densities of states diverge as rho ( omega ) varies as omega -1/2 and varies as omega -1/3 for p>q and p=q, respectively.
The work of this thesis concerns the problem of linear low energy excitations of vector spin glass models. An analytical and numerical study is carried out, considering a fully connected random-field Heisenberg model at zero temperature, a fully-connected vector p-spin glass model and a sparse random-field Heisenberg model. We test these models against the low temperature behavior of finite dimensional glassy systems, in particular we show that they posses phases where the density of states is gapless with quasi-localised modes. In the case of the sparse model, we show that the density of states follows a quartic law at low frequency, consistently with several recent measures of this quantity that can be found in the literature of computer glasses. In all the three models, the spin glass transition is characterised in terms of the behavior of the softest excitations. We found that in the fully connected models the zero temperature spin glass transition in a field is a delocalisation transition of the softest modes. In the sparse case, a weaker form of delocalisation appears at the transition. These results broaden our understanding of the zero temperature critical point, by showing how spin glass ordering affects the way the system responds to small magnetic perturbations.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.
We investigate the spectral and localization properties of unmagnetized Heisenberg-Mattis spin glasses, in space dimensionalities d=2d=2d=2 and 333, at T=0T=0T=0. We use numerical transfer-matrix methods combined with finite-size scaling to calculate Lyapunov exponents, and eigenvalue-counting theorems, coupled with Gaussian elimination algorithms, to evaluate densities of states. In d=2d=2d=2 we find that all states are localized, with the localization length diverging as ${\ensuremath{\omega}}^{\ensuremath{-}1}$, as energy $\ensuremath{\omega}\ensuremath{\rightarrow}0$. Logarithmic corrections to density of states behave in accordance with theoretical predictions. In d=3d=3d=3 the density-of-states dependence on energy is the same as for spin waves in pure antiferromagnets, again in agreement with theoretical predictions, though the corresponding amplitudes differ.
For the spin-glass chain in an external field hhh, a nonzero weight at the origin of the bond distribution $\ensuremath{\rho}(J)$ is known to induce a nonanalytical magnetization at zero temperature; for $\ensuremath{\rho}(J)\ensuremath{\sim}A{\ensuremath{\mid}J\ensuremath{\mid}}^{\ensuremath{\mu}\ensuremath{-}1}$ near $J\ensuremath{\rightarrow}0$, the magnetization follows the Chen-Ma scaling $M\ensuremath{\sim}{h}^{\ensuremath{\mu}∕(2+\ensuremath{\mu})}$. In this paper, we numerically revisit this model to obtain detailed statistical information on the ground-state configuration and on the low-energy two-level excitations that govern the low-temperature properties. The ground state consists of long unfrustrated intervals separated by weak frustrated bonds. We accordingly compute the strength distribution of these frustrated bonds, as well as the length and magnetization distributions of the unfrustrated intervals. We find that the low-energy excitations are of two types: (i) one frustrated bond of the ground state may have two positions that are nearly degenerate in energy and (ii) two neighboring frustrated bonds of the ground state may be annihilated or created with nearly zero energy cost. For each excitation type, we compute its probability density as a function of its length. Moreover, we show that the contributions of these excitations to various observables (specific heat, Edwards-Anderson order parameter, susceptibility) are in full agreement with the direct transfer matrix evaluations at low temperature. Finally, following the special bimodal case $\ifmmode\pm\else\textpm\fi{}J$, where a Ma-Dasgupta, RG procedure has been previously used to compute explicitly the above observables, we discuss the possibility of an extended RG procedure. We find that the ground state can be seen as the result of a hierarchical ``fragmentation'' procedure that we describe.
We determine the low-energy spectrum and Parisi replica-symmetry-breaking function for the spin-glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal ( h ) fields. We show that, for all h , the spin-glass state has full replica symmetry breaking and the local spin spectrum is gapless, with a spectral density that vanishes linearly with frequency. These results are obtained using an action functional—argued to yield exact results at low frequencies—that expands in powers of a spin-glass order parameter, which is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum p -rotor model at nonzero h : here, the spin-glass state has one-step replica symmetry breaking and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted. Published by the American Physical Society 2024