The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-
Hc), where
Hc is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=
kzf(\ensuremath{\Delta}/
kcphi), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors:
kz for \ensuremath{\Delta}\ensuremath{\ll}
kcphi and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}
kcphi, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.
The transverse dynamics of a vector spin glass chain at zero temperature is studied by a numerical scaling method which provides a new physical insight into the nature of the spin excitations. In the range of low energies considered we find that the states are all localized with the coupling between spins V decaying asymptotically for large lengths separating the sites l , like V ∼ exp[- l /ξ], where ξ measures the localization length. A dispersion relation is found between the energies and the associated inverse localization lengths of the form ω ∝ (1/ξ) z with the dynamic exponent z = 1.501 ± 0.044. This implies that the system at zero temperature has nontrivial dynamic critical behaviour, in agreement with our recent analytical results; it moreover gives for the first time a clear interpretation for the characteristic length of the spin fluctuations as the localization length.
The authors study the statics and long-time dynamics of a spin glass on a family of hierarchical lattices. The statics may be solved exactly on these lattices, and they treat the dynamics in a physically motivated low-temperature approximation, retaining only the slowest mode at each renormalization step. The results are generally in good agreement with Monte Carlo simulations in two and three dimensions. In particular, they find dynamical scaling at the spin glass transition in three dimensions, with a dynamical exponent z=6.5. The spin autocorrelation function at T g decays like t -x with x=0.05. A new result for statics which is used in the dynamics calculation is that for the magnetic exponent y ch at the critical point, which is found to be 2.65. Above T g , the autocorrelation function can be fitted in a wide intermediate time range by a Kohlrausch form, but in this model this is only a consequence of a crossover in the renormalization flow and has no fundamental dynamical significance.
The ±J Ising model in three dimensions is investigated using dynamical scaling analysis for a nonequilibrium relaxation process. The spin-glass (SG) transition is analyzed under an assumed dynamical exponent z proposed for the scaling analysis of the SG transition. To estimate dynamical exponent z precisely, we introduce an extrapolation scheme for the asymptotic value of the logarithmic derivative of the dynamical Binder parameter. We obtain the phase diagram in the temperature–concentration (p–T) space, as well as the p-dependence of critical exponents γ, ν, and z. For the static exponents of the SG transition, the universality is strongly indicated numerically along the whole region of the paramagnetic (PM)–SG phase boundary. A remarkable crossover behavior is observed for estimations of critical exponents at the boundary close to the multicritical point (MCP) of PM–ferromagnetic (FM)–SG phases. In addition, the dynamical assumption for z resulting from the slow relaxation in the SG region disappears in the FM region, and the crossover between them occurs around the Nishimori Line.
We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field
Γ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of
Γ and for system sizes
N≤16. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of
Γ. The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at
Γc≃1.5. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get
z=2.1±0.1 which is in contradiction with a recent estimate (
z=4). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for
Γ<Γc.