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.
We present results of a Monte Carlo simulation study of the zero-temperature quantum phase transition of a three-dimensional short-range spin-1/2 Ising spin glass, driven by a transverse magnetic field. This quantum transition is equivalent to the finite-temperature transition in the (3+1)-dimensional anisotropic classical random Ising model that is the path integral of the quantum system. The critical exponents are estimated using a finite-size scaling analysis. The uniform linear susceptibility is finite at the transition, while the nonlinear susceptibility diverges. The results are more consistent with conventional, rather than activated, dynamic scaling in the quantum system.
We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.
Anisotropic dipolar systems are considered. Such systems in an external magnetic field are expected to be a good experimental realization of the transverse field Ising model. With random interactions, this model yields a spin glass to paramagnet phase transition as a function of the transverse field. We show that the off-diagonal dipolar interaction, although effectively reduced, induces a finite correlation length and thus destroys the spin-glass order at any finite transverse field. We thus explain the behavior of the nonlinear susceptibility in the experiments on LiHo(x)Y(1-x)F(4), and argue that a crossover to the paramagnetic phase, and not quantum criticality, is observed.
We present a scaling theory for the low-temperature phase transition of the Ising spin glass in a transverse magnetic field (\ensuremath{\Gamma}). The theory provides relations and bounds on some critical exponents, which are supported by results from actual calculations. A renormalization-group analysis of the zero-temperature transition in the presence of a longitudinal symmetry-breaking field enables us to estimate the critical exponents \ensuremath{\beta} and \ensuremath{\gamma}, associated with the Edwards-Anderson order parameter and the nonlinear susceptibility, respectively. We have found no indications of the transition being of first order.
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.