The zero-temperature critical behaviours of the classical Heisenberg and XY ferromagnets in the uniaxial random field are studied in the arbitrary spatial dimension. Exact results are obtained for the magnetization, transversal to the random field direction, at the Gaussian and bimodal distributions of the random field. For the Gaussian distribution the critical behaviour in strong random fields is independent of the spatial dimension. The transversal magnetization, , where is the distribution width. For the bimodal distribution of the random field, the transversal magnetization obeys the law , where is a critical field, and H is the random field amplitude. The same critical behaviour is expected for related systems, for example random antiferromagnets in the uniform field.
We study numerically the zero-temperature random field Ising model on cubic lattices of various linear sizes L in three dimensions. For each random field configuration we vary the ferromagnetic coupling strength J . We find that in the infinite volume limit the magnetization is discontinuous in J . The energy and its first J derivative are continuous. The approach to the thermodynamic limit is slow, behaving like L − p with p ∼ 0.8 for the Gaussian distribution of the random field. We also study the bimodal distribution h i = ± h , and we find similar results for the magnetization but with a different value of the exponent p ∼ 0.6. This raises the question of the validity of universality for the random field problem.
The three-dimensional anisotropic classical XY ferromagnet has been investigated by extensive Monte Carlo simulation using the Metropolis single spin flip algorithm. The magnetisation (M) and the susceptibility (χ) are measured and studied as functions of the temperature of the system. For constant anisotropy, the ferro-para phase transition has been found to take place at a higher temperature than that observed in the isotropic case. The system gets ordered at higher temperatures for higher values of the strength of anisotropy. The opposite scenario is observed in the case of random anisotropy. For all three different kinds of statistical distributions (uniform, Gaussian, and bimodal) of random anisotropy, the system gets ordered at lower temperatures for higher values of the width of the distribution of anisotropy. We have provided the phase boundaries in the case of random anisotropy. The critical exponents for the scaling laws M∼L−βν and χ∼Lγν are estimated through the finite size analysis.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where
r is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to
L=219 via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength
h of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength
hc>0.
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 critical behaviour of the two-dimensional Ising model in annealed random fields is studied. Two different kinds of random field are considered and exact results obtained for the transition temperature when the field distribution is symmetric. In the first case the random field is unrestricted and the explicit results obtained for discrete and gaussian field distributions show the annealed field favours the ordering of the system. In the second case the random field satisfies a constraint and explicit results are also obtained for discrete and gaussian field distributions. In the discrete distribution the field assumes the values ±h0,0 and it is shown that there is a critical field hc such that for h0>hc the system presents a reentrant phase transition. In this case the ferromagnetic ground state becomes unstable and there is the appearance of a superantiferromagnetic phase. For h0<hc the ferromagnetic state is stable and the transition temperature decreases due to the presence of the random field.