We study the effect of quenched time-independent random fields coupled linearly to the order parameter on the dynamical critical behavior of spin systems. We assume that the dynamics is described by a Langevin equation without conservation of the order parameter. It is shown that the dominant fluctuations are those induced by the random fields and, therefore, thermal fluctuations are irrelevant. This allows us to establish a relation between this model and a quantum spin system in the presence of a quenched random field. Moreover, we find that only static exponents in
D dimensions are the same as those of the pure ($D\ensuremath{-}2$)-dimensional theory, but the dynamical exponent
z does not satisfy this relation. The quantum system in
D dimensions is studied through its (
D+1)-dimensional equivalent model where the quenched random fields are totally correlated in the additional imaginary-time ($\ensuremath{\tau}$) direction. The system is anisotropic, and there is a new exponent
zA associated with the scaling behavior in the $\ensuremath{\tau}$ direction. We find the relation
z=2zA to all orders in perturbation theory. For the zero-temperature quantum model we find that the static (zero-frequency) exponents are the same as those of the ($D\ensuremath{-}3$)-dimensional pure quantum model. At finite temperature, when the classical system is finite in the $\ensuremath{\tau}$ direction, we predict a crossover to
D-dimensional classical behavior in nonstatic response and correlation functions, with crossover exponent ${({z}_{A}{\ensuremath{\nu}}_{(D\ensuremath{-}2)})}^{\ensuremath{-}1}$, where ${\ensuremath{\nu}}_{(D\ensuremath{-}2)}$ is the exponent $\ensuremath{\nu}$ for the ($D\ensuremath{-}2$)-dimensional pure classical system. The static correlation functions do not have this cross-over behavior and are the same as those of the ($D\ensuremath{-}3$)-dimensional pure quantum model. The dimensional shift in static quantities for both quantum and classical dynamical systems is a consequence of a supersymmetry in the underlying field theory. The exponent ${z}_{A}=1+\ensuremath{\eta}+O(\frac{1}{{N}^{2}})$ in the large-
N limit or ${z}_{A}=1+c\ensuremath{\eta}+O({\ensuremath{\epsilon}}^{4})$ in the $\ensuremath{\epsilon}$ expansion where $c=1\ensuremath{-}\frac{3}{4}[\frac{(N+2)}{{(N+8)}^{2}}]\ensuremath{\epsilon}$ and $\ensuremath{\eta}$ is the same as in the ($D\ensuremath{-}2$)-dimensional pure classical system. We also study the above classical random-field Ising problem using the interface approach, but are unable to draw any definite conclusion about the dynamics at the lower critical dimension
D=3.
Using the generalized Ginzburg-Landau-Wilson Hamiltonian with a random field, we have shown the absence of classical to quantum crossover near zero temperature. Our finding suggests that the critical exponents of a quantum system with a random field at
T=0 can be obtained from those of the corresponding pure system by a dimensionality shift from
d to $d\ensuremath{-}(\ensuremath{\sigma}+z)$, where $\ensuremath{\sigma}$ is related to the range of interaction and
z is the characteristic exponent of a given quantum system.
The critical behavior of random-field systems is characterized by exponentially long relaxation times. They may differ by orders of magnitude and slow modes have to be considered quenched with respect to the faster ones. This requires a drastic modification of the renormalization process in which the degrees of freedom are integrated out. A new replica-space renormalization procedure, to carry out the coarse-graining of the time intervals, is presented. We relate by recursion-relations the effective reduced dimensions of consecutive time scales. Their stable fixed-points yield the apparent dimensionalities for the longest and shortest time scales, which are insensitive to the exact behavior on intermediate scales. For Ising systems we find that d = 2 is the lower critical dimension in both regimes. The thermal exponents for d = 3, in the short-time observable regime, are related to those of the pure system in d = 2. This is consistent with the observations in field-cooled random antiferromagnets of the correlation length exponent nu approx. = 1 (by neutron scattering) and of the symmetric logarithmic divergence in the specific-heat (by linear birefringence). 17 refs., 3 figs.
The effect of quenched random fields and local perturbations of critical temperature on the critical behavior at phase transitions is studied within the framework of an exactly solvable model that takes into account interaction of fluctuations with equal and opposite momenta. Using the replica method the dimensional reduction by 2 for systems with finite-range interaction and quenched random fields is explicitly shown. For interaction of the infinite range the model demonstrates the mean-field critical asymptotics independently of dimensionality or the presence of random fields. \textcopyright{} 1996 The American Physical Society.
The effects of short-range-correlated impurities on the critical behavior of some quantum models with a free-energy spectrum $\ensuremath{\sim}{k}^{\ensuremath{\sigma}}(0<\ensuremath{\sigma}<~2)$ are investigated by using the renormalization-group approach via the replica trick. At zero temperature, two nonsimultaneous expansion parameters ${\ensuremath{\epsilon}}_{q}={d}_{q}\ensuremath{-}d({d}_{q}=\ensuremath{\sigma}$ or $\frac{3}{2}\ensuremath{\sigma}$) and $\ensuremath{\epsilon}=2\ensuremath{\sigma}\ensuremath{-}d$ are involved in terms of which an unusual phase transition is predicted for $d<2\ensuremath{\sigma}$.