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.
The stochastic quantization of Nelson is extended to the finite-temperature equilibrium states and is applied to the study of the quantum effects in the critical dynamics of the time-dependent Ginzburg-Landau model in the large-
n limit. A temperature-dependent (non-Markovian) stochastic process for the order parameter is derived which contains the crossover from the ground-state dynamics dominated by the quantum fluctuations, to classical critical dynamics in the neighborhood of the finite-temperature critical points. The crossover is exhibited by means of the effective dynamical critical exponent.
We consider the influence of a quenched random field on the critical behavior of a quantum spherical model at the zero-temperature quantum phase transition. We find a complete solution of the model for arbitrary translationally invariant pair interactions. It turns out that the critical behavior for zero as well as finite temperatures is dominated by static random field fluctuations rather than by quantum or thermal fluctuations. Therefore the critical behavior close to the zero-temperature quantum phase transition is identical to that close to a finite-temperature transition. The system does not show a crossover from quantum to classical behavior. \textcopyright{} 1996 The American Physical Society.
Equilibrium properties of the phase transition of a ferromagnetic spin system under a quenched random magnetic field are briefly summarized. Based upon a path integral formulation for stochastic processes, the n→0 method of replica is extended to treat critical relaxation of quenched random spin systems. As an application, the dynamic scaling hypothesis is examined for the random field problem by means of 1/m-expansion method (m; the number of order parameter components) and the renormalization group method near 6 space dimensions. It is shown that d→d−2 rule does not hold for the dynamic critical exponent of the characteristic frequency.
Abstract The effect of quenched random fields on the zero‐temperature critical behaviour is studied by the means of the Wilson‐Fisher renormalisation‐group recursion relations to first order in the ϵ‐expansion. A number of quantum models is considered. The simultaneous influence of both, the random‐field and the classical‐to‐quantum dimensional crossover phenomena on the quantum critical behaviour is investigated. A special attention is paid to the properties of some reasonable particular forms of Gaussian random correlations which might be of relevance to the description of natural systems.
We consider two strategies for stochastic quantization. With the first, one posits an additional time dimension (fictitious time) and describes the evolution of classical fields by means of the Langevin equation. One then evaluates stochastic averages of the field functions. In the limit that the fictitious time goes to infinity, these approach the time ordered correlation functions of canonically quantized field theory. We conclude that, while this strategy successfully describes QED and other quantum field theories, it is contrived and probably lacks deep physical significance. With the second strategy for field quantization, one begins with a classical action in either one or two extra dimensions coupled to an external random source. We review a method of quantizing bosonic fields which uses this strategy. Furthermore, we present an analogous method for quantizing fermion fields and a possible new way of quantizing interacting fermion and boson fields. Finally, we discuss applications to quantum mechanics and stochastically quantize the simple harmonic oscillator.