Effects of quenched random magnetic fields on the critical relaxation of the time-dependent Ginzburg-Landau equation for m-component classical spin system are investigated by means of the dynamical renormalization group. In the static limit, the recursion relations near below 6 space dimensions found by Grinstein are recovered. Dynamic critical exponent for the characteristic frequency is obtained to be z = 2 + c η, c = 2 up to order ε2 (ε= 6 - d) on the basis of the dynamic scaling hypothesis. The static correlation exponent η is given by that of a pure system in d - 2 space dimensions. Since c does not agree with the result by Halperin-Hohenberg-Ma (c = 6 log (4/3) - 1), d →d - 2 rule does not hold for time-dependent phenomena.
We study the purely relaxational critical dynamics with non-conserved order parameter (model A critical dynamics) for three-dimensional magnets with disorder in a form of the random anisotropy axis.For the random axis anisotropic distribution, the static asymptotic critical behaviour coincides with that of random site Ising systems.Therefore the asymptotic critical dynamics is governed by the dynamical exponent of the random Ising model.However, the disorder effects considerably the dynamical behaviour in the non-asymptotic regime.We perform a field-theoretical renormalization group analysis within the minimal subtraction scheme in two-loop approximation to investigate asymptotic and effective critical dynamics of random anisotropy systems.The results demonstrate the non-monotonic behaviour of the dynamical effective critical exponent z eff .
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 problem of the lower critical dimension (LCD) of spin systems in random fields is discussed. The different theoretical predictions for the Ising case-i.e. LCD=2 from considerations of domain wall roughening but LCD=3 from diagrammatic renormalisation-are related to different results for the spectral function Delta (q) of the dressed static-field fluctuations-i.e. LCD=3 is obtained if Delta (q) diverges as xi eta with increasing thermal correlation length xi , while LCD=2 corresponds to a non-divergent Delta (q). This difference is analogous to the difference between the true value of the dynamical critical exponent z of a pure ferromagnetic spin system and the 'conventional approximation' z=2- eta . In this way, the weak point of the arguments leading to LCD=2 is discovered, namely the neglect of certain 'mode couplings' between fluctuations with different q or, more pictorially, the neglect of 'domains within domains', and thus the result LCD=2 is seen to be no longer credible. The domain argument leading to LCD=2 are modified by replacing the bare fields by 'dressed' ones; in this way, they are reconciled with the field theoretic result LCD=3.
This paper is a field theoretic description of the critical dynamics of spin systems with frozen nonmagnetic impurities. For three-dimensional systems the dynamical critical exponent is found directly by employing the three-loop approximation with the Padé-Borel summation technique. The results are compared with those obtained by calculating the dynamical exponent for homogeneous systems in the four-loop approximation, and with the values obtained by computer simulation of the critical dynamics by Monte Carlo methods. Calculations of the dynamical exponent for the two-dimensional Ising model in the four-loop approximation are also presented.
Critical behaviors in quenched random-spin systems with N-spin component are studied in the limit M → 0 of the non-random MN-component models by means of the renormalization group theory. As the static critical phenomena the stability of the fixed points is investigated and the critical exponents η[~ O(ε3); ε ≡ 4 – d], γ, α, and crossover index [Formula: see text] and the equation of state [~ O(ε)] are obtained. Within the approximation up to the order ε2, even the random-spin systems with N = 2 or 3 are unstable in the three dimensions and the pure systems are stable there.