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.
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.
We study the dynamic critical behavior of spin systems in the presence of quenched random magnetic fields with long-range correlations of the type ${[h(0)h(R)]}_{\mathrm{av}}\ensuremath{\sim}{R}^{\ensuremath{-}(d\ensuremath{-}\ensuremath{\theta})}$. We assume that the dynamics is described by a Langevin equation without conservation of the order parameter. It is shown in both the $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}$ expansion ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}=6+\ensuremath{\theta}\ensuremath{-}d$) to order ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}$ and
N1 expansion to leading order for $4+\ensuremath{\theta}<d<6+\ensuremath{\theta}$ that the dynamic critical exponent
z can be expressed in terms of static exponent $\ensuremath{\eta}$ as $z=2+g(\ensuremath{\theta})\ensuremath{\eta}$. The dynamic scaling form of the spin-spin correlation function at
T=Tc is obtained to order ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}^{2}$.
Earlier work on dynamical critical phenomena in the context of magnetic hysteresis for uniaxial (scalar) spins, is extended to the case of a multicomponent (vector) field. From symmetry arguments and a perturbative renormalization group approach (in the path integral formalism), it is found that the generic behavior at long time and length scales is described by the scalar fixed point (reached for a given value of the magnetic field and of the quenched disorder), with the corresponding Ising-like exponents. By tuning an additional parameter, however, a fully rotationally invariant fixed point can be reached, at which all components become critical simultaneously, with O(n)-like exponents. Furthermore, the possibility of a spontaneous non-equilibrium transverse ordering, controlled by a distinct fixed point, is unveiled and the associated exponents calculated. In addition to these central results, a didactic ``derivation'' of the equations of motion for the spin field are given, the scalar model is revisited and treated in a more direct fashion, and some issues pertaining to time dependences and the problem of multiple solutions within the path integral formalism are clarified.
Critical behaviors of a ferromagnetic spin system under quenched random magnetic fields are investigated near above four space dimensions. Critical temperature Tc(m, µ, ε) as a function of the number of spin components m(≥2) and strength µ of the random field is obtained in a power series of ε ≡ d-4 (>0) by means of the Migdal-Polyakov renormalization group theory. Critical exponents ν and η are also calculated in powers of ε. It is shown that they are in agreement with those of (2 + ε)-dimensional pure system for sufficiently close to Tc. In the limit m→∞, present results reduce to d = 4 + ε case of our previous results for the spherical model.
The critical dynamics of the time-dependent Ginzburg-Landau model for a system with quenched random impurities and nonconserved order parameter is studied in the framework of the $\ensuremath{\epsilon}$ expansion. In contrast to the situation in pure systems, the dynamic critical exponent
z deviates from its conventional value at first order in $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. The impurities cause an enhancement of the shape function ${f}_{x}(\ensuremath{\nu})$ at small frequencies $\ensuremath{\nu}$; ${f}_{x}(\ensuremath{\nu}=0)$ diverges as $T\ensuremath{\rightarrow}{T}_{c}$. Below
Tc the equation of state, static susceptibility $\ensuremath{\chi}$, and dynamic response function
G, are studied. A new, purely static correlated function,
C(s), whose existence is unique to the random system is introduced. The coexistence curve singularities of
C(s), $\ensuremath{\chi}$, and
G in systems with continuously broken symmetry are explored. The connection of the quenched-impurity model with "model
C" of Halperin, Hohenberg, and Ma is discussed.