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Progress of Theoretical Physics

Renormalized Field Theory of Quenched System under Gaussian Random Fields Conjugate to Anisotropic Spin Fields

M. Oku, R. Abe

Published Nov 1, 1980• 0 citations
Publisher

Abstract

The effects of Gaussian random fields applied only to the m spin fields of the anisotropic (n+m)-vector model are studied in the quenched case via the renormalized φ4 field theory by examining the stability of four fixed points and one fixed line in the large n and m limits. Regardless of the value of n/m, only a decoupled case is infrared stable, so that random fields prove to have no effects on the critical properties (e.g., critical indices) of the pure n spin fields. In such a decoupled case, as far as random fields are weak, longitudinal order corresponding to a random fixed point is observed remarkably. However, as the strength of random fields grows stronger, random fixed point moves to a Gaussian (trivial) fixed point and in consequence only transverse order gets conspicuous. The above results are physically consistent with Aharony's arguments.

DOI
10.1143/ptp.64.1577

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The effects of Gaussian random fields applied only to the m spin fields of the anisotropic (n+m)-vector model are studied in the quenched case via the renormalized φ4 field theory by examining the stability of four fixed points and one fixed line in the large n and m limits. Regardless of the value of n/m, only a decoupled case is infrared stable, so that random fields prove to have no effects on the critical properties (e.g., critical indices) of the pure n spin fields. In such a decoupled case, as far as random fields are weak, longitudinal order corresponding to a random fixed point is observed remarkably. However, as the strength of random fields grows stronger, random fixed point moves to a Gaussian (trivial) fixed point and in consequence only transverse order gets conspicuous. The above results are physically consistent with Aharony's arguments.
We consider the Ginzburg-Landau MN\mathrm{MN}MN model that describes MNMNMN-vector cubic models with O(M)O(M)O(M)-symmetric couplings. We compute the renormalization-group functions to six-loop order in d=3.d=3.d=3. We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model (M=1).(M=1).(M=1). We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For M> 2M>~2M> 2 we show that the O(M)O(M)O(M) fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.
The phase transitions in a large class of physical systems are described by $n\ensuremath{\ge}4$ component order parameters. Here, the critical behavior of quenched random $n\ensuremath{\ge}4$ vector models is studied by means of renormalization-group theory in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Recursion relations for average potentials are constructed following the methods derived by Lubensky. For several Hamiltonians describing homogeneous $n\ensuremath{\ge}4$ systems there exist no stable fixed points in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions which explains the first-order transitions actually observed in these systems. It is shown that the recursion relations for the corresponding quenched random systems are also unstable. However, the runaway in this case is of a fundamentally different nature. The fluctuations of the local mean-field transition temperature diverge, and this behavior is interpreted as a "smeared" transition. This interpretation is consistent with existing experiments. On the other hand, in the cases where the homogeneous Hamiltonian possesses a stable fixed point in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, this fixed point remains stable against random perturbations, so no change in the critical behavior is expected. For most of the models studied there is at least one fixed point of order ${\ensuremath{\epsilon}}^{\frac{1}{2}}$. These fixed points are all unstable. It is suggested that experiments should be performed to determine the critical behavior of random $n\ensuremath{\ge}4$ systems. Of particular interest are the systems with no stable fixed points, such as Cr, Eu, MnO, and UO2{\mathrm{UO}}_{2}UO2​, where crossover from first-order to a smeared transition is predicted.
We investigate the effect of a unidirectional quenched random field on the anisotropic quantum spin-1/2 XYXYXY model, which magnetizes spontaneously in the absence of the random field. We adopt a mean-field approach for this analysis. In general, the models considered have Ising symmetry, and as such they exhibit ferromagnetic order in two and three dimensions in the presence of not too large disorder. Even in the special case when the model without disorder has U(1)\text{U}(1)U(1) symmetry, a small U(1)\text{U}(1)U(1)-symmetry-breaking random field induces ferromagnetic long-range order in two dimensions. The mean-field approach, consequently, provides a rather good qualitative and even quantitative description when applied not too close to the criticality. We show that spontaneous magnetization persists even in the presence of the random field, but the magnitude of magnetization gets suppressed due to disorder, and the system magnetizes in the directions parallel and transverse to the random field. Our results are obtained via analytical calculations within a perturbative framework and by numerical simulations. Interestingly, we show that it is possible to enhance a component of magnetization in the presence of the disorder field provided that we apply an additional constant field in the XYXYXY plane. Moreover, we derive generalized expressions for the critical temperature and the scalings of the magnetization near the critical point for the XYXYXY spin system with arbitrary fixed quantum spin angular momentum.
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.
Non-symmetric multicritical points in random magnetic mixtures with competing spin anisotropies are studied by means of the renormalized field theory and the replica method. Nine β functions and two critical indices η are evaluated up to O(ε2)[ε=4-d(d: space dimensionality)]. The investigated model consists of two terms: the anisotropic exchange energy previously studied by Fishman-Aharony and the random off-diagonal anisotropy energy (RAM) recently proposed by Mukamel-Grinstein. Many randomly mixed magnets have the latter term due to the spin lattice coupling or the dipole-dipole interaction. Random off-diagonal anisotropy energy couples longitudinal and transverse spin components in this system. Owing to the existence of this term, the second-order tetracritical point predicted by Fishman-Aharony should be replaced by the unstable coupled tetracritical point, which corresponds to the first-order or smeared phase transition.
In this note, we clarify the stability of the large-N functional RG fixed points of the order/disorder transition in the random-field (RF) and random-anisotropy (RA) O(N) models. We carefully distinguish between infinite N, and large but finite N. For infinite N, the Schwarz-Soffer inequality does not give a useful bound, and all fixed points found in cond-mat/0510344 (Phys. Rev. Lett. 96, 197202 (2006)) correspond to physical disorder. For large but finite N (i.e. to first order in 1/N) the non-analytic RF fixed point becomes unstable, and the disorder flows to an analytic fixed point characterized by dimensional reduction. However, for random anisotropy the fixed point remains non-analytic (i.e. exhibits a cusp) and is stable in the 1/N expansion, while the corresponding dimensional-reduction fixed point is unstable. In this case the Schwarz-Soffer inequality does not constrain the 2-point spin correlation. We compute the critical exponents of this new fixed point in a series in 1/N and to 2-loop order.