We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, $2.8\ensuremath{\le}d\ensuremath{\le}4$, and number of spin components,
N. The investigation employs the scaling-field method developed by Golner and Riedel for Wilson's exact momentum-space renormalization-group equation. Fixed points studied include the isotropic and decoupled Ising ($\ensuremath{-}2\ensuremath{\le}N<\ensuremath{\infty}$), the face-and corner-ordered cubic ($1\ensuremath{\le}N<\ensuremath{\infty}$), and, via the replica method for $N\ensuremath{\rightarrow}0$, the quenched random Ising fixed point. Variations of
N and
d are used to link the results to exact results or results from other calculational methods, such as $\ensuremath{\epsilon}$ expansions near two and four dimensions. This establishes the consistency of the calculation for three dimensions. Specifically, truncated sets involving seven (twelve) scaling-field equations are derived for the cubic
N-vector model. A stable random Ising fixed point is found and shown to be distinct from the cubic fixed point and to connect, as a function of
d, with the Khmelnitskii ${\ensuremath{\epsilon}}^{\frac{1}{2}}$ fixed point. At
d=3, the short truncation yields $\ensuremath{\alpha}\ensuremath{\approx}0.11$ for the pure Ising and $\ensuremath{\alpha}\ensuremath{\approx}\ensuremath{-}0.09$ for the random Ising fixed point. A search for a random tricritical fixed point was inconclusive. For the
N-component cubic model, the spin dimensionality
Nc, at which the isotropic and cubic fixed points change stability, is determined as a function of
d. The results support
Nc>3 for three dimensions.
We consider the Ginzburg-Landau
MN model that describes
MN-vector cubic models with
O(M)-symmetric couplings. We compute the renormalization-group functions to six-loop order in
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). 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> 2 we show that the
O(M) fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.
The critical behavior of three-dimensional weakly diluted quenched Ising model is examined on the base of six-loop renormalization group expansions obtained within the minimal subtraction scheme in 4-ɛ space dimensions. For this purpose the ϕ^{4} field theory with cubic symmetry was analyzed in the replica limit n→0. Along with renormalization group expansions in terms of renormalized couplings the sqrt[ɛ] expansions of critical exponents are presented. Corresponding numerical estimates for the physical, three-dimensional system are obtained by means of different resummation procedures applied both to the sqrt[ɛ] series and to initial renormalization group expansions. The results given by the latter approach are in a good agreement with their counterparts obtained experimentally and within the Monte Carlo simulations, while resumming of sqrt[ɛ] series themselves turned out to be disappointing.
We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d , introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .
The critical behaviour of an m -vector model with a local anisotropy axis of random orientation is studied within the field-theoretical renormalization group approach for cubic distribution of anisotropy axis.Expressions for the renormalization group functions are calculated up to the two-loop order and investigated both by an ε = 4 -d expansion and directly at space dimension d = 3 by means of the Pad é-Borel resummation.One accessible stable fixed point indicating a 2nd order ferromagnetic phase transition with dilute Ising-like critical exponents is obtained.