Abstract The renormalization group technique is applied to the three‐dimensional classical Heisenberg spin system with modified dipole–dipole interaction. For this system, the values and dependence of several critical exponents are investigated. These exponents are: α, the specific heat exponent; β and δ, the magnetization exponents; γ, the susceptibility exponent; η, the correlation function exponent; and v, the correlation length exponent. All the critical exponents except β depend upon the potential range σ. When the potential range σ = 3/2, the modified dipole–dipole results agree with mean‐field values of the critical exponents.
Abstract The renormalization group technique is applied to the three‐dimensional classical Heisenberg spin system with modified dipole–dipole interaction. For this system, the values and dependence of several critical exponents are investigated. These exponents are: α, the specific heat exponent; β and δ, the magnetization exponents; γ, the susceptibility exponent; η, the correlation function exponent; and v, the correlation length exponent. All the critical exponents except β depend upon the potential range σ. When the potential range σ = 3/2, the modified dipole–dipole results agree with mean‐field values of the critical exponents.
Abstract Firstly the renormalization group technique is applied to an isotropic Heisenberg spin system with long‐range interaction and a set of critical exponents are calculated. These exponents are expressed in terms of a potential parameter s̀. All critical exponents except β, the magnetization exponent, depend on s̀. Secondly, the anisotropy is considered and the related crossover exponent φ is evaluated. This later parameter turns out to be equal to γ, the susceptibility exponent, as predicted by some authors.
We compute the critical exponents of three-dimensional magnets with strong dipole-dipole interactions using the functional renormalization group (FRG) within the local potential approximation including the wave function renormalization (LPA′). The system is governed by the Aharony fixed point, which is scale-invariant but lacks conformal invariance. Our nonperturbative FRG analysis identifies this fixed point and determines its scaling behavior. The resulting critical exponents are found to be close to those of the Heisenberg O (3) universality class, as computed within the same FRG/LPA′ framework. This proximity confirms the distinct yet numerically similar nature of the two universality classes.
Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.
We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to 1003 spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.
We have tested the theoretical values of critical exponents, predicted for the three--dimensional Heisenberg model, based on the published Monte Carlo (MC) simulation data for the susceptibility. Two different sets of the critical exponents have been considered - one provided by the usual (perturbative) renormalization group (RG) theory, and another predicted by grouping of Feynman diagrams in phi^4 model (our theory). The test consists of two steps. First we determine the critical coupling by fitting the MC data to the theoretical expression, including both confluent and analytical corrections to scaling, the values of critical exponents being taken from theory. Then we use the obtained value of critical coupling to test the agreement between theory and MC data at criticality. As a result, we have found that predictions of our theory (gamma=19/14, eta=1/10, omega=3/5) are consistent, whereas those of the perturbative RG theory (gamma=1.3895, eta=0.0355, omega=0.782) are inconsistent with the MC data. The seemable agreement between the RG prediction for eta and MC results at criticality, reported in literature, appears due to slightly overestimated value of the critical coupling. Estimation of critical exponents of 3D Ising model from complex zeroth of the partition function is discussed. A refined analysis yields the best estimate 1/nu=1.518. We conclude that the recent MC data can be completely explained within our theory (providing 1/nu=1.5 and omega=0.5) rather than within the conventional RG theory.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].