Anisotropic Spherical Model Research Articles

Semi-infinite anisotropic spherical model: Correlations atT>~Tc

The ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model (ASM), which is the limit $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty}$ of the $D$-component uniaxially anisotropic classical vector model. The bulk limit of the ASM is similar to that of the spherical model, apart from the role of the anisotropy stabilizing ordering for low lattice dimensions, $d<~2$, at finite temperatures. The correlation functions and the energy density profile in the semi-infinite ASM are calculated analytically and numerically for $T>~{T}_{c}$ and $1<~d<~\ensuremath{\infty}$. Since the lattice dimensionalities $d=1$, 2, 3, and 4 are special, a continuous spatial dimensionality ${d}^{\ensuremath{'}}=d\ensuremath{-}1$ has been introduced for dimensions parallel to the surface. However, preserving a discrete layer structure perpendicular to the surface avoids unphysical surface singularities and allows numerical solitions that reveal significant short-range features near the surface. The results obtained generalize the isotropic-criticality results for $2<d<4$ of Bray and Moore [Phys. Rev. Lett. 38, 735 (1977); J. Phys. A 10, 1927 (1977)].

Numerical Stability of a Family of Osipkov‐Merritt Models

We have investigated the stability of a set of nonrotating anisotropic spherical models with a phase-space distribution function of the Osipkov-Merritt type. The velocity distribution in these models is isotropic near the center and becomes radially anisotropic at large radii. The models are special members of the family studied by Dehnen and by Tremaine et al. in which the mass density has a power-law cusp ρ ∝ r-γ at small radii and decays as ρ ∝ r-4 at large radii. The radial-orbit instability of models with γ = 0, 1/2, 1, 3/2, and 2 was studied using an N-body code written by one of us and based on the self-consistent field method developed by Hernquist & Ostriker. These simulations have allowed us to delineate a boundary in the (γ, ra)-plane that separates the stable from the unstable models. This boundary is given by 2Tr/Tt = 2.31 ± 0.27 for the ratio of the total radial to tangential kinetic energy. We also found that the stability criterion df/dQ ≤ 0, recently raised by Hjorth, gives lower values compared with our numerical results. The stability to radial modes of some Osipkov-Merritt γ-models that fail to satisfy the Doremus-Feix criterion ∂f/∂E < 0 has been studied using the same N-body code but retaining only the l = 0 terms in the potential expansion. We have found no signs of radial instabilities for these models.

Bloch wall phase transition in the anisotropic spherical model (abstract)

The temperature-induced second-order phase transition from Bloch to linear (Ising-like) domain walls in uniaxial ferromagnets was predicted theoretically1 within the mean-field approximation and observed in the dynamic susceptibility experiments on Sr hexaferrite.2 Here the fluctuational effects at this phase transition are investigated for the exactly solvable anisotropic model of D-component classical spin vectors in the limit D→∞. This anisotropic spherical model is equivalent to the common spherical model in the homogeneous case, but deviates from it and is free from nonphysical behavior in a general inhomogeneous situation. It is shown that the thermal fluctuations of the transverse magnetization in the wall (the Bloch wall order parameter) result in the diminishing of the domain wall transition temperature TB in comparison to its mean-field value, thus favoring the existence of linear walls. For ensuring finite values of TB, an additional anisotropy in the basal plane x, y is required: In purely uniaxial ferromagnets a domain wall behaves like a two-dimensional system with a continuous spin symmetry and does not order into the Bloch wall at any nonzero temperature. The theory qualitatively explains the fact that the value TB=0.99Tc measured on the Sr hexaferrite2 is substantially more remote from Tc than the mean-field estimation TB=0.996Tc, as well as some other characteristics of domain wall relaxation in the critical region.

Susceptibilities and correlation functions of the anisotropic spherical model

The static transverse and longitudinal correlation functions (CF) of a 3-dimensional ferromagnet are calculated for the exactly solvable anisotropic spherical model (ASM) determined as the limit D \to \infty of the classical D-component vector model. The results are nonequivalent to those for the standard spherical model of Berlin and Kac even in the isotropic case. Whereas the transverse CF has the usual Ornstein-Zernike form for small wave vectors, the longitudinal CF shows a nontrivial behavior in the ordered region caused by spin-wave fluctuations. In particular, in the isotropic case below T_c one has S_{zz}(k) \propto 1/k (the result of the spin-wave theory) for k \lsim \kappa_m \propto T_c-T.