Semi-infinite Ising Model Research Articles

Early-time spinodal decomposition in the surface layer.

The early stages of spinodal decomposition in the surface layer of a semi-infinite Ising model are studied by using the theory of Langer, Bar-on, and Miller and the self-consistent method. It is shown that the speed of phase separation in the surface layer is faster than that in the bulk because of the surface enhancement, and the one-point distribution function in the surface is different from that of the bulk. It is inferred that the surface with a negative enhancement (Cg0) acts as a two-dimensional ``nucleus'' for the phase separation and for forming a ``wetting layer.''

Magnetic properties of the semi-infinite Ising model with a surface amorphization.

Magnetic properties of the semi-infinite spin-(1/2 Ising ferromagnet with a surface amorphization are investigated by the use of the effective-field theory with correlations. We find a number of characteristic behaviors for the surface magnetic properties, such as the possibility of surface reentrant phenomenon. The study of surfaces with amorphous layers may open a new field of surface magnetism.

Surface phase diagrams of a transverse Ising model with a surface amorphisation

Phase diagrams of a semi-infinite Ising model with surface amorphisation in the presence of a transverse field are studied within the framework of an effective-field theory. A number of interesting characteristics may be observed in the effect of a transverse field on the phase diagrams for amorphisation, namely the re-entrant phenomenon on the surface is obtained for appropriate values of the surface interaction and transverse field.

Mean-field theory of the semi-infinite spin-1 Ising model and adsorbed3He-4He mixtures

The semi-infinite spin-1 Blume-Emery-Griffiths model (1971) is investigated in the mean-field approximation. The model is used to study surface superfluidity in mixtures of 3He and 4He adsorbed on graphite. The various coupling constants are estimated from bulk experiments and atomic potentials, and representative phase diagrams for different couplings are discussed.

The semi-infinite Ising model with mixed ferromagnetic and antiferromagnetic interactions: a MFRG approach

The semi-infinite Ising model based on a simple cubic lattice with ferromagnetic couplings in the bulk and antiferromagnetic couplings on the surface is studied via the MFRG technique. The phase diagram exhibits a phase in which bulk ferromagnetic and surface antiferromagnetic order coexist. Estimates for the various phase boundaries are obtained.

Critical properties of the anisotropic Ising model on three-node hierarchical lattices

The criticality of the anisotropic Ising model on a three-node hierarchical lattice is investigated by an exact renormalisation group transformation. The phase diagram exhibits three physically different phases, namely a paramagnetic one and surface and bulk ferromagnetic ones. When J1 not=J/sub /2 not=J3, the system orders in the direction with the largest J before it orders in the bulk. The bulk para-ferromagnetic transition is separated into three different universality classes, i.e. one isotropic and two anisotropic. This phenomenon, which is quite different from that on anisotropic Bravais lattices and seems analogous to that of a semi-infinite Ising model, is analysed.

Semi-infinite Ising model

We study the equilibrium statistical mechanics of the semi-infinite Ising model, interpreted as a model of a binary system near a wall. In particular, the wetting transition is analyzed. In dimensionsd≧3 and at low temperature, we prove the existence of a layering transition which is of first-order.

Bulk, surface, and interface properties of the Ising model and conformal invariance.

By solving partial differential equations for the correlation functions implied by conformal invariance, we obtain exact results at the critical point for the bulk six-point function 〈\ensuremath{\sigma}(1)...\ensuremath{\sigma}(6)〉 of the two-dimensional Ising model and the three-point functions 〈\ensuremath{\sigma}(1)\ensuremath{\sigma}(2)\ensuremath{\sigma}(3)〉, 〈\ensuremath{\sigma}(1)\ensuremath{\mu}(2)\ensuremath{\mu}(3)〉 of the semi-infinite two-dimensional Ising model with fixed boundary spins. Here \ensuremath{\sigma} and \ensuremath{\mu} denote order (spin) and disorder variables, respectively. We also derive the explicit form of the magnetization profile at the bulk critical temperature in an Ising strip of infinite length and finite width, with spin-up boundary conditions on one edge and spin-down boundary conditions on the other.

Semi-infinite Ising model

Semi-infinite Ising model

Critical behavior of a semi-infinite Ising model with an edge.

A semi-infinite Ising model on a simple-cubic lattice, bounded by two perpendicular planes which meet at an edge, is studied with the Migdal-Kadanoff renormalization-group approach. It is found that the edge spins can order at a temperature higher than the bulk transition temperature if one of the surfaces is also ordered, whereas no edge ordering is possible if none of the surfaces nor the bulk are ordered. A complete phase diagram is presented in the parameter space of the model considered. Correlation-length critical exponents \ensuremath{\nu} and the crossover exponent \ensuremath{\varphi} are calculated and various universality classes are established.

Mean-field approximations to critical behaviour at surfaces

Improved mean-field-like approximations to critical behaviour at surfaces are considered. By using the high-temperature expansion of the 2D Ising model susceptibility, good estimates for critical values of the 3D semiinfinite Ising model; are obtained. As an example, the critical surface coupling enhancement is found to be Delta C=0.44, which must be compared with the standard mean-field approximation value Delta CMF=0.25 and the recent Monte Carlo result Delta CMC=0.50+or-0.03.

Finite-temperature surface properties of itinerant-electron ferromagnets

Magnetic properties of metallic surfaces at finite temperatures are discussed by using the single-site spin-fluctuation theory in which the effects of spin fluctuations both at the surface and in the bulk are taken into account on the same footing by means of the static functional-integral method. The temperature dependence of the layer-by-layer magnetic moments on the first time layers is calculated numerically for the semi-infinite simple cubic lattice with the (100) surface. Model calculations using the parameters appropriate to Ni show that the magnetic order at the 'clean' surface persists above the bulk Curie temperature, TCb, and that the amplitudes of local spin fluctuations on the first few layers are significantly enhanced near TCb compared with that in the bulk. Calculated results are compared with those obtained in the mean-field theory for the semi-infinite simple cubic Ising model and they are discussed with reference to relevant experimental data.

Dilute semi-infinite Ising model

The influence of bond and site dilution on the phase transitions of a semi-infinite three-dimensional Ising model is investigated. To determine, in particular, phase diagrams two different techniques have been used: an extension of Migdal's approach to disordered systems and a more tractable approximation introduced recently. Temperature-concentration phase diagrams for fixed values of the ratio of surface and bulk interactions can exhibit five different types of phase transitions.

Mean-field theory of magnetic transitions in semi-infinite Ising models.

Le modele semi infini de Ising, pour S=1/2 et avec un nombre arbitraire de couplages magnetiques de surface J mn , differents de ceux du volume, J, est resolu dans l'approximation du champ moyen. On obtient des expressions exactes pour les couplages critiques J mn,c , conduisant a une temperature de Curie en surface T cs superieure a la valeur T c en volume. On obtient la valeur de T cs dans le cristal semi infini par une methode de fraction continue. Application du modele a l'explication de resultats experimentaux recents obtenus sur la surface (0001) du gadolinium

Universal relations among critical amplitudes of surface quantities.

Universal relations among critical amplitudes of surface quantities are discussed for d-dimensional semi-infinite systems near their bulk critical temperature ${T}_{c}^{b}$. Based on the field-theoretic renormalization-group approach, a general derivation of the expected multi-scale-factor universality is given and various universal ratios involving critical amplitudes of the ordinary and special transitions both above and below ${T}_{c}^{b}$ are calculated to first order in \ensuremath{\epsilon}=4-d. In addition, an error in the published one-loop result for the universal order-parameter profile at the ordinary transition of the semi-infinite Ising model is corrected.

Critical behaviour at surfaces: variational approach for the free energy

The variational approach based on the Bogoliubov's inequality for the free energy is used in order to study the thermodynamical properties of the semi-infinite Ising model with modified exchange interactions Js=J(1+ Delta ) on a free surface. The critical temperature of the surface ordering, as well as the extrapolation length are obtained as a function of Delta for the two- and three-dimensional models.

Surface critical behavior of the smoothly inhomogeneous Ising model

We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large $m$, $m$ being the distance from the edge. The case $Al0$ of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when $Ag0$. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.

The Migdal-Kadanoff renormalization group scheme for the Ising model with a free surface

The Migdal-Kadanoff scheme is applied to the Ising model with a free surface. The resulting renormalization group transformation and the duality transformation commute in any dimension. Two simple recursion relations are obtained which reproduce the global phase diagram for the semi-infinite Ising model. The surface critical exponents calculated in this way are comparable to those obtained by more complex positionspace methods. In dimension d =2 + e', we find the exponents ~tl~sm--e' and y(hSm = 1 + e' for the multicritical surface-bulk transition. We also derive and discuss approximate differential recursion relations for the bulk and the surface free energies.

Renormalization-group approach to surface critical behavior in the Ising model

A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The $d=2$ square lattice and the $d=3$ bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for $d=3$ are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\ensuremath{\epsilon}$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.

Renormalization-group approach to the Ising model with a free surface

We use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.