Critical exponents for the O(n) spin model local field near a surface, an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or corner is formed by mutually perpendicular (d-1)-dimensional walls, each having von Neumann (special transition) or Dirichlet (ordinary transitions) boundary conditions. The combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order in ( in =4-d). The case of two or more von Neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities.
Critical exponents for the O(n) spin model local field near a surface, an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or corner is formed by mutually perpendicular (d-1)-dimensional walls, each having von Neumann (special transition) or Dirichlet (ordinary transitions) boundary conditions. The combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order in ( in =4-d). The case of two or more von Neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities.
The previously introduced method of mean-field renormalisation has provided a unified approach to bulk and surface critical behaviour. The method is used for the computation of the critical exponent associated with the order parameter at corners. Applications to the square and triangular Ising model are presented. The angle dependence of the corner critical exponent is qualitatively reproduced. The results for the bulk critical exponents are more accurate than in previously employed mean-field renormalisation schemes.
Using the strong disorder renormalization group method, we study numerically the critical behavior of the random transverse Ising model at a free surface, at a corner, and at an edge in two-, three-, and four-dimensional lattices. The surface magnetization exponents are found to be xs=1.60(2), 2.65(15), and 3.7(1) in D=2, 3, and 4, respectively, which do not depend on the form of disorder. We have also studied critical magnetization profiles in slab, pyramid, and wedge geometries with fixed-free boundary conditions and analyzed their scaling behavior.
Abstract. A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some ρ> 0, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility χ1,1 has been evaluated exactly. For ρ = 1 we find that χ1,1 is finite at the bulk critical temperature Tc, in contrast with the recently derived value γ1,1 = 1 in the case of just one global spherical constraint. The result γ1,1 = 1 is recovered only if ρ = ρc = 2 − (12Kc) −1, where Kc is the dimensionless critical coupling. When ρ> ρc, χ1,1 diverges exponentially as T → T + c. An effective hamiltonian which leads to an exactly solvable model with γ1,1 = 2, the value for the n → ∞ limit of the corresponding O(n) model, is proposed too.
The O(n) loop model on the honeycomb lattice with mixed ordinary and special boundary conditions is solved exactly by means of the Bethe ansatz. The calculation of the dominant finite-size corrections to the eigenspectrum yields the mixed boundary scaling index and the geometric scaling dimensions describing the universal surface critical behaviour. Exact results follow in the limit n=0 for the polymer adsorption transition with a mixed adsorbing and free boundary. These include the new configurational exponent gamma l =85/64.
With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane y>0 with different boundary conditions a and b on the negative and positive x axes. For ab=-+ and f+, they determined the one- and two-point averages of the spin σ and energy ε. Here +,-, and f stand for spin-up, spin-down, and free-spin boundaries, respectively. The case +-+-+⋯, where the boundary condition switches between + and - at arbitrary points, ζ_{1},ζ_{2},⋯ on the x axis was also analyzed. In the first half of this paper a similar study is carried out for the alternating boundary condition +f+f+⋯ and the case -f+ of three different boundary conditions. Exact results for the one- and two-point averages of σ,ε, and the stress tensor T are derived with conformal-invariance methods. From the results for 〈T〉, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is derived for mixed boundary conditions. In the second half of the paper, arbitrary two-dimensional critical systems with mixed boundary conditions are analyzed with boundary-operator expansions. Two distinct types of expansions-away from switching points of the boundary condition and at switching points-are considered. Using the expansions, we express the asymptotic behavior of two-point averages near boundaries in terms of one-point averages. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge. Finally we confirm the consistency of the predictions obtained with conformal-invariance methods and with boundary-operator expansions, in the the first and second halves of the paper.
The critical behaviour of a magnetic system with O(N) spin symmetry, bounded by two (d-1)-dimensional hyperplanes meeting at an angle alpha , is studied within mean field theory and in d=4- epsilon dimensions. New exponents emerge for correlation functions, and magnetisation and susceptibilities, for spins close to the edge. They can be expressed in terms of known bulk and surface exponents, together with a single new edge exponent which depends, however, on the angle alpha . This exponent is computed to first order in epsilon .