The analysis of published experimental data for seven different binary liquid mixtures allows us to test the concept of universality for critical phenomena at interfaces and to compare the values of three independent universal surface amplitudes with theoretical predictions. In spite of uncomfortable large experimental error bars in the case of one surface amplitude we find agreement for five different systems and disagreement for two. For the other two surface amplitudes we obtain a marginal agreement between theory and experiment which is impeded by the lack of experimental error bars. We also test the scaling functions for critical adsorption. In view of the complexity of the data analysis the agreement between theory and experiment is satisfactory and encouraging. Experimental improvements are suggested in order to obtain more stringent tests.

Universal amplitudes and profiles for critical adsorption

Monte Carlo simulations based on an integration scheme for free energy differences is used to compute critical Casimir forces for three-dimensional Ising films with various boundary fields. We study the scaling behavior of the critical Casimir force, including the scaling variable related to the boundary fields. Finite size corrections to scaling are taken into account. We pay special attention to that range of surface field strengths within which the force changes from repulsive to attractive upon increasing the temperature. Our data are compared with other results available in the literature.

Critical Casimir forces for Ising films with variable boundary fields.

In our previous paper (Kitanine et al 2017 J. Phys. A: Math. Theor. 50 224001) we have obtained, for the XXX spin-1/2 Heisenberg open chain, new determinant representations for the scalar products of separate states in the quantum separation of variables (SoV) framework. In this article we perform a similar study in a more complicated case: the XXZ open spin-1/2 chain with the most general integrable boundary terms. We first revisit the SoV solution of this model using a matrix version of the Vertex-IRF gauge transformation so as to reduce one of the boundary K-matrices to a diagonal form. As usual within the SoV approach, the scalar products of separate states are computed in terms of dressed Vandermonde determinants having an intricate dependency on the inhomogeneity parameters. We show that these determinants can be transformed into different ones in which the homogeneous limit can be taken straightforwardly. These representations generalize in a non-trivial manner to the trigonometric case the expressions found previously in the rational case. We also show that generically all scalar products can be expressed in a form which is similar to—although more cumbersome than—the well-known Slavnov determinant representation for the scalar products of the Bethe states of the periodic chain. Considering a special choice of the boundary parameters relevant in the thermodynamic limit to describe the half infinite chain with a general boundary, we particularize these representations to the case of one of the two states being an eigenstate. We obtain simplified formulas that should be of direct use to compute the form factors and correlation functions of this model.

https://iopscience.iop.org/article/10.1088/1751-8121/aa6cc9/pdf

The open XXX spin chain in the SoV framework: scalar product of separate states

The density profile ρ(z) and Gibbs adsorption Γ of a near-critical fluid confined between two identical planar walls is studied by means of Monte Carlo simulation and by density functional theory for a Lennard-Jones fluid By reducing the strength of wall–fluid interactions relative to fluid–fluid interactions we observe a crossover from behavior characteristic of the normal surface universality class, strong critical adsorption, to behavior characteristic of a “neutral” wall The crossover is reminiscent of that which occurs near the ordinary surface transition in Ising films subject to vanishing surface fields For the “neutral” wall ρ(z), away from the walls, is almost constant throughout the slit capillary and gives rise to a Γ that is constant along the critical isochore The same “neutral” wall yields a line of capillary coexistence that is almost identical to the bulk coexistence line In the crossover regime we observe features in ρ(z) similar to those found in the magnetization profile of the critical Ising film subject to weak surface fields, namely two smooth maxima, located away from the walls, which merge into a single maximum at midpoint as the strength of the wall–fluid interaction is reduced or as the distance between walls is decreased We discuss similarities and differences between the surface critical behavior of fluids and of Ising magnets

Effects of Weak Surface Fields on the Density Profiles and Adsorption of a Confined Fluid Near Bulk Criticality

We study the transfer matrix spectral problem for the cyclic representationsof the trigonometric 6-vertex reflection algebra associated to theBazhanov-Stroganov Lax operator. The results apply as well to the spectralanalysis of the lattice sine-Gordon model with integrable open boundaryconditions. This spectral analysis is developed by implementing the method ofseparation of variables (SoV). The transfer matrix spectrum (both eigenvaluesand eigenstates) is completely characterized in terms of the set of solutionsto a discrete system of polynomial equations in a given class of functions.Moreover, we prove an equivalent characterization as the set of solutions to aBaxter's like T-Q functional equation and rewrite the transfer matrixeigenstates in an algebraic Bethe ansatz form. In order to explain our methodin a simple case, the present paper is restricted to representations containingone constraint on the boundary parameters and on the parameters of theBazhanov-Stroganov Lax operator. In a next article, some more technical tools(like Baxter's gauge transformations) will be introduced to extend our approachto general integrable boundary conditions.

/pdf/transfer-matrix-spectrum-for-cyclic-representations-of-the-6-k447tllbod.pdf

R. Z. Bariev

Papers

Correlation functions of semi-infinite two-dimensional ising model. III. influence of a “fixed” boundary