The Szego-Widom theorem provides an expression for the determinant of block Toeplitz matrices in the asymptotic limit of large matrix dimension n. We show that the presence of zero modes, i.e, eigenvalues that vanish as n, ||<1, when n, requires a modification of the Szego-Widom theorem. A new asymptotic expression for the determinant of a certain class of block Toeplitz matrices with one pair of zero modes is derived. The result is inspired by one-dimensional topological superconductors, and the relation with the latter systems is discussed.

VL - 174 UR - http://link.springer.com/10.1007/s10955-018-2177-8 JO - J Stat Phys ER - TY - JOUR T1 - Conformal field theory of critical Casimir forces between surfaces with alternating boundary conditions in two dimensions JF - Journal of Statistical Mechanics: Theory and Experiment Y1 - 2017 A1 - Dubail, Jerome A1 - Santachiara, Raoul A1 - Emig, Thorsten KW - Casimir effect; conformal field theory; critical exponents and amplitudes AB -Systems as diverse as binary mixtures and inclusions in biological membranes, and many more, can be described effectively by interacting spins. When the critical fluctuations in these systems are constrained by boundary conditions, critical Casimir forces (CCF) emerge. Here we analyze CCF between boundaries with alternating boundary conditions in two dimensions, employing conformal field theory (CFT). After presenting the concept of boundary changing operators, we specifically consider two different boundary configurations for a strip of critical Ising spins: (I) alternating equi-sized domains of up and down spins on both sides of the strip, with a possible lateral shift, and (II) alternating domains of up and down spins of different size on one side and homogeneously fixed spins on the other side of the strip. Asymptotic results for the CCF at small and large distances are derived. We introduce a novel modified Szego formula for determinants of real antisymmetric block Toeplitz matrices to obtain the exact CCF and the corresponding scaling functions at all distances. We demonstrate the existence of a surface renormalization group flow between universal force amplitudes of different magnitude and sign. The Casimir force can vanish at a stable equilibrium position that can be controlled by parameters of the boundary conditions. Lateral Casimir forces assume a universal simple cosine form at large separations.

UR - http://stacks.iop.org/1742-5468/2017/i=3/a=033201?key=crossref.e58e04fb8593248b573c53589bed0f1d JO - J. Stat. Mech. ER - TY - JOUR T1 - Critical Casimir force between inhomogeneous boundaries JF - EPL (Europhysics Letters) Y1 - 2015 A1 - Dubail, Jerome A1 - Santachiara, Raoul A1 - Emig, Thorsten AB -To study the critical Casimir force between chemically structured boundaries immersed in a binary mixture at its demixing transition, we consider a strip of Ising spins subject to alternating fixed spin boundary conditions. The system exhibits a boundary induced phase transition as function of the relative amount of up and down boundary spins. This transition is associated with a sign change of the asymptotic force and a diverging correlation length that sets the scale for the crossover between different universal force amplitudes. Using conformal field theory and a mapping to Majorana fermions, we obtain the universal scaling function of this crossover, and the force at short distances.

VL - 112 IS - 6 JO - EPL ER -