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Physica A Statistical Mechanics and its Applications

Critical wall perturbations and a local free energy functional

Michael Ellis Fisher, Helen Au-Yang

Published Apr 1, 1980• 62 citations
Publisher
DOI
10.1016/0378-4371(80)90112-0

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The space-fractional Schrödinger equation for a local potential is difficult to solve because the fractional Riesz operator in it is nonlocal. In fractional quantum mechanics, the infinite wall is a basic problem and has not been solved yet. In this paper, we consider a free particle in an infinite wall region. Making use of the Lévy path integral method, we derive the Lévy path integral amplitude of the particle and take advantage of it to get the solution of the fractional Schrödinger equation for an infinite wall. Then, the infinite wall potential with a delta-function perturbation is also studied. According to Fox’s H-function, we get an equation of bound state energies of a free particle moving in an infinite wall region perturbed by the δ-function for two situations, E < 0 and E > 0. We give a transcendental equation that determines energy levels when E < 0 and prove that there is no bound states when E > 0. The asymptotic behaviors of the center of the delta function approaching to the origin and the infinite are also shown. The results of this paper include special cases in standard quantum mechanics.
Domain-wall free-energy, the difference in the free-energy between the system with the periodic boundary condition (BC) and that with the anti-periodic BC, is measured in the 4 dimensional ±J Ising spin glass. Temperature dependence of the domain-wall free-energy for each sample shows that the favorable BC changes with temperature, suggesting the existence of temperature chaos. We also investigate fragility of the equilibrium state against small changes of couplings { Jij } (bond chaos) by measuring the domain-wall free-energy. As a result, we find scaling behavior which suggests that the chaos exponent and the scaling function for temperature perturbation are the same as those for bond perturbation, meaning that temperature chaos and bond chaos belong to the same universarity class on fragility.
The O ( n ) ϕ 4 model on a slab bounded by free surfaces is studied for 2 < d < 4 in the limit n → ∞ . The self-consistent potential V ( z ) which the exact n → ∞ solution of the model involves is analysed by means of boundary operator expansions. Building on the known exact n → ∞ solution for V ( z ) in the semi-infinite case L = ∞ at the bulk critical point, we exactly determine two types of corrections to this potential: (i) those linear in the temperature scaling field t at L = ∞ , and (ii) the leading L -dependent (distant-wall) corrections at the critical point. From (i) exact analytical results at d = 3 are obtained for the leading temperature singularity of the excess surface free energy and the implied asymptotic behaviours of the scaling functions Θ 3 ( x ) and ϑ 3 ( x ) of the residual free energy f res = L 1 − d Θ d ( tL ) and the critical Casimir force in the limit x → 0 ±. The second derivative is computed exactly.