Casimir amplitudes in a quantum spherical model with long-range interaction

A $d$-dimensional quantum model system confined to a general hypercubical geometry with linear spatial size $L$ and ``temporal size'' $1/T$ ($T$ - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. For a film geometry in different space dimensions $\frac 12\sigma <d<\frac 32\sigma$, where $0<\sigma \leq 2$ is a parameter controlling the decay of the long-range interaction, the free energy and the Casimir amplitudes are given. We have proven that, if $d=\sigma$, the Casimir amplitude of the model, characterizing the leading temperature corrections to its ground state, is $\Delta =-16\zeta(3)/[5\sigma(4\pi)^{\sigma/2}\Gamma (\sigma /2)]$. The last implies that the universal constant $\tilde{c}=4/5$ of the model remains the same for both short, as well as long-range interactions, if one takes the normalization factor for the Gaussian model to be such that $\tilde{c}=1$. This is a generalization to the case of long-range interaction of the well known result due to Sachdev. That constant differs from the corresponding one characterizing the leading finite-size corrections at zero temperature which for $d=\sigma=1$ is $\tilde c=0.606$.Comment: 10 pages latex, no figures, to appear in EPJB (2000

Low-temperature regimes and finite-size scaling in a quantum spherical model

A $d$--dimensional quantum model in the spherical approximation confined to a general geometry of the form $L^{d-d^{\prime}} \times\infty^{d^{\prime}}\times L_{\tau}^{z}$ ($L$--linear space size and $L_{\tau}$--temporal size) and subjected to periodic boundary conditions is considered. Because of its close relation with the quantum rotors model it can be regarded as an effective model for studying the low-temperature behavior of the quantum Heisenberg antiferromagnets. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1<d<3$ and $0\leq d^{\prime}\leq d$ a detailed analysis, in terms of the special functions of classical mathematics, for the free energy, the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.Comment: 36 pages, Revtex+epsf, 3 figures included. Some minor corrections are don

On the finite-size behavior of systems with asymptotically large critical shift

Exact results of the finite-size behavior of the susceptibility in three-dimensional mean spherical model films under Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions are presented. The corresponding scaling functions are explicitly derived and their asymptotics close to, above and below the bulk critical temperature $T_c$ are obtained. The results can be incorporated in the framework of the finite-size scaling theory where the exponent $\lambda$ characterizing the shift of the finite-size critical temperature with respect to $T_c$ is smaller than $1/\nu$, with $\nu$ being the critical exponent of the bulk correlation length.Comment: 24 pages, late

Rings Additively Generated by Periodic Elements

In the present paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major cases, additively periodic rings remain periodic too; this includes, for instance, algebraic algebras, group rings, and matrix rings over commutative rings. Moreover, we also obtain some independent results for the new class of rings; for example, the triangular matrix rings retain that property.Comment: 16 page

Non-universal size dependence of the free energy of confined systems near criticality

The singular part of the finite-size free energy density $f_s$ of the O(n) symmetric $\phi^4$ field theory in the large-n limit is calculated at finite cutoff for confined geometries of linear size L with periodic boundary conditions in 2 < d < 4 dimensions. We find that a sharp cutoff $\Lambda$ causes a non-universal leading size dependence $f_s \sim \Lambda^{d-2} L^{-2}$ near $T_c$ which dominates the universal scaling term $\sim L^{-d}$. This implies a non-universal critical Casimir effect at $T_c$ and a leading non-scaling term $\sim L^{-2}$ of the finite-size specific heat above $T_c$.Comment: RevTex, 4 page