Nicholay S. Tonchev

Generalized Mittag–Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long-range interaction

The difficulties arising in the investigation of finite-size scaling in d-dimensional O(n) systems with strong anisotropy and/or long-range interaction, decaying with the interparticle distance r as r−d−σ(0 < σ ≤ 2), are discussed. Some integral representations aiming at the simplification of the investigations are presented for the classical and quantum lattice sums that take place in the theory. Special attention is paid to a more general form allowing to treat both cases on an equal footing and in addition cases with strong anisotropic interactions and different geometries. The analysis is simplified further by expressing this general form in terms of a generalization of the Mittag–Leffler special functions. This turned out to be very useful for the extraction of asymptotic finite-size behaviours of the thermodynamic functions.

CRITICAL BEHAVIOR OF SYSTEMS WITH LONG-RANGE INTERACTION IN RESTRICTED GEOMETRY

The present review is devoted to the problems of finite-size scaling due to the presence of long-range interaction decaying at large distance as 1/rd+σ, σ>0. The attention is focused mainly on the renormalization group results in the framework of -theory for systems with fully finite (block) geometry under periodic boundary conditions. Some bulk critical properties and Monte Carlo results are also reviewed. The role of the cut-off effects as well as their relation with those originating from the long-range interaction is additionally discussed. Special attention is paid to the description of an adequate mathematical technique that allows for the treatment of the long-range and short-range interactions equally. The review concludes with a short discussion of some open problems.

Finite-size scaling investigations in the quantum varphi4-model with long-range interaction

In this paper, we study in detail the critical behaviour of the (n ) quantum 4 -model with long-range interaction decaying with distances r by a power law as r -d - in the large-n limit. The zero-temperature critical behaviour is discussed. Its alteration by finite temperature and/or finite sizes in the space is studied. The scaling behaviours are studied in different regimes depending upon whether the finite temperature or the finite sizes of the system lead. A number of results for the correlation length, critical amplitudes and the finite-size shift, for different dimensionalities between the lower d = 3 /2 critical dimensions, are calculated.

Exact results for some Madelung-type constants in the finite-size scaling theory

A general formula is obtained from which the madelung type constant:

On the finite-temperature generalization of the C-theorem and the interplay between classical and quantum fluctuations

C(d|\nu)=\int_0^\infty dx x^{d/2-\nu-1}[(\sum_{l=-\infty}^\infty e^{-xl^2})^d-1-(\frac\pi x)^{d/2}]

Finite-size scaling in disordered systems.

extensively used in the finite-size scaling theory is computed analytically for some particular cases of the parameters

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

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Quantum critical scaling and the Gross-Neveu model in 2+1 dimensions

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