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arXiv:cond-mat/9910508 · Journal of Physics A Mathematical and General

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

Hassan Chamati, N. S. Tonchev

Published Feb 1, 2000• 15 citations
arXivHTMLPDF

Abstract

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 < = /2 and the upper d > = 3 /2 critical dimensions, are calculated.

DOI
10.1088/0305-4470/33/5/305

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Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry Ld−d′×∞d′, periodic boundary conditions, and long-range interactions decaying like R−d−σ at large distances R, with arbitrary real parameters 0<σ≤2, σ≤d≤2σ, d′≥0. The analytical technique used is based on integral transformations with kernels of Mittag–Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the Ld−d′×∞d′ geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the ε-expansion when ε→0+ neither at d=σ+ε, nor at d=2σ−ε. Moreover, the applicability of the ε-expansion is tested in the specific finite-size case of d′=σ±ε. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.
We address the problem of the definition of the finite-volume correlation length. First, we study the large-N limit of the N-vector model, and we show the existence of several constraints on the definition if regularity of the finite-size scaling functions and correct anomalous behaviour above the upper critical dimension are required. Then, we study in detail a model in which the zero mode is prohibited. Such. a model is a generalization of the fixed-magnetization Ising model which is equivalent to the lattice gas. Also in this case, we find that the finite-volume correlation length must satisfy appropriate constraints in order to obtain regular finite-size scaling functions, and, above the upper critical dimension, an anomalous scaling behaviour. The large-N results are confirmed by a one-loop calculation in the lattice phi (4) theory.