Theory of Critical Phenomena in Finite-Size Systems - Scaling and Quantum Effects

Abstract

Library of Congress Cataloging-in-PublicationAll rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. PrefaceIt is hard to find a field in theoretical physics which can compete with the theory of phase transitions and critical phenomena with respect to the number of written textbooks, reviews and monographs. In our opinion, writing a new book can be justified only by treating in a different manner new trends existing in the area. What do we have in mind in our case? There are just a few books on critical phenomena in systems with confined geometry: a collection of reprints [Cardy, ed. (1988)], a collection of reviews [Privman, ed. (1990)], and the monograph [Krech (1994)] on the Casimir effect. Indeed, in some modern texbooks on critical phenomena one can find special chapters devoted to this topic, see, e.g., [Cardy (1996)], [Domb (1996)], [Zinn-Justin (1996)], [Henkel (1999)]. Against the background of the numerous papers that appear annually, the gap in the monographic literature on the subject is obvious. The present book attempts to partially fill up this gap. We hope also to give our modest contribution in spreading the scaling ideas for fruitful interpretation and analysis of phase transitions in classical and quantum systems of finite volume. It is a well known fact that the volume is an irrelevant parameter for the local properties of a macroscopic system and, therefore, can be chosen arbi trary large. The conventional statistical mechanical theory studies abstract systems, consisting of infinitely many particles in an infinite volume, due to the essential simplifications that occur in their description. Moreover, it becomes possible to describe phase transitions mathematically in terms of discontinuous or singular behavior of some thermodynamic functions. In constructing the above, so called thermodynamic limit [Van Hove (1949) VU1Preface tities. For example, in the canonical Gibbs ensemble the increase in the number of particles N has to be accompanied by a proportional increase in the volume V, so that the density p = N/V be constant. Any intensive quantity ay of a finite system can be written in the form ay = a,oo + Say, where a^ is the bulk value and Say is a finite-size correction which tends to zero as V -► oo. The finite-size correction Say contains a more detailed information about the shape of the system and the boundary conditions. Usually, the correction term becomes essential under rather special condi tions, e.g., in the vicinity of a second-order phase transition. When the relevant thermodynamic parameters approach a critical point, t...

Universality of the thermodynamic Casimir effect

“…(1.7) implies a leading nonuniversal (cutoff dependent) non-scaling term ∼ L −2 in the behavior of the Casimir force, because the scaling function X sr (x) ∼ exp(−x) when x ≫ 1 [7,41,42,43,45]. Therefore Eqs.…”

“…Finite-size scaling asserts that near the bulk critical temperature T c the influence of a finite sample size L on critical phenomena is governed by universal finite-size scaling functions which depend on the ratio L/ξ, so that the rounding of the thermodynamic singularities sets in for L/ξ ≃ O(1) [7,41,42,43,44,45].…”

“…As a case study we quote the expression for the total free energy density of a fully finite mean spherical model (the n → ∞ limit of an O(n) model) with nearestneighbor interactions of strength J on a hypercubic lattice, which is given by [7] βf (K, h|L,…”

“…(2.2) and, c.f., Eq.(2.13)). In order to provide a general description of finite-size scaling of the free energy, we will therefore consider the quantity [7] (see also Eq. (14) in Ref.…”

“…The confinement of critical fluctuations of an order parameter also induces long-ranged forces between the system boundaries [5,6,7]. This is known as the statisticalmechanical (thermodynamic) Casimir effect.…”