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Berichte der Bunsengesellschaft für physikalische Chemie

On the quantum finite‐size scaling

Elka Korutcheva, N. S. Tonchev

Published Mar 1, 1994• 0 citations
Publisher

Abstract

Abstract We verify the finite‐size scaling hypothesis in the presence of quantum fluctuations up to one‐loop order in the ϵ‐expansion, using the formalism of Brézin [1], Brézin and Zinn‐Justin [2] and Rudnick et al. [3]. The analysis is performed within the framework of three models commonly used in the theory of structural phase transitions, superconducting phase transitions and Bose systems.

DOI
10.1002/bbpc.19940980349

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Abstract We verify the finite‐size scaling hypothesis in the presence of quantum fluctuations up to one‐loop order in the ϵ‐expansion, using the formalism of Brézin [1], Brézin and Zinn‐Justin [2] and Rudnick et al. [3]. The analysis is performed within the framework of three models commonly used in the theory of structural phase transitions, superconducting phase transitions and Bose systems.
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For systems with infinite-order phase transitions, in which an order parameter smoothly becomes nonzero, a new observable for finite-size scaling analysis is suggested. By construction this new observable has the favourable property of diverging at the critical point. Focussing on the example of the F-model we compare the analysis of this observable with that of another observable, which is also derived from the order parameter but does not diverge, as well as that of the associated susceptibility. We discuss the difficulties that arise in the finite-size scaling analysis of such systems. In particular we show that one may reach incorrect conclusions from large-system size extrapolations of observables that are not known to diverge at the critical point. Our work suggests that one should base finite-size scaling analyses for infinite-order phase transitions only on observables that are guaranteed to diverge.
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In this paper, I give an overview over a recently developed rigorous theory of finite-size scaling near first order transitions. Leaving out details of the mathematical proofs, the main emphasis is put on the underlying physical ideas and the discussion of the validity of the results for regions which are, in the sense of mathematical rigour, not covered by the original papers. I present both the finite-size scaling for cubic systems and for long cylinders, discussing also a recent controversy which stems from our work on the finite-size scaling of the mass gap in long cylinders.