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Physical review. B, Condensed matter

Superconductivity in an external magnetic field as a finite-size system

Denjoe O’Connor, Christopher Rhodes Stephens

Published Feb 1, 1991• 5 citations
Publisher

Abstract

Based on a general formulation of finite-size systems, we study superconductivity in the presence of an external magnetic field. We show that such a system can be thought of as being of finite size, then following Br\'ezin exhibit finite-size scaling for the correlation functions when d4, as the transition is approached from above. The finite-size interpretation gives an alternative explanation of why the phase transition becomes first order when the field is turned on.

DOI
10.1103/physrevb.43.3652

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Based on a general formulation of finite-size systems, we study superconductivity in the presence of an external magnetic field. We show that such a system can be thought of as being of finite size, then following Br'ezin exhibit finite-size scaling for the correlation functions when d4, as the transition is approached from above. The finite-size interpretation gives an alternative explanation of why the phase transition becomes first order when the field is turned on.
A scaling hypothesis is set up for the magnetization m(T,H;L) and susceptibility \ensuremath{\chi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\mathcal{'}}$(2d4,d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in both m and \ensuremath{\chi} are predicted, in the region of the first-order phase transition (TTc{T}_{c}Tc​) as well as in the region of the second-order phase transition (T\ensuremath{\simeq}Tc{T}_{c}Tc​), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out.
The analysis of a previous study (Allen and Pathria. Can. J. Phys. 67, 952 (1989)) on finite-size effects in systems with O(n) symmetry [Formula: see text], confined to geometry Ld−d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions, is extended (i) to include the region of first-order phase transition (T < Tc) as well as the region of second-order phase transition [Formula: see text] and (ii) to allow the presence of an external field H > 0. Predictions, involving both amplitudes and exponents, are made on the magnetization m and susceptibility χ in different regimes of the variables T, H, and L. Analytical verification of the predicted results is carried out in the case of the spherical model of ferromagnetism (n = ∞), and complete agreement is found.
The explicit analytical expression for the effective correlation length ξ as a function of reduced temperature , external field h and system size L for the Ising-like system near the phase transition point Tc is obtained. The role of these quantities in the formation of the correlation length value is ascertained. It is shown that the irregular increase of the correlation length exists only in the case of τ → 0, h → 0 and L → ∞. With deviation from these values the essential slowing down exists for the increasing . The criterium of the permissible range of temperature values (field values), where the correlation length behaviour is defined by temperature (or field) variable, is established for the fixed size of the system. Beyond this range τ < τc, h < hcr the system size becomes crucial for forming the correlation length.
The explicit analytical expression for the effective correlation length ξ as a function of reduced temperature τ , external field h and system size L for the Ising-like system near the phase transition point Tc is obtained.The role of these quantities in the formation of the correlation length value is ascertained.It is shown that the irregular increase of the correlation length exists only in the case of τ → 0, h → 0 and L → ∞.With deviation from these values the essential slowing down exists for the increasing ξ.The criterium of the permissible range of temperature values (field values), where the correlation length behaviour is defined by temperature (or field) variable, is established for the fixed size of the system.Beyond this range τ < τc, h < hcr the system size becomes crucial for forming the correlation length.
A conceptual and computational framework is developed for the general investigation of the infrared behaviour of finite size systems. In particular a definition of a finite size system is presented. It is shown that there are many systems which satisfy this definition and are therefore finite size, but whose finite size nature has not been previously uncovered. A discussion of the conditions under which the effects of finite size are important is given. The one and two particle irreducible effective actions are shown to provide powerful analytical tools for extracting information from these systems, with superconductivity in the presence of an external magnetic field chosen as a prototypical example with which to illustrate the concepts and formalism herein.