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Physical review. B, Condensed matter

Universal size dependence of the free energy of finite systems near criticality

Vladimir Privman A

Published Nov 1, 1988• 60 citations
Publisher

Abstract

A scaling framework is developed for the emergence of the universal finite-size correction $\frac{(u\mathrm{ln}L)}{{L}^{2}}$, recently found by Cardy and Peschel in the free-energy density of the two-dimensional (2D) conformally invariant models at ${T}_{c}$. Our approach applies in the critical region $T\ensuremath{\simeq}{T}_{\mathrm{cZ}}$, and accounts for the universality of $u$, but not its value. We predict a similar universal term, $\frac{(u\mathrm{ln}L)}{{L}^{3}}$, in 3D. We also find that the "corner" free energy in the bulk ($L=\ensuremath{\infty}$) has the leading singular behavior ($\ensuremath{-}\ensuremath{\nu}u\mathrm{ln}|t|$).

DOI
10.1103/physrevb.38.9261

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A scaling framework is developed for the emergence of the universal finite-size correction (ulnL)L2\frac{(u\mathrm{ln}L)}{{L}^{2}}L2(ulnL)​, recently found by Cardy and Peschel in the free-energy density of the two-dimensional (2D) conformally invariant models at Tc{T}_{c}Tc​. Our approach applies in the critical region T\ensuremath≃TcZT\ensuremath{\simeq}{T}_{\mathrm{cZ}}T\ensuremath≃TcZ​, and accounts for the universality of uuu, but not its value. We predict a similar universal term, (ulnL)L3\frac{(u\mathrm{ln}L)}{{L}^{3}}L3(ulnL)​, in 3D. We also find that the "corner" free energy in the bulk (L=\ensuremath∞L=\ensuremath{\infty}L=\ensuremath∞) has the leading singular behavior (\ensuremath−\ensuremathνuln∣t∣\ensuremath{-}\ensuremath{\nu}u\mathrm{ln}|t|\ensuremath−\ensuremathνuln∣t∣).
Abstract Based on the results published recently (Hucht 2017 J. Phys. A: Math. Theor . 50 065201), the universal finite-size contributions to the free energy of the square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit L , M → ∞ , T → T c , with fixed temperature scaling variable x ∝ ( T / T c − 1 ) M and fixed aspect ratio ρ ∝ L / M . We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point T = T c we confirm predictions from conformal field theory (Cardy and Peschel 1988 Nucl. Phys . B 300 377, Kleban and Vassileva 1991 J. Phys. A: Math. Gen . 24 3407). The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.
The free energy of a two-dimensional system at criticality has in general a universal part proportional to the logarithm of the system size. This term was shown by Cardy and Peschel to be related to the curvature of the system, with smooth metrics and singular points contributing in distinct ways. In this paper we present a numerical study of the effect for the Ising model on lattices with various topologies ranging sphere to genus two surfaces. The nature of this term for specific lattice models is an open problem because the distinction between the two kinds of contributions involves an interchange of the limit of singular curvature with the thermodynamic limit. For all the lattices studied we found precise agreement with the conformal field theory prediction for conical singularities.
We show that the leading finite-size correction to lnZ for a two-dimensional system at a conformally invariant critical point on a strip of length L width \ensuremath{\beta}(\ensuremath{\beta}\ensuremath{\ll}L) is (\ensuremath{\pi}/6)c(L/\ensuremath{\beta}), where c is the conformal anomaly. Equivalently, the leading low-temperature correction to the free energy of a one-dimensional quantum system is -(\ensuremath{\pi}/6)cL(kT)2{)}^{2})2\ensuremath{\Elzxh}v, where v is the effective ``velocity of light.'' The latter formula is used to check recently derived critical theories of spin-s quantum chains against Bethe-Ansatz solutions.
The structure of finite-size scaling within the framework of the field-theoretic renormalization-group approach is examined. New size-dependent ultraviolet divergences are introduced into the theory. It is shown, nonetheless, that ``two-scale-factor'' universality reflects itself in the finite-size free energy and produces a scaling form such as that discussed by Privman and Fisher. A consequence is that the singular part of the finite-system free energy at the bulk critical point is universal. Extensions to surface and interfacial free energies, first proposed from the results of Ising simulations, are supported qualitatively.