Physical review. B, Condensed matter

Universal critical amplitudes in finite-size scaling

Vladimir Privman A, Michael Ellis Fisher

Published Jul 1, 1984• 539 citations

Abstract

It is argued that there is no nonuniversal, system-dependent, multiplicative metric factor in the finite-size scaling relation for the singular part of the free energy near a bulk critical point. New universality properties for various critical-point amplitudes follow: one universal ratio is $\frac{L}{{\ensuremath{\xi}}_{\ensuremath{\parallel}}({T}_{c})}$ in finite-size transfer-matrix calculations.

DOI: 10.1103/physrevb.30.322

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Physical review. B, Condensed matter

Universal critical amplitudes in finite-size scaling

Vladimir Privman A, Michael Ellis Fisher

Published Jul 1, 1984• 539 citations

Publisher

Abstract

\frac{L}{{\ensuremath{\xi}}_{\ensuremath{\parallel}}({T}_{c})}

in finite-size transfer-matrix calculations.

DOI: 10.1103/physrevb.30.322

Related papers

Hyperuniversality and the renormalization group for finite systems

Hong Min Guo, David Jasnow

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.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.

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

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\frac{L}{{\ensuremath{\xi}}_{\ensuremath{\parallel}}({T}_{c})}

in finite-size transfer-matrix calculations.

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 ($\ensuremath{-}\ensuremath{\nu}u\mathrm{ln}|t|$).

Universal relations among critical amplitudes of surface quantities are discussed for d-dimensional semi-infinite systems near their bulk critical temperature

{T}_{c}^{b}

. Based on the field-theoretic renormalization-group approach, a general derivation of the expected multi-scale-factor universality is given and various universal ratios involving critical amplitudes of the ordinary and special transitions both above and below

{T}_{c}^{b}

are calculated to first order in \ensuremath{\epsilon}=4-d. In addition, an error in the published one-loop result for the universal order-parameter profile at the ordinary transition of the semi-infinite Ising model is corrected.