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Journal of the Physical Society of Japan

Finite Size Correction and Conformal Anomaly for O(n) Spin System

Junji Suzuki

Published Sep 15, 1988• 23 citations
Publisher

Abstract

The finite size correction to the free energy of the critical O( n ) model is calculated. With the use of the formula obtained by Blöte et al. , the present result can predict the conformal anomaly of the model. In this way, the parameter dependence of the conformal anomaly predicted by Dotsenko and Fateev is confirmed in a certain parameter region.

DOI
10.1143/jpsj.57.2966

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The finite size correction to the free energy of the critical O( n ) model is calculated. With the use of the formula obtained by Blöte et al. , the present result can predict the conformal anomaly of the model. In this way, the parameter dependence of the conformal anomaly predicted by Dotsenko and Fateev is confirmed in a certain parameter region.
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 critical O(n)\mathrm{O}(n)O(n) model on a finite honeycomb lattice is solved by the Bethe-Ansatz method. Amplitudes of the dominant finite-size corrections to part of the eigenspectrum are obtained analytically. A comparison with Cardy's predictions from the theory of conformal invariance leads to the exact results for the conformal anomaly and scaling dimensions.
The effect of anisotropic interactions on the eigenvalue spectrum of the row-to-row transfer matrix of critical lattice spin models is investigated. It is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta =(a y /a x ) sin theta where a y and a x are lattice spacings and theta is an angle describing the anisotropy. For exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices. This is done for the eight-vertex model, hard hexagons and interacting hard squares and it is found that theta = pi u/ lambda , 10u/3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter. For each of these models, the amplitude of the finite-size corrections to the free energy at criticality is found to be of the form pi zeta c/6N 2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c=1, 4/5 and 7/10 respectively. This is an analytic result for the eight-vertex model. For the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to N=48. Finally, the authors argue that the anisotropy angle for magnetic hard squares and the q-state Potts models is also given by theta = pi u/ lambda , so this result is quite general.
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 $T\ensuremath{\simeq}{T}_{\mathrm{cZ}}$, 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 ($L=\ensuremath{\infty}$) has the leading singular behavior ($\ensuremath{-}\ensuremath{\nu}u\mathrm{ln}|t|$).