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Physical Review Letters

Universal term in the free energy at a critical point and the conformal anomaly

Ian K. Affleck

Published Feb 17, 1986• 1,219 citations
Publisher

Abstract

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${)}^{2}$\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.

DOI
10.1103/physrevlett.56.746

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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.
We show that for conformally invariant two-dimensional systems, the amplitude of the finite-size corrections to the free energy of an infinitely long strip of width L at criticality is linearly related to the conformal anomaly number c, for various boundary conditions. The result is confirmed by renormalization-group arguments and numerical calculations. It is also related to the magnitude of the Casimir effect in an interacting one-dimensional field theory, and to the low-temperature specific heat in quantum chains.
The finite-size effects in the spectrum of the integrable XXZ Heisenberg chain with arbitrary spin-S are studied analytically on the basis of the string hypothesis for bound states and numerically by solving the associated Bethe ansatz equations. The finite-size corrections to the spectrum and to the low-temperature behaviour of the free energy are found to be similar to the universal predictions for conformally invariant theories, although the model is, in general, not Lorentz invariant, since it can have an arbitrary number of branches for low-energy excitations with different Fermi velocities depending on the value of S and the anisotropy parameter gamma .
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 ($\ensuremath{-}\ensuremath{\nu}u\mathrm{ln}|t|$).
The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}\ensuremath{\pi}/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.
The two-dimensional Ising model with a slit is studied. The slit free energy is defined, in which the bulk term, edge terms, and corner terms other than that of the slit are canceled. The bond propagation algorithm is used to calculate the free energy, internal energy, and heat capacity numerically. At the critical point, the slit free energy has a logarithmic correction L^{-1}lnL, in addition to the correction lnL, where L is the typical length of the system. The fitted coefficients of these terms agree with the conformal field theory very well. The integral constant and extrapolation length, which are pending in the conformal field theory, are obtained. In the slit internal energy, the leading term is proportional to L. There is a correction term, ln^{2}L, that has a geometry-independent coefficient, the exact value of which is conjectured to be -sqrt[2]/(2π^{2}). The leading term in the slit heat capacity is proportional to L^{2}. There is also a ln^{2}L term; however, its coefficient depends on the geometry. In the infinitely long strip limit, its exact value is conjectured to be 5/π^{2}. Near the critical point, for different size lattices, the rescaled slit free energies, internal energies, and heat capacities collapse onto the corresponding universal curves. According to the scaling and conformal field theory, the leading term of the slit free energy should be -c/8ln|t| for L|t|≫1 and |t|≪1, where t is the reduced temperature. The amplitude c/8 is universal, and c is the central charge. Our results validate these predictions qualitatively. For L|t|≫1 and |t|≪1, the slit internal energy and heat capacity scale as t^{-1} and t^{-2}, respectively.