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arXiv:cond-mat/9511001 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Finite-size scaling and the toroidal partition function of the critical asymmetric six-vertex model

Jae Dong Noh, Doochul Kim

Published Apr 1, 1996• 29 citations
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Abstract

Finite-size corrections to the energy levels of the asymmetric six-vertex model transfer matrix are considered using the Bethe ansatz solution for the critical region. The nonuniversal complex anisotropy factor is related to the bulk susceptibilities. The universal Gaussian coupling constant g is also related to the bulk susceptibilities as g=2${\mathit{H}}^{\mathrm{\ensuremath{-}}1/2}$/\ensuremath{\pi}, H being the Hessian of the bulk free energy surface viewed as a function of the two fields. The modular covariant toroidal partition function is derived in the form of the modified Coulombic partition function which embodies the effect of incommensurability through two mismatch parameters. The effect of twisted boundary conditions is also considered. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.53.3225

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Finite-size corrections to the energy levels of the asymmetric six-vertex model transfer matrix are considered using the Bethe ansatz solution for the critical region. The nonuniversal complex anisotropy factor is related to the bulk susceptibilities. The universal Gaussian coupling constant g is also related to the bulk susceptibilities as g=2H\ensuremath−1/2{\mathit{H}}^{\mathrm{\ensuremath{-}}1/2}H\ensuremath−1/2/\ensuremath{\pi}, H being the Hessian of the bulk free energy surface viewed as a function of the two fields. The modular covariant toroidal partition function is derived in the form of the modified Coulombic partition function which embodies the effect of incommensurability through two mismatch parameters. The effect of twisted boundary conditions is also considered. \textcopyright{} 1996 The American Physical Society.
The Bethe ansatz equations are derived for the six-vertex model with general boundary weights on a lattice in a diagonal orientation. These are solved in the thermodynamic limit. Finite-size corrections to the free energy (for a restricted class of boundary weights, including those corresponding to a Potts model with free boundaries) are calculated in the critical region. The first-order term gives the surface free energy of the model. The second-order term is found to be - pi okT tan( pi nu /2 mu )c/48N' 2 where c=(1-6 mu 2 /( pi 2 - pi mu )) is the conformal anomaly. This can be compared to - pi kT sin( pi nu / mu )c/6N' 2 for a calculation on the lattice in the standard orientation and with periodic boundary conditions. This difference can be explained geometrically using conformal invariance.
The authors study finite-size corrections to the free energy in a two-dimensional random tiling model. This model is equivalent to an asymmetric six-vertex model and exhibits commensurate-incommensurate phase transitions of the Pokrovsky-Talapov type. They calculate finite-size scaling amplitudes of the free energy inside the incommensurate phase, analytically and numerically, by employing a transfer matrix method. Both periodic and free boundary conditions are considered. These amplitudes are consistent with predictions of the theory of conformal invariance with the conformal charge c=1.
The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size N as and their amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. A similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as . The partition functions for this case are found to be that of a non-relativistic free fermion system. From the symmetry of the lattice model under rotation, several identities between the partition functions are found. The scaling at zero vertical field is interpreted as a feature arising from viewing the Pokrovsky - Talapov transition with the space and time coordinates interchanged.
The critical Ising model in two dimensions with a specific defect line is analyzed to deliver the first exact solution with twisted boundary conditions. We derive exact expressions for the eigenvalues of the transfer matrix and obtain analytically the partition function and the asymptotic expansions of the free energy and inverse correlation lengths for an infinitely long cylinder of circumference L x . We find that finite-size corrections to scaling are of the form ak/Lx2k−1a_k/L^{2k-1}_xak​/Lx2k−1​ for the free energy f and bk(p)/Lx2k−1b_k(p)/L_x^{2k-1}bk​(p)/Lx2k−1​ and ck(p)/Lx2k−1c_k(p)/L_x^{2k-1}ck​(p)/Lx2k−1​ for inverse correlation lengths xip−1\\xi^{-1}_pxip−1​ and xiL−p−1\\xi^{-1}_{L-p}xiL−p−1​ , respectively, with integer values of k. By exact evaluation we find that the amplitude ratios bk(p)/akb_k(p)/a_kbk​(p)/ak​ and ck(p)/akc_k(p)/a_kck​(p)/ak​ are universal and verify this universal behavior using a perturbative conformal approach.
The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a Hermitian one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the six-vertex model with DWBC in the different phases. In particular, in the antiferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions.