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Physical review. B, Condensed matter

Critical behavior of a 19-vertex model with full frustration

Yasushi Honda, Tsuyoshi Horiguchi

Published Nov 1, 1998• 7 citations
Publisher

Abstract

We apply the singular value decomposition method to matrices made from eigenvectors of the transfer matrix of a 19-vertex model on a square lattice with full frustration. The conformal anomaly c is estimated to be close to 3/2. This result suggests that the 19-vertex model with full frustration has a single transition due to two kinds of degrees of freedom, each of which has U(1) symmetry and ${Z}_{2}$ symmetry. Moreover, we find that the value of index $\ensuremath{\eta}$ is smaller than that at the Berezinskii-Kosterlitz-Thouless (BKT) transition by considering a logarithmic correction to the finite size analysis. This result for $\ensuremath{\eta}$ suggests that there is another universality class which is different from that of either the Onsager type or the BKT type.

DOI
10.1103/physrevb.58.11501

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We apply the singular value decomposition method to matrices made from eigenvectors of the transfer matrix of a 19-vertex model on a square lattice with full frustration. The conformal anomaly c is estimated to be close to 3/2. This result suggests that the 19-vertex model with full frustration has a single transition due to two kinds of degrees of freedom, each of which has U(1) symmetry and Z2{Z}_{2}Z2​ symmetry. Moreover, we find that the value of index \ensuremathη\ensuremath{\eta}\ensuremathη is smaller than that at the Berezinskii-Kosterlitz-Thouless (BKT) transition by considering a logarithmic correction to the finite size analysis. This result for \ensuremathη\ensuremath{\eta}\ensuremathη suggests that there is another universality class which is different from that of either the Onsager type or the BKT type.
We embody the density-matrix renormalization-group (DMRG) method for the 19-vertex model on a square lattice in order to investigate the Berezinskii-Kosterlitz-Thouless transition. Elements of the transfer matrix of the 19-vertex model are classified in terms of the total value of arrows in one layer of the square lattice. By using this classification, we succeed in reducing enormously the dimension of the matrix that has to be diagonalized in the DMRG method. We apply our method to the 19-vertex model with the interaction K=1.0866K=1.0866K=1.0866 and obtain c=1.006(1)c=1.006(1)c=1.006(1) for the conformal anomaly.
We use extensive Monte Carlo transfer-matrix calculations on infinite strips of widths L up to 30 lattice spacing and a finite-size scaling analysis to obtain critical exponents and conformal anomaly number c for the two-dimensional XY Ising model. This model is expected to describe the critical behavior of a class of systems with simultaneous U(1) and Z2{\mathit{Z}}_{2}Z2​ symmetries of which the fully frustrated XY model is a special case. The effective values obtained for c show a significant decrease with L at different points along the line where the transition to the ordered phase takes place in a single transition. Extrapolations based on power-law corrections give values consistent with c=3/2 although larger values cannot be ruled out. Critical exponents are obtained more accurately and are consistent with previous Monte Carlo simulations suggesting critical behavior and with recent calculations for the frustrated XY model.
Classical XY spins on a two dimensional triangular lattice with antiferromagnetic interactions are reconsidered. We find that the Kosterlitz-Thouless transition associated to the U(1) symmetry appears at a temperature 0.0020(2) below the Ising transition at 0.5122(1) associated to the Z_2 symmetry. The Ising transition has critical exponents different from the standard ones. Using extensive Monte Carlo simulations for equilibrium and dynamical properties we show that the lack of universality observed in previous studies is due to finite size corrections not taken account. Likewise the Kosterlitz-Thouless transition has a critical exponent η\approx0.36 larger than the corresponding standard value 0.25. Also the helicity jump at the critical temperature is smaller than in the ferromagnetic case in disagreement with theoretical predictions. We try using the concept of an "quasi Universality class" to reconcile the standard critical behavior observable at higher temperatures with the different quasi universal one close to the critical region.
We consider the two-dimensional (2D) ${J}_{1}\ensuremath{-}{J}_{2}$ classical XY\mathrm{XY}XY model on a square lattice. In the frustrated phase corresponding to J2>J1/2,{J}_{2}>{J}_{1}/2,J2​>J1​/2, an Ising-like order parameter emerges by an ``order due to disorder'' effect. This leads to a discrete Z2{Z}_{2}Z2​ symmetry plus the U(1)U(1)U(1) global one. Using a powerful algorithm we show that the system undergoes two transitions at different but still very close temperatures, one of Kosterlitz-Thouless (KT) type and another one which does not belong to the expected Ising universality class. A new analysis of the KT transition has been developed in order to avoid the use of the nonuniversal helicity jump and to allow the computation of the exponents without a precise determination of the critical temperature. Moreover, our huge number of data enables us to exhibit the existence of large finite size effects explaining the dispersed results found in the literature concerning the more studied frustrated 2D, XY\mathrm{XY}XY models.
Critical properties of the fully frustrated XY models on square and triangular lattices are investigated by means of the nonequilibrium relaxation (NER) method. We examine the validity of the conclusion on the universality class of the chiral transition previously obtained by the NER method, in which that belongs to a non-Ising-type one. To clarify it, we analyze the NER of fluctuations for longer Monte Carlo steps on lager lattices than those performed previously. The calculation is made at the chiral transition temperatures estimated carefully by the use of recently improved dynamical scaling analysis. The result indicates that the asymptotic behavior of the time-dependent exponents, which should converge to the critical exponents, show the same tendency as those obtained previously in shorter times. We also apply the improved dynamical scaling analysis to the estimation of Kosterlitz-Thouless (KT) transitions and confirm the existence of the double transitions for the chiral and KT phases with more reliable estimations.