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Physical review. B, Condensed matter

19-vertex version of the fully frustratedXYmodel

Yolanda M. M. Knops, Bernard Nienhuis, H. J. F. Knops, Henk W. J. Blöte

Published Jul 1, 1994• 55 citations
Publisher

Abstract

We investigate a 19-vertex version of the two-dimensional fully frustrated XY (FFXY) model. We construct Yang-Baxter equations for this model and show that there is no solution. Therefore we have chosen a numerical approach based on the transfer matrix. The results show that a coupled XY Ising model is in the same universality class as the FFXY model. We find that the phase coupling over an Ising wall is irrelevant at criticality. This leads to a correction of earlier determinations of the dimension ${\mathit{x}}_{\mathit{h},\mathrm{I}\mathrm{s}}^{\mathrm{*}}$ of the Ising disorder operator. We find ${\mathit{x}}_{\mathit{h},\mathrm{I}\mathrm{s}}^{\mathrm{*}}$=0.123(5) and a conformal anomaly c=1.55(5). These results are consistent with the hypothesis that the FFXY model behaves as a superposition of an Ising model and an XY model. However, the dimensions associated with the energy, ${\mathit{x}}_{\mathit{t}}$=0.77(3), and with the XY magnetization ${\mathit{x}}_{\mathit{h},}$XY\ensuremath{\approxeq}0.17, refute this hypothesis.

DOI
10.1103/physrevb.50.1061

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We investigate a 19-vertex version of the two-dimensional fully frustrated XY (FFXY) model. We construct Yang-Baxter equations for this model and show that there is no solution. Therefore we have chosen a numerical approach based on the transfer matrix. The results show that a coupled XY Ising model is in the same universality class as the FFXY model. We find that the phase coupling over an Ising wall is irrelevant at criticality. This leads to a correction of earlier determinations of the dimension xh,Is∗{\mathit{x}}_{\mathit{h},\mathrm{I}\mathrm{s}}^{\mathrm{*}}xh,Is∗​ of the Ising disorder operator. We find xh,Is∗{\mathit{x}}_{\mathit{h},\mathrm{I}\mathrm{s}}^{\mathrm{*}}xh,Is∗​=0.123(5) and a conformal anomaly c=1.55(5). These results are consistent with the hypothesis that the FFXY model behaves as a superposition of an Ising model and an XY model. However, the dimensions associated with the energy, xt{\mathit{x}}_{\mathit{t}}xt​=0.77(3), and with the XY magnetization xh,{\mathit{x}}_{\mathit{h},}xh,​XY\ensuremath{\approxeq}0.17, refute this hypothesis.
We study the phase diagram and critical behaviour of the two-dimensional square-lattice fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau–Ginzburg–Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising XY model. We present a finite-size-scaling analysis of the results of high-precision Monte Carlo simulations on L × L square lattices, up to L = O (10 3 ). In the FFXY model and in the other models, when the transitions are continuous, there are two very close but separate transitions. There is an Ising chiral transition characterized by the onset of chiral long-range order while spins remain paramagnetic. Then, as temperature decreases, the systems undergo a Kosterlitz–Thouless spin transition to a phase with quasi-long-range order. The FFXY model and the other models, in a rather large parameter region, show a crossover behaviour at the chiral and spin transitions that is universal to some extent. We conjecture that this universal behaviour is due to a multicritical point. The numerical data suggest that the relevant multicritical point is a zero-temperature transition. A possible candidate is the O(4) point that controls the low-temperature behaviour of the 4-vector model.
We discuss the critical behavior of several models, all related to the 19-vertex model. These models combine Ising and Gaussian degrees of freedom and have central charge c=3/2 at the critical point. We present arguments why the frustrated XY model, which also has Gaussian and Ising degrees of freedom, might fall into a different universality class. We use the Monte Carlo transfer-matrix method of Bl\"ote and Nightingale to obtain accurate numerical results for the central charge and exponents of the frustrated XY model. In particular, we test the prediction for the central charge c to be equal to 3/2 also in this model. We find c=1.66\ifmmode\pm\else\textpm\fi{}0.04 for the central charge of the frustrated XY model at the critical point. It is not clear at this stage to which universality class this critical behavior belongs.
We study the phase diagram of the two-dimensional fully frustrated XY model (FFXY) and of two related models, a lattice discretization of the Landau-Ginzburg-Wilson Hamiltonian for the critical modes of the FFXY model, and a coupled Ising-XY model. We present Monte Carlo simulations on square lattices L × L , L ≲ 10 3 . We show that the low-temperature phase of these models is controlled by the same line of Gaussian fixed points as in the standard XY model. We find that, if a model undergoes a unique transition by varying temperature, then the transition is of first order. In the opposite case we observe two very close transitions: a transition associated with the spin degrees of freedom and, as temperature increases, a transition where chiral modes become critical. If they are continuous, they belong to the Kosterlitz-Thouless and to the Ising universality class, respectively. Ising and Kosterlitz-Thouless behavior is observed only after a preasymptotic regime, which is universal to some extent. In the chiral case, the approach is nonmonotonic for most observables, and there is a wide region in which finite-size scaling is controlled by an effective exponent ν eff ≈ 0.8. This explains the result ν ≈ 0.8 of many previous studies using smaller lattices.
Following an earlier suggestion, we show how the symmetry of the fully frustrated XY ( FFXY ) model on a square lattice can be accounted for in the framework of the m -reduction procedure developed for a Quantum Hall system at 'paired states' fillings ν = 1. The resulting twisted conformal field theory (CFT) with central charge c = 2 is shown to well describe the physical properties of the FFXY model. In particular, the whole phase diagram is recovered by analysing the flow from the Z 2 degenerate vacuum of the c = 2 CFT to the infrared fixed point unique vacuum of the CFT. The last theory is known to successfully describe the critical behaviour of the system at the overlap temperature for the Ising and vortex-unbinding transitions.
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.
The issue of the number, nature and sequence of phase transitions in the fully frustrated XY (FFXY) model is a highly non trivial one due to the complex interplay between its continuous and discrete degrees of freedom. In this contribution we attack such a problem by means of a twisted conformal field theory (CFT) approach and show how it gives rise to the U (1)⊗Z2\otimes Z_{2}⊗Z2​ symmetry and to the whole spectrum of excitations of the FFXY model.