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Physical review. B, Condensed matter

Monte Carlo transfer-matrix study of the frustratedXYmodel

Joseph Marie Thijssen, H. J. F. Knops

Published Aug 1, 1990• 55 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.42.2438

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arXiv:cond-mat/9402100•Physical review. B, Condensed matter•Apr 1, 1994•87 citations
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.
Thermal and chiral critical exponents of the fully frustrated XY model on a square lattice are obtained from a finite-size scaling analysis of the free energy of chiral domain walls. Data were obtained by extensive Monte Carlo transfer-matrix computations for infinite strips of widths up to 14 lattice spacings. Two transfer matrices were implemented, one for each of two principal lattice directions. The results of both are consistent, but the critical exponents differ significantly from the pure Ising values. This is in agreement with other recent Monte Carlo simulations. Our results also support the identification of the critical behavior of this model with that along the line of transitions of simultaneous ordering or becoming critical of Ising and planar rotor degrees of freedom in the XY-Ising model studied recently.
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 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.
A one-dimensional quantum version of the frustrated XY model is introduced which can be physically realized as a ladder of Josephson junctions at half of a flux quantum per plaquette. From a fluctuation effective action, the zero-temperature (superconductor-insulator) transition is predicted to be in the universality class of the two-dimensional classical XY-Ising model. A Monte Carlo transfer matrix is used to calculate critical exponents and central charge. A finite-size-scaling analysis of extensive calculations on small system sizes supports the prediction. The same critical behavior has recently been found for the two-dimensional classical version. Together, the results strongly support an XY-Ising-like critical behavior for these systems.
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.
We present a detailed study of the critical properties of the two-dimensional (2D) XY model with maximal frustration in a square lattice. We use extensive Monte Carlo simulations to study the thermodynamics of the spin and chiral degrees of freedom, concentrating on their correlation functions. The gauge-invariant spin-spin correlation functions are calculated close to the critical point for lattice sizes up to 240\ifmmode\times\else\texttimes\fi{}240; the chiral correlation functions are studied on lattices up to 96\ifmmode\times\else\texttimes\fi{}96. We find that the critical exponents of the spin-phase transition are \ensuremath{\nu}=0.3069 and \ensuremath{\eta}=0.1915, which are to be compared with the unfrustrated XY model exponents \ensuremath{\nu}=1/2 and \ensuremath{\eta}=0.25. We also find that the critical exponents of the chiral transition are ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\chi}}}$=0.875, 2\ensuremath{\beta}=0.1936, 2\ensuremath{\gamma}=1.82, and 2\ensuremath{\gamma}'=1.025, which are different from the expected 2D Ising critical exponents. The spin-phase transition occurs at TU(1){\mathit{T}}_{\mathit{U}(1)}TU(1)​=0.446, which is about 7% above the estimated chiral critical temperature TZ2{\mathit{T}}_{\mathit{Z}2}TZ2​=0.4206. However, because of the size of the statistical errors, it is difficult to decide with certainty whether the transitions occur at the same or at slightly different temperatures. Finally, the jump in the helicity modulus in the fully frustrated system is found to be about 23% below the unfrustrated universal value. The most important consequence of these results is that the fully frustrated XY model appears to be in a novel universality class. Recent successful comparisons of some of these results with experimental data are also briefly discussed.