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arXiv:hep-th/0606170 · Journal of Statistical Mechanics Theory and Experiment

CFT description of the fully frustratedXYmodel and phase diagram analysis

Gerardo Cristofano, Vincenzo Marotta, Petter Minnhagen, Adele Naddeo, Giuliano Niccoli

Published Nov 13, 2006• 12 citations
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Abstract

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.

DOI
10.1088/1742-5468/2006/11/p11009

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arXiv:cond-mat/9402100•Physical review. B, Condensed matter•Apr 1, 1994•87 citations
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.
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.
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 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.
The 2D Fully Frustrated XY(FFXY) class of models is shown to contain a new groundstate in addition to the checkerboard groundstate of the standard 2D XY model. The spin configuration of this additional groundstate is obtained and its connection to a broken Z2-symmetry explained. This means that the class of 2D FFXY models belongs within a U(1) ⊗ Z2 ⊗ Z2-symmetry phase-transition representation. The phase diagram is reviewed and the central charges of the four multicritical points described. The implications for the standard 2D FFXY-model are discussed and elucidated, in particular with respect to the long standing controversy concerning the phase transitions of the standard 2D FFXY-model.
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.