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arXiv:0811.3840 · Physical Review B

Symmetry-allowed phase transitions realized by the two-dimensional fully frustrated XY class

Petter Minnhagen, Beom Jun Kim, Sebastian Bernhardsson, Gerardo Cristofano

Published Nov 21, 2008• 4 citations
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Abstract

A two-dimensional (2D) fully frustrated $XY$ $(\text{FF}XY)$ class of models is shown to contain a different ground state in addition to the checkerboard ground states of the standard 2D $\text{FF}XY$ model. The spin configuration of this additional ground state is obtained. Associated with this ground state there are additional phase transitions. An order parameter accounting for these transitions is proposed. The transitions associated with this order parameter are suggested to be similar to a 2D liquid-gas transition which implies ${Z}_{2}$ Ising-type transitions. This suggests that the class of 2D $\text{FF}XY$ models belongs within a $U(1)\ensuremath{\bigotimes}{Z}_{2}\ensuremath{\bigotimes}{Z}_{2}$ designation of possible transitions, which implies that there are seven different possible single and combined transitions. Monte Carlo (MC) simulations for the generalized fully frustrated $XY$ model on a square lattice are used to investigate which of these possibilities can be realized in practice: five of the seven are encountered. Four critical points are deduced from the MC simulations: three are consistent with central charge $c=3/2$ and one is consistent with $c=1$. The implications for the standard 2D $\text{FF}XY$ model are discussed in particular with respect to the long-standing controversy concerning the characteristics of its phase transitions.

DOI
10.1103/physrevb.78.184432

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A two-dimensional (2D) fully frustrated XYXYXY (FFXY)(\text{FF}XY)(FFXY) class of models is shown to contain a different ground state in addition to the checkerboard ground states of the standard 2D FFXY\text{FF}XYFFXY model. The spin configuration of this additional ground state is obtained. Associated with this ground state there are additional phase transitions. An order parameter accounting for these transitions is proposed. The transitions associated with this order parameter are suggested to be similar to a 2D liquid-gas transition which implies Z2{Z}_{2}Z2​ Ising-type transitions. This suggests that the class of 2D FFXY\text{FF}XYFFXY models belongs within a U(1)\ensuremath⨂Z2\ensuremath⨂Z2U(1)\ensuremath{\bigotimes}{Z}_{2}\ensuremath{\bigotimes}{Z}_{2}U(1)\ensuremath⨂Z2​\ensuremath⨂Z2​ designation of possible transitions, which implies that there are seven different possible single and combined transitions. Monte Carlo (MC) simulations for the generalized fully frustrated XYXYXY model on a square lattice are used to investigate which of these possibilities can be realized in practice: five of the seven are encountered. Four critical points are deduced from the MC simulations: three are consistent with central charge c=3/2c=3/2c=3/2 and one is consistent with c=1c=1c=1. The implications for the standard 2D FFXY\text{FF}XYFFXY model are discussed in particular with respect to the long-standing controversy concerning the characteristics of its phase transitions.
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 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.
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.
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 study the critical behavior of the fully frustrated XY\mathrm{XY}XY model on a square lattice by means of extensive histogram Monte Carlo simulations. We find a single transition in contrast to the scenario of two distinct transitions. From a finite-size scaling we determine the critical exponents associated with the Ising-like order parameter and find ${T}_{c}(\ensuremath{\infty})=0.4552(2)$, $\ensuremath{\nu}=0.852(2),$ and $\ensuremath{\gamma}=1.531(3)$. These exponents, different from the pure two-dimensional Ising critical exponents, are in agreement with the ones obtained for a mixed XY\mathrm{XY}XY Ising system.