A new critical point is established through Monte Carlo simulations for a class of two dimensional XY models with a modified nearest-neighbor spin interaction. The critical exponents found are \ensuremath{\nu}\ensuremath{\approxeq}1.4, \ensuremath{\beta}\ensuremath{\approxeq}0.6, and an \ensuremath{\alpha}0, consistent with hyperscaling. The phase transition is interpreted as a vortex-unbinding transition. Comparisons are made with the phase diagram of the two dimensional Coulomb gas and with the spin-1/2 XY model.
A new critical point is established through Monte Carlo simulations for a class of two dimensional XY models with a modified nearest-neighbor spin interaction. The critical exponents found are \ensuremath{\nu}\ensuremath{\approxeq}1.4, \ensuremath{\beta}\ensuremath{\approxeq}0.6, and an \ensuremath{\alpha}0, consistent with hyperscaling. The phase transition is interpreted as a vortex-unbinding transition. Comparisons are made with the phase diagram of the two dimensional Coulomb gas and with the spin-1/2 XY model.
A class of two-dimensional XY models, constructed by inserting one-dimensional spin chains between the spins of the usual XY model on a square lattice, is studied with the aid of Monte Carlo simulations. The one-dimensional fluctuations associated with the chains drive the phase transition to zero temperature in the limit of infinite chains. The modifications due to the one-dimensional fluctuations of the vortex unbinding causing the phase transition are obtained. It is found that the transition is of the Kosterlitz-Thouless type also in the limit of infinite chains. The variation of the critical temperature with the average number of nearest-neighboring spins is determined and is interpreted with the aid of an approximate translation to the two-dimensional Coulomb-gas model. Coulomb-gas scaling is found to be obeyed to a good approximation.
The short time dynamic behavior of the second-order phase transition in the two-dimensional fully frustrated XY model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings is investigated using Monte Carlo simulations. When x<1/2(x is the ratio between NNN and NN couplings), the transition temperature Tc and the dynamic and static critical exponents z, $2\ensuremath{\beta}/\ensuremath{\nu},$ and $\ensuremath{\nu}$ are estimated using the short-time dynamic scaling analysis. Except x close to the critical value 1/2, the critical exponents $2\ensuremath{\beta}/\ensuremath{\nu}$ and $\ensuremath{\nu}$ are found to be almost independent of the NNN coupling, indicating that the second-order phase transitions in the FFXY model, with or without NNN coupling, are in the same universality class. When x>1/2, the short-time dynamic study suggests that there is no second-order phase transition, in agreement with the previous result [Phys. Rev. B 62, R9287 (2000)].
We have performed Monte Carlo (MC) simulations on systems of L\ifmmode\times\else\texttimes\fi{}L classical planar unit spins on square lattices, for L=6, 15, 30, 60, 90, and 200. The interaction between any two given spins S${\ensuremath{\rightarrow}}_{1}$ and S${\ensuremath{\rightarrow}}_{2}$ is given by -JS${\ensuremath{\rightarrow}}_{1}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{2}$ if S1 and S2 are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature (Tc). We have found that the diffusion constant for vortex motion is given at Tc by D\ensuremath{\simeq}0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically 105 MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of Tc=0.89J/kB, the value of \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, and the value of ${\ensuremath{\eta}}_{c}$=0.24\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory.
The critical properties of the three-dimensional XY model are investigated through measurements of the two-point correlation functions via a worm-algorithm Monte Carlo calculation, which automatically crunches the correlation functions while tracing worms, in large systems. This method has the advantage of directly measuring the order parameter, as well as the correlation length, the susceptibility, and their associated critical exponents. From the finite-size scaling behavior, we find 2 β/ν = 1.033(5) and γ/ν = 1.967(5), confirming the hyperscaling relations.
We study the two-dimensional XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
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)=0.446, which is about 7% above the estimated chiral critical temperature 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.