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arXiv:1805.08387 · Physical review. B./Physical review. B

Correlation length in a generalized two-dimensional XY model

Duong Xuan Nui, Le Tuan, Nguyen Duc Trung Kien, Phạm Thành Huy, Hung T. Dang, Dao Xuan Viet

Published Oct 15, 2018• 34 citations
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Abstract

The measurements of the magnetic and nematic correlation lengths in a generalization of the two-dimensional XY model on the square lattice are presented using classical Monte Carlo simulation. The full phase diagram is reexamined based on these correlation lengths, demonstrating their power in studying generalized XY models. The ratio between the correlation length and the lattice size has distinctive behaviors which can be used to distinguish different types of phase transition. More importantly, the magnetic correlation length give more insights into the tricritical region where the paramagnetic, nematic, and quasi-long-range phases meet. It shows signatures for the intermediate region starting from the tricritical point, where the transition line is neither of the same physics as the Ising transition below nor the Berezinskii-Kosterlitz-Thouless transition far above the tricritical point.

DOI
10.1103/physrevb.98.144421

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The measurements of the magnetic and nematic correlation lengths in a generalization of the two-dimensional XY model on the square lattice are presented using classical Monte Carlo simulation. The full phase diagram is reexamined based on these correlation lengths, demonstrating their power in studying generalized XY models. The ratio between the correlation length and the lattice size has distinctive behaviors which can be used to distinguish different types of phase transition. More importantly, the magnetic correlation length give more insights into the tricritical region where the paramagnetic, nematic, and quasi-long-range phases meet. It shows signatures for the intermediate region starting from the tricritical point, where the transition line is neither of the same physics as the Ising transition below nor the Berezinskii-Kosterlitz-Thouless transition far above the tricritical point.
We performed Monte Carlo simulation for a two-dimensional generalized XY model and calculated the magnetic and nematic Binder parameters. The phase diagram is re-examined based on these Binder parameters, demonstrating their power in studying generalized XY models. The Binder parameter has distinctive behaviors, resulting in different types of phase transition. More importantly, the magnetic Binder parameter gives more insights into the tricritical region where the Kosterlitz-Thouless, Ising, and 1/2 Kosterlitz-Thouless (KT) transition lines meet. It shows signatures for the intermediate region starting from the tricritical point, where the transition line is neither the same physics as the Ising transition below nor the KT transition far above the tricritical point.
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Critical properties of the two-dimensional XY model involving solely nematic-like terms of the second and third orders are investigated by spin-wave analysis and Monte Carlo simulation. It is found that, even though neither of the nematic-like terms alone can induce magnetic ordering, their coexistence and competition leads to an extended phase of the magnetic quasi-long-range-order phase, wedged between the two nematic-like phases induced by the respective couplings. Thus, except for the multicritical point, at which all the phases meet, for any finite value of the coupling parameters ratio there are two phase transition: one from the paramagnetic phase to one of the two nematic-like phases followed by another one at lower temperatures to the magnetic phase. The finite-size scaling analysis indicates that the phase transitions between the magnetic and nematic-like phases belong to the Ising and three-state Potts universality classes. Inside the competition-induced algebraic magnetic phase, the spin-pair correlation function is found to decay even much more slowly than in the standard XY model with purely magnetic interactions. Such a magnetic phase is characterized by an extremely low vortex-antivortex pair density attaining a minimum close to the point at which the two couplings are of about equal strength.
We study the two-dimensional XY\mathrm{XY}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=0T=0T=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 study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.
A generalisation of the two-dimensional XY model is studied using Monte Carlo simulation. In this model, two-component classical spins interact both ferromagnetically and nematically. Three phases occur as temperature and the two interaction strengths are varied: a high-temperature, disordered phase, and two low-temperature phases with algebraic correlations in, respectively, the ferromagnetic and nematic order parameters. The critical behaviour at the phase boundaries is examined: the high-temperature phase is entered via Kosterlitz-Thouless transitions, whilst the low-temperature phases are separated by an Ising transition.