Effects of higher-order exchange interactions in classical XY model
Department of Theoretical Physics and Astrophysics
P. J. Šafárik University in Košice
A standard two-dimensional (2D) XY model is well known to show a topological Berezinskii-Kosterlitz-Thouless (BKT) phase transition with the quasi-long-range-order (QLRO) BKT phase characterized by an algebraically decaying correlation function. It will be shown by spin-wave theory and Monte Carlo simulations that including higher-order terms can lead to dramatic changes in its critical behavior, including phase transition order or universality class and even the phase diagram topology. This points to a significant lack of universality of 2D systems and raises a troubling question: How much can we really deduce about the thermodynamics of 2D systems from the form of their coarse grained Hamiltonian.
In the first part, the XY model is generalized by inclusion of up to an infinite number of higher-order pairwise interactions with an exponentially decreasing strength. We demonstrate that at low temperatures the model displays QLRO phase with the exponent η = T/[2πJ(p,α)], nonlinearly dependent on parameters p and α that control the number of higher-order terms and the decay rate of their intensity, respectively. At higher temperatures the system shows a crossover from the continuous BKT to the first-order transition for the parameter values corresponding to a highly nonlinear shape of the potential well. The role of topological excitations (vortices) in changing the nature of the transition will be discussed.
In the second part, we will focus on critical properties of the XY model with solely nematic-like terms, cos(qφ) with q = 2, 3, and 4. We will demonstrate that, even though neither of the terms alone can induce ferromagnetic (FM) ordering, their coexistence and competition can lead to a complex phase diagram including the FM phase at low temperatures. In particular, in the model involving the first two terms the FM phase appears wedged between the two nematic-like phases induced by the respective couplings. The phase transitions between the FM and nematic-like phases belong to the Ising and three-state Potts universality classes. Inside the FM phase the spin pair correlation function decays even much more slowly than in the standard XY model and the vortex-antivortex pair density is extremely low. In the model including all three terms it is possible to observe up to three successive phase transitions: the order-disorder transition of the BKT type is followed by two more transitions, as the system passes through two nematic-like phases to the FM phase at low temperatures that are both of Ising type.