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.
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.
It is well known that the 2D XY model exhibits an unusual infinite order phase transition belonging to the Kosterlitz-Thouless (KT) universality class. Introduction of a nematic coupling into the XY Hamiltonian leads to an additional phase transition in the Ising universality class [D. H. Lee and G. Grinstein, Phys. Rev. Lett. 55, 541 (1985)]. Using a combination of extensive Monte Carlo simulations and finite size scaling, we show that the higher order harmonics lead to a qualitatively different phase diagram, with additional ordered phases originating from the competition between the ferromagnetic and pseudonematic couplings. The new phase transitions belong to the 2D Potts, Ising, or KT universality classes.
We propose generalized variants of the XY model capable of exhibiting an arbitrary number of phase transitions only by varying temperature. They are constructed by supplementing the magnetic coupling with n_{t}-1 nematic terms of exponentially increasing order with the base q=2,3,4, and 5, and increasing interaction strength. It is found that for q=2,3, and 4 with sufficiently large coupling strength of the final term, the models exhibit a number of phase transitions equal to the number of the terms in the generalized Hamiltonian. Starting from the paramagnetic phase, the system transitions through the cascade of n_{t}-1 nematic phases of the orders q^{k}, k=n_{t}-1,n_{t}-2,⋯,1, that are characterized by q^{k} preferential spin directions symmetrically disposed around the circle, to the ferromagnetic (FM) phase at the lowest temperatures. Besides the Berezinskii-Kosterlitz-Thouless (BKT) transition from the paramagnetic phase, all the remaining transitions have a non-BKT nature: depending on the value of q they belong to either the Ising (q=2 and 4) or the three-state Potts (q=3) universality class. For q=5, due to the interplay between different terms, the phase transitions between the ordered phases observed for q<5 split into two, and the number of the ordered phases increases to 2n_{t}-1. These phases are characterized by a domain structure with the gradually increasing short-range FM ordering within domains that extends to different kinds of FM ordering in the last two low-temperature phases. The respective transitions do not seem to obey any universality.
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.
Critical properties of an Ising bilayer system, consisting of two triangular planes with antiferromagnetic (AF) JA and ferromagnetic (FM) JB exchange interactions, coupled by the interlayer exchange JAB, are studied by Monte Carlo simulation. The FM plane is found to induce in the AF one some effects similar to those resulting from the joint presence of a next-nearest-neighbor FM coupling and an external magnetic field in the single triangular lattice Ising antiferromagnet. The similarities include the appearance of the quasi-long-range-order BKT phase with algebraically decaying correlations for smaller JAB, as well as the phase transitions belonging to the three-state Potts universality class for larger JAB. On the other hand, no first-order phase transitions are observed. A new phenomenon is a cross-over from the BKT to the Wannier state at higher temperatures, that can be stabilized over a relatively wide temperature range. In the strong JAB limit, the critical behavior of the entire bilayer is governed by the behavior of the plane with a dominant intraplane interaction. If JA is dominant, then the strong JAB coupling makes the system kinetically freeze at lower temperatures.
The interplay of topology and competing interactions can induce enriched phases and phase transitions at finite temperatures. We consider a weakly coupled two-dimensional hexatic-nematic XY model with a relative Z3 Potts degrees of freedom, and apply the matrix product state method to solve this model rigorously. Since the partition function is expressed as a product of two-legged one-dimensional transfer matrix operator, an entanglement entropy of the eigenstate corresponding to the maximal eigenvalue of this transfer operator can be used as a stringent criterion to determine various phase transitions precisely. At low temperatures, the intercomponent Z3 Potts long-range order (LRO) exists, indicating that the hexatic and nematic fields are locked together and their respective vortices exhibit quasi-LRO. In the hexatic regime, below the BKT transition of the hexatic vortices, the intercomponent Z3 Potts LRO appears, accompanying the binding of nematic vortices. In the nematic regime, however, the intercomponent Z3 Potts LRO undergoes a two-stage melting process. An intermediate Potts liquid phase emerges between the Potts ordered and disordered phases, characterized by an algebraic correlation with formation of charge-neutral pairs of both hexatic and nematic vortices. These two-stage phase transitions are associated with the proliferation of the domain walls and vortices of the relative Z3 Potts variable, respectively. Our results thus provide a prototype example of two-stage melting of a two-dimensional LRO driven by multiple topological defects.
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.