Competing nematic interactions in a generalized XY model in two and three dimensions

Generalized XY Models with Arbitrary Number of Phase Transitions

We propose spin models that can display an arbitrary number of phase transitions. The models are based on the standard XY model, which is generalized by including higher-order nematic terms with exponentially increasing order and linearly increasing interaction strength. By employing Monte Carlo simulation we demonstrate that under certain conditions the number of phase transitions in such models is equal to the number of terms in the generalized Hamiltonian and, thus, it can be predetermined by construction. The proposed models produce the desirable number of phase transitions by solely varying the temperature. With decreasing temperature the system passes through a sequence of different phases with gradually decreasing symmetries. The corresponding phase transitions start with a presumably BKT transition that breaks the U(1) symmetry of the paramagnetic phase, and they proceed through a sequence of discrete Z2 symmetry-breaking transitions between different nematic phases down to the lowest-temperature ferromagnetic phase.

Generalized XY model with competing antiferromagnetic and antinematic interactions

I study effects of q-order antinematic (AN_{q}) interactions on the critical behavior of the antiferromagnetic (AF) XY model on a square lattice. It is found that the evolution of the phase diagram topology of such AF-AN_{q} models with the parameter q does not follow the same line as for the corresponding FM-N_{q} models with the ferromagnetic (FM) and q-order nematic (N_{q}) interactions. Their phase diagrams are similar only for odd values of the parameter q. In such cases the respective phases reported in the FM-N_{q} models are observed in the AF-AN_{q} models on each of the two AF-coupled sublattices and the corresponding phase transitions are concluded to be of the same kind. On the other hand, for even values of q the phase diagrams of the AF-AN_{q} models are different from the FM-N_{q} models and their topology does not change with q. Besides the pure AF and AN_{q} phases, observed at higher temperatures in the regions of the dominant respective couplings, at low temperatures there is a new canted (C)AF phase, which results from the competition between the AF and AN_{q} ordering tendencies and has no counterpart in the FM-N_{q} model. The phase transitions to the CAF phase from both AF and AN_{q} phases appear to be of the Berezinskii-Kosterlitz-Thouless nature.

Ordering kinetics and steady states of XY-model with ferromagnetic and nematic interaction

Previous studies on the generalized XY model have concentrated on the equilibrium phase diagram and the equilibrium nature of distinct phases under varying parameter conditions. We direct our attention towards examining the system's evolution towards equilibrium states across different parameter values, specifically by varying the relative strengths of ferromagnetic and nematic interactions. We study the kinetics of the system, using the temporal annihilation of defects at varying temperatures and its impact on the coarsening behavior of the system. For both pure polar and pure nematic systems, we observe temperature-dependent decay of the exponent, leading to a decelerated growth of domains within the system. At parameter values where both ferromagnetic and nematic interactions are simultaneously present, we show a phase diagram highlighting three low-temperature phases-polar, nematic, and coexistence-along- side a high-temperature disordered phase. Our study provides valuable insights into the complex interplay of interactions, offering a comprehensive understanding of the system's behavior during its evolution towards equilibrium.

XY model with competing higher-order interactions

We study effects of competing pairwise higher-order interactions (HOI) with alternating signs and exponentially decreasing intensity on critical behavior of the XY model. It is found that critical properties of such a generalized model can be very different from the standard XY model and can strongly depend on whether the number of HOI terms is odd or even. Inclusion of any odd number of HOI terms results in two consecutive phase transitions to distinct ferromagnetic quasi-long-range order phases. Even number of HOI terms leads to two phase transitions only if the decay of the HOI intensities is relatively slow. Then the high-temperature transition to the ferromagnetic phase is followed by another transition to a peculiar competition-induced canted ferromagnetic phase. In the limit of an infinite number of HOI terms only one phase transition is confirmed, and under the conditions of fierce competition between the even and odd terms the transition temperature can be suppressed practically to zero.

Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology

We use persistent homology and persistence images as an observable of three variants of the two-dimensional XY model to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbor models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behavior and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.

Phase behavior of hard circular arcs

By using Monte Carlo numerical simulation, this work investigates the phase behavior of systems of hard infinitesimally thin circular arcs, from an aperture angle θ→0 to an aperture angle θ→2π, in the two-dimensional Euclidean space. Except in the isotropic phase at lower density and in the (quasi)nematic phase, in the other phases that form, including the isotropic phase at higher density, hard infinitesimally thin circular arcs autoassemble to form clusters. These clusters are either filamentous, for smaller values of θ, or roundish, for larger values of θ. Provided the density is sufficiently high, the filaments lengthen, merge, and straighten to finally produce a filamentary phase while the roundels compact and dispose themselves with their centers of mass at the sites of a triangular lattice to finally produce a cluster hexagonal phase.

Phase diagrams of the antiferromagnetic XY model on a triangular lattice with higher-order interactions

We study the effects of higher-order antinematic interactions on the critical behavior of the antiferromagnetic (AFM) XY model on a triangular lattice using Monte Carlo simulations. The parameter q of the generalized antinematic (ANq) interaction is found to have a pronounced effect on the phase diagram topology by inducing new quasi-long-range ordered phases due to competition with the conventional AFM interaction as well as geometrical frustration. For values of q divisible by 3, the ground-state competition between the two interactions results in a frustrated canted AFM phase appearing at low temperatures wedged between the AFM and ANq phases. For q nondivisible by 3, with the increase of q one can observe the evolution of the phase diagram topology featuring two (q=2), three (q=4,5), and four (q≥7) ordered phases. In addition to the two phases previously found for q=2, the first new phase with solely AFM ordering arises for q=4 in the limit of strong AFM coupling and higher temperatures by separating from the phase with the coexisting AFM and ANq orderings. For q=7, another phase with AFM ordering but multimodal spin distribution in each sublattice appears at intermediate temperatures. All these algebraic phases also display standard and generalized chiral long-range orderings, which decouple at higher temperatures in the regime of dominant ANq (AFM) interaction for q≥4 (q≥7) preserving only the generalized (standard) chiral ordering.

New ordered phase in geometrically frustrated generalized XY model

Critical properties of a geometrically frustrated generalized XY model with antiferromagnetic (AFM) and third-order antinematic (AN3) couplings on a triangular lattice are studied by Monte Carlo simulation. It is found that such a generalization leads to a phase diagram consisting of three different quasi-long-range ordered (QLRO) phases. Compared to the model with the second-order antinematic (AN2) coupling, besides the AFM and AN3 phases which appear in the limits of relatively strong AFM and AN3 interactions, respectively, it includes an additional complex canted antiferromagnetic (CAFM) phase. It emerges at lower temperatures, wedged between the AFM and AN3 phases as a result of the competition between the AFM and the AN3 couplings, which is absent in the model with the AN2 coupling. The AFM-CAFM and AN3-CAFM phase transitions are concluded to belong to the weak Ising and weak three-state Potts universality classes, respectively. Additionally, all three QLRO phases also feature true LRO of the standard and generalized chiralities, which both vanish simultaneously at second-order phase transitions with non-Ising critical exponents and the critical temperatures slightly higher than the magnetic and nematic order-disorder transition temperatures.

XY model with antinematic interaction

We consider the XY model with ferromagnetic (FM) and antinematic (AN) nearest-neighbor interactions on a square lattice for a varying interaction strength ratio. Besides the expected FM and AN quasi-long-range order (QLRO) phases we identify at low temperatures another peculiar canted ferromagnetic (CFM) QLRO phase, resulting from the competition between the collinear FM and noncollinear AN ordering tendencies. In the CFM phase neighboring spins that belong to different sublattices are canted by a nonuniversal (dependent on the interaction strength ratio) angle and the ordering is characterized by a fast-decaying power-law intrasublattice correlation function. Compared to the FM phase, in the CFM phase correlations are significantly diminished by the presence of zero-energy domain walls due to the inherent degeneracy caused by the AN interactions. We present the phase diagram as a function of the interaction strength ratio and discuss the character of the respective phases as well as the transitions between them.

Machine learning of phase transitions in the percolation and XY models

In this paper, we apply machine learning methods to study phase transitions in certain statistical mechanical models on the two-dimensional lattices, whose transitions involve nonlocal or topological properties, including site and bond percolations, the XY model, and the generalized XY model. We find that using just one hidden layer in a fully connected neural network, the percolation transition can be learned and the data collapse by using the average output layer gives correct estimate of the critical exponent ν. We also study the Berezinskii-Kosterlitz-Thouless transition, which involves binding and unbinding of topological defects, vortices and antivortices, in the classical XY model. The generalized XY model contains richer phases, such as the nematic phase, the paramagnetic and the quasi-long-range ferromagnetic phases, and we also apply machine learning method to it. We obtain a consistent phase diagram from the network trained with only data along the temperature axis at two particular parameter Δ values, where Δ is the relative weight of pure XY coupling. Aside from using the spin configurations (either angles or spin components) as the input information in a convolutional neural network, we devise a feature engineering approach using the histograms of the spin orientations in order to train the network to learn the three phases in the generalized XY model and demonstrate that it indeed works. The trained network by using system size L×L can be used to the phase diagram for other sizes (L^{'}×L^{'}, where L^{'}≠L) without any further training.

Magnetic quasi-long-range ordering in nematic systems due to competition between higher-order couplings

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.

XY model with higher-order exchange

An XY model, generalized by inclusion of up to an infinite number of higher-order pairwise interactions with an exponentially decreasing strength, is studied by spin-wave theory and Monte Carlo simulations. At low temperatures the model displays a quasi-long-range-order phase characterized by an algebraically decaying correlation function with the exponent η=T/[2πJ(p,α)], nonlinearly dependent on the parameters p and α that control the number of the higher-order terms and the decay rate of their intensity, respectively. At higher temperatures the system shows a crossover from the continuous Berezinskii-Kosterlitz-Thouless 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 is discussed.