Classical XY Research Articles

Phase diagram of generalized XY model using the tensor renormalization group

We use the higher-order tensor renormalization group method to study the two-dimensional generalized XY model that admits integer and half-integer vortices. This model is the deformation of the classical XY model and has a rich phase structure consisting of nematic, ferromagnetic, and disordered phases and three transition lines belonging to the Berezinskii-Kosterlitz-Thouless and Ising class. We explore the model for a wide range of temperatures, T, and the deformation parameter, Δ, and compute specific heat along with integer and half-integer magnetic susceptibility, finding both Berezinskii-Kosterlitz-Thouless-like and Ising-like transitions and the region where they meet. Published by the American Physical Society 2024

Study of the Berezinskii–Kosterlitz–Thouless transition: an unsupervised machine learning approach

The Berezinskii–Kosterlitz–Thouless (BKT) transition in magnetic systems is an intriguing phenomenon, and estimating the BKT transition temperature is a long-standing problem. In this work, we explore anisotropic classical Heisenberg XY and XXZ models with ferromagnetic exchange on a square lattice and antiferromagnetic exchange on a triangular lattice using an unsupervised machine learning approach called principal component analysis (PCA). The earlier PCA studies of the BKT transition temperature ( TBKT ) using the vorticities as input fail to give any conclusive results, whereas, in this work, we show that the proper analysis of the first principal component-temperature curve can estimate TBKT which is consistent with the existing literature. This analysis works well for the anisotropic classical Heisenberg with a ferromagnetic exchange on a square lattice and for frustrated antiferromagnetic exchange on a triangular lattice. The classical anisotropic Heisenberg antiferromagnetic model on the triangular lattice has two close transitions: the TBKT and Ising-like phase transition for chirality at Tc , and it is difficult to separate these transition points. It is also noted that using the PCA method and manipulation of their first principal component not only makes the separation of transition points possible but also determines transition temperature.

Scalar susceptibility of a diluted classical XY model

Scalar susceptibility of a diluted classical XY model

Quantum criticality in chains of planar rotors with dipolar interactions.

In this work, we perform a density matrix renormalization group study of chains of planar rotors interacting via dipolar interactions. By exploring the ground state from weakly to strongly interacting rotors, we find the occurrence of a quantum phase transition between a disordered and a dipole-ordered quantum state. We show that the nature of the ordered state changes from ferroelectric to antiferroelectric when the relative orientation of the rotor planes varies and that this change requires no modification of the overall symmetry. The observed quantum phase transitions are characterized by critical exponents and central charges, which reveal different universality classes ranging from that of the (1 + 1)D Ising model to the 2D classical XY model.

Asymptotic Freedom at the Berezinskii-Kosterlitz-Thouless Transition without Fine-Tuning Using a Qubit Regularization

We propose a two-dimensional hard-core loop-gas model as a way to regularize the asymptotically free massive continuum quantum field theory that emerges at the Berezinskii-Kosterlitz-Thouless transition. Without fine-tuning, our model can reproduce the universal step-scaling function of the classical lattice XY model in the massive phase as we approach the phase transition. This is achieved by lowering the fugacity of Fock-vacuum sites in the loop-gas configuration space to zero in the thermodynamic limit. Some of the universal quantities at the Berezinskii-Kosterlitz-Thouless transition show smaller finite size effects in our model as compared to the traditional XY model. Our model is a prime example of qubit regularization of an asymptotically free massive quantum field theory in Euclidean space-time and helps understand how asymptotic freedom can arise as a relevant perturbation at a decoupled fixed point without fine-tuning. Published by the American Physical Society 2024

Monte Carlo study of the phase transitions in the classical XY ferromagnets with random anisotropy

ABSTRACT The three-dimensional anisotropic classical XY ferromagnet has been investigated by extensive Monte Carlo simulation using the Metropolis single spin flip algorithm. The magnetisation (M) and the susceptibility (χ) are measured and studied as functions of the temperature of the system. For constant anisotropy, the ferro-para phase transition has been found to take place at a higher temperature than that observed in the isotropic case. The system gets ordered at higher temperatures for higher values of the strength of anisotropy. The opposite scenario is observed in the case of random anisotropy. For all three different kinds of statistical distributions (uniform, Gaussian, and bimodal) of random anisotropy, the system gets ordered at lower temperatures for higher values of the width of the distribution of anisotropy. We have provided the phase boundaries in the case of random anisotropy. The critical exponents for the scaling laws and are estimated through the finite size analysis.

A universal training scheme and the resulting universality for machine learning phases

Abstract An autoencoder (AE) and a generative adversarial network (GAN) are trained only once on a one-dimensional (1D) lattice of 200 sites. Moreover, the AE contains only one hidden layer consisting of two neurons, and both the generator and the discriminator of the GAN are made up of two neurons as well. The training set employed to train both the considered unsupervised neural networks (NNs) is composed of two artificial configurations. Remarkably, despite their simple architectures, both the built AE and GAN have precisely determined the critical points of several models, including the three-dimensional classical O(3) model, the two-dimensional generalized classical XY model, the two-dimensional two-state Potts model, and the one-dimensional Bose–Hubbard model. In addition, a factor of several thousands in the speed of calculation is gained for the built AE and GAN when they are compared with the conventional unsupervised NN approaches. The results presented here, as well as those shown previously in the literature, suggest that when phase transitions are considered, an elegant universal neural network that is extremely efficient and is applicable to broad physical systems can be constructed with ease. In particular, since an NN trained with two configurations can be applied to many models, it is likely that when machine learning is concerned, the majority of phase transitions belong to a class having two elements, i.e. the Ising class.

Fractional vortices, Z2 gauge theory, and the confinement-deconfinement transition

In this Letter we discuss the classical three-dimensional XY model whose nearest-neighbor interaction is a mixture of $cos({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j})$ (ferromagnetic) and $cos2({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j})$ (nematic). This model is dual to a theory with integer and half-integer vortices. While both types of vortices interact with a noncompact $U(1)$ gauge field, the half-integer vortices interact with an extra interaction mediated by a ${\mathbb{Z}}_{2}$ gauge field. We shall discuss the confinement-deconfinement transition of the half-integer vortices, the Wilson and the `t Hooft loops, and their mutual statistics in path integral language. In addition, we shall present a quantum version of the classical model which exhibits these physics.

Continuous symmetry breaking along the Nishimori line

We prove continuous symmetry breaking in three dimensions for a special class of disordered models described by the Nishimori line. The spins take values in a group, such as S1, SU(n) or SO(n). Our proof is based on a theorem about group synchronization proved by Abbe et al. [Math. Stat. Learn. 1(3), 227–256 (2018)]. It also relies on a gauge transformation acting jointly on the disorder and the spin configurations due to Nishimori [Prog. Theor. Phys. 66(4), 1169–1181 (1981)]. The proof does not use reflection positivity. The correlation inequalities of Messager et al. [Commun. Math. Phys. 58(1), 19–29 (1978)] imply symmetry breaking for the classical XY model without disorder.

The N-Clock Model: Variational Analysis for Fast and Slow Divergence Rates of N

We study a nearest neighbors ferromagnetic classical spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of N equi-spaced vectors, also known as the N-clock model. We find a fast rate of divergence of N with respect to the lattice spacing for which the N-clock model has the same discrete-to-continuum variational limit as the classical XY model (also known as planar rotator model), in particular concentrating energy on topological defects of dimension 0. We prove the existence of a slow rate of divergence of N at which the coarse-grain limit does not detect topological defects, but it is instead a BV-total variation. Finally, the two different types of limit behaviors are coupled in a critical regime for N, whose analysis requires the aid of Cartesian currents.

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.

Berezinskii–Kosterlitz–Thouless transition – A universal neural network study with benchmarks

Using a supervised neural network (NN) trained once on a one-dimensional lattice of 200 sites, we calculate the Berezinskii–Kosterlitz–Thouless phase transitions of the two-dimensional (2D) classical XY and the 2D generalized classical XY models. In particular, both the bulk quantities Binder ratios and the spin states of the studied systems are employed to construct the needed configurations for the NN prediction. By applying semiempirical finite-size scaling to the relevant data, the critical points obtained by the NN approach agree well with the known results established in the literature. This implies that for each of the considered models, the determination of its various phases requires only a little information. The outcomes presented here demonstrate convincingly that the employed universal NN is not only valid for the symmetry breaking related phase transitions, but also works for calculating the critical points of the phase transitions associated with topology. The efficiency of the used NN in the computation is examined by carrying out several detailed benchmark calculations.

Universal Dynamics of Magnetic Monopoles in Two-Dimensional Kagomé Ice

A magnetic monopole in spin ice is a novel quasiparticle excitation in condensed matter physics, and we found that the ac frequency dependent magnetic susceptibility $\chi(\omega)$ in the two-dimensional (2D) spin ice (so-called kagom\'{e} ice) of Dy$_2$Ti$_2$O$_7$ shows a single scaling form. This behavior can be understood in terms of the dynamical scaling law for 2D Coulomb gas (CG) systems [Phys. Rev. B 90, 144428 (2014)], characterized by the charge correlation length $\xi (\propto1/\sqrt{\omega_1})$, where $\omega_{1}$ is a characteristic frequency proportional to the peak position of the imaginary part of $\chi(\omega)$. It is a generic behavior among a wide variety of models such as the vortex dynamics of 2D superconductors, 2D superfluids, classical XY magnets, and dynamics of melting of Wigner crystals.

Neural network architectures based on the classical XY model

Classical XY model is a lattice model of statistical mechanics notable for its universality in the rich hierarchy of the optical, laser, and condensed matter systems. We show how to build complex structures for machine learning based on the XY model's nonlinear blocks. The final target is to reproduce the deep learning architectures, which can perform complicated tasks usually attributed to such architectures: speech recognition, visual processing, or other complex classification types with high quality. We developed a robust and transparent approach for the construction of such models, which has universal applicability (i.e., does not strongly connect to any particular physical system) and allows many possible extensions, while at the same time preserving the simplicity of the methodology.

Self-Organized Vortex and Antivortex Patterns in Laser Arrays

Recently it is shown that dissipatively coupled laser arrays simulate the classical XY model. We show that phase-locking of laser arrays can give rise to the spontaneous formation of vortex and antivortex phase patterns that are analogous to topological defects of the XY model. These patterns are stable although their formation is less likely in comparison to the ground state lasing mode. In addition, we show that small ratios of photon-to-gain lifetime destabilizes vortex and antivortex phase patterns. These findings are important for studying topological effects in optics as well as for designing laser array devices.

Resolving the Berezinskii-Kosterlitz-Thouless transition in the two-dimensional XY model with tensor-network-based level spectroscopy

Berezinskii-Kosterlitz-Thouless transition of the classical XY model is re-investigated, combining the Tensor Network Renormalization (TNR) and the Level Spectroscopy method based on the finite-size scaling of the Conformal Field Theory. By systematically analyzing the spectrum of the transfer matrix of the systems of various moderate sizes which can be accurately handled with a finite bond dimension, we determine the critical point removing the logarithmic corrections. This improves the accuracy by an order of magnitude over previous studies including those utilizing TNR. Our analysis also gives a visualization of the celebrated Kosterlitz Renormalization Group flow based on the numerical data.

Localization of the Higgs mode at the superfluid–Mott glass transition

The amplitude (Higgs) mode near the two-dimensional superfluid-Mott glass quantum phase transition is studied. We map the Bose-Hubbard Hamiltonian of disordered interacting bosons onto an equivalent classical XY model in (2+1) dimensions and compute the scalar susceptibility of the order parameter amplitude via Monte Carlo simulation. Analytic continuation of the scalar susceptibilities from imaginary to real frequency to obtain the spectral densities is performed by a modified maximum entropy technique. Our results show that the introduction of disorder into the system leads to unconventional dynamical behavior of the Higgs mode that violates naive scaling,despite the underlying thermodynamics of the transition being of conventional power-law type. The computed spectral densities exhibit a broad, non-critical response for all energies, and a momentum-independent dispersion for long-wavelengths, indicating strong evidence for the localization of the Higgs mode for all dilutions.

Topological persistence machine of phase transitions.

The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass-liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed "topological persistence machine," to construct the shape of data from correlations in states, so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis. We demonstrate the efficacy of the approach in detecting the Berezinskii-Kosterlitz-Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose-Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide practical insights for exploring the phases of experimental physical systems.

Multiple pseudo-plateaux states and antiferromagnetic pair selection in the XY model in the highly frustrated honeycomb lattice

We present a study of the low temperature magnetic phases of the classical XY model with third nearest neighbor interactions on the honeycomb lattice at the maximally frustrated point under an external magnetic field by extensive Monte Carlo simulations. We focus on the characterization of the emergent low temperature phases, which are a direct consequence of the unusually high numbers of spins per plaquette in the model. Specifically, we show that, since thermal fluctuations partially lift the ground-state degeneracy and select the most collinear states, the selected states are those with the highest number of ‘antiferromagnetic pairs’ (AFp) compatible with the external magnetic field. These AFp are formed in such a way that they maximize the degeneracy of the selected submanifold of ground states. Moreover, two collinear pseudoplateaux emerge at M = 1/3 and M = 2/3. To characterize the magnetization process, we employ Monte Carlo simulations and calculate relevant order parameters to construct the complete temperature vs magnetic field phase diagram.

Hamiltonian chaos and differential geometry of configuration space–time

This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. More precisely, a Hamiltonian flow is identified with a geodesic flow on configuration space–time endowed with a suitable metric due to Eisenhart. Until now, this framework has never been given attention to describe chaotic dynamics. A gap that is filled in the present work. In a Riemannian-geometric context, the stability/instability of the dynamics depends on the curvature properties of the ambient manifold and is investigated by means of the Jacobi–Levi-Civita (JLC) equation for geodesic spread. It is confirmed that the dominant mechanism at the ground of chaotic dynamics is parametric instability due to curvature variations along the geodesics. A comparison is reported of the outcomes of the JLC equation written also for the Jacobi metric on configuration space and for another metric due to Eisenhart on an extended configuration space–time. This has been applied to the Hénon–Heiles model, a two-degrees of freedom system. Then the study has been extended to the 1D classical Heisenberg XY model at a large number of degrees of freedom. Both the advantages and drawbacks of this geometrization of Hamiltonian dynamics are discussed. Finally, a quick hint is put forward concerning the possible extension of the differential–geometric investigation of chaos in generic dynamical systems, including dissipative ones, by resorting to Finsler manifolds.