Classical Spin Systems Research Articles

A Frustrated Antipolar Phase Analogous to Classical Spin Liquids.

The study of magnetic frustration in classical spin systems is motivated by the prediction and discovery of classical spin liquid states. These uncommon magnetic phases are characterized by a massive degeneracy of their ground state implying a finite magnetic entropy at zero temperature. While the classical spin liquid state is originally predicted in the Ising triangular lattice antiferromagnet in 1950, this state has never been experimentally observed in any triangular magnets. The discovery of an electric analogue of classical spin liquids on a triangular lattice of uniaxial electric dipoles in EuAl12O19 is reported here. This new type of frustrated antipolar phase is characterized by a highly-degenerate state at low temperature implying an absence of long-range antiferroelectric order, despite short-range antipolar correlations. Its dynamics are governed by a thermally activated process, slowing down upon cooling toward a complete freezing at zerotemperature.

Learning energy-based representations of quantum many-body states

Efficient representation of quantum many-body states on classical computers is a problem of practical importance. An ideal representation of a quantum state combines a succinct characterization informed by the structure and symmetries of the system along with the ability to predict the physical observables of interest. Several machine-learning approaches have been recently used to construct such classical representations, which enable predictions of observables and account for physical symmetries. However, the structure of a quantum state typically gets lost unless a specialized is employed based on prior knowledge of the system. Moreover, most such approaches give no information about what states are easier to learn in comparison with others. Here, we propose a generative energy-based representation of quantum many-body states derived from Gibbs distributions used for modeling the thermal states of classical spin systems. Based on the prior information on a family of quantum states, the energy function can be specified by a small number of parameters using an explicit low-degree polynomial or a generic parametric family such as neural nets and can naturally include the known symmetries of the system. Our results show that such a representation can be efficiently learned from data using exact algorithms in a form that enables the prediction of expectation values of physical observables. Importantly, the structure of the learned energy function provides a natural explanation for the difficulty of learning an energy-based representation of a given class of quantum states when measured in a certain basis. Published by the American Physical Society 2024

Sparse autoregressive neural networks for classical spin systems

Efficient sampling and approximation of Boltzmann distributions involving large sets of binary variables, or spins, are pivotal in diverse scientific fields even beyond physics. Recent advances in generative neural networks have significantly impacted this domain. However, these neural networks are often treated as black boxes, with architectures primarily influenced by data-driven problems in computational science. Addressing this gap, we introduce a novel autoregressive neural network architecture named TwoBo, specifically designed for sparse two-body interacting spin systems. We directly incorporate the Boltzmann distribution into its architecture and parameters, resulting in enhanced convergence speed, superior free energy accuracy, and reduced trainable parameters. We perform numerical experiments on disordered, frustrated systems with more than 1000 spins on grids and random graphs, and demonstrate its advantages compared to previous autoregressive and recurrent architectures. Our findings validate a physically informed approach and suggest potential extensions to multivalued variables and many-body interaction systems, paving the way for broader applications in scientific research.

Accounting for quantum effects in atomistic spin dynamics

Atomistic spin dynamics (ASD) is a standard tool to model the magnetization dynamics of a variety of materials. The fundamental dynamical model underlying ASD is entirely classical. In this paper, we present two approaches to effectively incorporate quantum effects into ASD simulations, thus enhancing their low-temperature predictions. The first allows to simulate the magnetic behavior of a quantum spin system by solving the equations of motion of a classical spin system at an effective temperature relative to the critical temperature. This effective temperature is determined from the microscopic properties of the system. The second approach is based on a semiclassical model where classical spins interact with an environment with a quantumlike power spectrum. The parameters that characterize this model can be calculated or extracted from experiments. This semiclassical model quantitatively reproduces the absolute temperature behavior of a magnetic system, thus accounting for the quantum mechanical aspects of its dynamics, even at low temperature. The methods presented here can be readily implemented in current ASD simulations with no additional complexity cost. Published by the American Physical Society 2024

Exotic magnetization curves in classical square-kagomé spin lattices

Classical spin systems with non-coplanar ground states typically exhibit nonlinear magnetization curves characterized by kinks and jumps. Our article briefly summarizes the most important related analytical results. In a comprehensive case study, we then address AF-square kagomé and AF/FM-square kagomé spin lattices equipped with additional cross-plaquette interactions. It is known that these systems have non-coplanar ground states that assume a cuboctahedral structure in the absence of a magnetic field. When a magnetic field H is switched on, a rich variety of different phases develops from the cuboctahedral ground state, which are studied in their dependence on H and a cross-plaquette coupling constant J3>0 . For the AF square-kagomé spin lattice, we carefully identify and describe seven phases that appear in a phase diagram with five triple points. The transitions between these phases are predominantly discontinuous, although two cases exhibit continuous transitions. In contrast, the phase diagram of the AF/FM square-kagomé model shows only four phases with a single triple point, but these also lead to exotic magnetization curves. Here, too, there are two types of phase boundaries belonging to continuous and discontinuous transitions.

Nonlinear Filtering of Classical and Quantum Spin Systems

Nonlinear Filtering of Classical and Quantum Spin Systems

Thermodynamic limit of spin systems on random graphs

We utilize the graphon—a continuous mathematical object which represents the limit of convergent sequences of dense graphs—to formulate a general, continuous description of quantum spin systems in thermal equilibrium when the average coordination number grows extensively in the system size. Specifically, we derive a closed set of coupled nonlinear Fredholm integral equations which governs the properties of the system. The graphon forms the kernel of these equations, and their solution yields exact expressions for the macroscopic observables in the system in the thermodynamic limit. We analyze these equations for both quantum and classical spin systems, recovering known results and providing analytical solutions for a range of more complex cases. We supplement this with controlled, finite-size numerical calculations using Monte Carlo and tensor network methods, showing their convergence towards our analytical results with increasing system size. Published by the American Physical Society 2024

Efficient Algorithms for Approximating Quantum Partition Functions at Low Temperature

We establish an efficient approximation algorithm for the partition functions of a class of quantum spin systems at low temperature, which can be viewed as stable quantum perturbations of classical spin systems. Our algorithm is based on combining the contour representation of quantum spin systems of this type due to Borgs, Kotecký, and Ueltschi with the algorithmic framework developed by Helmuth, Perkins, and Regts, and Borgs et al.

Antiferromagnet on Pyrochlore Lattice in Spherical Approximation

"We develop an 1/n expansion method for the systematic approximation of the solutions of classical spin systems (tri-dimensional vectors) which can describe general spin-spin interactions and phases with complex magnetization pattern. This method is applied to study the phase transitions observed in pyrochlore crystals (e.g. Gd2Ti2O7) and the results have qualitative agreement with the experimental ones."

Estimating Patterns of Classical and Quantum Skyrmion States

In this review we discuss the latest results concerning development of the machine learning algorithms for characterization of the magnetic skyrmions that are topologically-protected magnetic textures originated from the Dzyaloshinskii-Moriya interaction that competes Heisenberg isotropic exchange in ferromagnets. We show that for classical spin systems there is a whole pool of machine approaches allowing their accurate phase classification and quantitative description on the basis of few magnetization snapshots. In turn, investigation of the quantum skyrmions is a less explored issue, since there are fundamental limitations on the simulation of such wave functions with classical supercomputers. One needs to find the ways to imitate quantum skyrmions on near-term quantum computers. In this respect, we discuss implementation of the method for estimating structural complexity of classical objects for characterization of the quantum skyrmion state on the basis of limited number of bitstrings obtained from the projective measurements.

Collective Monte Carlo updates through tensor network renormalization

We introduce a Metropolis-Hastings Markov chain for Boltzmann distributions of classical spin systems. It relies on approximate tensor network contractions to propose correlated collective updates at each step of the evolution. We present benchmark computations for a wide variety of instances of the two-dimensional Ising model, including ferromagnetic, antiferromagnetic, (fully) frustrated and Edwards-Anderson spin glass instances, and we show that, with modest computational effort, our Markov chain achieves sizeable acceptance rates, even in the vicinity of critical points. In each of the situations we have considered, the Markov chain compares well with other Monte Carlo schemes such as the Metropolis or Wolff’s algorithm: equilibration times appear to be reduced by a factor that varies between 4040 and 20002000, depending on the model and the observable being monitored. We also present an extension to three spatial dimensions, and demonstrate that it exhibits fast equilibration for finite ferro- and antiferromagnetic instances. Additionally, and although it is originally designed for a square lattice of finite degrees of freedom with open boundary conditions, the proposed scheme can be used as such, or with slight modifications, to study triangular lattices, systems with continuous degrees of freedom, matrix models, a confined gas of hard spheres, or to deal with arbitrary boundary conditions.

A closure for the master equation starting from the dynamic cavity method

We consider classical spin systems evolving in continuous time with interactions given by a locally tree-like graph. Several approximate analysis methods have earlier been reported based on the idea of Belief Propagation / cavity method. We introduce a new such method which can be derived in a more systematic manner using the theory of Random Point Processes. Within this approach, the master equation governing the system’s dynamics is closed via a set of differential equations for the auxiliary cavity probabilities. The numerical results improve on the earlier versions of the closure on several important classes of problems. We re-visit here the cases of the Ising ferromagnet and the Viana–Bray spin-glass model.

Dynamics of mixed quantum–classical spin systems *

Mixed quantum–classical spin systems have been proposed in spin chain theory and, more recently, in magnon spintronics. However, current models of quantum–classical dynamics beyond mean-field approximations typically suffer from long-standing consistency issues, and, in some cases, invalidate Heisenberg’s uncertainty principle. Here, we present a fully Hamiltonian theory of quantum–classical spin dynamics that appears to be the first to ensure an entire series of consistency properties, including positivity of both the classical and the quantum density, so that Heisenberg’s principle is satisfied at all times. We show how this theory may connect to recent energy-balance considerations in measurement theory and we present its Poisson bracket structure explicitly. After focusing on the simpler case of a classical Bloch vector interacting with a quantum spin observable, we illustrate the extension of the model to systems with several spins, and restore the presence of orbital degrees of freedom.

Nematic ordering in the Heisenberg spin-glass system in three dimensions.

Nematic ordering, where the spins globally align along a spontaneously chosen axis irrespective of direction, occurs in spin-glass systems of classical Heisenberg spins in d=3. In this system where the nearest-neighbor interactions are quenched randomly ferromagnetic or antiferromagnetic, instead of the locally randomly ordered spin-glass phase, the system orders globally as a nematic phase. This nematic ordering of the Heisenberg spin-glass system is dramatically different from the spin-glass ordering of the Ising spin-glass system. The system is solved exactly on a hierarchical lattice and, equivalently, Migdal-Kadanoff approximately on a cubic lattice. The global phase diagram is calculated, exhibiting this nematic phase, and ferromagnetic, antiferromagnetic, disordered phases. The nematic phase of the classical Heisenberg spin-glass system is also found in other dimensions d>2: We calculate nematic transition temperatures in 24 different dimensions in 2<d≤4.

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.

Phase diagram and thermodynamic properties of the frustrated ferro-antiferromagnetic spin system on the octahedral lattice

The influence of anisotropy generated by the competition between the ferromagnetic and antiferromagnetic nearest-neighbor interactions on the magnetic and thermodynamic properties of the classical spin system on the octahedral lattice is investigated in the framework of the exactly solvable Ising-like model on the recursive octahedral lattice. The phase diagram of the model is found and the nature of all phase transitions is determined. It is shown that the model exhibits the existence of the intermediate phase that separates the ferromagnetic phase from the antiferromagnetic one. Moreover, it is also shown that while the intermediate phase and the antiferromagnetic one are separated by the critical curve of the second-order phase transitions the transitions between the ferromagnetic phase and the intermediate one have the first-order nature on the coexistence line. The magnetization and entropy properties of all phases of the model are studied. Depending on the value of the frustration parameter, four different ground states are identified with formation of a strict residual entropy hierarchy. Due to the frustration nature of the model the specific heat capacity exhibits anomalous (Schottky) behavior at low temperatures in the vicinity of the value of the frustration parameter, for which two highly macroscopically degenerated single-point-like ground states are formed. The presence of the intermediate phase in the model also leads to the existence of the series of even three successive second-order phase transitions in the temperature dependence of the magnetization as well as of the specific heat capacity.

Complexity as information in spin-glass Gibbs states and metastates: Upper bounds at nonzero temperature and long-range models.

In classical finite-range spin systems, especially those with disorder such as spin glasses, a low-temperature Gibbs state may be a mixture of a number of pure or ordered states; the complexity of the Gibbs state has been defined in the past roughly as the logarithm of this number, assuming the question is meaningful in a finite system. As nontrivial pure-state structure is lost in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)2470-004510.1103/PhysRevE.101.042114] Höller and the author introduced a definition of the complexity of an infinite-size Gibbs state as the mutual information between the pure state and the spin configuration in a finite region, and applied this also within a metastate construction. (A metastate is a probability distribution on Gibbs states.) They found an upper bound on the complexity for models of Ising spins in which each spin interacts with only a finite number of others, in terms of the surface area of the region, for all T≥0. In the present paper, the complexity of a metastate is defined likewise in terms of the mutual information between the Gibbs state and the spin configuration. Upper bounds are found for each of these complexities for general finite-range (i.e., short- or long-range, in a sense we define) mixed p-spin interactions of discrete or continuous spins (such as m-vector models), but only for T>0. For short-range models, the bound reduces to the surface area. For long-range interactions, the definition of a Gibbs state has to be modified, and for these models we also prove that the states obtained within the metastate constructions are Gibbs states under the modified definition. All results are valid for a large class of disorder distributions.

Thermodynamics of the classical spin triangle

Abstract The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper, we explicitly calculate thermodynamic quantities such as density of states, specific heat, susceptibility and spin autocorrelation functions. These calculations are performed (semi-)analytically and shown to agree with corresponding Monte Carlo simulations. It is shown that the thermodynamic functions depend qualitatively on the character of the system in terms of its frustration, especially w. r. t. their low temperature limit. For the long-time autocorrelation function, we find, for certain values of the coupling constants, a decay to constant values in the form of an 1/t damped harmonic oscillation and propose a theoretical explanation.

Cooling classical many-spin systems using feedback control

We propose a technique for polarizing and cooling finite many-body classical systems using feedback control. The technique requires the system to have one collective degree of freedom conserved by the internal dynamics. The fluctuations of other degrees of freedom are then converted into the growth of the conserved one. The proposal is validated using numerical simulations of classical spin systems in a setting representative of nuclear magnetic resonance experiments. In particular, we were able to achieve 90% polarization for a lattice of 1000 classical spins starting from an unpolarized infinite temperature state.

Magnetic Structures of Electron Systems on the Extended Spatially Completely Anisotropic Triangular Lattice near Quantum Critical Points

We examine magnetic structures of electron systems on an extended triangular lattice that consists of two types of bond triangles with electron transfer energies t_l and t'_l (l = 1, 2, and 3), respectively. We examine the ground state in the mean-field theory when t_1 = t'_1, focusing on collinear states with two sublattices. It is shown that when the imbalance of the spatial anisotropies of the two triangles is large, up-up-down-down (uudd) phases are stable, and the most likely ground states of the lambda-(BETS)_2FeCl_4 system are the Neel state with the modulation vector (\pi/c,\pi/a) and a uudd state, where c = a_1 = a'_1 and a = (a_2+a'_2)/2, with a_1, a_1', a_2, and a_2' being the lattice constants of the bonds with t_1, t'_1, t_2, and t'_2, respectively. These results are consistent with those from the classical spin system. In addition, this study reveals behaviors near the quantum critical point, which cannot be reproduced in the localized spin model. As the imbalance of the spatial anisotropies increases, the U_c of the Neel state increases, and that of the uudd state decreases. In the phase diagrams containing areas of the paramagnetic state, the Neel state with (\pi/c,\pi/a), and a uudd state, their boundaries terminate at a triple point, near which all the transitions are of the first order. The phase boundary between the antiferromagnetic phases does not depend on U, and the transition is of the first order everywhere on the boundary. By contrast, the transitions from the two antiferromagnetic phases to the paramagnetic phase are of the second order, unless the system is close to the triple point.