Classical XY Spin Research Articles

Dynamics of dissipative topological defects in coupled phase oscillators

The dynamics of topological defects in a system of coupled phase oscillators, arranged in one and two-dimensional arrays, was numerically investigated using the Kuramoto model. After a rapid decay of the number of topological defects, a long-time quasi steady state with few topological defects was detected. Two competing time scales governed the dynamics corresponding to the dissipation rate and the coupling quench rate. The density of topological defects scales as a power law function of the coupling quench rate ρ ∼ Cν with ν = 0.25. Reducing the number of topological defects improves the long time coherence and order parameter of the system, enhancing the probability to reach a global minimal loss state that can be mapped to the ground state of a classical XY spin Hamiltonian.

Loop Correlations in Random Wire Models

We introduce a family of loop soup models on the hypercubic lattice. The models involve links on the edges, and random pairings of the link endpoints on the sites. We conjecture that loop correlations of distant points are given by Poisson–Dirichlet correlations in dimensions three and higher. We prove that, in a specific random wire model that is related to the classical XY spin system, the probability that distant sites form an even partition is given by the Poisson–Dirichlet counterpart.

Linear magnetic susceptibility of a finite XY chain model with magnetic helices as metastable states

Abstract A finite-size ferromagnetic chain of classical XY spins, where each of the two edge spins is exchange-coupled to a fictitious fully-pinned spin, is investigated in the presence of a weak in-plane perturbation (external magnetic field or uniaxial anisotropy). Within a macrospin approximation, the above chain model can be representative of a soft ferromagnetic film consisting of N parallel atomic layers, sandwiched between two magnetically hard films. In zero field, the model has a ferromagnetic collinear ground state and admits a class of metastable helical states containing an integer number of twists. Analytical expressions are obtained for the linear corrections to the helical magnetic configuration and for the local magnetic susceptibility tensor. The dependence of the integral susceptibility of the chain, χ , on the number of spins in the chain, N, is also calculated. It is found that χ ∝ N 3 for small values of the helix mode ( m ≪ N ), while χ ∝ N 2 for large values of the helix mode ( m ≈ N + 1 4 ).

Magnetic helices as metastable states of finite XY ferromagnetic chains: An analytical study

We investigated a simple but non trivial model, consisting of a chain of N classical XY spins with nearest neighbor ferromagnetic interaction, where each of the two end-point spins is assumed to be exchange-coupled to a fully-pinned fictitious spin. In the mean field approximation, the system might be representative of a soft ferromagnetic film sandwiched between two magnetically hard layers. We show that, while the ground state is ferromagnetic and collinear, the system can attain non-collinear metastable states in the form of magnetic helices. The helical solutions and their stability were studied analytically in the absence of an external magnetic field. There are four possible classes of solutions. Only one class is metastable, and its helical states contain an integer number of turns. Among the remaining unstable classes, there is a class of helices which contain an integer number of turns. Therefore, an integer number of turns in a helical configuration is a necessary, but not a sufficient, condition for metastability. These results may be useful to devise future applications of metastable magnetic helices as energy-storing elements.

Topological defect formation in 1D and 2D spin chains realized by network of optical parametric oscillators

A network of optical parametric oscillators (OPOs) is used to simulate classical Ising and XY spin chains. The collective nonlinear dynamics of this network, driven by quantum noise rather than thermal fluctuations, seeks out the Ising/XY ground state as the system transitions from below to above the lasing threshold. We study the behavior of this “Ising machine” for three canonical problems: a 1D ferromagnetic spin chain, a 2D square lattice and problems where next-nearest-neighbor couplings give rise to frustration. If the pump turn-on time is finite, topological defects form (domain walls for the Ising model, winding number and vortices for XY) and their density can be predicted from a numerical model involving a linear “growth stage” and a nonlinear “saturation stage”. These predictions are compared against recent data for a 10,000-spin 1D Ising machine.

Persistent current in a 2D Josephson junction array wrapped around a cylinder

We study persistent currents in a Josephson junction array wrapped around a cylinder. The T = 0 quantum statistical mechanics of the array is equivalent to the statistical mechanics of a classical xy spin system in 2+1 dimensions at the effective temperature , with J being the Josephson energy of the junction and U being the charging energy of the superconducting island. It is investigated analytically and numerically on lattices containing over one million sites. For weak disorder and the dependence of the persistent current on disorder and computed numerically agrees quantitatively with the analytical result derived within the spin-wave approximation. The high- and/or strong-disorder behavior is dominated by instantons corresponding to the vortex loops in 2 + 1 dimensions. The current becomes destroyed completely at the quantum phase transition into the Cooper-pair insulating phase.

Magnetic properties of a classical XY spin dimer in a “planar” magnetic field

Single-molecule magnetism originates from the strong intra-molecular magnetic coupling of a small number of interacting spins. Such spins generally interact very weakly with the neighboring spins in the other molecules of the compound, therefore, inter-molecular spin couplings are negligible. In certain cases the number of magnetically coupled spins is as small as a dimer, a system that can be considered the smallest nanomagnet capable of storing non-trivial magnetic information on the molecular level. Additional interesting patterns arise if the spin motion is confined to a two-dimensional space. In such a scenario, clusters consisting of spins with large-spin values are particularly attractive since their magnetic interactions can be described well in terms of classical Heisenberg XY spins. In this work we calculate exactly the magnetic properties of a nanomagnetic dimer of classical XY spins in a “planar” external magnetic field. The problem is solved by employing a mathematical approach whose idea is the introduction of auxiliary spin variables into the starting expression of the partition function. Results for the total internal energy, total magnetic moment, spin–spin correlation function and zero-field magnetic susceptibility can serve as a basis to understand the magnetic properties of large-spin dimer building blocks.

Metastable configurations of a finite-size chain of classical spins within the one-dimensional chiral XY-model

The metastable states of a finite-size chain of N classical spins described by the chiral XY-model on a discrete one-dimensional lattice are calculated by means of a general theoretical method recently developed by one of us. This method allows one to determine all the possible equilibrium magnetic states in an accurate and systematic way. The ground state of a chain consisting of N classical XY spins is calculated in the presence of (i) a symmetric ferromagnetic exchange interaction, favoring parallel alignment of nearest neighbor spins, (ii) a uniaxial anisotropy, favoring a given direction in the film plane, and (iii) an antisymmetric Dzyaloshinskii–Moriya interaction (DMI), favoring perpendicular alignment of nearest neighbor spins. In addition to the ground state with a non-uniform helical spin arrangement, which originates from the energy competition in the finite-size chain with open boundary conditions, we have found a considerable number of higher-energy equilibrium states. In the investigated case of a chain with N=10 spins and a DMI much smaller than the in-plane uniaxial anisotropy, it turns out that a metastable (unstable) state of the finite chain is characterized by a configuration where none (at least one) of the inner spins is nearly parallel to the hard axis. The role of the DMI is to establish a unique rotational sense for the helical ground state. Moreover, the number of both metastable and unstable equilibrium states is doubled with respect to the case of zero DMI. This produces modifications in the Peierls–Nabarro potential encountered by a domain wall during its displacement along the discrete spin chain.

Observing Geometric Frustration with Thousands of Coupled Lasers

Geometric frustration, the inability of an ordered system to find a unique ground state plays a key role in a wide range of systems. We present a new experimental approach to observe large-scale geometric frustration with 1500 negatively coupled lasers arranged in a kagome lattice. We show how dissipation drives the lasers into a phase-locked state that directly maps to the classical XY spin Hamiltonian ground state. In our system, frustration is manifested by the lack of long range phase ordering. Finally, we show how next-nearest-neighbor coupling removes frustration and restores order.

Frequency and Size Dependences of Superfluidity in Low-Dimensional 4He Fluids

We have studied superfluidity of 4He fluids in two- and one-dimensional states. In the 2D state of the 4He films on flat substrate, superfluidity was observed in the normal fluid state above the 2D Kosterlitz-Thouless temperature at high measurement frequencies. The superfluidity in 2D also depends on the system size, e.g. pore diameter of porous glasses and grain size of powder. The 1D state was realized for 4He fluid nanotubes formed in 1D nanopores 1.8–4.7 nm in diameter and about 300 nm in length. Superfluidity in the 1D state was observed by the torsional oscillator experiment. The results are qualitatively well reproduced by the Monte Carlo calculation for a classical XY spin system modeled on the present 4He nanotubes. Although the 1D state is in the normal fluid state at any finite low temperatures due to the 1D phonon fluctuation, the superfluid frequency shift Δf of the oscillator can be observed. Above a temperature, Δf decreases due to another kind of 1D thermal fluctuation of which excitations destroy the phase coherence in the nanotubes. The excitations depend on the tube length as well as the tube diameter.

Duality argument for chiral-nematic phase of planar spins

A duality argument for the recently discovered chiral-nematic phase of the XY model in a triangular lattice is presented. We show that a new Ising variable naturally emerges in mapping the antiferromagnetic J1-J2 classical XY spin Hamiltonian onto an appropriate Villain model on a triangular lattice. The new variable is the chirality degree of freedom, which exists in addition to the usual vortex variables, in the dual picture. Elementary excitations and the associated phase transition of the Ising degrees of freedom are discussed in some detail.

Low-temperature long-range ordering of a classical XY spin system with bilinear-biquadratic exchange Hamiltonian

We investigate low-temperature long-range ordering of a classical XY spin system with bilinear and biquadratic exchange interactions, J 1 and J 2, respectively, on stacked triangular lattice by histogram Monte Carlo simulations. Focus is laid on determination of phase boundaries, nature of transitions, and magnetic structures in the respective phases. Negative J 1 and/or J 2 induce lattice geometry frustration that can be either relaxed or enhanced, depending on the signs and strengths of both exchanges. Moreover, the exchanges can also mutually compete. Hence, the resulting low-temperature phase diagram in J 1– J 2 parameter space features several peculiar phases, including highly disordered ones in the regions of the coexistence of a non-frustrated exchange with a frustration inducing one.

Off equilibrium dynamics in the 2d-XY system

We study the non-equilibrium time evolution of the classical XY spin model in two dimensions. The two-time autocorrelation and linear response functions are considered for systems initially prepared in a high temperature state and in a completely ordered state. After a quench into the critical phase, we determine, via Monte Carlo simulations, the time-evolution of these quantities and extract the temperature dependence of the slope of the parametric plot susceptibility/correlation in the asymptotic regime. This slope is usually identified with the infinite fluctuation-dissipation ratio which measures the violation to the equilibrium fluctuation-dissipation theorem. However, a direct measure of this ratio leads to a vanishing value.

Field-induced transitions and spatial chaos in the classical XY spin chain

We study the lowest-lying excitation of a classical ferromagnetic XY spin chain, in the presence of a symmetry breaking magnetic field. Extremizing the energy of this system leads to a two-dimensional nonlinear map, whose allowed phase space shrinks with increasing field in a nontrivial manner. The orbits of the map represent the set of extremum energy spin configurations. For each field, we compute the energy of the members of this set and find the lowest energy among them, excluding the obvious ground state configuration with all spins parallel along the field direction. This state turns out to be the unstable fixed point of the map. We show that up to a certain (primary) critical field, a separatrixlike 2pi soliton configuration is the lowest-energy excitation, with an energy very close to the ground state energy. For any field beyond this critical field, the soliton disappears and lowest excitation is a librational configuration corresponding to the outermost orbit in the phase plot at that field. Further, its energy is found to be much higher than the ground state energy, leading to a sharp jump in the difference in energy between the former and the latter at this field. With further increase in the field, sharp jumps in the excitation energy arise at certain secondary critical fields as well. We show that these appear when the corresponding spin configurations become commensurate. This complex behavior of the energy is interpreted and its effect on the magnetization and static susceptibility of the system is also studied.

Surface spin-flop and discommensuration transitions in antiferromagnets

Phase diagrams as a function of anisotropy $D$ and magnetic field $H$ are obtained for discommensurations and surface states for an antiferromagnet in which $H$ is parallel to the easy axis, by modeling it using the ground states of a one-dimensional chain of classical XY spins. A surface spin-flop phase exists for all $D$, but the interval in $H$ over which it is stable becomes extremely small as $D$ goes to zero. First-order transitions, separating different surface states and ending in critical points, exist inside the surface spin-flop region. They accumulate at a field $H'$ (depending on $D$) significantly less than the value $H_{SF}$ for a bulk spin-flop transition. For $H' < H < H_{SF}$ there is no surface spin-flop phase in the strict sense; instead, the surface restructures by, in effect, producing a discommensuration infinitely far away in the bulk. The results are used to explain in detail the phase transitions occurring in systems consisting of a finite, even number of layers.

Phase diagram of XY antiferromagnetic stacked triangular lattices.

The phase transition in antiferromagnetic stacked triangular lattices with classical XY spins interacting via antiferromagnetic nearest- and next-nearest-neighbor bonds, J 1 and J 2, is studied by means of extensive histogram Monte Carlo simulations. When J 250, the transition is of second order with critical exponents slightly different from those given by other authors. It is shown that in a range of J 2, the transition is of first order. The general phase diagram in the ( J 2 ,T) space (T, temperature! is shown and discussed. @S0163-1829~96!03530-8#

Onsager reaction field theory of Josephson junctions arrays

Josephson junctions arrays, in the limit where the order parameter has a fixed modulus, can be described by the classical XY model. We calculate the transition temperature for a model of classical XY spin chains coupled to their nearest neighbor chains. The interchain interaction is treated within the Onsager reaction field formalism, whereas each chain, in the effective field caused by the other chains, is treated exactly.

Monte Carlo simulation of high Tc superconductive 3D-glass model

We studied Monte Carlo simulations of three dimensional(3D) classical anisotropic XY-spin glass model with bonding disorder and frustration. We argued the relation between 3D array, with weakly coupled superconducting grains and 3D classical anisotropic XY-spin glass model. The physical quantities such as susceptibility and specific heat were obtained. We also investigated phase diagram of intrinsic critical current and vortex pinning of 3D Josephson-Junction Arrays(JJA).

First-order transition in antiferromagnetic stacked triangular lattices with vector spins

In this article, we show by Monte Carlo simulations that the phase transition is definitely of first order in the antiferromagnetic stacked triangular lattices with both classical XY and Heisenberg spins in a range of the antiferromagnetic next-nearest neighbor interaction J2. The phase diagram in the (J2/J1, T) space (J1: nearest-neighbor interaction,T: temperature) is shown and discussed.

Frustrated classical XY spin systems in the two-dimensional Penrose tiling

Magnetic ground states of frustrated classical XY spin systems in a finite two-dimensional Penrose tiling are determined by Monte Carlo simulations using the simulated annealing method. The interactions are limited to the third neighbours ( J 1, J 2, J 3) and the next nearest neighbour interaction is assumed to be antiferromagnetic. If frustration is low, the ground state is antiferromagnetic. If J 1 or J 3 is positive, increasing the absolute value of the negative exchange integral ( J 3 or J 1, respectively) which increases the local frustration does not break up the magnetic order but only broadens the two peaks of the spin angular distribution. There is evidence for a Kosterlitz-Thouless transition, as in periodic systems, in the thermal variation of the specific heat. This transition temperature is found to be independent of the nearest neighbour interaction, J 1, but varies linearly with the third neighbour one, J 3.