XY Model In Random Field Research Articles

Machine learning analysis of dimensional reduction conjecture for nonequilibrium Berezinskii-Kosterlitz-Thouless transition in three dimensions.

We investigate the recently proposed dimensional reduction conjecture in driven disordered systems using a machine learning technique. The conjecture states that a static snapshot of a disordered system driven at a constant velocity is equal to a space-time trajectory of its lower-dimensional pure counterpart. This suggests that the three-dimensional (3D) random field XY model exhibits the Berezinskii-Kosterlitz-Thouless transition when driven out of equilibrium. To verify the conjecture directly by observing configurations of the system, we utilize the capacity of neural networks to detect subtle features of images. Specifically, we train neural networks to differentiate snapshots of the 3D driven random field XY model from space-time trajectories of the two-dimensional pure XY model. Our results demonstrate that the network cannot distinguish between the two, confirming the dimensional reduction conjecture.

Spin cones in random-field XY models.

We determine the arrangement of spins in the ground state of the XY model with quenched, random fields, on a fully connected graph. Two types of disordered fields are considered, namely, randomly oriented magnetic fields and randomly oriented crystal fields. Orientations are chosen from a uniformly isotropic distribution, but disorder fluctuations in each realization of a finite system lead to a breaking of rotational symmetry. The result is an interesting pattern of spin orientations found by solving a system of coupled, nonlinear equationswithin perturbation theory and also by exact numerical continuation. All spins lie within a cone for small enough ratio of field to coupling strength, with an interesting distribution of spin orientations, with peaks at the cone edges. The orientation of the cone depends strongly on the realization of disorder, but the opening angle does not. In the case of random magnetic fields, the cone angle widens as the ratio increases till a critical value at which there is a first-order phase transition and the cone disappears. With random crystal fields, there is no phase transition and the cone angle approaches 180^{∘} for large values of the ratio. At finite low temperatures, Monte Carlo simulations show that the formation of a cone and its subsequent alignment along the equilibrium direction occur on two different timescales.

Domain growth and aging in the random field XY model: A Monte Carlo study.

We use large-scale Monte Carlo simulations to obtain comprehensive results for domain growth and aging in the random field XY model in dimensions d=2,3. After a deep quench from the paramagnetic phase, the system orders locally via annihilation of topological defects, i.e., vortices and antivortices. The evolution morphology of the system is characterized by the correlation function and the structure factor of the magnetization field. We find that these quantities obey dynamical scaling, and their scaling function is independent of the disorder strength Δ. However, the scaling form of the autocorrelation function is found to be dependent on Δ, i.e., superuniversality is violated. The large-t behavior of the autocorrelation function is explored by studying aging and autocorrelation exponents. We also investigate the characteristic growth law L(t,Δ) in d=2,3, which shows an asymptotic logarithmic behavior: L(t,Δ)∼Δ^{-φ}(lnt)^{1/ψ}, with exponents φ,ψ>0.

Behavior of the random-field XY model on simple cubic lattices at h_{r}=1.5.

We have performed studies of the three-dimensional random-field XY model on 32 samples of L×L×L simple cubic lattices with periodic boundary conditions, with a random field strength of h_{r} = 1.5, for L= 128, using a parallelized Monte Carlo algorithm. We present results for the sample-averaged magnetic structure factor S(k[over ⃗]) over a range of temperature, using both random hot start and ferromagnetic cold start initial states, and k[over ⃗] along the [1,0,0] and [1,1,1] directions. At T= 1.875, S(k[over ⃗]) shows a broad peak near |k[over ⃗]|=0, with a correlation length which is limited by thermal fluctuations, rather than the lattice size. As T is lowered, this peak grows and sharpens. By T= 1.5, it is clear that the correlation length is larger than L= 128. The lowest temperature for which S(k[over ⃗]) was calculated is T= 1.421875, where the hot start and cold start initial conditions usually do not find the same local minimum in the phase space. Our results are consistent with the idea that there is a finite value of T below which S(k[over ⃗]) diverges slowly as |k[over ⃗]| goes to zero. This divergence would imply that the relaxation time of the spins is also diverging. That is the signature of an ergodicity-breaking phase transition.

The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

The ferromagnetic XY model on sparse random graphs in a randomly oriented field is analyzed via the belief propagation algorithm. At variance with the fully connected case and with the random field Ising model on the same topology, we find strong evidence of a tiny region with replica symmetry breaking (RSB) in the limit of very low temperatures. This RSB phase is robust against different choices of the external field direction, while it rapidly vanishes when increasing the graph mean degree, the temperature or the directional bias in the external field. The crucial ingredients to have such a RSB phase seem to be the continuous nature of vector spins, mostly preserved by the -invariant random field, and the strong spatial heterogeneity, due to graph sparsity. We also uncover that the ferromagnetic phase can be marginally stable despite the presence of the random field. Finally, we study the proper correlation functions approaching the critical points to identify the ones that become more critical.

Nonequilibrium Kosterlitz-Thouless transition in a three-dimensional driven disordered system

We demonstrate a three-dimensional Kosterlitz-Thouless (KT) transition in the random field XY model driven out of thermal equilibrium. By employing the spin-wave approximation and functional renormalization group approach, in the weak disorder regime, the three-dimensional driven random field XY model is found to exhibit a quasi-long-range order phase, wherein the correlation function shows power-law decay with a non-universal exponent that depends on the disorder strength. This result is consistent with that reported in a previous numerical study. We further develop a phenomenological theory of the three-dimensional KT transition by taking into account the effect of vortices. The point of this theory is that the cross-section of the system with respect to a plane perpendicular to the driving direction is essentially identical to the two-dimensional pure XY model.

Synchronization and Spin-Flop Transitions for a Mean-Field XY Model in Random Field

We characterize the phase space for the infinite volume limit of a ferromagnetic mean-field XY model in a random field pointing in one direction with two symmetric values. We determine the stationary solutions and detect possible phase transitions in the interaction strength for fixed random field intensity. We show that at low temperature magnetic ordering appears perpendicularly to the field. The latter situation corresponds to a spin-flop transition.

Quantum correlations in quenched disordered spin models: Enhanced order from disorder by thermal fluctuations.

We investigate the behavior of quantum correlations of paradigmatic quenched disordered quantum spin models, viz., the XY spin glass and random-field XY models. We show that quenched averaged quantum correlations can exhibit the order-from-disorder phenomenon for finite-size systems as well as in the thermodynamic limit. Moreover, we find that the order-from-disorder can become more pronounced in the presence of temperature by suitable tuning of the system parameters. The effects are found for entanglement measures as well as for information-theoretic quantum correlation ones, although the former show them more prominently. We also observe that the equivalence between the quenched averages and their self-averaged cousins--for classical and quantum correlations--is related to the quantum critical point in the corresponding ordered system.

Random Field Induced Order in Low Dimension I

Consider the classical XY model in a weak random external field pointing along the Y axis with strength \({\epsilon}\). We prove that the model defined on \({\mathbb{Z}^3}\) with nearest neighbor coupling exhibits residual magnetic order in the horizontal direction for arbitrarily weak random field strengths and, depending on field strength, sufficiently low temperature.

The role of vortices in the three-dimensional random-field xy model

We study vortex states in a 3d random-field xy model of up to one billion lattice spins at T = 0. Starting with random spin orientations, the sample freezes into the vortex-glass state with a stretched-exponential decay of spin correlations, having short correlation length and a low susceptibility, compared to vortex-free states. In a field opposite to the initial magnetization, peculiar topological objects —walls of spins still opposite to the field— emerge along the hysteresis curve. On increasing the field strength, the walls develop cracks bounded by vortex loops. The loops then grow in size and eat the walls away. Applications to magnets and superconductors are discussed.

Exact form of the exponential correlation function in the glassy super-rough phase

We consider the random-phase sine-Gordon model in two dimensions. It describes two-dimensional elastic systems with random periodic disorder, such as pinned flux-line arrays, random field XY models, and surfaces of disordered crystals. The model exhibits a super-rough glass phase at low temperature $T<T_{c}$ with relative displacements growing with distance $r$ as $\bar{\langle [\theta(r)-\theta(0)]^2\rangle} \simeq A(\tau) \ln^2 (r/a)$, where $A(\tau) = 2 \tau^2- 2 \tau^3 +\mathcal{O}(\tau^4)$ near the transition and $\tau=1-T/T_{c}$. We calculate all higher cumulants and show that they grow as $\bar{\langle[\theta(r)-\theta(0)]^{2n}\rangle}_c \simeq [2 (-1)^{n+1} (2n)! \zeta(2n-1) \tau^2 + \mathcal{O}(\tau^3) ] \ln(r/a)$, $n \geq 2$, where $\zeta$ is the Riemann zeta function. By summation, we obtain the decay of the exponential correlation function as $\bar{\langle e^{iq\left[\theta(r)-\theta(0)\right]}\rangle} \simeq (a/r)^{\eta(q)} \exp\boldsymbol(-\frac{1}{2}\mathcal{A}(q)\ln^2(r/a)\boldsymbol)$ where $\eta(q)$ and ${\cal A}(q)$ are obtained for arbitrary $q \leq 1$ to leading order in $\tau$. The anomalous exponent is $\eta(q) = c q^2 - \tau^2 q^2 [2\gamma_E+\psi(q)+\psi(-q)]$ in terms of the digamma function $\psi$, where $c$ is non-universal and $\gamma_E$ is the Euler constant. The correlation function shows a faster decay at $q=1$, corresponding to fermion operators in the dual picture, which should be visible in Bragg scattering experiments.

Smectic-Ato -Cphase transition in isotropic disordered environments

We study theoretically the smectic-A to -C phase transition in isotropic disordered environments. Surprisingly, we find that, as in the clean smectic-A to -C phase transition, smectic layer fluctuations do not affect the nature of the transition, in spite of the fact that they are much stronger in the presence of the disorder. As a result, we find that the universality class of the transition is that of the "random field XY model" (RFXY).

Phase dynamics of a Josephson junction ladder driven by modulated currents

Phase dynamics of disordered Josephson junction ladders (JJLs) driven by external currents which are spatially and temporally modulated is studied using a numerical simulation based on a random field XY model. This model is considered theoretically as an effective model of JJLs with structural disorder in a magnetic field. The spatiotemporal modulation of external currents causes peculiar dynamical effects of phases in the system under certain conditions, such as the directed motion of phases and the mode-locking in the absence of dc currents. We clarify the details of effects of the spatiotemporal modulation on the phase dynamics.

Critical hysteresis in random-fieldXYand Heisenberg models

We study zero-temperature hysteresis in random-field XY and Heisenberg models in the zero-frequency limit of a cyclic driving field. We consider three distributions of the random field and present exact solutions in the mean-field limit. The results show a strong effect of the form of disorder on critical hysteresis as well as the shape of hysteresis loops. A discrepancy with an earlier study based on the renormalization group is resolved.

On Random Field Induced Ordering in the Classical XY Model

Consider the classical XY model in a weak random external field pointing along the $Y$ axis with strength $\epsilon$. We study the behavior of this model as the range of the interaction is varied. We prove that in any dimension $d \geq 2$ and for all $\epsilon$ sufficiently small, there is a range $L=L(\epsilon)$ so that whenever the inverse temperature $\beta$ is larger than some $\beta(\epsilon)$, there is strong residual ordering along the $X$ direction.

Liquid Crystal Cells with “Dirty” Substrates

We explore liquid crystal order in a cell with a "dirty" substrate imposing a random surface pinning. Modeling such systems by a random-field xy model with surface heterogeneity, we find that orientational order in the three-dimensional system is marginally unstable to such surface pinning. We compute the Larkin length scale, and the corresponding surface and bulk distortions. On longer scales we calculate correlation functions using the functional renormalization group and matching methods, finding a universal logarithmic and double-logarithmic roughness in two and three dimensions, respectively. For a finite thickness cell, we explore the interplay of homogeneous-heterogeneous substrate pair and detail crossovers as a function of disorder strength and cell thickness.

Unified Picture of Ferromagnetism, Quasi-Long-Range Order, and Criticality in Random-Field Models

By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long-range order (QLRO), and criticality in the d-dimensional random-field O(N) model in the whole (N, d) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure O(N) model, with, however, no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely, a topologically ordered "Bragg glass" phase, is absent in the 3-dimensional random-field XY model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around d = 4.

Numerical study on “plastic flow - moving solid” transition of charge density waves in nonequilibrium steady state under external drive

Dynamical phase transition of charge density waves between plastic flow and moving solid phases controlled by electric field is numerically studied based on the three dimensional driven random field XY model. We analyze a spatial configuration of local velocity and find a percolation transition of a co-moving domain. The critical driving force, however, logarithmically diverges with the observation time. The moving solid phase has a finite life time that grows exponentially with the driving force.

Possibility of Solid–Fluid Transition in Moving Periodic Systems

The steady sliding state of periodic structures such as charge density waves and flux line lattices is numerically studied based on two and three dimensional driven random field XY models. We focus on the dynamical phase transition between plastic flow and moving solid phases controlled by the magnitude of the driving force. By analyzing the connectivity of co-moving clusters, we find that they percolate the system within a finite observation time under driving forces larger than a certain critical force. The critical force, however, logarithmically diverges with the observation time, i.e. the moving solid phase exhibits only within a certain finite time, which exponentially grows with the driving force.

Quasi Long Range Ordered Ground-State of the Random Field XY Model

We show that in the simple Imry-Ma type argument of random field XY model, additional vortex energy is missing which is crucial to explain the quasi-long-range ordered state experimentally observed in the so-called Bragg glass phase of superconductors. We also show numerical results which indicate that power-law correlated state is easily obtained by simple local relaxation scheme at zero-temperature. The same scheme is also applied to random field 0(4) spin model, in which there are no topological defects, and we show that the obtained states have only short-ranged correlation. These results indicate that the topological defects are crucial in determining the low-temperature properties of random field spin models.