Weak Random Fields Research Articles

Strong law of large numbers for functionals of random fields with unboundedly increasing covariances

The paper proves the Strong Law of Large Numbers for integral functionals of random fields with unboundedly increasing covariances. The case of functional data and increasing domain asymptotics is studied. Conditions to guarantee that the Strong Law of Large Numbers holds true are provided. The considered scenarios include wide classes of non stationary random fields. The discussion about application to weak and long-range dependent random fields and numerical examples are given.

Ferroelectric phase transitions of uniaxial Sr1-xBaxNb2O6 and their composition variation

Elastic properties of uniaxial ferroelectric strontium barium niobate, SrxBa1-xNb2O6 (SBN100x), crystals were studied by Brillouin scattering. With the increase of x, the random fields (RFs) strength becomes strong and the relaxor ferroelectric nature is enhanced. SBN26 (x = 0.26) with very weak RFs undergoes a normal ferroelectric phase transition at the Curie temperature, TC = 464 K. On cooling from a high temperature, the sound velocity of the longitudinal acoustic mode along the ferroelectric c-axis shows a remarkable softening towards TC. The sound attenuation of this mode increases markedly towards TC. These temperature dependences are well fitted by power laws above TC. The relaxation time of a central peak (CP) shows the critical slowing down towards TC. In contrast, SBN80 (x = 0.80) with strong RFs shows only a diffusive elastic anomaly in the vicinity of TC = 289 K and the temperature dependence of a CP is very small. These differences between the observed results of SBN26 and SBN80 can be attributed to the difference of RFs strength.

Universal Scaling Theory of the Boundary Geometric Tensor in Disordered Metals.

We investigate the finite-size scaling of the boundary quantum geometric tensor (QGT) numerically close to the Anderson localization transition in the presence of small external magnetic fields. The QGT exhibits universal scaling and reveals the crossover between the orthogonal and unitary critical states in weak random magnetic fields. The flow of the QGT near the critical points determines the critical exponents. Critical distributions of the QGT are universal and exhibit a remarkable isotropy even in a homogeneous magnetic field. We predict universal and isotropic Hall conductance fluctuations at the metal-insulator transition in an external magnetic field.

Vibrational Dynamics of Ferroelectric K(Ta1-xNbx)O3 Studied by Inelastic Light Scattering

The Fano resonance of non-doped KTa1-xNbxO3 (x = 0.32) single crystals was investigated by the Raman spectroscopy. The intensity of central peak (CP) and parameters of the Fano resonance revealed a significant anomaly in the vicinity of the Curie temperature, TC = 259 K. The vibrational dynamics and the symmetry breaking in the cubic phase are discussed in the framework of weak random fields. It is concluded that the coupling between continuum states induced by polar nanoregions (PNRs) and a transverse optic (TO2) phonon at about 194 cm−1 is the origin of a Fano resonance in non-doped KTN crystals.

Topological Order Generated by a Random Field in a 2D Exchange Model

We study a 2D exchange model with a weak static random field on lattices containing over 10^{8} spins. Ferromagnetic correlations persist on the Imry-Ma scale inversely proportional to the random-field strength and decay exponentially at greater distances. We find that the average energy of the correlated area is close to the ground-state energy of a Skyrmion, while the topological charge of the area is close to ±1. The correlation function of the topological charge density changes sign at a distance determined by the ferromagnetic correlation length, while its Fourier transform exhibits a maximum. These findings suggest that static randomness transforms a 2D ferromagnetic state into a Skyrmion-anti-Skyrmion glass.

Electric field effect on polar nanoregions of uniaxial ferroelectric SrxBa1−xNb2O6 with weak random fields studied by Brillouin scattering

The electric field effects on the acoustic properties of uniaxial ferroelectric SrxBa1−xNb2O6 (x = 0.40, SBN40) crystals with weak random fields were studied by Brillouin scattering. On field cooling, the minimum of longitudinal acoustic (LA) velocity shifted to a higher value and a marked increase in LA velocity was observed below the Curie temperature TC. This signifies the enhancement of the long-range ferroelectric order due to the alignment of polar nanoregions (PNRs) along the field direction. After poling the crystal with a high electric field for 60 min above TC, the alignment of all PNRs along the field direction was observed. By decreasing the poling time, LA mode splitting was observed below TC owing to the back switching of PNRs. In a field-dependent measurement below TC, a mixed state consisting of macro- and nanodomains persisted up to a very high electric field because of the incomplete switching of nanodomains due to the interaction among PNRs.

Ferroelectric phase transition of Li-doped KTa1-xNbxO3 single crystals with weak random fields: Inelastic light scattering study

The broadband gigahertz dynamics of a ferroelectric phase transition of the 2.5%Li-doped KTa1-xNbxO3 crystals with x = 0.38 (KLTN/2.5/38) was investigated by inelastic light scattering. Its phase transition was found to be slightly diffused type due to weak random fields (RFs) caused by disorder in B-site of perovskite structure. The microscopic origin of the central peak (CP) and the local symmetry of the polar nanoregions (PNRs) in the cubic phase were studied by Raman scattering. In Brillouin scattering, the remarkable elastic anomaly of the longitudinal acoustic (LA) mode was observed near paraelectric cubic to ferroelectric tetragonal phase transition temperature, TC-T = 46 °C, which is attributed to the coupling between fluctuating local polarization in PNRs and the LA mode. The intermediate temperature, T* ∼ 97 °C, at which a dynamic to static transition of PNRs occurs, was determined from the significant increase of damping and intensity of the LA mode and the CP upon cooling. In the vicinity of the TC-T, a slightly stretched slowing down of the relaxation time, τCP, a feature of order-disorder nature of the ferroelectric phase transition with weak RFs, was clearly observed.

Role of polar nanoregions with weak random fields in Pb-based perovskite ferroelectrics

In relaxor ferroelectrics, the role of randomly orientated polar nanoregions (PNRs) with weak random fields (RFs) is one of the most puzzling issues of materials science. The relaxation time of polarization fluctuations of PNRs, which manifests themselves as a central peak (CP) in inelastic light scattering, is the important physical quantity to understand the dynamics of PNRs. Here, the angular and temperature dependences of depolarized and polarized CPs in 0.44Pb(Mg1/3Nb2/3)O3-0.56PbTiO3 single crystals with weak RFs have been studied by Raman and Brillouin scattering, respectively. The CPs observed in Raman scattering show the very clear angular dependence which is consistent with the local tetragonal symmetry. It is different from the well-known local rhombohedral symmetry with strong RFs for Pb(Mg1/3Nb2/3)O3. In Brillouin scattering, depolarized and polarized CPs show two relaxation processes corresponding to transverse and longitudinal fluctuations of PNRs. The remarkable slowing down towards the Curie temperature was observed for transverse fluctuations in local tetragonal symmetry.

Thermal transport in disordered one-dimensional spin chains

We study a one-dimensional anisotropic XXZ Heisenberg spin-$\frac{1}{2}$ chain with weak random fields ${h}_{i}^{z}{S}_{i}^{z}$ by means of Jordan-Wigner transformation to spinless Luttinger liquid with disorder and bosonization technique. First, we reinvestigate the phase diagram of the system in terms of dimensionless disorder $\ensuremath{\gamma}=\ensuremath{\langle}{h}^{2}\ensuremath{\rangle}/{J}^{2}\ensuremath{\ll}1$ and anisotropy parameter $\mathrm{\ensuremath{\Delta}}={J}_{z}/{J}_{xy}$, we find the range of these parameters where disorder is irrelevant in the infrared limit and spin-spin correlations are described by power laws, and compare it with previously obtained numerical and analytical results. Then we use the diagram technique in terms of plasmon excitations to study the low-temperature $(T\ensuremath{\ll}J)$ behavior of heat conductivity $\ensuremath{\kappa}$ and spin conductivity $\ensuremath{\sigma}$ in this power-law phase. The obtained Lorentz number $L\ensuremath{\equiv}\ensuremath{\kappa}/\ensuremath{\sigma}T$ differs from the value derived earlier by means of the memory function method. We argue also that in the studied region inelastic scattering is strong enough to suppress quantum interference in the low-temperature limit.

From classical to quantum non-equilibrium dynamics of Rydberg excitations in optical lattices

The glass phase and its quantum analog are prominent challenges of current non-equilibrium statistical mechanics and condensed matter physics. As a model system to study the transition from classical to quantum glassy dynamics, we propose a setup of laser driven three-level atoms trapped in an optical lattice. Tuning the strength of the laser driving to the intermediate level allows one to study the transition from a classical kinetically constrained model to the coherent regime. For strong driving, Rydberg excitations evolve analogously to defects in the one-spin facilitated model, a minimal model known to exhibit glassy dynamics. In our setup, the constraints result from the interplay between Rydberg interactions and the laser detuning from the Rydberg state. The emerging heterogeneous relaxation timescales are tuneable over several orders of magnitudes. In the opposite limit of weak driving of the intermediate level, we find an effective cluster model which describes the dynamics in a reduced subspace of the allowed number and positions of Rydberg excitations. This subspace is uniquely determined by the initial state and is characterized by a fixed number of clusters of Rydberg excitations. In addition, we investigate the influence of random fields on the classical relaxation. We find that the glassy dynamics can relax faster in the presence of weak random fields.

HYSTERESIS BETWEEN DISTINCT MODES OF TURBULENT DYNAMOS

Nonlinear mean-field models of the solar dynamo show long-term variability, which may be relevant to different states of activity inferred from long-term radiocarbon data. This paper is aimed to probe the dynamo hysteresis predicted by the recent mean-field models of Kitchatinov \& Olemskoy (2010) with direct numerical simulations. We perform three-dimensional simulations of large-scale dynamos in a shearing box with helically forced turbulence. As initial condition, we either take a weak random magnetic field or we start from a snapshot of an earlier simulation. Two quasi-stable states are found to coexist in a certain range of parameters close to the onset of the large-scale dynamo. The simulations converge to one of these states depending on the initial conditions. When either the fractional helicity or the magnetic Prandtl number is increased between successive runs above the critical value for onset of the dynamo, the field strength jumps to a finite value. However, when the fractional helicity or the magnetic Prandtl number is then decreased again, the field strength stays at a similar value (strong field branch) even below the original onset. We also observe intermittent decaying phases away from the strong field branch close to the point where large-scale dynamo action is just possible. The dynamo hysteresis seen previously in mean-field models is thus reproduced by 3D simulations. Its possible relation to distinct modes of solar activity such as grand minima is discussed.

Effect of a dilute random field on a continuous-symmetry order parameter

XY and Heisenberg spins, subjected to strong random fields acting at few points in space with concentration $c_r \ll 1$, are studied numerically on 3d lattices containing over four million sites. Glassy behavior with strong dependence on initial conditions is found. Beginning with a random initial orientation of spins, the system evolves into ferromagnetic domains inversely proportional to $c_r$ in size. The area of the hysteresis loop, $m(H)$, scales as $c_r^2$. These findings are explained by mapping the effect of strong dilute random field onto the effect of weak continuous random field. Our theory applies directly to ferromagnets with magnetic impurities, and is conceptually relevant to strongly pinned vortex lattices in superconductors and pinned charge density waves.

Critical slowing down and elastic anomaly of uniaxial ferroelectricCa0.28Ba0.72Nb2O6crystals with tungsten bronze structure

The ferroelectric phase transition of uniaxial ${\mathrm{Ca}}_{0.28}{\mathrm{Ba}}_{0.72}{\mathrm{Nb}}_{2}{\mathrm{O}}_{6}$ single crystals with a moderate effective charge disorder was investigated by Brillouin scattering to clarify the dynamic properties. In the tetragonal paraelectric phase a remarkable softening of the sound velocity of the longitudinal acoustic mode and a significant increase in the sound attenuation were observed close to the Curie temperature ${T}_{\mathrm{C}}=527\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. The intermediate temperature ${T}^{*}$ $\ensuremath{\sim}640\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and the Burns temperature ${T}_{\mathrm{B}}$ $\ensuremath{\sim}790\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ were determined from the temperature variation in the sound attenuation. The intense broad central peak (CP) caused by polarization and strain fluctuations due to polar nanoregions was clearly observed in the vicinity of ${T}_{\mathrm{C}}$. The relaxation time determined by the CP width clearly shows critical slowing down towards ${T}_{\mathrm{C}}$, reflecting a weakly first-order phase transition under weak random fields.

Ground state of one-dimension repulsing particles on disordered lattice

The ground state (GS) of interacting particles on a disordered one-dimensional (1D) host-lattice is studied by a new numerical method. It is shown that if the concentration of particles is small, then even a weak disorder of the host-lattice breaks the long-range order of Generalized Wigner Crystal (GWC), replacing it by the sequence of blocks (domains) of particles with random lengths. The mean domains length as a function of the host-lattice disorder parameter is also found. It is shown that the domain structure can be detected by a weak random field, whose form is similar to that of the ground state but has fluctuating domain walls positions. This is because the generalized magnetization corresponding to the field has a sufficiently sharp peak as a function of the amplitude of fluctuations for small amplitudes.

Non-linear permittivity study of the crossover from ferroelectric to relaxor and cluster glass in BaTi1−xSnxO3 (x = 0.175–0.30)

The electric susceptibilities χ1, χ2, and χ3 of BaTi1−xSnxO3 ceramics with 0.175 ≤ x ≤ 0.30 are measured at 90 ≤ T ≤ 350 K and 37 ≤ f ≤ 106 Hz. Crossover from ferroelectric (x = 0.175) to relaxor behavior (x = 0.30) via coexistence of both (x = 0.25) is indicated by increasing polydispersivity and suppression of χ2 intensity. The relaxor properties are due to weak random fields acting on dipolar Ti4+ clusters in non-polar Sn4+ environment. Upon cooling frustrated interaction between the blocked polar nanoregions yields a cluster glass ground state for x = 0.30 as verified by the dynamic criticality of χ1, χ3, and the scaled non-linear susceptibility a3.

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.

Random Fields, Topology, and the Imry-Ma Argument

We consider an n-component fixed-length order parameter interacting with a weak random field in d=1, 2, 3 dimensions. Relaxation from the initially ordered state and spin-spin correlation functions are studied on lattices containing hundreds of millions of sites. At n ≤ d the presence of topological defects leads to strong metastability and glassy behavior, with the final state depending on the initial condition. At n=d+1, when topological structures are nonsingular, the system possesses a weak metastability. At n>d+1, when topological objects are absent, the final, lowest-energy state is independent of the initial condition. It is characterized by the exponential decay of correlations that agrees quantitatively with the theory based upon the Imry-Ma argument.

Random fieldxymodel in three dimensions

We study random-field xy spin model at T=0 numerically on lattices of up to 1000 x 1000 x 1000 spins with the accent on the weak random field. Our numerical method is physically equivalent to slow cooling in which the system is gradually losing the energy and relaxing to an energy minimum. The system shows glass properties, the resulting spin states depending strongly on the initial conditions. Random initial condition for the spins leads to the vortex glass (VG) state with short-range spin-spin correlations defined by the average distance between vortex lines. Collinear and some other vortex-free initial conditions result in the vortex-free ferromagnetic (F) states that have a lower energy. The energy difference between the F and VG states correlates with vorticity of the VG state. Correlation functions in the F states agree with the Larkin-Imry-Ma theory at short distances. Hysteresis curves for weak random field are dominated by topologically stable spin walls raptured by vortex loops. We find no relaxation paths from the F, VG, or any other states to the hypothetical vortex-free state with zero magnetization.

Crossover from ferroelectric to relaxor and cluster glass in BaTi1−xZrxO3 (x = 0.25–0.35) studied by non-linear permittivity

The electric susceptibilities χ1, χ2, and χ3 of BaTi1−xZrxO3 ceramics with 0.25 ≤ x ≤ 0.35 are measured at 90 ≤ T ≤ 350 K and 37 ≤ f ≤ 106 Hz. Crossover from ferroelectric (x = 0.25) to relaxor behavior (x = 0.35) via coexistence of both (x = 0.30) is indicated by increasing polydispersivity and suppression of χ2 intensity and negative χ3 tails. The relaxor properties are due to weak random fields acting on dipolar Ti4+ clusters in non-polar Zr4+ environment. Frustrated interaction between blocked polar nanoregions yields a cluster glass ground state for x = 0.35.

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.