Fluctuations Of Order Parameter Research Articles

Dynamics of finite width Kernel and prediction fluctuations in mean field neural networks*

Abstract We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Starting from a dynamical mean field theory description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the O ( 1 / width ) fluctuations of the dynamical mean field theory order parameters over random initializations of the network weights. Our results, while perturbative in width, unlike prior analyses, are non-perturbative in the strength of feature learning. We find that once the mean field/µP parameterization is adopted, the leading finite size effect on the dynamics is to introduce initialization variance in the predictions and feature kernels of the networks. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with a variance that can be computed self-consistently. In two layer networks, we show how feature learning can dynamically reduce the variance of the final tangent kernel and final network predictions. We also show how initialization variance can slow down online learning in wide but finite networks. In deeper networks, kernel variance can dramatically accumulate through subsequent layers at large feature learning strengths, but feature learning continues to improve the signal-to-noise ratio of the feature kernels. In discrete time, we demonstrate that large learning rate phenomena such as edge of stability effects can be well captured by infinite width dynamics and that initialization variance can decrease dynamically. For convolutional neural networks trained on CIFAR-10, we empirically find significant corrections to both the bias and variance of network dynamics due to finite width.

Disordered non-Fermi liquid fixed point for two-dimensional metals at Ising-nematic quantum critical points

Understanding the influence of quenched random potential is crucial for comprehending the exotic electronic transport of non-Fermi liquid metals near metallic quantum critical points. In this study, we identify a stable fixed point governing the quantum critical behavior of two-dimensional non-Fermi liquid metals in the presence of a random potential disorder. By performing renormalization group analysis on a dimensional-regularized field theory for Ising-nematic quantum critical points, we systematically investigate the interplay between random potential disorder for electrons and Yukawa-type interactions between electrons and bosonic order-parameter fluctuations in a perturbative epsilon expansion. At the one-loop order, the effective field theory lacks stable fixed points, instead exhibiting a runaway flow toward infinite disorder strength. However, at the two-loop order, the effective field theory converges to a stable fixed point characterized by finite disorder strength, termed the “disordered non-Fermi liquid (DNFL) fixed point”. Our investigation reveals that two-loop vertex corrections induced by Yukawa couplings are pivotal in the emergence of the DNFL fixed point, primarily through screening disorder scattering. Additionally, the DNFL fixed point is distinguished by a substantial anomalous scaling dimension of fermion fields, resulting in pseudogap-like behavior in the electron’s density of states. These findings shed light on the quantum critical behavior of disordered non-Fermi liquid metals, emphasizing the indispensable role of higher-order loop corrections in such comprehension.

Fluctuation effects in superconducting nanowires under electric field

This study applies the time-dependent Ginzburg–Landau (TDGL) theory with thermal noise to analyze the thermoelectric and transport properties of superconducting Sn nanowires, focusing on the thermopower (αxx) and the phase transition characteristics in the S-shaped J-E curves. We observe significant contributions from superconducting Cooper pairs, which remain nonzero above the critical temperature (Tc), indicating residual superconductivity due to strong thermal fluctuations. The S-like shape in the J-E curves is attributed to a dynamical instability transition temperature (T∗) at approximately 2.9 K, where thermal fluctuations dominate. Furthermore, we compare the resistance in the linear response to experimental data for Sn nanowires both below and above Tc. Below Tc, the resistance sharply decreases, reflecting the robust superconducting state, while above Tc, it increases, aligning with normal state behavior. In nonlinear response case, our results indicate that high electric fields can be effectively used to suppress order-parameter fluctuations and the electrical conductivity in superconducting nanowires. The findings provide critical insights into the thermoelectric behavior and phase transitions in Sn nanowires, highlighting the importance of Cooper pair dynamics in shaping the transport properties of one-dimensional superconductors. This understanding is essential for the development of advanced nanoelectronic devices leveraging these unique superconducting properties.

Evidence of a distinct collective mode in Kagome superconductors

The collective modes of the superconducting order parameter fluctuation can provide key insights into the nature of the superconductor. Recently, a family of superconductors has emerged in non-magnetic kagome materials AV3Sb5 (A = K, Rb, Cs), exhibiting fertile emergent phenomenology. However, the collective behaviors of Cooper pairs have not been studied. Here, we report a distinct collective mode in CsV3-xTaxSb5 using scanning tunneling microscope/spectroscopy. The spectral line-shape is well-described by one isotropic and one anisotropic superconducting gap, and a bosonic mode due to electron-mode coupling. With increasing x, the two gaps move closer in energy, merge into two isotropic gaps of equal amplitude, and then increase synchronously. The mode energy decreases monotonically to well below 2Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{2}}{{\\Delta }}$$\\end{document} and survives even after the charge density wave order is suppressed. We propose the interpretation of this collective mode as Leggett mode between different superconducting components or the Bardasis-Schrieffer mode due to a subleading superconducting component.

Topological Kondo effect with spinful Majorana fermions

Motivated by the importance of studying topological superconductors beyond the mean-field approximation, we here investigate mesoscopic islands of time-reversal-invariant topological superconductors. We characterize the spectrum in the presence of strong order-parameter fluctuations in the presence of an arbitrary number of Kramers pairs of Majorana edge states and study the effect of coupling the Coulomb blockaded island to external leads. In the case of an odd fermionic parity on the island, we derive an unconventional Kondo Hamiltonian in which metallic leads couple to both topological Majorana degrees of freedom (which keep track of the parity in different leads) and the overall spin 12 in the island. For the simplest case of a single wire (two pairs of Majorana edge states), we demonstrate that anisotropies are irrelevant in the weak coupling renormalization group flow. This permits us to solve the Kondo problem in the vicinity of a Toulouse-type point using Abelian bosonization. We demonstrate a residual ground-state entropy of ln(2), which is protected by spin-rotation symmetry, but reduced to ln(2) (as in the spinless topological Kondo effect) by symmetry-breaking perturbations. In the symmetric case, we further demonstrate the simultaneous presence of both Fermi-liquid and non-Fermi-liquid-like thermodynamics (depending on the observable) and derive charge and spin transport signatures of the Coulomb blockaded island. Published by the American Physical Society 2024

Landau free energy of small clusters beyond mean-field approach.

Landau free energy determines the landscape of order parameter fluctuations that occur in a physical system at thermal equilibrium and, in particular, characterizes the critical phenomena. We propose a semianalytical approach based on the fluctuating local field method, which allows us to estimate Landau free energy for small clusters with discrete (Ising model) and continuous (Heisenberg model) order parameter.

Mpemba effect in pure spin systems : A universal picture of the role of spatial correlations at initial states.

The quicker freezing of hotter water, than a colder sample, when quenched to a common lower temperature, is referred to as the Mpemba effect (ME). While this counter-intuitive fact remains a surprize since long, efforts have begun to identify similar effect in other systems. We investigate the ME in a rather general context concerning magnetic phase transitions. From Monte Carlo simulations of model systems, viz., the Ising model and the q-state Potts model, with varying range of interaction and space dimension, we assert that hotter paramagnets undergo ferromagnetic ordering faster than the colder ones. This conclusion we have arrived at following the analyses of the simulation results on decay of energy and growth in ordering following quenches from different starting temperatures, to fixed final temperatures below the Curie points. The general observation, in all the considered models, without any element of frustration, is a crucial and important fact of our study. Furthermore, we have obtained an important scaling picture, on the strength of the effect, with respect to the variation in spatial correlation in the initial states. This behavior appears true irrespective of the nature of order-parameter fluctuation and even order of transition. The observations are expected to be relevant to the understanding of ME in a rather general class of systems.

Exceptionally Slow, Long-Range, and Non-Gaussian Critical Fluctuations Dominate the Charge Density Wave Transition.

(TaSe_{4})_{2}I is a well-studied quasi-one-dimensional compound long-known to have a charge-density wave (CDW) transition around 263K. We argue that the critical fluctuations of the pinned CDW order parameter near the transition can be inferred from the resistance noise on account of their coupling to the dissipative normal carriers. Remarkably, the critical fluctuations of the CDW order parameter are slow enough to survive the thermodynamic limit and dominate the low-frequency resistance noise. The noise variance and relaxation time show rapid growth (critical opalescence and critical slowing down) within a temperature window of ϵ≈±0.1, where ϵ is the reduced temperature. This is very wide but consistent with the Ginzburg criterion. We further show that this resistance noise can be quantitatively used to extract the associated critical exponents. Below |ϵ|≲0.02, we observe a crossover from mean-field to a fluctuation-dominated regime with the critical exponents taking anomalously low values. The distribution of fluctuations in the critical transition region is skewed and strongly non-Gaussian. This non-Gaussianity is interpreted as the breakdown of the validity of the central limit theorem as the diverging coherence volume becomes comparable to the macroscopic sample size. The large magnitude critical fluctuations observed over an extended temperature range, as well as the crossover from the mean-field to the fluctuation-dominated regime highlight the role of the quasi-one-dimensional character in controlling the phase transition.

Thermodynamic, structural, surface and transport properties of Au-Cu melt

Thermodynamic, structural, surface and transport properties of Au-Cu liquid alloy were calculated on the basis of different theoretical modeling equations. The thermodynamic properties, such as excess Gibbs free energy of mixing, enthapy of mixing and activity were estimated and compared with the available experimental and literature data on the basis of simple theory of mixing (STM). The best fit values of model parameters were estimated using the experimental values of excess Gibbs free energy of mixing and enthalpy of mixing of the system at 1550 K. Using the same model parameters and frame, the concentration fluctuation in long wave-length limit and Warren-Cowley short range order parameter were calculated and analysed to understand the local arrangement of atoms in the liquid mixture. The surface tension and extent of surface segregation of atoms in the initial melt were computed using Butler’s model. In transport properties, the ratio of mutual to intrinsic diffusion coefficients was calculated using STM and viscosity was calculated using Kaptay equation. The system showed complete ordering tendency at its melting temperature.

Marginal Fermi liquid behavior at the onset of 2kF density wave order in two-dimensional metals with flat hot spots

We analyze quantum fluctuation effects at the onset of incommensurate 2kF charge- or spin-density-wave order in two-dimensional metals for a model in which the ordering wave vector Q connects a single pair of hot spots on the Fermi surface with a vanishing Fermi surface curvature. The tangential momentum dependence of the bare dispersion near the hot spots is proportional to |kt|α with α>2. We first compute the order parameter susceptibility and the fermion self-energy in a random phase approximation (RPA). Logarithmic divergences are subsequently treated by a renormalization-group analysis. The coupling between the order parameter fluctuations and the fermions vanishes logarithmically in the low-energy limit. As a consequence, the logarithmic divergences found in the RPA do not sum up to anomalous power laws. Instead, only logarithmic corrections to Fermi liquid behavior are obtained. In particular, the quasiparticle weight and the Fermi velocity vanish logarithmically at the hot spots. Published by the American Physical Society 2024

Thermodynamic information geometry of criticality, fluctuation anomaly and hyperscaling breakdown in the spin-3/2 chain

Equilibrium state space geometry is worked out for the exactly solved one-dimensional spin-3/2 lattice with a Blume–Emery–Griffiths (BEG) Hamiltonian as well as a more general one with a non-zero field coupled to the octopole moments. The phase behaviour of the spin-3/2 chain is revisited and novel observations of anomaly in the hyperscaling relation and in the decay of fluctuations are reported for the first time. Using the method of constrained fluctuations developed earlier in Sahay (2017); Sanwari and Sahay (2022) three appropriate state space sectional curvatures and a 3d curvature are shown to separately encode dipolar, quadrupolar and octopolar correlations both near and away from pseudo-criticality. In all observations of hyperscaling breakdown the 3d scalar curvature is found to encode the correlation length while the relevant sectional curvature equals the inverse of singular free energy. For parameter values where the order parameter fluctuation anomalously decays despite a divergence in its correlation length the relevant scalar curvature undergoes a sign change to positive values, signalling a possible change in the nature of the underlying statistical interactions.

Emergent gauge symmetry in active Brownian matter.

We investigate a two-dimensional system of interacting active Brownian particles. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we built up the generating functional for correlation functions. We study in detail the hydrodynamic regime with a constant density stationary state. Our findings reveal that, within a small density fluctuations regime, an emergent U(1) gauge symmetry arises, originated from the conservation of fluid vorticity. Consequently, the interaction between the orientational order parameter and density fluctuations can be cast into a gauge theory, where the concept of "electric charge density" aligns with the local vorticity of the original fluid. We study in detail the case of a microscopic local two-body interaction. We show that, upon integrating out the gauge fields, the stationary states of the rotational degrees of freedom satisfy a nonlocal Frank free energy for a nematic fluid. We give explicit expressions for the splay and bend elastic constants as a function of the Péclet number (Pe) and the diffusion interaction constant (k_{d}).

Dynamics and Topology of Symmetry Breaking with Skyrmions.

We observe that pretransitional order parameter fluctuations of a skyrmion-forming chiral nematic liquid crystal are slowed down for 4 orders of magnitude, if confined to ≲100 nm thin layers. Fluctuating fragments of half-skyrmions are observed in a narrow temperature interval and are explained by thermally activated hopping between the various energy states. Skyrmion fluctuations are accompanied by imbalanced topological charge: positive charges appear at higher temperatures and dominate in the fluctuating region until skyrmions fully condense and negative charges appear at lower temperatures.

Direct interconnection of seismicity with variations of the Earth's electric and magnetic field before major earthquakes

Upon employing the new concept of time, termed natural time, the analysis of seismicity reveals that, before major earthquakes, the variations of the Earth's electric and/or magnetic field are accompanied by increase of the fluctuations of the entropy change of seismicity under time reversal as well as by decrease of the fluctuations of the seismicity order parameter. Hence, natural time analysis reveals that before major earthquakes independent datasets of different geophysical observables (seismicity, Earth's magnetic and/or electric field) exhibit changes, which are observed simultaneously. To the memory of the Academician Seiya Uyeda.

The breaking of Mermin-Wagner-Hohenberg's theorem for fractional systems

The Mermin-Wagner-Hohenberg theorem forbids the existence of long range order in two dimensional systems, since continuous symmetries cannot be spontaneously broken in d≤2. However, one of the premises for the validity of the theorem is that the energy-momentum dispersion of the quanta responsible for the order parameter fluctuations has a quadratic form, ε(k)=Ak2. In this paper we address the emergence of a fractional order energy-momentum dispersion in d-dimensional systems, leading to important consequences for the existence of long-range order in the case d=2. A first order renormalization group analysis in the ϵ-expansion is also performed in the fractional regime. It is shown that for fractional systems the universality classes are characterized by the number of spatial dimensions d and the number of components n of the order parameter, as in the case of integer order systems, but the labeling by the fractional order α is also required.

Hefei Advanced Light Facility: Empowering research of correlated electron systems

The Hefei Advanced Light Facility is the fourth-generation diffraction-limited storage ring light source, scheduled to begin operation in 2028. With its high-brightness and highly coherent X-rays, it will break through the current spatiotemporal resolution bottlenecks of X-ray techniques in studying correlated electron systems, providing crucial information for understanding the nature and microscopic origins of novel physical properties in these materials. This article introduces the main scientific goals and technical advantages of the Hefei Advanced Light Facility, focusing on the application perspectives of advanced technologies such as angle-resolved photoemission spectroscopy, magnetic circular dichroism, coherent X-ray scattering, and coherent X-ray imaging in researches of quantum materials and correlated electron systems. These techniques will enable the detailed analysis of the distribution and dynamics of electronic/spin/orbital states, reveal various novel quantum phenomena, and elucidate the fluctuations of order parameters in correlated electron systems. The completion of the Hefei Advanced Light Facility will provide advanced technical supports for decoding complex quantum states and non-equilibrium properties, ultimately promoting the application of quantum materials and correlated electron systems in frontier fields such as energy and information.

Statistical field theory of mechanical stresses in Coulomb fluids: general covariant approach vs Noether’s theorem

In this paper, we introduce a statistical field theory that describes the macroscopic mechanical forces in inhomogeneous Coulomb fluids. Our approach employs the generalization of Noether’s first theorem for the case of a fluctuating order parameter to calculate the stress tensor for Coulomb fluids. This tensor encompasses the mean-field stress tensor and fluctuation corrections derived through the one-loop approximation. The correction for fluctuations includes a term that accounts for the thermal fluctuations of the local electrostatic potential and field in the vicinity of the mean-field configuration. This correlation stress tensor determines how electrostatic correlation affects local stresses in a nonuniform Coulomb fluid. We also use a previously formulated general covariant methodology (Brandyshev and Budkov 2023 J. Chem. Phys. 158 174114) in conjunction with a functional Legendre transformation method and derive within it the same total stress tensor. We would like to emphasize that our general approaches are applicable not only to Coulomb fluids but also to nonionic simple or complex fluids, for which the field-theoretic Hamiltonian is known as a function of the relevant scalar order parameters.

Evidence for charge delocalization crossover in the quantum critical superconductor CeRhIn5.

The nature of charge degrees-of-freedom distinguishes scenarios for interpreting the character of a second order magnetic transition at zero temperature, that is, a magnetic quantum critical point (QCP). Heavy-fermion systems are prototypes of this paradigm, and in those, the relevant question is where, relative to a magnetic QCP, does the Kondo effect delocalize their f-electron degrees-of-freedom. Herein, we use pressure-dependent Hall measurements to identify a finite-temperature scale Eloc that signals a crossover from f-localized to f-delocalized character. As a function of pressure, Eloc(P) extrapolates smoothly to zero temperature at the antiferromagnetic QCP of CeRhIn5 where its Fermi surface reconstructs, hallmarks of Kondo-breakdown criticality that generates critical magnetic and charge fluctuations. In 4.4% Sn-doped CeRhIn5, however, Eloc(P) extrapolates into its magnetically ordered phase and is decoupled from the pressure-induced magnetic QCP, which implies a spin-density-wave (SDW) type of criticality that produces only critical fluctuations of the SDW order parameter. Our results demonstrate the importance of experimentally determining Eloc to characterize quantum criticality and the associated consequences for understanding the pairing mechanism of superconductivity that reaches a maximum Tc in both materials at their respective magnetic QCP.

Mesoscopic critical fluctuations.

We investigate the magnetic fluctuations in a mesoscopic critical region formed at the interface due to smooth time-independent spatial variations of a control parameter around its critical value. In the proximity of the spatial critical point, the order parameter fluctuations exhibit a mesoscopic nature, characterized by their significant size compared to the lattice constant, while gradually decaying away from the critical region. To explain this phenomenon, we present a minimal model that effectively captures this behavior and demonstrates its connection to the integrable Painlevé-II equationgoverning the local order parameter. By leveraging the well-established mathematical properties of this equation, we gain valuable insights into the nonlinear susceptibilities exhibited within this region.

Effect of hidden geometry and higher-order interactions on the synchronization and hysteresis behavior of phase oscillators on 5-clique simplicial assemblies.

The hidden geometry of simplicial complexes can influence the collective dynamics of nodes in different ways depending on the simplex-based interactions of various orders and competition between local and global structural features. We study a system of phase oscillators attached to nodes of four-dimensional simplicial complexes and interacting via positive/negative edges-based pairwise K_{1} and triangle-based triple K_{2}≥0 couplings. Three prototypal simplicial complexes are grown by aggregation of 5-cliques, controlled by the chemical affinity parameter ν, resulting in sparse, mixed, and compact architecture, all of which have 1-hyperbolic graphs but different spectral dimensions. By changing the interaction strength K_{1}∈[-4,2] along the forward and backward sweeps, we numerically determine individual phases of each oscillator and a global order parameter to measure the level of synchronization. Our results reveal how different architectures of simplicial complexes, in conjunction with the interactions and internal-frequency distributions, impact the shape of the hysteresis loop and lead to patterns of locally synchronized groups that hinder global network synchronization. Remarkably, these groups are differently affected by the size of the shared faces between neighboring 5-cliques and the presence of higher-order interactions. At K_{1}<0, partial synchronization is much higher in the compact community than in the assemblies of cliques sharing single nodes, at least occasionally. These structures also partially desynchronize at a lower triangle-based coupling K_{2} than the compact assembly. Broadening of the internal frequency distribution gradually reduces the synchronization level in the mixed and sparse communities, even at positive pairwise couplings. The order-parameter fluctuations in these partially synchronized states are quasicyclical with higher harmonics, described by multifractal analysis and broad singularity spectra.