Nonlocal Order Research Articles

M-shaped, W-shaped and dark soliton propagation in optical fiber for nonlocal fourth order dispersive nonlinear Schrödinger equation under distinct conditions

The formation of soliton in optical fiber governed by nonlocal nonlinear Schrödinger (NLS) equation with fourth order dispersion is studied. The model of nonlocal NLS equation with fourth order dispersion is solved using the new extended auxiliary method and the solutions are obtained. The solutions are in the form of hyperbolic and trigonometric functions which are based on Jacobi elliptic function m. Shape changing soliton in optical fiber for nonlocal fourth order dispersive NLS equation is discussed by suitably choosing the values of kerr and quintic nonlinearities and by varying fourth order dispersion term. The effect of fourth order dispersion on soliton in fiber for different conditions of kerr and quintic nonlinearity is also discussed. In addition, the phase portraits of the system have been investigated and the stability of wave in optical fiber for nonlocal NLS equation is discussed using fourth order Runge-Kutta algorithm. This paper addresses a significant gap in the current literature by examining the impact of fourth order dispersion on the nonlocal NLS equation in optical fiber.

Nodal solutions for a nonlocal fourth order equation of Kirchhoff type

We study the bifurcation behavior of nodal solutions for the Kirchhoff type beam equation u′′′′−M(∫01|u′|2dx)u′′=λf(x,u),x∈(0,1),u(0)=u(1)=u′′(0)=u′′(1)=0, where λ∈R is a parameter, M:[0,∞)→[0,∞) and f:[0,1]×R→R are smooth functions. We obtain the existence of nodal solutions under some suitable conditions. The proof of our main result is based upon bifurcation techniques.

Rayleigh wave in nonlocal piezo-thermo-electric semiconductor medium with fractional MGT model

This paper addresses a critical gap in Rayleigh wave propagation studies within piezoelectric semiconductors. Existing research often overlooks the combined effects of piezoelectricity, semiconductivity, and thermal behavior. This work presents analytical solutions for Rayleigh waves in a novel nonlocal thermo-piezo-photo-electric semiconductor half-space, employing the fractional Moore-Gibson-Thompson model. The wave-mode method and Durand-Kerner technique are employed to extract relevant solutions from the characteristic equation, ensuring surface wave decay. Analysis is conducted for waves propagating in a stress-free, photo-excited, and isothermal boundary condition. Numerical simulations unveil phase velocity, attenuation coefficient, and particle motion, leading to a deeper understanding of wave behavior. The influence of nonlocal and fractional order parameters is meticulously investigated. These findings reveal unique phenomena that significantly advance the comprehension of Rayleigh waves in these intricate materials.

Algebraic Hastatic Order in One-Dimensional Two-Channel Kondo Lattice.

The two-channel Kondo lattice likely hosts a rich array of phases, including hastatic order, a channel symmetry breaking heavy Fermi liquid. We revisit its one-dimensional phase diagram using density matrix renormalization group and, in contrast to previous work, find algebraic hastatic orders generically for stronger couplings. These are heavy Tomonaga-Luttinger liquids with nonanalyticities at Fermi vectors captured by hastatic density waves. We also find a predicted additional nonlocal order parameter due to interference between hastatic spinors, not present at large N, and residual repulsive interactions at strong coupling suggesting non-Fermi-liquid physics in higher dimensions.

Modeling of nonlocal Caputo–Fabrizio integral models in a nanoscale resonator

In the present work, the nonlocal integral constitutive laws for application to nano-beams, are investigated in a general setting. Evidence of boundary effects are enlightened by theoretical analysis and numerical computations. The proposed compensation procedures efficiently analyzes the thermoelastic interaction in a nanoscale resonator in the context of two-temperature three-phase lag model of thermoelasticity involving Caputo Fabrizio (CF) derivative where both the ends of the nanoscale beam is clamped. Employing the Laplace transform as a tool, the solutions for conductive temperature, thermodynamic temperature, stress and deflection have been determined. The corresponding solutions in the space-time domain is obtained by incorporating the numerical inversion of the Laplace transform using Riemann-sum approximation technique. According to the graphical representations corresponding to the numerical results, effectiveness of the recently proposed theory is demonstrated due to the presence of nonlocal CF order and the nonsingular kernel.

Fundamental role of nonlocal orders in 1D extended Bose-Hubbard model.

Nonlocal order parameters capture the presence of correlated fluctuations between specific degrees of freedom, in otherwise disordered quantum matter. Here, we provide a further example of their fundamental role, deriving the ground state phase diagram of the filling one extended Bose-Hubbard model, exclusively in terms of their ordering. By means of a density matrix renormalization group numerical analysis, we show that in addition to the (even) parity order characteristic of the Mott insulating phase and the string order nonvanishing in the Haldane insulator, the recently proposed odd parity order completes the picture, becoming nonvanishing at the transition from the normal superfluid to the paired superfluid phase. The above three nonlocal parameters capture all the distinct phases, including the density wave phase, in which the local order is seen as the simultaneous presence of correlated fluctuations in different channels. They provide a unique tool for the experimental observation of the full phase diagram of strongly correlated quantum matter, by means of local density measurements.

Detecting hidden order in fractional Chern insulators

Topological phase transitions go beyond Ginzburg and Landau's paradigm of spontaneous symmetry breaking and occur without an associated local order parameter. Instead, such transitions can be characterized by the emergence of nonlocal order parameters, which require measurements on extensively many particles simultaneously—an impossible venture in real materials. On the other hand, quantum simulators have demonstrated such measurements, making them prime candidates for experimental confirmation of nonlocal topological order. Here, building upon the recent advances in preparing few-particle fractional Chern insulators using ultracold atoms and photons, we propose a realistic scheme for detecting the hidden off-diagonal long-range order (HODLRO) characterizing Laughlin states. Furthermore, we demonstrate the existence of this hidden order in fractional Chern insulators, specifically for the ν=1/2-Laughlin state in the isotropic Hofstadter-Bose-Hubbard model. This is achieved by large-scale numerical density matrix renormalization group (DMRG) simulations based on matrix product states, for which we formulate an efficient sampling procedure providing direct access to HODLRO in close analogy to the proposed experimental scheme. We confirm the characteristic power-law scaling of HODLRO, with an exponent 1/ν=2, and show that its detection requires only a few thousand snapshots. This makes our scheme realistically achievable with current technology and paves the way for further analysis of nonlocal topological orders, e.g., in topological states with non-Abelian anyonic excitations. Published by the American Physical Society 2024

Axi-Symmetric Problem in the Thermoelastic Medium under Moore-Gibson-Thompson Heat Equation with Hyperbolic Two Temperature, Non-Local and Fractional Order

Axi-Symmetric Problem in the Thermoelastic Medium under Moore-Gibson-Thompson Heat Equation with Hyperbolic Two Temperature, Non-Local and Fractional Order

Twice hidden string order and competing phases in the spin-1/2 Kitaev–Gamma ladder

Finding the Kitaev spin liquid in candidate materials involves understanding the entire phase diagram, including other allowed interactions. One of these interactions, called the Gamma (Γ) interaction, causes magnetic frustration and its interplay with the Kitaev (K) interaction is crucial to comprehend Kitaev materials. Due to the complexity of the combined KΓ model, quasi-one-dimensional models have been investigated. While several disordered phases are found in the 2-leg ladder, the nature of the phases are yet to be determined. Here we focus on the disordered phase near the antiferromagnetic Γ limit (denoted by AΓ phase) next to the ferromagnetic Kitaev phase. We report a distinct non-local string order parameter characterizing the AΓ phase, different from the string order parameter in the Kitaev phase. This string order parameter becomes evident only after two unitary transformation, referred to as a twice hidden string order parameter. The related entanglement spectrum, edge states, magnetic field responses, and the symmetry protecting the phase are presented, and its relevance to the two-dimensional Kitaev materials is discussed. Two newly identified disordered phases in the phase diagram of KΓ ladder is also reported.

Exploring topological phase transition in Pt2Hg1−xTlxSe3

The transition from trivial to non-trivial phase in two-dimensional materials are called a topological phase transition (TPT). The Berry phase, non-local string order parameter, and edge states define the topological nature of the system. A newly discovered jacutingaite ma- terial Pt2HgSe3 is a layered material which occurs naturally in the form of minerals. The material can be exfoliated and was predicted as a quantum spin Hall insulator. Here, on the basis of density functional theory and tight-binding calculations, we explore Pt2Hg1−xTlxSe3 (x = 0.25, 0.50, 0.75, 1) to understand the electronic and topological properties. We start with the parent material Pt2HgSe3 wherein Hg is replaced partially with x amount of Tl, to tune the topological phases. From the electronic structure calculations, Pt2HgSe3 is found to be a non-trivial semimetal in it’s bulk. Upon electron doping, the material transforms to strong topological metallic phase. The topological Z2 invariant calculation shows TPT in Pt2Hg1−xTlxSe3 with weak topological insulating state (0;001) for x=0, to strong topological metal (1;000) for x=1, respectively.

Propagation and reflection of thermoelastic wave in a rotating nonlocal fractional order porous medium under Hall current influence

This investigation relates to the research on Hall current on propagation and reflection of elastic waves through non-local fractional-order thermoelastic rotating medium with voids. The system is split up into longitudinal and transverse components using the Helmholtz vector rule. It is observed that, through the frequency dispersion relation four coupled quasi-waves exist in the medium. The rotating solid modifies the nature of purely longitudinal and transverse waves toward the quasi-type waves. All the propagating waves are dispersive as they depend upon angular frequency. The quasi-longitudinal wave qP and quasi-transverse wave qSV faces cut-off frequencies. The nonlocal parameter affect all the waves except the quasi void wave. Analytically, the reflection coefficients of the wave are computed using suitable boundary conditions. MATLAB software is used to perform numerical computations for a chosen solid material. The amplitude ratios and the speed of propagation of the wave are plotted graphically for rotational frequency, nonlocal, fractional order, and Hall current parameter. The significant effect of the physical parameters on the computed results has been observed. The cut-off frequency of the waves is also presented graphically. The energy conservation law is proved in the form of energy ratios. The earlier findings in the literature are obtained as special case in the absence of rotation, Hall current parameter and porous voids.

Thermoelastic response of an elastic rod under the action of a moving heat source based on fractional order strain theory considering nonlocal effects

In this paper, based on fractional order strain theory and nonlocal heat conduction theory, the thermoelastic response of an elastic rod rigidly fixed at both ends and subjected to a moving heat source is investigated. The nonlocal fractional order strain control equation is established, and the analytical expressions of dimensionless temperature, displacement, and stress are solved using the Laplace integral transformation and numerical inverse transformation. The distribution law of physical quantities such as temperature, displacement, and stress when the fractional parameters, heat source moving speed, and thermal nonlocal parameters change is obtained. The results show that strain parameters and thermal nonlocal parameters have less influence on thermal–mechanical waves, while heat source velocity has a significant influence on thermal–mechanical waves.

Non-Local Q-Tensor Approach for Description of Elastic Deformations of Nematic Liquid Crystals at Sub-Micron Scale

Within the framework of phenomenological approach, a generalized concept of a non-local order parameter, that have the form of a traceless correlation tensor function or a tensor integral operator, is introduced. The main relationships, which determine the equilibrium orientational states and the phase transitions of a deformed nematic liquid crystal, are described. The resulting expression for the free energy is quadratic with respect to the gradients of the order parameter tensor. The limiting transition of the non-local order parameter to the local one leads to an expression that complements the classical Oseen–Frank theory of elasticity. The considered model includes eight independent elastic constants. Three of them relate to "bulk" constants analogous to Frank constants, another three ones relate to "surface" constants, and the remaining two constants determine the anisotropic effect of the gradient of the scalar order parameter on the director field. The constants are necessary for the consideration of a liquid crystal state near defects, as well as for the description of orientation effects caused by the inhomogeneity of order parameter. It is shown that in the free energy expression, the "surface" terms contribute to the director orientation in the case of a non-constant scalar order parameter, even in the approximation of rigid boundary conditions.

Entanglement entropy as an order parameter for strongly coupled nodal line semimetals

Topological semimetals are a class of many-body systems exhibiting novel macroscopic quantum phenomena at the interplay between high energy and condensed matter physics. They display a topological quantum phase transition (TQPT) which evades the standard Landau paradigm. In the case of Weyl semimetals, the anomalous Hall effect is a good non-local order parameter for the TQPT, as it is proportional to the separation between the Weyl nodes in momentum space. On the contrary, for nodal line semimetals (NLSM), the quest for an order parameter is still open. By taking advantage of a recently proposed holographic model for strongly-coupled NLSM, we explicitly show that entanglement entropy (EE) provides an optimal probe for nodal topology. We propose a generalized c-function, constructed from the EE, as an order parameter for the TQPT. Moreover, we find that the derivative of the renormalized EE with respect to the external coupling driving the TQPT diverges at the critical point, signaling the rise of non-local quantum correlations. Finally, we show that these quantum information quantities are able to characterize not only the critical point but also features of the quantum critical region at finite temperature.

A Numerical Algorithm for Solving Nonlocal Nonlinear Stochastic Delayed Systems with Variable-Order Fractional Brownian Noise

A numerical technique was developed for solving nonlocal nonlinear stochastic delayed differential equations driven by fractional variable-order Brownian noise. Error analysis of the proposed technique was performed and discussed. The method was applied to the nonlocal stochastic fluctuations of the human body and the Nicholson’s blowfly models, and its accuracy and computational time were assessed for different values of the nonlocal order parameters. A comparison with other techniques available in the literature revealed the effectiveness of the proposed scheme.

On the three-point order parameters of chiral symmetry breaking

The nonlocal order parameters of the QCD chiral symmetry breaking, namely the nonanomalous three-point Green functions of the quark bilinears belonging to the odd intrinsic parity sector, are studied within the framework of the Resonance chiral theory. The general form of these correlators is then matched with various high energy constraints: the leading and sub-leading OPE for all three momenta simultaneously large, the leading OPE for two momenta large and the leading and sub-leading Brodsky-Lepage limit for the corresponding transition form factor. In order to satisfy these constraints, the Resonance chiral theory has to be enlarged with additional resonance multiplets and with higher derivative operators as well. We discuss a minimal extension of this kind in detail and present corresponding correlators in a closed form. The remaining free parameters are then constrained from lattice data. As a phenomenological application, we discuss the pion-pole contribution to the muon g − 2 and the decay π0→ e+e−.

Variational approach to quantum spin liquid in a Rydberg atom simulator

Recently the Rydberg blockade effect has been utilized to realize quantum spin liquid (QSL) on a kagome lattice. Evidence of QSL has been obtained experimentally by directly measuring non-local string order. In this paper, we report a Bardeen–Cooper–Schrieffer (BCS)-type variational wave function study of the spin liquid state in this model. This wave function is motivated by mapping the Rydberg blockade model to a lattice gauge theory, where the local gauge conservations replace the role of constraints from the Rydberg blockade. We determine the variational parameter from the experimental measurement of the Rydberg atom population. Then we compare the predictions of this deterministic wave function with the experimental measurements of non-local string order. Combining the measurements on both open and closed strings, we extract the fluctuations only associated with the closed-loop as an indicator of the topological order. The prediction from our wave function agrees reasonably well with the experimental data, with only one fitting parameter determined by measurement of Rydberg atom population. Our variational wave function provides a simple and intuitive picture of the QSL in this system that can be generalized to similar spin liquid phases in other lattice geometry.

Rigorous formalism for unconventional symmetry breaking in Fermi liquid theory and its application to nematicity in FeSe

Unconventional symmetry breaking due to nonlocal order parameters has attracted considerable attention in many strongly correlated metals. Famous examples are the nematic order in Fe-based superconductors and the star-of-David charge density order in kagome metals. Such exotic symmetry breaking in metals is a central issue of modern condensed matter physics, while its theoretical foundation is still unclear in comparison with the well-established theory of superconductivity. To overcome this difficulty, here we introduce the "form factor" that generalizes the nonlocal order parameter into the Luttinger-Ward (LW) Fermi liquid theory. We then construct a rigorous formalism of the "density-wave equation" that gives the thermodynamically stable form factor, similarly to the superconducting-gap equation. In addition, a rigorous expression of the Ginzburg-Landau free-energy for the unconventional order is presented to calculate various thermodynamic properties. In the next stage, we apply the derived formalism to a typical Fe-based superconductor FeSe, by using the one-loop LW function that represents the free-energy gain due to the interference among paramagnons. The following key experiments are naturally explained: (i) Lifshitz transition (=disappearance of an electron-pocket) due to the bond+orbital order below $T_c$. (ii) Curie-Weiss behavior of the nematic susceptibility at higher T, and the deviation from the Curie-Weiss behavior at lower T near the nematic quantum-critical-point. (iii) Scaling relation of the specific heat jump at $T_c$, $\Delta C/T_c \propto T_c^b$ with $b \sim 3$. (Note that b=0 in the BCS theory.) These results lead to a conclusion that the nematicity in FeSe is the bond+orbital order due to the "paramagnon interference mechanism". The present theory paves the way for solving various unconventional phase transition systems.

Driven one-dimensional noisy Kitaev chain

We study a one-dimensional Kitaev chain driven by the anisotropy parameter ${J}_{\ensuremath{-}}=t/{\ensuremath{\tau}}_{Q}$, in the presence of weak Gaussian noise in the parameter ${J}_{+}$. The system is prepared in the ground state of the Hamiltonian at the initial time $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$. The defect density and the residual energy at the end of the drive protocol reveals anti-Kibble Zurek (AKZ) behavior which deviates from the earlier reported linear in quench time dependence for the defect density. The entropy density at the end of the protocol is found to exhibit the signature of the AKZ behavior. In the context of the Kitaev chain, the two-point spin correlators are short ranged. The hidden topological order across the quantum phase transition point is probed via the nonlocal string order parameters. The two-point Majorana correlator and the hidden string correlator calculated in the final decohered state at the end of the noisy drive protocol is consistent with the AKZ picture. We have analyzed the kink statistics in the dual spin space, the first three cumulants at the end of the noisy drive also exhibit the AKZ scaling behavior for slower sweeps similar to the defect density. The kink distribution function is well approximated with the normal distribution with nonuniversal noise dependent mean and variance.

Non-local order parameters for fermion chains via the partial transpose

In the last two decades, a vast variety of topological phases have been described, predicted, classified, proposed, and measured. While there is a certain unity in method and philosophy, the phenomenology differs wildly. This work deals with the simplest such case: fermions in one spatial dimension, in the presence of a symmetry group G, which contains anti-unitary symmetries. A complete classification of topological phases, in this case, is available. Nevertheless, these methods are to some extent lacking as they generally do not allow to determine the class of a given system easily. This paper will take up proposals for non-local order parameters defined through anti-unitary symmetries. They are shown to be homotopy invariants on a suitable set of ground states. For matrix product states, an interpretation of these invariants is provided: in particular, for a particle–hole symmetry, the invariant determines a real division super algebra D such that the bond algebra is a matrix algebra over D.