Discrete Lattice Research Articles

Quantum coherent states of interacting Bose-Fermi mixtures in one dimension

We study two-component atomic gas mixtures in one dimension involving both bosons and fermions. When the inter-species interaction is attractive, we report a rich variety of coherent ground-state phases that vary with the intrinsic and relative strength of the interactions. We avoid any artifacts of lattice discretization by developing a novel implementation of a continuous matrix product state ansatz for mixtures and priorly demonstrate the validity of our approach on the integrable point that exists for mixtures with equal masses and interactions (Lai-Yang model) where we find that the ansatz correctly and systematically converges towards the exact results.

Thermal control of nucleation and propagation transition stresses in discrete lattices with non-local interactions and non-convex energy

Non-local and non-convex energies represent fundamental interacting effects regulating the complex behavior of many systems in biophysics and materials science. We study one-dimensional, prototypical schemes able to represent the behavior of several biomacromolecules and the phase transformation phenomena in solid mechanics. To elucidate the effects of thermal fluctuations on the non-convex non-local behavior of such systems, we consider three models of different complexity relying on thermodynamics and statistical mechanics: (i) an Ising-type scheme with an arbitrary temperature-dependent number of interfaces between different domains, (ii) a zipper model with a single interface between two evolving domains, and (iii) an approximation based on the stationary phase method. In all three cases, we study the system under both isometric condition (prescribed extension, matching with the Helmholtz ensemble of the statistical mechanics) and isotensional condition (applied force, matching with the Gibbs ensemble). Interestingly, in the Helmholtz ensemble the analysis shows the possibility of interpreting the experimentally observed thermal effects with the theoretical force–extension relation characterized by a temperature-dependent force plateau (Maxwell stress) and a force peak (nucleation stress). We obtain explicit relations for the configurational properties of the system as well (expected values of the phase fractions and number of interfaces). Moreover, we are able to prove the equivalence of the two thermodynamic ensembles in the thermodynamic limit. We finally discuss the comparison with data from the literature showing the efficiency of the proposed model in describing known experimental effects.

Helicity locking of a square skyrmion crystal in a centrosymmetric lattice system without vertical mirror symmetry

We theoretically investigate the stability of a square skyrmion crystal (SkX) in a centrosymmetric tetragonal lattice structure with the emphasis on the role of the magnetic anisotropy arising from the absence of vertical mirror symmetry. Our analysis is based on an effective bilinear and biquadratic model in momentum space, which is a canonical model for itinerant magnets in a weak-coupling regime. By performing the simulated annealing for the model on the two-dimensional square lattice, we find that the off-diagonal spin component in the interaction, which becomes nonzero when the vertical mirror symmetry is broken, gives rise to the square SkX with a definite helicity in an external magnetic field. We show that the helicity of the centrosymmetric SkXs is determined by the competition between the off-diagonal and diagonal anisotropic interactions, the latter of which appears in the discrete fourfold-rotational lattice structure. Furthermore, we discuss helicity-dependent physical phenomena by introducing odd-parity multipoles, where electric (magnetic) and electric (magnetic) toroidal multipoles are sources of an antisymmetric spin polarization and an Edelstein effect (a magnetoelectric effect). We also discuss the stability of the SkXs with different helicities in a magnetic field rotation. Our results provide a way of engineering the helicity-locked SkXs by the symmetric anisotropic interaction in centrosymmetric magnets, which is distinct from that by the antisymmetric Dzyaloshinskii-Moriya interaction in noncentrosymmetric magnets.

Hybrid static potentials in SU(3) lattice gauge theory at small quark-antiquark separations

We compute the ${\mathrm{\ensuremath{\Pi}}}_{u}$ and ${\mathrm{\ensuremath{\Sigma}}}_{u}^{\ensuremath{-}}$ hybrid static potentials in SU(3) lattice gauge theory using four different lattice spacings ranging from $a=0.040\text{ }\text{ }\mathrm{fm}$ to $a=0.093\text{ }\text{ }\mathrm{fm}$. We provide lattice data points for quark-antiquark separations as small as 0.08 fm, where the $a$-dependent self-energy as well as lattice discretization errors at tree level of perturbation theory and at leading order in ${a}^{2}$ have been removed. We also investigate and exclude possibly present systematic errors from topological freezing, due to the finite spatial lattice volume and from glueball decays. Moreover, we provide corresponding parametrizations of the potentials, which can e.g. be used for Born-Oppenheimer predictions of heavy hybrid mesons.

Diffraction by a Dirichlet right angle on a discrete planar lattice

A problem of scattering by a Dirichlet right angle on a discrete square lattice is studied. The waves are governed by a discrete Helmholtz equation. The solution is looked for in the form of the Sommerfeld integral. The Sommerfeld transformant of the field is built as an algebraic function. The paper is a continuation of the work by A. V. Shanin and A. I. Korolkov, Sommerfeld-type integrals for discrete diffraction problems, Wave Motion 97 (2020).

Multistability and transitions between spatiotemporal patterns through versatile Notch-Hes signaling

The Delta-Notch-Hes signaling pathway is involved in various developmental processes ranging from the formation of somites to the dynamic fine-grained patterns of cell types in developing or regenerating tissues. Such broad patterning capabilities rely in part on the versatile and tunable dynamics of the Notch-Hes feedback circuit eliciting both pulsatile and switching behaviors. This raises the theoretical issue of which specific spatiotemporal features emerge from lateral inhibition between cells that can display and transit between monostable, oscillatory and bistable regimes. To address this issue, I consider a discrete cell lattice model where intracellular dynamics is described by a phase-like variable and displays a typical cross-shaped phase diagram. Model analysis determines how the existence and stability of many spatially inhomogeneous and temporally synchronized states depends on key intracellular and intercellular parameters. It reveals a parameter-dependent multistability between those diverse spatiotemporal patterns, giving rise to tunable and robust developmental transition scenarios ensuring defect-free spatial patterns. Such broad repertoire and multistability of spatiotemporal patterns is corroborated with regulatory network modeling of the Delta-Notch-Hes pathway.

Collective elastic properties of mechanical metamaterials

Recent studies of the unusual frequency–wavenumber dependences and the related phenomena of the negative effective mass and stiffness in phononic metamaterials unveiled many futuristic opportunities for the acoustical and heat transfer engineering [1], [2], [3], [4], [5]. They also facilitated investigations of quasistatic responses of architected material systems in a quest for negative elastic moduli [6], [7], [8], [9], [10] and other unusual basic properties [11], [12], [13], [14], [15] expected at uniform loading conditions. In this letter, we show that materials with discrete internal structure may also demonstrate strong dependences of their effective (continuum equivalent) mechanical properties on the wavenumber/spatial frequency of static sinusoidal pressure waves. The Fourier mode stiffness, a ratio of the periodic load amplitude and a response amplitude is a quadratic function of the load’s wavenumber in continuum materials. However, in discrete periodic lattices, we show it to be a squared sine function of the wavenumber, which flattens down when the wavelength becomes comparable with a unit cell size. As a result, comparing similarly loaded continuum and lattice materials leads to a varying dependence of the effective longitudinal and shear moduli on the wavenumber. Practically, this means that both material design and static load patterning can influence an effective elastic modulus of a mechanical metamaterial, and it can be seen from collective responses of a large group of unit cells.

Biskyrmion fractionalization in frustrated ferromagnetic nanotracks

A stable fusion of two skyrmions is called biskyrmion, which has a topological charge equal to 2. Here, we investigate this object with two cores in frustrated ferromagnetic racetracks. Biskyrmion dynamics induced by a spin-polarized current is analyzed, and the biskyrmion Hall effect is compared with the usual skyrmion Hall effect. We also study the chance of biskyrmion fractionalization in a single skyrmion, a phenomenon that reduces the topological charge of the system by one-half. From the topological perspective, a change in the topological sector is a drastic phenomenon allowed only in discrete lattices. In general, such a change leads to the total annihilation of the topological structure. Here, we show that the biskyrmion splits into a skyrmion and a burst of spin waves (emerging from the destroyed skyrmion). Such dissociation is possible by impelling the biskyrmion against a peculiar hole-defect, which must be able to destroy only one skyrmion, also impeding that the generated spin waves strike the other skyrmion. Our results suggest practical ways to manipulate the shape and chirality of spin textures via lattice defects, which can be intentionally inserted in magnetic thin films.

Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms

We consider the problem of the continuation with respect to a small parameter ɛ of spatially localized and time periodic solutions in 1-dimensional dNLS lattices, where ɛ represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal form algorithm in order to investigate the continuation and the linear stability of degenerate localized periodic orbits on lower and full dimensional invariant resonant tori. We recover results already existing in the literature and provide new insightful ones, both for discrete solitons and for invariant subtori.

A three-dimensional non-local lattice bond model for fracturing behavior prediction in brittle solids

In this paper, a 3D non-local lattice bond model is proposed to model fracturing behaviors of materials. First, the formulations and detailed derivation for three-dimensional non-local lattice bond models are obtained by comparing the strain energy stored in a discrete lattice with the classical continuum strain energy. Then, the capabilities of three-dimensional non-local lattice bond models are verified using benchmarks. To further assess the performance of the non-local lattice bond model, fracturing behaviors in brittle solids are predicted. Compared with the previous numerical results, the proposed model demonstrates better performances, which are more consistent with the experimental observations.

Snowmass 2021 Whitepaper: Proton Structure at the Precision Frontier

An overwhelming number of theoretical predictions for hadron colliders require parton distribution functions (PDFs), which are an important ingredient of theory infrastructure for the next generation of high-energy experiments. This whitepaper summarizes the status and future prospects for determination of high-precision PDFs applicable in a wide range of energies and experiments, in particular in precision tests of the Standard Model and in new physics searches at the high-luminosity Large Hadron Collider and Electron-Ion Collider. We discuss the envisioned advancements in experimental measurements, QCD theory, global analysis methodology, and computing that are necessary to bring unpolarized PDFs in the nucleon to the N2LO and N3LO accuracy in the QCD coupling strength. Special attention is given to the new tasks that emerge in the era of the precision PDF analysis, such as those focusing on the robust control of systematic factors both in experimental measurements and theoretical computations. Various synergies between experimental and theoretical studies of the hadron structure are explored, including opportunities for studying PDFs for nuclear and meson targets, PDFs with electroweak contributions or dependence on the transverse momentum, for incisive comparisons between phenomenological models for the PDFs and computations on discrete lattice, and for cross-fertilization with machine learning/AI approaches. [Submitted to the US Community Study on the Future of Particle Physics (Snowmass 2021).]

Discrete lattices on the single bearing spiral: “Musical crystallography”

Discrete lattices on the single bearing spiral: “Musical crystallography”

Propagation Characteristics of Flexural Wave in One-Dimensional Phononic Crystals Based on Lattice Dynamics Model

In this paper, we establish discrete flexural lattice chain models of Bragg and locally resonant phononic crystals by setting mass defect atoms and local resonant elements on the flexural lattice chain. The bandgap characteristics of flexural wave in phononic crystals are studied by establishing the governing equations of the model. The results from models show that with the change of the mass ratio of defective atoms to normal atoms, the bandgap of the flexural wave produced by Bragg scattering shows a certain rule. When the local resonant bandgap and Bragg scattering bandgap are close to each other, the two bandgaps will be coupled to form a wider flexural wave bandgap. The effect of axial strain on bending wave propagation is only the shift of bandgap position. The effect of material damping on the propagation of a bending wave is only energy dissipation at high frequency. In addition, we use finite element simulation to calculate the bandgap of flexural wave in phononic crystals with mass defects, and the results are consistent with lattice chain model. This shows that lattice chain model can effectively guide the bandgap design of phononic crystals. This comprehensive study may help to elucidate the rule of bandgap generation of flexural wave in one-dimensional phononic crystals.

Solitary Wave Solutions to the Modified Zakharov–Kuznetsov and the (2 + 1)‐Dimensional Calogero–Bogoyavlenskii–Schiff Models in Mathematical Physics

The modified Zakharov–Kuznetsov (mZK) and the (2 + 1)‐dimensional Calogero–Bogoyavlenskii–Schiff (CBS) models convey a significant role to instruct the internal structure of tangible composite phenomena in the domain of two‐dimensional discrete electrical lattice, plasma physics, wave behaviors of deep oceans, nonlinear optics, etc. In this article, the dynamic, companionable, and further broad‐spectrum exact solitary solitons are extracted to the formerly stated nonlinear models by the aid of the recently enhanced auxiliary equation method through the traveling wave transformation. The implication of the soliton solutions attained with arbitrary constants can be substantial to interpret the involuted phenomena. The established soliton solutions show that the approach is broad‐spectrum, efficient, and algebraic computing friendly and it may be used to classify a variety of wave shapes. We analyze the achieved solitons by sketching figures for distinct values of the associated parameters by the aid of the Wolfram Mathematica program.

НЕЛИНЕЙНАЯ ЛОКАЛИЗОВАННАЯ ВОЛНА В МЕТАМАТЕРИАЛЕ, МАТЕМАТИЧЕСКАЯ МОДЕЛЬ КОТОРОГО ПОЛУЧЕНА МЕТОДОМ АЛЬТЕРНАТИВНОЙ КОНТИНУАЛИЗАЦИИ

A metamaterial is defined as a class of substances with a complex internal structure and unique physical and mechanical properties. As a rule, such a material is a complex periodic system, in the nodes of which there are not material points, but bodies of small but finite sizes, which have internal degrees of freedom. To describe metamaterials, gradient continuums are often used, which are obtained by continuumizing the equations of motion of discrete lattices consisting of identical masses and springs of different stiffness. However, it should be noted that the gradient continuum model must be dynamically consistent, i.e. stable and providing a finite rate of energy transfer, while in most gradient models the group velocity of waves increases indefinitely with frequency. To achieve dynamic consistency of the gradient continuum model, the continuum method proposed by A.V. Metrikine and H. Askes, the essence of which is the assumption of a nonlocal connection between the displacements of the lattice nodes and the resulting continuum (the method of alternative continualization). In this paper, this method is generalized to the case of finite deformations and applied to obtain a nonlinear dynamically consistent model of a metamaterial (gradient elastic medium). Within the framework of the obtained model, the formation of spatially localized nonlinear waves, which are strain solitons and their periodic analogs, in gradient-elastic media is studied. The sign of the dimensionless parameter, which is the ratio of the nonlinear addition to the spring stiffness to its linear stiffness, affects the polarity of the soliton. For positive values of the parameter (hard nonlinearity), the soliton has a negative polarity. For negative values of the parameter (soft nonlinearity), the soliton has a positive polarity. The magnitude of the nonlinearity does not affect the speed of wave propagation and their width, but affects their amplitude.

Dark discrete breather modes in a monoaxial chiral helimagnet with easy-plane anisotropy

Nonlinearity and discreteness are two pivotal factors for an emergence of discrete breather excitations in various media. We argue that these requirements are met in the forced ferromagnetic phase of the monoaxial chiral helimagnet $\mathrm{Cr}{\mathrm{Nb}}_{3}\mathrm{S}{}_{6}$ due to the specific domain structure of the compound. The stationary, time-periodic breather modes appear as the discrete breather lattice solutions whose period mismatches with a system size. Thanks to easy-plane single-ion anisotropy intrinsic to $\mathrm{Cr}{\mathrm{Nb}}_{3}\mathrm{S}{}_{6}$, these modes are of the dark type with frequencies lying within the linear spin-wave band, close to its bottom edge. They represent cnoidal states of magnetization, similar to the well-known soliton lattice ground state, with differing but limited number of embedded $2\ensuremath{\pi}$ kinks. The linear stability of these dark breather modes is verified by means of Floquet analysis. Their energy, which is controlled by two parameters, namely, the breather lattice period and amplitude, falls off linearly with a growth of the kink number. These results may pave a path to design spintronic resonators on the base of chiral helimagnets.

Locking of skyrmion cores on a centrosymmetric discrete lattice: Onsite versus offsite

A magnetic skyrmion crystal (SkX) with a swirling spin configuration, which is one of topological spin crystals as a consequence of an interference between multiple spin density waves, shows a variety of noncoplanar spin patterns depending on a way of superposing the waves. By focusing on a phase degree of freedom among the constituent waves in the SkX, we theoretically investigate a position of the skyrmion core on a discrete lattice, which is relevant with the symmetry of the SkX. The results are obtained for the double exchange (classical Kondo lattice) model on a discrete triangular lattice by the variational calculations. We find that the skyrmion cores in both two SkXs with the skyrmion number of one and two are locked at the interstitial site on the triangular lattice, while it is located at the onsite by introducing a relatively large easy-axis single-ion anisotropy. The variational parameters and the resultant Fermi surfaces in each SkX spin texture are also discussed. The different symmetry of the Fermi surfaces depending on the core position is obtained when the skyrmion crystal is commensurate with the lattice. The different Fermi-surface topology is directly distinguished by an electric probe of angle-resolved photoemission spectroscopy. Furthermore, we show that the SkXs obtained by the variational calculations are also confirmed by numerical simulations on the basis of the kernel polynomial method and the Langevin dynamics for the double exchange model and the simulated annealing for an effective spin model.

Pearson equations for discrete orthogonal polynomials: I. Generalized hypergeometric functions and Toda equations

AbstractThe Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre–Freud structure semiinfinite matrix that models the shifts by ±1 in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre–Freud matrix is banded. From the well‐known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff–Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous 1D Toda equation for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It is also shown that the Kadomtesev–Petviashvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case, the deformation does not satisfy a Pearson equation.

The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the O(3)O(3) non-linear sigma model in 1+11+1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3+13+1 dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an SU(2)SU(2) symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to \chi_E^\text{eff} \sim 1500χEeff∼1500, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite-TT transition and asymptotic freedom, though with a slight preference for the second.

Topological wave energy harvesting in bistable lattices

In this paper, we present an input-independent energy harvesting mechanism exploiting topological solitary waves. This class of medium transforming solitons, or transition waves, entails energy radiation in the form of trailing phonons in discrete bistable lattices. We observe numerically and experimentally that the most dominant frequencies of these phonons are invariant to the input excitations as long as transition waves are generated. The phonon energy at each unit cell is clustered around a single invariant frequency, enabling input-independent resonant harvesting with conventional energy transduction mechanisms. The presented mechanism fundamentally breaks the link between the unit cell size and the metamaterial’s operating frequencies, offering a broadband solution to energy harvesting that is particularly robust for low-frequency input sources. We further investigate the effect of lattice discreteness on the energy harvesting potential, observing two performance gaps and a topological wave harvesting pass band where the potential for energy conversion increases almost monotonically. The observed frequency-invariant phonons are intrinsic to the discrete bistable lattices, enabling broadband energy harvesting to be an inherent metamaterial property.