Harmonic Lattice Research Articles

Unprecedented Integral-Free Debye Temperature Formulas: Sample Applications to Heat Capacities of ZnSe and ZnTe

Detailed analytical and numerical analyses are performed for combinations of several complementary sets of measured heat capacities, for ZnSe and ZnTe, from the liquid-helium region up to 600 K. The isochoric (harmonic) parts of heat capacities, CVh(T), are described within the frame of a properly devised four-oscillator hybrid model. Additional anharmonicity-related terms are included for comprehensive numerical fittings of the isobaric heat capacities, Cp(T). The contributions of Debye and non-Debye type due to the low-energy acoustical phonon sections are represented here for the first time by unprecedented, integral-free formulas. Indications for weak electronic contributions to the cryogenic heat capacities are found for both materials. A novel analytical framework has been constructed for high-accuracy evaluations of Debye function integrals via a couple of integral-free formulas, consisting of Debye’s conventional low-temperature series expansion in combination with an unprecedented high-temperature series representation for reciprocal values of the Debye function. The zero-temperature limits of Debye temperatures have been detected from published low-temperature Cp(T) data sets to be significantly lower than previously estimated, namely, 270 (±3) K for ZnSe and 220 (±2) K for ZnTe. The high-temperature limits of the “true” (harmonic lattice) Debye temperatures are found to be 317 K for ZnSe and 262 K for ZnTe.

An asymptotic formula for displacement field in triangular lattice with vacancy

An infinite harmonic triangular lattice with a vacancy under imposed volumetric strain is investigated. Displacement field around the vacancy is considered. In previous works of the authors, an exact displacement field is obtained in the integral form. The integral contains large parameter, notably particle index proportional to distance from the vacancy. In the present paper, behavior of the displacement field far from the vacancy is considered. Asymptotic expansion of the exact solution yields simple expression for the displacement field. The expression has reasonable accuracy for all lattice nodes and it rapidly converges to the exact solution with increasing distance from the vacancy. Strain concentration factor, defined as the ratio of the maximal deformation of the bonds adjacent to the vacancy to the deformations of bonds at infinity, is calculated. It is shown that the asymptotic formula predicts the strain concentration factor with 4% accuracy. The results are compared with predictions of continuum elasticity theory. In continuum formulation, the lattice with vacancy is modelled by an elastic plate with circular hole. It is shown that the continuum displacement field has the same form as the asymptotic expression for the discrete displacement field. Comparison of discrete and continuum results yields an effective diameter of the vacancy. The comparison also shows that influence of vacancies on strength of the lattice can not be accurately modeled using continuum theory.

Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice.

We predict the existence of gap solitons in the nonlinear fractional Schrödinger equation (NLFSE) with an imprinted optically harmonic lattice. Symmetric/antisymmetric nonlinear localized modes bifurcate from the lower/upper edge of the first/second band in defocusing/focusing Kerr media. A unique feature we revealed is that, in focusing Kerr media, stable solitons appear in the finite bandgaps with the decrease of the Lévy index, which is in sharp contrast to the standard NLSE with a focusing nonlinearity. Nonlinear bound states composed by in-phase and out-of-phase soliton units supported by the NLFSE are also uncovered. Our work may pave the way for the study of spatial lattice solitons in fractional dimensions.

The Klein–Gordon equation on [formula omitted] and the quantum harmonic lattice

The discrete Klein–Gordon equation on a two-dimensional square lattice satisfies an ℓ1↦ℓ∞ dispersive bound with polynomial decay rate |t|−3/4. We determine the shape of the light cone for any choice of the mass parameter and relative propagation speeds along the two coordinate axes. Fundamental solutions experience the least dispersion along four lines interior to the light cone rather than along its boundary, and decay exponentially to arbitrary order outside the light cone. The overall geometry of the propagation pattern and its associated dispersive bounds are independent of the particular choice of parameters. In particular there is no bifurcation of the number or type of caustics that are present. The dispersive bounds imply global well-posedness for small solutions of a nonlinear discrete Klein–Gordon equation.The discrete Klein–Gordon equation is a classical analogue of the quantum harmonic lattice. In the quantum setting, commutators of time-shifted observables experience the same decay rates as the corresponding Klein–Gordon solutions, which depend in turn on the relative location of the observables' support sets.

From chirald-wave to nodal line superconductivity in the harmonic honeycomb lattices

Motivated by the recent realization of the three-dimensional hyperhoneycomb and stripyhoneycomb lattices in lithium iridate (Li$_2$IrO$_3$), we study the possible spin-singlet superconducting states on the whole series of harmonic honeycomb lattices. Beginning with an isolated out-of-plane twist making the honeycomb lattice three-dimensional, we find that the chiral d$\pm$id state, well-known from the honeycomb lattice, is realized in the largest members of the series at low to intermediate doping. Along the twist, four chiral edge states form a two-dimensional dispersive band. Reducing the distance between the twists to form the smaller members of the harmonic honeycomb lattices, the degeneracy between the d-wave states is lifted, which finally destroys the chiral state. By analyzing the hyper- and stripyhoneycomb lattices and generalizing using the $D_{2h}$ point group of all the harmonic honeycomb lattices, we show that the superconducting state often belongs to the trivial irreducible representation. This state has nodal lines at low to intermediate doping, which is possible because the full lattice symmetry allows sign changes between different sets of bonds. We also find time-reversal symmetry broken states, which are either fully gapped or feature nodal points, in certain parts of the phase diagram. Finally, we draw a comparison between the states classified in terms of the $D_{2h}$ symmetries and those observed on the $D_{6h}$ honeycomb lattice.

Entanglement negativity in a two dimensional harmonic lattice: area law and corner contributions

We study the logarithmic negativity and the moments of the partial transpose in the ground state of a two dimensional massless harmonic square lattice with nearest neighbour interactions for various configurations of adjacent domains. At leading order for large domains, the logarithmic negativity and the logarithm of the ratio between the generic moment of the partial transpose and the moment of the reduced density matrix at the same order satisfy an area law in terms of the length of the curve shared by the adjacent regions. We give numerical evidence that the coefficient of the area law term in these quantities is related to the coefficient of the area law term in the Rényi entropies. Whenever the curve shared by the adjacent domains contains vertices, a subleading logarithmic term occurs in these quantities and the numerical values of the corner function for some pairs of angles are obtained. In the special case of vertices corresponding to explementary angles, we provide numerical evidence that the corner function of the logarithmic negativity is given by the corner function of the Rényi entropy of order 1/2.

Inductive intrinsic localized modes in a one-dimensional nonlinear electric transmission line.

The experimental properties of intrinsic localized modes (ILMs) have long been compared with theoretical dynamical lattice models that make use of nonlinear onsite and/or nearest-neighbor intersite potentials. Here it is shown for a one-dimensional lumped electrical transmission line that a nonlinear inductive component in an otherwise linear parallel capacitor lattice makes possible a new kind of ILM outside the plane wave spectrum. To simplify the analysis, the nonlinear inductive current equations are transformed to flux transmission line equations with analog onsite hard potential nonlinearities. Approximate analytic results compare favorably with those obtained from a driven damped lattice model and with eigenvalue simulations. For this mono-element lattice, ILMs above the top of the plane wave spectrum are the result. We find that the current ILM is spatially compressed relative to the corresponding flux ILM. Finally, this study makes the connection between the dynamics of mass and force constant defects in the harmonic lattice and ILMs in a strongly anharmonic lattice.

Thermal properties of the orthorhombic CaSnO3 perovskite under pressure from ab initio quasi-harmonic calculations

Structural, elastic and thermodynamic properties of the orthorhombic CaSnO3 perovskite are theoretically investigated at the ab initio level as a function of temperature and pressure. Harmonic and quasi-harmonic lattice dynamical calculations are performed with the Crystal program, by explicitly accounting for thermal expansion effects and by exploring the effect of several DFT functionals. The anisotropic, directional elastic response of the system is characterized up to 20 GPa of pressure. The thermal lattice expansion and elastic bulk modulus are described at simultaneous temperatures up to 2000 K and pressures up to 20 GPa. The Gibbs free energy of formation of CaSnO3 from CaO and SnO2 as a function of temperature is also addressed by means of fully converged phonon dispersion calculations on the three systems.

A formalism for calculating the modal contributions to thermal interface conductance

A formalism termed the interface conductance modal analysis (ICMA) method is presented, which allows for calculations of the modal contributions to thermal interface conductance within the context of molecular dynamics (MD) simulations, which inherently include anharmonicity to full order. The eigen modes of vibration in this study are calculated from harmonic lattice dynamics (LD) calculations, however the generality of ICMA formalism also allows for incorporation of anharmonic LD results into the calculations. ICMA can be implemented in both equilibrium MD (EMD) and non-equilibrium MD (NEMD) simulations and both methods show qualitative agreement as validated through study of a simple system of Lennard-Jones solids. The formalism itself is based on a modal decomposition of the heat flow across an interface, which is then substituted into expressions for the conductance either based on EMD or NEMD. As a MD based method, it not only includes anharmonicity, but also naturally includes the atomic level details of the interface quality. The ICMA method now enables more in-depth study of various effects such as temperature, anharmonicity, interdiffusion, roughness, imperfections, dislocations, stress, changes in crystal structure through a single unified model, as it can essentially treat any material or object where the atoms vibrate around equilibrium sites (e.g., ordered or disordered solids and molecules). This formalism therefore serves as an important step forward toward better understanding of heat flow at interfaces.

Interfacial thermal conduction and negative temperature jump in one-dimensional lattices

We study the thermal boundary conduction in one-dimensional harmonic and ϕ^{4} lattices, both of which consist of two segments coupled by a harmonic interaction. For the ballistic interfacial heat transport through the harmonic lattice, we use both theoretical calculation and molecular dynamics simulation to study the heat flux and temperature jump at the interface as to gain insights into the Kapitza resistance at the atomic scale. In the weak coupling regime, the heat current is proportional to the square of the coupling strength for the harmonic model as well as anharmonic models. Interestingly, there exists a negative temperature jump between the interfacial particles in particular parameter regimes. A nonlinear response of the boundary temperature jump to the externally applied temperature difference in the ϕ^{4} lattice is observed. To understand the anomalous result, we then extend our studies to a model in which the interface is represented by a relatively small segment with gradually changing spring constants and find that the negative temperature jump still exists. Finally, we show that the local velocity distribution at the interface is so close to the Gaussian distribution that the existence or absence of a local equilibrium state is unable to be determined by numerics in this way.

Theory of Raman response in three-dimensional Kitaev spin liquids: Application toβ- andγ−Li2IrO3compounds

We calculate the Raman response for the Kitaev spin model on the $\mathcal{H}$-$0$, $\mathcal{H}$-$1$, and $\mathcal{H}$-$\infty$ harmonic honeycomb lattices. We identify several quantitative features in the Raman spectrum that are characteristic of the spin liquid phase. Unlike the dynamical structure factor, which probes both the Majorana spinons and flux excitations that emerge from spin fractionalization, the Raman spectrum in the Kitaev models directly probes a density of states of pairs of fractional, dispersing Majorana spinons. As a consequence, the Raman spectrum in all these models is gapless for sufficiently isotropic couplings, with a low-energy power law that results from the Fermi lines (or points) of the dispersing Majorana spinons. We show that the polarization dependence of the Raman spectrum contains crucial information about the symmetry of the ground state. We also discuss to what extent the features of the Raman response that we find reflect generic properties of the spin liquid phase, and comment on their possible relevance to $\alpha-$, $\beta-$ and $\gamma-$Li$_2$IrO$_3$ compounds.

An efficient Laplace transform-wave packet method hybrid with substructure technique

By virtue of substructure technique, the wave packet method can be applied to efficiently evaluate the thermal conduction of each phonon mode in the large periodic harmonic lattice. The substructures are connected by the inter-belts due to the long-range interatomic interaction. A parallel dynamic substructure method is presented in the Laplace transform domain, without any approximation in the condensation process. The validity and efficiency of the proposed method is demonstrated by the one- and three-dimensional cases. By launching a wide band wave packet, the whole given phonon branch can be studied by a one-time calculation. Defined as the ratio between square of Fourier transform of the transmitted and incident wave packets, the resulted transmission coefficient agrees with that obtained by the atomistic Green’s function method. Moreover the time evolution of the wave field can be given by the method, and the multilevel substructure technique can also be used.

Soliton solutions of cubic–quintic nonlinear Schrödinger equation with combined time-dependent magnetic–optical potentials

We investigate the explicit matter-wave soliton solutions of the cubic–quintic nonlinear Schrodinger equation with spatiotemporal modulation of the nonlinearities and potentials. With a systematic way, we construct some integrable systems with localized cubic–quintic nonlinearities and an infinite number of potentials, including optical lattice potential and combined time-dependent magnetic–optical potentials in the form of linear–lattice, harmonic–lattice and harmonic–linear–lattice ones. Also, corresponding analytical localized soliton solutions in terms of Mathieu and elliptic functions are studied, such as snake solitons, moving breathing solitons and oscillating solitons. Finally, some stable solitons are found by means of the stability analysis of the exact solutions with the split-step Fourier transform method.

A fully quantum description of the free-energy in high pressure hydrogen

Hydrogen is the lightest and most abundant element in the universe. Ever since Wigner and Huntington's prediction that pressure induced metallization might happen in solid hydrogen, understanding the hydrogen phase diagram has become one of the greatest challenges in condensed matter and high pressure physics. The light mass of hydrogen means that the nuclear quantum effects could be important in describing this phase diagram under high pressures. Numerical evaluations of their contributions to the structural, vibrational, and energetic properties, however, are difficult and up to now most of the theoretical simulations still remain classical. This is particularly true for the energetic properties. When the free-energies of different phases are compared in determining the ground state structure of the system at a given pressure and temperature, most of the theoretical simulations remain classical. When nuclear quantum effects must be taken into account, one often resorts to the harmonic approximation. In the very rare case, the anharmonic contributions from the nuclear statistical effects are considered by using a combination of the thermodynamic integration and the at initio molecular dynamics methods, which helps to include the classical nuclear anharmonic effects. Quantum nuclear anharmonic effects, however, are completely untouched. Here, using a self-developed combination of the thermodynamic integration and the at initio path-integral molecular dynamics methods, we calculated the free-energies of the high pressure hydrogen at 100 K from 200 GPa to 300 GPa. The harmonic lattice was taken as the reference and the Cmca phase of the solid hydrogen was chosen. When the bead number of the path-integral (P) equals one, our approach reaches the so-called classical limit. Upon increasing P until the results are converged, our approach reaches the limit when both classical and quantum nuclear anharmonic effects are included. Therefore, by comparing the free-energy of the harmonic lattice and the thermodynamic integration results at P equals one, we isolate the classical nuclear anharmonic effects. By comparing the thermodynamic integration results at P equals one and with those when they are converged with respect to P, we isolate the quantum nuclear anharmonic effects in a very clean manner. Our calculations show that the classical nuclear anharmonic contributions to the free-energy are negligible at this low temperature. Those contributions from the quantum nuclear anharmonic effects, however, are as large as ～15 meV per atom. This value also increases with pressure. This study presents an algorithm to quantitatively calculate the quantum contribution of the nuclear motion to free-energy beyond the often used harmonic approximation. The large numbers we got obtained also indicate that such quantum nuclear anharmonic effects are important in describing the phase diagram of hydrogen, at/above the pressures studied.

Entanglement dynamics in harmonic-oscillator chains

We study the long-time evolution of the bipartite entanglement in translationally invariant one-dimensional harmonic lattice systems. We show that for Gaussian states, in quadratic interactions with periodic boundary conditions, there exists a lower bound for the von Neumann entropy which increases linearly in time. This implies that the dynamics of harmonic lattice systems can in general not efficiently be simulated by algorithms based on matrix-product decompositions of the quantum state, and interactions are needed to suppress the entanglement growth with time.

A spectral multiscale method for wave propagation analysis: Atomistic–continuum coupled simulation

In this paper, we present a new multiscale method which is capable of coupling atomistic and continuum domains for high frequency wave propagation analysis. The problem of non-physical wave reflection, which occurs due to the change in system description across the interface between two scales, can be satisfactorily overcome by the proposed method. We propose an efficient spectral domain decomposition of the total fine scale displacement along with a potent macroscale equation in the Laplace domain to eliminate the spurious interfacial reflection. We use Laplace transform based spectral finite element method to model the macroscale, which provides the optimum approximations for required dynamic responses of the outer atoms of the simulated microscale region very accurately. This new method shows excellent agreement between the proposed multiscale model and the full molecular dynamics (MD) results. Numerical experiments of wave propagation in a 1D harmonic lattice, a 1D lattice with Lennard-Jones potential, a 2D square Bravais lattice, and a 2D triangular lattice with microcrack demonstrate the accuracy and the robustness of the method. In addition, under certain conditions, this method can simulate complex dynamics of crystalline solids involving different spatial and/or temporal scales with sufficient accuracy and efficiency.

Nanoscale transport of phonons: Dimensionality, subdiffusion, molecular damping, and interference effects

We examine heat transport carried by acoustic phonons in the systems composed of nanoscale chains of masses coupled to two thermal baths of different temperatures. Thermal conductance is obtained by using linearized Landauer-type formula for heat flux with phonon transmission probability calculated within atomistic Green's functions (AGF) method. AGF formalism is extended onto dissipative chains of masses with harmonic coupling beyond nearest-neighbor approximation, while atomistic description of heat reservoirs is also included into computational scheme. In particular, the phonon lifetimes and the phonon frequency shifts are discussed for harmonic lattices of different dimensions. Further, resonant structure of phonon transmission spectrum is analyzed with respect to reservoir-induced effects, molecular damping, and mass-to-mass harmonic coupling. Analysis of transmission zeros (antiresonances) and their accompanied Fano-shape resonances are discussed as a result of interference effects between different vibrational modes. Finally, we also predict subdiffusive transport regime for low-frequency ballistic phonons propagated along a linear chain of harmonically coupled masses.

Numerical studies on vortices in rotating dipolar Bose-Einstein condensates

We study the formation and dynamics of vortices in quasi two-dimensional dipolar Bose-Einstein condensates (BECs). We solve the time dependent two-dimensional Gross-Pitaevskii equation by using a combined split-step Crank-Nicolson (SSCN) and Fast Fourier-Transform (FFT) based numerical scheme and investigate the vortex dynamics in dipolar BECs trapped with harmonic and optical lattice potentials. The consequence of dipole-dipole interaction on vortex nucleation and number of vortices has been analysed. We observe that the breaking of symmetry due to anisotropic dipolar interaction enormously increase the speed of creation of vortices.

DYNAMICAL PROPERTIES OF AN ATOMIC INTERFACE BETWEEN bcc LATTICES

A study of the phonon contribution to the interface properties between two bcc dissimilar solids is presented. The model system is obtained by the juxtaposition of two semi-infinite harmonic bcc lattices. The interface observables are numerically calculated for different cases of masses and elastic softening to hardening, to investigate how the local dynamics can respond to many environmental microscopic changes in the field interfacial domain. The theoretical formalism using simultaneously the Green's functions and the matching method is employed to describe the dynamics of the bcc system, the complete evanescent and the propagating fields. A calculation is presented for the vibration localized states, the coherent phonon transmission and the density of states (DOS), as element of a Landauer–Büttiker type scattering matrix. The system dynamics, the phonon scattering and the transmission spectra via the interface domain between bcc lattices and the DOS are analyzed as function of the atomic masses and the elastic force constants occurring in the nanojunction zone of the model system. Our results show that the interface zone is an effective phonon splitter and suggest that its characteristics may be controlled by varying its nanometric parameters. The observed fluctuations are due to the coherent coupling between continuum and discrete states induced by the interface domain.

Quantum particle in a parabolic lattice in the presence of a gauge field

We analyze the eigenstates of a two-dimensional lattice with additional harmonic confinement in the presence of an artificial magnetic field. While the softness of the confinement makes a distinction between bulk and edge states difficult, the interplay of harmonic potential and lattice leads to a different classification of states in three energy regions: In the low-energy regime, where lattice effects are small, all states are transporting topologically non-trivial states. For large energies above a certain critical value, the periodic lattice causes localization of all states through a mechanism similar to Wannier-Stark localization. In the intermediate energy regime transporting, topologically non- trivial states coexist with topologically trivial counter-transporting chaotic states. The character of the eigenstates, in particular their transport properties are studied numerically and are explained using a semiclassical analysis.