Triangular Lattice Research Articles

Wave propagation on hexagonal lattices

Abstract We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem.

Hidden orders and phase transitions for the fully packed quantum loop model on the triangular lattice

Quantum loop and dimer models are prototypical correlated systems with local constraints, which are not only intimately connected to lattice gauge theories and topological orders but are also widely applicable to the broad research areas of quantum materials and quantum simulation. Employing our sweeping cluster quantum Monte Carlo algorithm, we reveal the complete phase diagram of the triangular-lattice fully packed quantum loop model. Apart from the known lattice nematic (LN) solid and the even Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} quantum spin liquid (QSL) phases, we discover a hidden vison plaquette (VP) phase, which had been overlooked and misinterpreted as a QSL for more than a decade. Moreover, the VP-to-QSL continuous transition belongs to the (2 + 1)D cubic* universality class, which offers a lattice realization of the (fractionalized) cubic fixed point that had long been considered as irrelevant towards the O(3) symmetry until corrected recently by conformal bootstrap calculations. Our results are therefore of relevance to recent developments in both experiments and theory, and facilitate further investigations of hidden phases and transitions.

SPT extension of Z2 quantum Ising model's ferromagnetic phase

This paper focuses on the creation of a model with explicitly defined symmetry-protected topological (SPT) phases on a triangular lattice as an extension of Z2 Ising model's ferromagnetic phase. Unlike in previously known similar works, this model is based on an initially interacting system which is known to describe experimentally realizable physical systems. The Hamiltonian for these edge states contains four-point spin interactions between next-to-next nearest neighbors. As an initially interacting A generic technique for creating SPT models is developed, allowing for the construction of translation-invariant edge models.

Designing edge states from fractional polarization insulators

We theoretically investigated disconnected dispersive edge states in an anisotropic honeycomb lattice without chiral symmetry. When both mirror and chiral symmetries are present, this system is defined by a topological quantity known as fractional polarization (FP) term and exhibits a bulk band gap, classifying it as an FP insulator. While the FP insulator accommodates robust, flat topological edge states (TES), it also offers the potential to engineer these edge states by deliberately disrupting a critical symmetry that safeguards the underlying topology. These symmetry-breaking terms allow the edge states to become dispersive and generate differing configurations along the open boundaries. Furthermore, disconnected helical-like and chiral-like edge states analogous to TES seen in quantum spin and anomalous hall effect are achieved by the finite size effect, not possible from the symmetry-breaking terms alone. The demonstration of manipulating these edge states from a FP insulator can open up new avenues in constructing devices that utilize topological domain walls.

4H silicon carbide bulk acoustic wave gyroscope with ultra-high Q-factor for on-chip inertial navigation

Inertial navigation on a chip has long been constrained by the noise and stability issues of micromechanical Coriolis gyroscopes, as silicon, the dominant material for microelectromechanical system devices, has reached the physical limits of its material properties. To address these challenges, this study explores silicon carbide, specifically its monocrystalline 4H polytype, as a substrate to improve gyroscope performance due to its low phonon Akhiezer dissipation and its isotropic hexagonal crystal lattice. We report on low-noise electrostatic acoustic resonant gyroscopes with mechanical quality factors exceeding several millions, fabricated on bonded 4H silicon carbide-on-insulator wafers. These gyroscopes operate using megahertz frequency bulk acoustic wave modes for large open-loop bandwidth and are tuned electrostatically using capacitive transducers created by wafer-level deep reactive ion etching. Experimental results show these gyroscopes achieve superior performance under various conditions and demonstrate higher quality factors at increased temperatures, enabling enhanced performance in an ovenized or high-temperature stabilized configuration.

Inverse design of crystals and quasicrystals in a non-additive binary mixture of hard disks.

The development of new materials typically involves a process of trial and error, guided by insights from past experimental and theoretical findings. The inverse design approach for soft-matter systems has the potential to optimize specific physical parameters, such as particle interactions, particle shape, or composition and packing fraction. This optimization aims to facilitate the spontaneous formation of specific target structures through self-assembly. In this study, we expand upon a recently introduced inverse design protocol for monodisperse systems to identify the required conditions and interactions for assembling crystal and quasicrystal phases within a binary mixture of two distinct species. This method utilizes an evolution algorithm to identify the optimal state point and interaction parameters, enabling the self-assembly of the desired structure. In addition, we employ a convolutional neural network (CNN) that classifies different phases based on their diffraction patterns, serving as a fitness function for the desired structure. Using our protocol, we successfully inverse design two-dimensional crystalline structures, including a hexagonal lattice and a dodecagonal quasicrystal, within a non-additive binary mixture of hard disks. Finally, we introduce a symmetry-based order parameter that leverages the encoded symmetry within the diffraction pattern. This order parameter circumvents the need for training a CNN and is used as a fitness function to inverse design an octagonal quasicrystal.

An Exploration of the Phonon Frequency Spectrum and Born–von Karman Periodic Boundary Conditions in 1D and 2D Lattice Systems Using a Computational Approach

Periodic boundary conditions (PBCs) are a pivotal concept in the treatment of ideal lattices of infinite extent as a finite lattice. Most undergraduate texts that delve into the analytical treatment of lattice dynamics solve the problem in the reciprocal space with an implicit assumption of PBCs. While the traditional approach centered in the reciprocal or [Formula: see text]-space is elegant for simple one-dimensional lattice structures, the method becomes mathematically cumbersome for more complex lattices with real-world applications, such as a honeycomb lattice involving a [Formula: see text] secular determinant. In this work, we have explored the solution of lattice dynamics in direct space numerically, by constructing unit cells through an explicit implementation of Born–von Karman PBCs and solving the lattice dynamical equations in the displacement–time domain using the fourth-order Runge–Kutta algorithm. The Fast Fourier Transform technique is then used to obtain the phonon frequency spectrum corresponding to the computed instantaneous displacements. The fidelity of the model is validated by the agreement of the computational results obtained for the phonon spectrum at high symmetry points in the Brillouin zone with the analytical values for the given problem. Initially, the dynamics and the phonon spectrum are explored for monatomic and diatomic linear lattices which serve to act as introductory cases for the approach before investigating the 2D monatomic square and honeycomb lattices using the nearest and next-nearest neighbor approximations. A short-range force constant model employing the central forces and angular forces is used for the monatomic square lattice to account for the interatomic interactions. Our computational approach is expected to be physically more intuitive for a student for visualizing the periodicity of a given lattice and understanding its related dynamics. The approach demonstrates the usage of Born–von Karman PBCs in constructing a unit cell that effectively captures the dynamics of a lattice system, complementing the traditional secular determinant formalism. The work also illustrates an instance of how the computer programming can be used as a tool to empower the students to explore abstract concepts in physics. A honing of programming and computational skills of students enables them to interact with dynamic models in physics and gain deeper insight into the real-world applications of physics principles. Target group of this work is the students and educators at undergraduate level who seek a thorough and didactic comprehension of lattice dynamics.

Evaluating graphene nanoribbons for miniaturization of field effect transistors: A density functional theory study

For the past 60 years, the number of silicon transistors the building blocks of integrated circuits that can be packed onto a microchip has doubled every two years. This phenomenon, famously termed Moores Law, has driven technological progress. However, this trajectory is approaching a barrier known as the Silicon limit. At sizes below a certain threshold, the efficiency of silicon diminishes, posing a challenge to continued miniaturization. Thus, the development of novel transistor materials has become a critical step towards realizing beyond-silicon nano-electronics. Graphene, a two-dimensional material composed of carbon atoms arranged in a hexagonal lattice, has risen to prominence as a potential way forward in the field of nano-electronic devices due to its numerous advantages, including high electron and hole mobilities, an atom-thin structure, and the ease of doping to enhance conductivity. However, the lack of a band gap in graphene poses a significant challenge in designing efficient nano-electronic devices. While cutting graphene into nanoribbons can open a band gap, further scaling down graphene nanoribbons will be difficult and costly. Therefore, alternative approaches to further enlarging the band gap based on the current size scale are essential. In this study, we propose a novel method to successfully increase the band gap of 7-armchair nanoribbons by introducing the pre-designed shape of cutting. Additionally, by further manipulating the cutting shape, we also propose a method to increase carriers mobility while retaining the band gap. These findings represent a significant advancement in optimizing the electrical performance of future carbon-based transistors. The utilization of pre-designed cutting shapes offers a flexible approach to tailor device performance according to specific requirements, thereby enhancing the versatility and functionality of carbon-based electronic devices.

Programmable order by disorder effect and underlying phases through dipolar quantum simulators

In this work, we study two different quantum simulators composed of molecules with dipole-dipole interaction through various theoretical and numerical tools. Our first result provides knowledge of the quantum order by disorder effect of the S=1/2 system, which is programmable in a quantum simulator composed of circular Rydberg atoms in the triangular optical lattice with a controllable diagonal anisotropy. When the numbers of up spins and down spins are equal, a set of subextensive degenerate ground states is present in the classical limit, composed of continuous strings whose configuration enjoys a large degree of freedom. Among all possible configurations, we focus on the stripe (up and down spins aligning straightly) and kinked (up and down spins forming zigzag spin chains) patterns. Adopting the the real space perturbation theory, we estimate the leading order energy correction when the nearest-neighbor spin exchange coupling, J, is considered, and the overall model becomes an effective XXZ model with a spatial anisotropy. Our calculation demonstrates a lifting of the degeneracy, favoring the stripe configuration. When J becomes larger, we adopt the infinite projected entangled-pair state (iPEPS) and numerically check the effect of degeneracy lifting. The iPEPS results show that even when the spin exchange coupling is strong the stripe pattern is still favored. Next, we study the dipolar bosonic model with tilted polar angle which can be realized through a quantum simulator composed of cold atomic gas with dipole-dipole interaction in an optical lattice. By placing the atoms in a triangular lattice and tilting the polar angle, the diagonal anisotropy can also be realized in the bosonic system. With our cluster mean-field theory calculation, we provide various phase diagrams with different tilted angles, showing the abundant underlying phases including the supersolid. Our proposal indicates realizable scenarios through quantum simulators in studying the quantum effect as well as extraordinary phases. We believe that our results indicated here can also become a good benchmark for two-dimensional quantum simulators. Published by the American Physical Society 2024

Emergent Physics in Metal−Organic Frameworks

AbstractMany‐body interactions in condensed matter could lead to emergent phenomena spanning superconductivity, ferromagnetism, exciton condensation, etc. The emergence of these phenomena often requires highly ordered spatial arrangements of the interacting species to enforce specific space symmetries and interacting strengths. Metal−organic frameworks (MOFs), crystalline materials formed by self‐assembly of metal ions and organic ligands, allow for precise design of their crystal structures and sophisticated tuning of Coulombic interaction or magnetic coupling among lattice sites. Such atomic‐level designability combined with high crystallinity and versatile types of lattices (e.g., kagome and honeycomb lattices) render MOFs as a great platform to investigate emergent physics. In this Emerging Topic, we summarize recent studies evidencing emergent phenomena in MOFs including strong correlations, superconductivity, charge density wave, long‐range magnetic order, and quantum spin liquid. We highlight the great potential of MOFs as quantum materials and discuss challenges including growth of high‐quality single crystals and in‐depth physical characterizations to reveal insights into the nature of physical properties of MOFs. Key Scientists

Phase- and temperature-driven chiral topological superfluids on a honeycomb lattice

The correlated spinful Haldane model exhibits rich topological phases consisting of chiral topological superfluids (TSFs) and topological spin density waves. However, most of previous studies mainly focus on the case with the fixed hopping phase or at zero temperature. In this paper, we study the attractive spinful Haldane model with arbitrary phase at finite temperature. The chiral TSFs with Chern number C = 2 and 4 emerge driven by the phase and temperature. In particular, the temperature can drive a C = 2 topological superfluid from a trivial normal insulator phase at an appropriate interaction. The bulk topology of all TSFs is uncovered by the Wilson loop method, and confirmed by the responses of edge dislocations.

Discovering Pyramidal Treasures: Multi-Scale Design of High Strength-Ductility Titanium Alloys.

Mechanical properties of titanium alloys, one of humankind's most essential structural materials, suffer from the lack of 〈c + a〉 dislocations on pyramidal slip planes, failing homogeneous plastic strain accommodation. This mechanical treasure is not easily accessible in titanium alloys because of the required excessively high stress levels. The present work demonstrates that such a dilemma may be overcome by meticulously tuning the c/a ratio, the simplest crystallographic parameter of the hexagonal close-packed lattice, through Sn alloying. Combining this lattice-scale design concept with a cross-rolling based polycrystal-scale design solution, this study showcases a facile route to bimodal (α + β) titanium alloys with exceptional strength-ductility synergy.

Utilizing topological invariants for encoding and manipulating chiral phonon devices

As a fundamental degree of freedom, phonon chirality is expected to promote the development of quantum information technology just like electron spin. Currently, central to this area is the realization of efficient transmission and control of chiral information. In this paper, we propose an approach by integrating topological theory, leveraging topologically invariant Chern numbers, to encode hexagonal lattice systems. Our investigation reveals the presence of topologically protected chiral interface states within the shared band gaps of both trivial and non-trivial system units. By precisely modulating the magnetic field distribution within the encoding system, we can effectively manipulate the topological pathways. Building upon this framework, we design and implement a chiral phonon three-port device. Through dynamic calculations, we demonstrate the transmission process of chiral information, showcasing the chiral phonon switching effect and logical OR operation. Our findings not only establish a fundamental mechanism for the manipulation and control of phonon chiral information but also provide a promising direction for research in harnessing chirality degrees of freedom in practical applications.

SO(5) Deconfined Phase Transition under the Fuzzy-Sphere Microscope: Approximate Conformal Symmetry, Pseudo-Criticality, and Operator Spectrum

The deconfined quantum critical point (DQCP) is an example of phase transitions beyond the Landau symmetry-breaking paradigm that attracts wide interest. However, its nature has not been settled after decades of study. In this paper, we apply the recently proposed fuzzy-sphere regularization to study the SO(5) nonlinear sigma model with a topological Wess-Zumino-Witten term, which serves as a dual description of the DQCP with an exact SO(5) symmetry. We demonstrate that the fuzzy sphere functions as a powerful microscope, magnifying and revealing a wealth of crucial information about the DQCP, ultimately paving the way toward its final answer. In particular, through exact diagonalization, we provide clear evidence that the DQCP exhibits approximate conformal symmetry. The evidence includes the existence of a conserved SO(5) symmetry current, a stress tensor, and integer-spaced levels between conformal primaries and their descendants. Most remarkably, we identify 23 primaries and 76 conformal descendants. Furthermore, by examining the renormalization group flow of the lowest symmetry singlet as well as other primaries, we provide numerical evidence in favor of DQCP being pseudo-critical, with the approximate conformal symmetry plausibly emerging from nearby complex fixed points. The primary spectrum we compute also has important implications, including the conclusion that the SO(5) DQCP cannot describe a direct transition from the Néel to valence bond solid phase on the honeycomb lattice. Published by the American Physical Society 2024

Tunable wave localization at the Dirac frequency in a metallic photonic crystal cavity.

In this study, the two-dimensional (2D) triangular lattice metallic photonic crystals (PCs) in visible and infrared bands have been utilized to achieve light confinement at the Dirac frequency. Distinct from the traditional bandgap or total internal reflection cavity modes, the unique photonic localization mechanism leads to an unusual algebraic decay of state and a unique frequency located beyond any bandgaps. This investigation delves into the band structure analysis of 2D metallic PCs, specifically focusing on their distinctive features, such as photonic bandgaps and Dirac cones. The plane wave expansion (PWE) method, enhanced with a linearization technique, is employed for band structure calculations, considering both the frequency-dependent dielectric properties and the intrinsic lossy nature of metallic materials described by the Drude model. The study provides a comprehensive derivation of the PWE equations for metallic PCs and investigates their band characteristics under both TM and TE polarizations. Focusing on TM modes in triangular lattice metallic PCs, it reveals zero density of states (DOS) at K points of the Brillouin corner and the existence of Dirac cones with linearly dispersion and linearly vanishing DOS. The study extends to exploring localized modes at Dirac frequencies, employing a relativistic quantum mechanics approach analogous to graphene's charge carriers. Theoretical predictions are corroborated by numerical simulations, and the potential for tunable Dirac localized modes is highlighted. This research not only deepens the understanding of Dirac properties in graphene-like systems but also lays the groundwork for further exploration of the practical quasi-2D devices, which will provide assistance in the integration of micro- and nano- devices, especially in applications requiring long-range coupling, given the critical importance of optical cavities in contemporary optical technologies.

Crystal structure, properties and pressure-induced insulator-metal transition in layered kagome chalcogenides

Layered materials with kagome lattice have attracted a lot of attention due to the presence of nontrivial topological bands and correlated electronic states with tunability. In this work, we investigate a unique van der Waals (vdW) material system, A 2 M 3 X 4 (A = K, Rb, Cs; M = Ni, Pd; X = S, Se), where transition metal kagome lattices, chalcogen honeycomb lattices and alkali metal triangular lattices coexist simultaneously. A notable feature of this material is that each Ni/Pd atom is positioned in the center of four chalcogen atoms, forming a local square-planar environment. This crystal field environment results in a low spin state S= 0 of Ni2+/Pd2+. A systematic study of the crystal growth, crystal structure, magnetic and transport properties of two representative compounds, Rb2Ni3S4 and Cs2Ni3Se4, has been carried out on powder and single crystal samples. Both compounds exhibit nonmagnetic p-type semiconducting behavior, closely related to the particular chemical environment of Ni2+ ions and the alkali metal intercalated vdW structure. Additionally, Cs2Ni3Se4 undergoes an insulator-metal transition (IMT) in transport measurements under pressure up to 87.1 GPa without any structural phase transition, while Rb2Ni3S4 shows the tendency to be metalized.

Back-focal plane scanning spectroscopy for investigating the optical dispersion of large-area two-dimensional photonic crystal fabricated by capillary force lithography

In this article, we introduce our custom-built back-focal plane (BFP) scanning spectroscopy to explore an angle-resolved optical dispersion in two-dimensional (2D) photonic crystal (PhC) constructed with hexagonal lattice of nano-scaled dielectric rods. We fabricated a uniformly large-area photonic crystal measuring 1 cm by 0.5 cm, featuring a polymer-based hexagonal lattice on a gold layer, using capillary force lithography. This precision enables the effective confinement of photonic modes, leading to enhanced optical interactions. We successfully map out the angle-resolved reflectance spectra by directly scanning BFP, revealing the structure's angle dependent optical response and providing insights into its iso-frequency contours. Our approach simplifies the exploration of advanced optical materials, highlighting the role of precise fabrication and measurement techniques in understanding and utilizing the optical properties of structured materials for various technological applications.

Sublattice Structure and Topology in Spontaneously Crystallized Electronic States.

The prediction and realization of the quantum anomalous Hall effect are often intimately connected to honeycomb lattices in which the sublattice degree of freedom plays a central role in the nontrivial topology. Two-dimensional Wigner crystals, on the other hand, form triangular lattices without sublattice degrees of freedom, resulting in a topologically trivial state. Here, we discuss the possibility of spontaneously formed honeycomb-lattice crystals that exhibit the quantum anomalous Hall effect. Starting from a single-band system with nontrivial quantum geometry, we derive the mean-field energy functional of a class of crystal states and express it as a model of sublattice pseudospins in momentum space. We find that nontrivial quantum geometry leads to extra terms in the pseudospin model that break an effective "time-reversal symmetry" and favor a topologically nontrivial pseudospin texture. When the effects of these extra terms dominate over the ferromagnetic exchange coupling between pseudospins, the anomalous Hall crystal state becomes energetically favorable over the trivial Wigner crystal state.

Many-body Chern insulator in the Kondo lattice model on a triangular lattice

Many-body Chern insulator in the Kondo lattice model on a triangular lattice

Study on optical properties of hexagonal lattice photonic crystal fibers infiltrated with heavy water

In this paper, we analyzed a PCF made from fused silica glass, with a core filled with heavy water. The guiding properties of proposed fibers in terms of effective refractive index, attenuation, and dispersion of the fundamental mode were studied and optimized setups were selected and analyzed in detail. After 25 simulations, we determined two structures possessing optimal dispersion with the lattice constant (Ʌ) and the filling factor as follows: Ʌ = 1.1 µm, d/Ʌ = 0.92 for #F1 and Ʌ = 1.4 µm, d/Ʌ = 0.92 for #F2. Besides, high nonlinearity and low confinement loss are also outstanding points in our model. Thanks to these advantages, the proposed fibers have been targeted for flat and smooth broadband supercontinuum (SC) generation for near-infrared applications.