Dimensional Lattice Research Articles

Contractibility of the Rips complexes of Integer lattices via local domination

We prove that for each positive integer n n , the Rips complexes of the n n -dimensional integer lattice in the d 1 d_1 metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above n 2 ( 2 n − 1 ) n^2(2n-1) , with the bounds arising from the Jung constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension n n and a fixed scale r r , this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for n = 1 , 2 , 3 n=1,2,3 , the Rips complex of the n n -dimensional integer lattice at scale greater or equal to n n is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions n n .

Memory adds no cost to lattice sieving for computers in 3 or more spatial dimensions

The security of lattice-based crytography (LWE, NTRU, and FHE) depends on the hardness of the shortest-vector problem (SVP). Sieving algorithms give the lowest asymptotic runtime to solve SVP, but depend on exponential memory. Memory access costs much more in reality than in the RAM model, so we consider a computational model where processors, memory, and meters of wire are in constant proportions to each other. While this adds substantial costs to route data during lattice sieving, we modify existing algorithms to amortize these costs and find that, asymptotically, a classical computer can achieve the previous RAM model cost of 2 0.2925 d + o ( d ) to sieve a d -dimensional lattice for a computer existing in 3 or more spatial dimensions, and can reach 2 0.3113 d + o ( d ) in 2 spatial dimensions, where “spatial dimensions” are the dimensions of the physical geometry in which the computer exists. Since this result implies an increased cost in 2 spatial dimensions, we make several assumptions about the constant terms of memory access and lattice attacks so that we can give bit security estimates for Kyber and Dilithium. These estimates support but do not increase the security categories claimed in the Kyber and Dilithium specifications, at least with respect to known attacks.

One dimensional staggered bosons, clock models, and their noninvertible symmetries

We study systems of staggered boson Hamiltonians in a one dimensional lattice and in particular how the translation symmetry by one unit in these systems is in reality a noninvertible symmetry closely related to T-duality. We also study the simplest systems of clock models derived from these staggered boson Hamiltonians. We show that the noninvertible symmetries of these lattice models together with the discrete ZN symmetry predict that these are critical points with a U(1) current algebra at c=1 and radius 2N whenever N>4. We also present an independent computation of this value that arises directly from the staggered boson variables and does not use these additional symmetries. We also present a theoretical estimate of the values of critical coupling constants away from the self-dual symmetry point in these clock models. Published by the American Physical Society 2024

Role of grain-level chemo-mechanics in composite cathode degradation of solid-state lithium batteries

Solid-state Li-ion batteries, based on Ni-rich oxide cathodes and Li-metal anodes, can theoretically reach a high specific energy of 393 Wh kg−1 and hold promise for electrochemical storage. However, Li intercalation-induced dimensional changes can lead to crystal defect formation in these cathodes, and contact mechanics problems between cathode and solid electrolyte. Understanding the interplay between cathode microstructure, operating conditions, micromechanics of battery materials, and capacity decay remains a challenge. Here, we present a microstructure-sensitive chemo-mechanical model to study the impact of grain-level chemo-mechanics on the degradation of composite cathodes. We reveal that crystalline anisotropy, state-of-charge-dependent Li diffusion rates, and lattice dimension changes drive dislocation formation in cathodes and contact loss at the cathode/electrolyte interface. These dislocations induce large lattice strain and trigger oxygen loss and structural degradation preferentially near the surface area of cathode particles. Moreover, contact loss is caused by the micromechanics resulting from the crystalline anisotropy of cathodes and the mechanical properties of solid electrolytes, not just operating conditions. These findings highlight the significance of grain-level cathode microstructures in causing cracking, formation of crystal defects, and chemo-mechanical degradation of solid-state batteries.

Condensate formation in a chiral lattice gas

Abstract We investigate the formation of condensates in a binary lattice gas in the presence of chiral interactions. These interactions differ between a given microscopic configuration and its mirror image. We consider a two dimensional lattice gas with nearest-neighbour interactions to which we add interactions involving favoured local structures that are chiral. We focus on favoured local structures that have the shape of the letter L and explore condensate formation through simulations and analytical calculations. At low temperature, this model can exhibit four different phases that are characterised by different periodic tiling patterns, depending on the strength of interactions and the chemical potential. When particle numbers are conserved, some of these phases can coexist. We analyse the structure and surface tension of interfaces between coexisting phases and determine the shapes of minimal free energy of crystalline condensates. We show that these shapes can be quadrilaterals or octagons of different orientation and symmetry.

The crossover from a dynamical percolation class to a directed percolation class on a two dimensional lattice

The crossover from a dynamical percolation class to a directed percolation class on a two dimensional lattice

Certified algorithms for equilibrium states of local quantum Hamiltonians

Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision ε, local observables can be approximated by a convex optimization program of quasi-polynomial size in 1/ε.

Loading conditions for self‐organization in the BML model with stochastic direction choice

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension according to a version of the rule of particle movement in Biham–Middleton–Levine traffic model. We introduce a stochastic case with direction choice for particles. Particles of the first type move along rows, and the particles of the second type move along columns. The goal is to find conditions of self‐organization system for any lattice dimension. We have proved that the BML model as a dynamical system is a special case of Buslaev nets. This equivalence allows us to use of Buslaev net analysis techniques to investigate the BML model. In Buslaev nets conception, the self‐organization property of the system corresponds to the existence of velocity single point spectrum equal to 1. In the paper, we consider the model version when one notable aspect is that a particle may change its type. Exactly, we assume a constant probability that a particle changes type at each step. In the case where , the system corresponds to the classical version of the BML model. We define a state of the system where all particles continue to move indefinitely, in both the present and the future, as a state of free movement. A sufficient condition for the system to result in a state of free movement from any initial state (condition for self‐organization) has been found. This condition is that the number of particles be not greater than half the greatest common divisor of the numbers . It has been proved that, if , and whether or and there are both at least one particle of the first type and at least particle of the second type, then a necessary condition for a state of free movement to exist is the greatest common divisor of and be not less than 3. The theorems are formulated in terms of the algebraic structures and terms of the dynamical system. The spectrum of particle velocities has been found for the net . This approach allows us to hope that the spectrum of dynamical system can be studied for an arbitrary dimension of the net.

Characterization of ferromagnetic semiconductors and valley polarization in janus VBXS2(X = N, P) monolayers

The manipulation of the valley degree of freedom has attracted increasing attention in both fundamental scientific research and emerging applications. Here, we employ first-principles calculations to investigate the structural stability and electronic properties of Janus monolayers of VBXS2 (X=N, P). These materials exhibit characteristics of ferromagnetic semiconductors, with their valence band maximum located at the K/K′ point. Due to the combined effects of inversion symmetry breaking and time reversal symmetry breaking, VBNS2 and VBPS2 exhibit exotic spontaneous valley polarization in their top valence bands, measured at magnitudes of 48.6 meV and 47.6 meV, respectively. Consequently, this phenomenon potentially enables the observation of the anomalous valley Hall effect (AVHE). The unique electronic and valleytronic properties exhibited by Janus VBXS2 suggest a feasible experimental avenue for exploring ferrovalley (FV) and valley-related Hall effects within a two dimensional lattice.

Functional renormalization group for fermions on a one dimensional lattice at arbitrary filling

Functional renormalization group for fermions on a one dimensional lattice at arbitrary filling

Design of All-Optical D Flip Flop Memory Unit Based on Photonic Crystal.

This paper proposes a unique configuration for an all-optical D Flip Flop (D-FF) utilizing a quasi-square ring resonator (RR) and T-Splitter, as well as NOT and OR logic gates within a 2-dimensional square lattice photonic crystal (PC) structure. The components realizing the all-optical D-FF comprise of optical waveguides in a 2D square lattice PC of 45 × 23 silicon (Si) rods in a silica (SiO2) substrate. The utilization of these specific materials has facilitated the fabrication process of the design, diverging from alternative approaches that employ an air substrate, a method inherently unattainable in fabrication. The configuration underwent examination and simulation utilizing both plane-wave expansion (PWE) and finite-difference time-domain (FDTD) methodologies. The simulation outcomes demonstrate that the designed waveguides and RR effectively execute the operational principles of the D-FF by guiding light as intended. The suggested configuration holds promise as a logic block within all-optical arithmetic logic units (ALUs) designed for digital computing optical circuits. The design underwent optimization for operation within the C-band spectrum, particularly at 1550 nm. The outcomes reveal a distinct differentiation between logic states '1' and '0', enhancing robust decision-making on the receiver side and minimizing logic errors in the photonic decision circuit. The D-FF displays a contrast ratio (CR) of 4.77 dB, a stabilization time of 0.66 psec, and a footprint of 21 μm × 12 μm.

Lattice holography on a quantum computer

We explore the potential application of quantum computers to the examination of lattice holography, which extends to the strongly coupled bulk theory regime. With adiabatic evolution, we compute the ground state of a spin system on a (2+1)-dimensional hyperbolic lattice, and measure the spin-spin correlation function on the boundary. Notably, we observe that with achievable resources for coming quantum devices, the correlation function demonstrates an approximate scale-invariant behavior, aligning with the pivotal theoretical predictions of the anti–de Sitter/conformal field theory correspondence. Published by the American Physical Society 2024

Numerical signatures of ultra-local criticality in a one dimensional Kondo lattice model

Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo coupling J_KJK at fixed doping x. At large positive J_KJK, we confirm the expected conventional Luttinger liquid phase with 2k_F=\frac{1+x}{2}2kF=1+x2 (in units of 2\pi2π), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In the J_K ≤ 0JK≤0 side, our simulation finds the existence of a fractional Luttinger liquid (LL\star⋆) phase with 2k_F=\frac{x}{2}2kF=x2, accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL\star⋆) phase in higher dimensions. The LL\star⋆ phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positive J_KJK. Then we mainly focus on the “critical regime” between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) of J_KJK, the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around 0.035 J0.035J) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponent z=+z=+. The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference.

Study on multifunctional magnesium copper zinc spinel ferrite nanoparticles prepared by hydrothermal route; Physical, electrical, and anti-microbial investigation

Polycrystalline spinel ferrite nanoparticles of Mg0.70-XCuXZn0.30Fe2O4(X=0.0,0.15,0.30,0.45,0.60) were prepared by hydrothermal approach. The structural modification by X-ray diffraction of Mg-Zn spinel ferrite with Cu2+ substitution was recorded through crystallite size (11–14 nm), Lattice dimension (8.34–8.38 A°), strain (7×10−3-9×10−3) and dislocation density (4×1015-7×1015) respectively. The formation of spinel structure (Fe-O) was identified by FTIR with a vibration band around 400–600 cm−1. Also, the Rietveld refined XRD pattern confirmed the presence of a cubical spinel structure. The electron microscopic (SEM) analysis revealed sphere-cubical agglomerates of the prepared material. TEM analysis revealed that the particles were inside the nanoscale range, with an average particle size of 15.97 nm determined using a lognormal histogram. The existence of the expected elements is confirmed by EDS spectra. From BET data, surface parameters including surface area (39.938–92.643 m2/gm), volume (2.355 × 10−1 to 2.52 × 10−1 gm/cm3), pore radius (5.0831–12.650 nm), and pore size distribution were determined. The magnetic parameters Ms (20–34 emu/g), Hc (30–39 Oe), and Mr (1.3–2.8 emu/g) obtained by VSM measurements indicate the single domain, pseudo single domain and multi-domain nature of as prepared nanoparticles and lower dielectric parameters at high frequency obtain from dielectric spectroscopy indicate the suitability of the material for switching devices, power application energy storage, and high-frequency applications. The antibacterial efficacy of the synthesized nanoparticles (NPs) was evaluated against different harmful bacteria and demonstrated the desired zone of inhibition (ZOI).

On difference equations with ‘[formula omitted]’-type solitons on three dimensional lattice

In this paper we discuss an example of classical integrable equation with rather unusual ‘B’-type Kadomtsev–Petviashvili (KP) soliton hierarchy.

Renormalization-group approach to ordered phases in music.

The organization of disordered sounds into the ordered structures of music can be understood through an analogy to the emergent ordering of physical systems undergoing phase transitions. This work builds off a prior mean-field model for pitch in music [J. Berezovsky, Sci. Adv. 5, eaav8490 (2019)2375-254810.1126/sciadv.aav8490] by using renormalization-group techniques to study the effects of dimensionality and local correlations. We corroborate the results of the mean-field model by showing convergence of the phase diagram as lattice dimension is increased, while also uncovering new phases which the mean-field model does not reveal. We also compute the nearest-neighbor correlations and provide comparisons to the mean-field model, as well as historical tuning systems used by different groups of musicians. The new phases and resulting correlations revealed in this work suggest a number of possible avenues for further exploration, including generating new music using the pitch distributions suggested by the model.

Bethe M-layer construction on the Ising model

In statistical physics, one of the standard methods to study second order phase transitions is the renormalization group that usually leads to an expansion around the corresponding fully connected solution. Unfortunately, often in disordered models, some important finite dimensional second-order phase transitions are qualitatively different or absent in the corresponding fully connected model: in such cases the standard expansion fails. Recently, a new method, the M-layer one, has been introduced that performs an expansion around a different soluble mean field model: the Bethe lattice one. This new method has been already used to compute the upper critical dimension DU of different disordered systems such as the Random Field Ising model or the Spin glass model with field. If then one wants to go beyond and construct an expansion around DU to understand how critical quantities get renormalized, the actual computation of all the numerical factors is needed. This next step has still not been performed, being technically more involved. In this paper we perform this computation for the ferromagnetic Ising model without quenched disorder, in finite dimensions: we show that, at one-loop order inside the M-layer approach, we recover the continuum quartic field theory and we are able to identify the coupling constant g and the other parameters of the theory, as a function of macroscopic and microscopic details of the model such as the lattice spacing, the physical lattice dimension and the temperature. This is a fundamental step that will help in applying in the future the same techniques to more complicated systems, for which the standard field theoretical approach is impracticable.

Z3 lattice gauge theory as a toy model for dense QCD

We propose the (3+1)-dimensional Z3 lattice gauge theory coupled with the 2-flavor Wilson-Dirac fermion as a toy model for studying quantum chromodynamics (QCD) at nonzero density. We study its phase diagram in the space of the lattice gauge couplings g2 and the quark chemical potentials μ and discuss the similarity and difference compared with anticipated behaviors of actual QCD. This model also provides a testing ground for various algorithms of the numerical Hamiltonian formalism as its Hilbert space is finite-dimensional in a finite box. Published by the American Physical Society 2024

Fusion surface models: 2+1d lattice models from fusion 2-categories

We construct (2+1)-dimensional lattice systems, which we call fusion surface models. These models have finite non-invertible symmetries described by general fusion 2-categories. Our method can be applied to build microscopic models with, for example, anomalous or non-anomalous one-form symmetries, 2-group symmetries, or non-invertible one-form symmetries that capture non-abelian anyon statistics. The construction of these models generalizes the construction of the 1+1d anyon chains formalized by Aasen, Fendley, and Mong. Along with the fusion surface models, we also obtain the corresponding three-dimensional classical statistical models, which are 3d analogues of the 2d Aasen-Fendley-Mong height models. In the construction, the “symmetry TFTs” for fusion 2-category symmetries play an important role.

Strong-coupling mean-field and classical percolation approach to produce the phase diagram of disordered Bose–Hubbard model at fixed n = 1 filling

We perform strong-coupling mean-field along with classical percolation analysis to produce the phase diagram of disordered Bose–Hubbard model with chemical potential disorder in two dimensional square lattice. The phase diagram in the disorder strength (Δ) and the on-site repulsion (U), for two and three dimensional system at fixed n = 1 filling show interesting re-entrant superfluid phase sandwiched between Bose-glass phases, as observed by the previous QMC results. We produce the phase diagram exhibiting the re-entrant phenomenon as well as phenomenologically demonstrate the robustness of the re-entrant phenomena within this method. We also discuss the importance of the method and applicability in further analysis of the same phase diagram.