Low-dimensional Lattices Research Articles

Giophantus distinguishing attack is a low dimensional learning with errors problem

In this paper, we attack the recent NIST submission Giophantus, a public key encryption scheme. We find that the complicated structure of Giophantus's ciphertexts leaks information via a correspondence from a low dimensional lattice. This allows us to distinguish encrypted data from random data by the LLL algorithm. This is a more efficient attack than previous proposed attacks.

Heat current fluctuations and anomalous transport in low-dimensional carbon lattices

Molecular dynamics simulations and nonequilibrium importance sampling are used to study the heat transport of low dimensional carbon lattices. For both carbon nanotubes and graphene sheets heat transport is found to be anomalous, violating Fourier's law of conduction with a system size dependent thermal conductivity and concomitant nonlinear temperature profiles. For carbon nanotubes, the thermal conductivity is found to increase as the square root of the length of the nanotube, while for graphene sheets the thermal conductivity is found to increase as the logarithm of the length of the sheet. The particular length dependence and nonlinear temperature profiles place carbon lattices into a universality class with nonlinear lattice models, and suggest that heat transport through carbon nano-structures is better described by a Levy walk rather than simple diffusion.

On Fundamental Limitations of Dynamic Feedback Control in Regular Large-Scale Networks

In this paper, we study fundamental performance limitations of distributed feedback control in large-scale networked dynamical systems. Specifically, we address the question of whether dynamic feedback controllers perform better than static (memoryless) ones when subject to locality constraints. We consider distributed linear consensus and vehicular formation control problems modeled over toric lattice networks. For the resulting spatially invariant systems, we study the large-scale asymptotics (in network size) of global performance metrics that quantify the level of network coherence. With static feedback from relative state measurements, such metrics are known to scale unfavorably in lattices of low spatial dimensions, preventing, for example, a one-dimensional string of vehicles to move like a rigid object. We show that the same limitations in general apply also to dynamic feedback control that is locally of first order. This means that the addition of one local state to the controller gives a similar asymptotic performance to the memoryless case. This holds unless the controller can access noiseless measurements of its local state with respect to an absolute reference frame, in which case the addition of controller memory may fundamentally improve performance. In simulations of platoons with 20–200 vehicles, we show that the performance limitations we derive manifest as unwanted accordionlike motions. Similar behaviors are to be expected in any network that is embeddable in a low-dimensional toric lattice, and the same fundamental limitations would apply. To derive our results, we present a general technical framework for the analysis of stability and performance of spatially invariant systems in the limit of large networks.

Giophantus distinguishing attack is a low dimensional learning with errors problem

In this paper, we attack the recent NIST submission Giophantus, a public key encryption scheme. We find that the complicated structure of Giophantus's ciphertexts leaks information via a correspondence from a low dimensional lattice. This allows us to distinguish encrypted data from random data by the LLL algorithm. This is a more efficient attack than previous proposed attacks.

Low Dimensional Magnetic Lattice and Room Temperature Magneto(di)electric Effect in Polyanion Ruddlesden-Popper Iron Oxides.

We have shown a new design strategy which exploits different oxyanions in a Ruddlesden-Popper (RP)-type phase to modulate the local crystal structure and magnetic lattice. Material (Sr4Fe2(SO4)0.5O6.5) with the larger voluminous oxyanion (SO4, S-O distance = 1.49 Å) as separating blocks between magnetic FeO layers shows a two-dimensional magnetic lattice. A three-dimensional magnetic lattice and spin reorientation transition is observed for the Sr4Fe2(CO3)O6, having CO3 (C-O distance = 1.25 Å), a smaller oxyanion, as a separating layer. Using mixed oxyanions (SO4 and CO3) in the central perovskite block of the RP3 phase, we have demonstrated a facile strategy to modulate the local crystal structure. The modulated displacement of the magnetic cations, which can break the local centrosymmetry, is suggested to originate the magnetodielectric effect near the magnetic ordering temperatures (higher than room temperature). Further, all CO3 containing samples show magnetodielectric coupling below room temperature due to the spin reorientation transition. The room temperature magnetodielectric effect coupled to the targeted local modulation of the crystal structure by oxyanions (in the absence of second-order Jahn-Teller active "distortion centers") opens a new door to the design of new multifunctional materials with the possibility for the room temperature application.

Tetrahedral chain ordering and low dimensional magnetic lattice in a new brownmillerite Sr2ScFeO5.

We report the synthesis, structure and physical properties of a hitherto unreported brownmillerite compound Sr2ScFeO5. We have shown a new ordering sequence of the interlayer iron tetrahedral chains. Reduced dimensionality of the magnetic lattice and the frustration in the two dimensional iron tetrahedral chains originate complex magnetic and magneto-dielectric effects. Our study highlights a novel approach to tailor the magnetic lattice in bulk oxides.

Twisted states in low-dimensional hypercubic lattices

Twisted states with non-zero winding numbers composed of sinusoidally coupled identical oscillators have been observed in a ring. The phase of each oscillator in these states constantly shifts, following its preceding neighbor in a clockwise direction, and the summation of such phase shifts around the ring over $2\pi$ characterizes the winding number of each state. In this work, we consider finite-sized $d$-dimensional hypercubic lattices, namely square ($d=2$) and cubic ($d=3$) lattices with periodic boundary conditions. For identical oscillators, we observe new states in which the oscillators belonging to each line (plane) for $d=2$ ($d=3$) are phase synchronized with non-zero winding numbers along the perpendicular direction. These states can be reduced into twisted states in a ring with the same winding number if we regard each subset of phase-synchronized oscillators as one single oscillator. For nonidentical oscillators with heterogeneous natural frequencies, we observe similar patterns with slightly heterogeneous phases in each line $(d=2)$ and plane $(d=3)$. We show that these states generally appear for random configurations when the global coupling strength is larger than the critical values for the states.

Effects of Interaction Range on the Behavior of Opinion Consensus

We have frequently encountered the rapid changes that prevalent opinion of the social community is toppled by a new and opposite opinion against the pre-exiting one. To understand this interesting process, mean-field model with infinite-interaction range has been mostly considered in previous studies S. A. Marvel et al., Phys. Rev. Lett. 110, 118702(2012). However, the mean-field interaction range is lack of reality in the sense that any individual cannot interact with all of the others in the community. Based on it, in the present work, we consider a simple model of opinion consensus so-called basic model on the low-dimensional lattices ($d$=1,2) with finite interaction range. The model consists of four types of subpopulations with different opinions: $A, B, AB$, and the zealot of $A$ denoted by $A_c$, following the basic model shown in the work by S. A. Marvel et al.. Comparing with their work, we consider the finite range of the interaction, and particularly reconstruct the lattice structure by adding new links when the two individuals have the distance $<\sigma$. We explore how the interaction range $\sigma$ affects the opinion consensus process on the reconstructed lattice structure. We find that the critical fraction of population for $A_c$ required for the opinion consensus on $A$ shows different behaviors in the small and large interaction ranges. Especially, the critical fraction for $A_c$ increases with the size of $\sigma$ in the region of small interaction range, which is counter-intuitive: When the interaction range is increased, not only the number of nodes affected by $A_c$ but also that affected by $B$ grows, which is believed to cause the increasing behavior of the critical fraction for $A_c$. We also present the difference of dynamic process to the opinion consensus between the regions of small and large interaction ranges.

Aging-induced continuous phase transition

Aging is considered as the property of the elements of a system to be less prone to change states as they get older. We incorporate aging into the noisy voter model, a stochastic model in which the agents modify their binary state by means of noise and pair-wise interactions. Interestingly, due to aging the system passes from a finite-size discontinuous transition between ordered (ferromagnetic) and disordered (paramagnetic) phases to a second order phase transition, well defined in the thermodynamic limit, belonging to the Ising universality class. We characterize it analytically by finding the stationary solution of an infinite set of mean field equations. The theoretical predictions are tested with extensive numerical simulations in low dimensional lattices and complex networks. We finally employ the aging properties to understand the symmetries broken in the phase transition.

Information Graph Flow: A Geometric Approximation of Quantum and Statistical Systems

Given a quantum (or statistical) system with a very large number of degrees of freedom and a preferred tensor product factorization of the Hilbert space (or of a space of distributions) we describe how it can be approximated with a very low-dimensional field theory with geometric degrees of freedom. The geometric approximation procedure consists of three steps. The first step is to construct weighted graphs (we call information graphs) with vertices representing subsystems (e.g. qubits or random variables) and edges representing mutual information (or the flow of information) between subsystems. The second step is to deform the adjacency matrices of the information graphs to that of a (locally) low-dimensional lattice using the graph flow equations introduced in the paper. (Note that the graph flow produces very sparse adjacency matrices and thus might also be used, for example, in machine learning or network science where the task of graph sparsification is of a central importance.) The third step is to define an emergent metric and to derive an effective description of the metric and possibly other degrees of freedom. To illustrate the procedure we analyze (numerically and analytically) two information graph flows with geometric attractors (towards locally one- and two-dimensional lattices) and metric perturbations obeying a geometric flow equation. Our analysis also suggests a possible approach to (a non-perturbative) quantum gravity in which the geometry (a secondary object) emerges directly from a quantum state (a primary object) due to the flow of the information graphs.

Quench dynamics of a dissipative Rydberg gas in the classical and quantum regimes

Understanding the non-equilibrium behavior of quantum systems is a major goal of contemporary physics. Much research is currently focused on the dynamics of many-body systems in low-dimensional lattices following a quench, i.e., a sudden change of parameters. Already such a simple setting poses substantial theoretical challenges for the investigation of the real-time post-quench quantum dynamics. In classical many-body systems the Kolmogorov-Mehl-Johnson-Avrami model describes the phase transformation kinetics of a system that is quenched across a first order phase transition. Here we show that a similar approach can be applied for shedding light on the quench dynamics of an interacting gas of Rydberg atoms, which has become an important experimental platform for the investigation of quantum non-equilibrium effects. We are able to gain an analytic understanding of the time-evolution following a sudden quench from an initial state devoid of Rydberg atoms and identify strikingly different behaviors of the excitation growth in the classical and quantum regimes. Our approach allows us to describe quenches near a non-equilibrium phase transition and provides an approximate analytic solution deep in the quantum domain.

Probabilistic low-rank factorization accelerates tensor network simulations of critical quantum many-body ground states.

We provide evidence that randomized low-rank factorization is a powerful tool for the determination of the ground-state properties of low-dimensional lattice Hamiltonians through tensor network techniques. In particular, we show that randomized matrix factorization outperforms truncated singular value decomposition based on state-of-the-art deterministic routines in time-evolving block decimation (TEBD)- and density matrix renormalization group (DMRG)-style simulations, even when the system under study gets close to a phase transition: We report linear speedups in the bond or local dimension of up to 24 times in quasi-two-dimensional cylindrical systems.

Near-Integrability of Low-Dimensional Periodic Klein-Gordon Lattices

The low-dimensional periodic Klein-Gordon lattices are studied for integrability. We prove that the periodic lattice with two particles and certain nonlinear potential is nonintegrable. However, in the cases of up to six particles, we prove that their Birkhoff-Gustavson normal forms are integrable, which allows us to apply KAM theory in most cases.

Cr3 Triangles induced competing magnetic interactions in the new metal boride TiCrIr2B2: An NMR and DFT study

The presence of one-dimensional trigonal arrangements of chromium atoms (Cr3) in the structure of TiCrIr2B2 leads to a magnetic transition near ambient temperature. Herein we report an investigation of the nature of electronic and magnetic properties of TiCrIr2B2 via ab initio calculations together with 11B NMR Knight shift (K) and spin-lattice relaxation rate (1/T1) analysis. The presence of a characteristic rectangular powder pattern below 280K, absence of a Korringa relation, strong enhancement of 1/T1.T at low temperatures and weak temperature dependence of K indicate competition between antiferromagnetic and ferromagnetic spin fluctuations in the itinerant d-band electrons, in agreement with ab initio calculations. One-dimensional trigonal arrangements of magnetically active elements in intermetallic compounds, as is found in TiCrIr2B2, are quite rare and could lead to exotic phenomena such as spin-chirality and quantum criticality in low-dimensional frustrated lattices.

The complexity of translationally invariant low-dimensional spin lattices in 3D

In this theoretical paper, we consider spin systems in three spatial dimensions and consider the computational complexity of estimating the ground state energy, known as the local Hamiltonian problem, for translationally invariant Hamiltonians. We prove that the local Hamiltonian problem for 3D lattices with face-centered cubic unit cells and 4-local translationally invariant interactions between spin-3/2 particles and open boundary conditions is QMAEXP-complete, where QMAEXP is the class of problems which can be verified in exponential time on a quantum computer. We go beyond a mere embedding of past hard 1D history state constructions, for which the local spin dimension is enormous: even state-of-the-art constructions have local dimension 42. We avoid such a large local dimension by combining some different techniques in a novel way. For the verifier circuit which we embed into the ground space of the local Hamiltonian, we utilize a recently developed computational model, called a quantum ring machine, which is especially well suited for translationally invariant history state constructions. This is encoded with a new and particularly simple universal gate set, which consists of a single 2-qubit gate applied only to nearest-neighbour qubits. The Hamiltonian construction involves a classical Wang tiling problem as a binary counter which translates one cube side length into a binary description for the encoded verifier input and a carefully engineered history state construction that implements the ring machine on the cubic lattice faces. These novel techniques allow us to significantly lower the local spin dimension, surpassing the best translationally invariant result to date by two orders of magnitude (in the number of degrees of freedom per coupling). This brings our models on par with the best non-translationally invariant construction.

Compact flat band states in optically induced flatland photonic lattices

We realize low-dimensional tight-binding lattices that host flat bands in their dispersion relation and demonstrate the existence of optical compact flat band states. The lattices are resembled by arrays of optical waveguides fabricated by the state-of-the-art spatio-temporal Bessel beam multiplexing optical induction in photorefractive media. We work out the decisive details of the transition from the discrete theory to the real optical system ensuring that the experimental lattices stand up to numerical scrutiny exhibiting well-approximated band structures. Our highly flexible system is a promising candidate for further experimental investigation of theoretically studied disorder effects in flat band lattices.

Preclinical safety and efficacy evaluation of ‘BioCaS’ bioactive calcium sulfate bone cement

A new bioactive calcium sulfate-based formulation (named ‘BioCaS’) has been developed for bone filler applications. This is a self-setting injectable cement where the preset form comprises bassanite obtained from the uniform submicron-sized precursor crystals of gypsum, modified with hydrogen orthophosphate ions. The results of the safety and efficacy evaluation of BioCaS cement, done as per the International Standards and guidelines, are presented in this paper. The study plan consisted of in vitro screening tests of cytotoxicity and haemolysis and in vivo biocompatibility evaluation, including an acute systemic toxicity test (in mice), an intracutaneous reactivity test (in rabbits), a pyrogen test (in rabbits) and a maximization sensitization test (in guinea pigs). The efficacy of the material in healing bone defects was investigated by implanting it in artificially created defects in rabbit femora, with clinically established hydroxyapatite porous ceramic as the control, followed by histological analysis at 12, 26 and 52 weeks. Set BioCaS cement consisted of hydrogen orthophosphate incorporating low-dimensional gypsum crystal lattices, the bioactivity of which has been identified by immersion in simulated body fluid. BioCaS was proved to be non-cytotoxic and non-haemolytic in the screening tests. In the live/dead assay, human osteoblast-like human osteosarcoma cells adhered well and spread on the surface of the material, attaining typical morphology and affirming the bone cell compatibility of the material. In the biocompatibility evaluation there were no acute systemic effects and the material proved non-pyrogenic. There was no intracutaneous erythemic or oedematous reactivity and no hypersensitivity observed in the Magnusson–Kligman method. The material satisfied the biocompatibility requirements. Bone implantation study revealed BioCaS to be osteoconductive and its efficacy of healing the experimental bone defects in rabbit femora is on a par with that of hydroxyapatite ceramic. The material resorbed at a pace matching that of new bone formation. This property of osteotransductivity will help the defect to heal and gain strength faster.

Low-Dimensional Shaping for High-Dimensional Lattice Codes

We propose two low-complexity lattice code constructions that have competitive coding and shaping gains. The first construction, named systematic Voronoi shaping , maps short blocks of integers to the dithered Voronoi integers, which are dithered integers that are uniformly distributed over the Voronoi region of a low-dimensional shaping lattice. Then, these dithered Voronoi integers are encoded using a high-dimensional lattice retaining the same shaping and coding gains of low- and high-dimensional lattices. A drawback to this construction is that there is no isomorphism between the underlying message and the lattice code, preventing its use in applications such as compute-and-forward. Therefore, we propose a second construction, called mixed nested lattice codes , in which a high-dimensional coding lattice is nested inside a concatenation of low-dimensional shaping lattices. This construction not only retains the same shaping/coding gains as first construction but also provides the desired algebraic structure. We numerically study these methods, for point-to-point channels as well as compute-and-forward using low-density lattice codes as coding lattices and $E_{8}$ and Barnes–Wall as shaping lattices. Numerical results indicate a shaping gain of up to 0.86 dB, compared with the state-of-the-art of 0.4 dB; furthermore, the proposed method has lower complexity than the state-of-the-art approaches.

Interaction-energy functional of the Hubbard model: Local formulation and application to low-dimensional lattices

The interaction energy $W[\ensuremath{\gamma}]$ of the Hubbard model is regarded as a functional of the single-particle density matrix $\ensuremath{\gamma}$ in the framework of lattice density-functional theory. The local character of the Hubbard interaction is exploited to express $W$ as a sum of local contributions ${\ensuremath{\omega}}_{i}[\ensuremath{\gamma}]$, for which a simple semilocal scaling approximation is proposed. The method is applied to the ionic Hubbard model on one- and two-dimensional lattices with homogeneous and inhomogeneous Coulomb repulsions. Results are given for the kinetic and Coulomb energies, interatomic charge transfers, local magnetic moments, and charge gaps. Goals and limitations of the functional are discussed by comparison with exact results.

Efimov-Like Behaviour in Low-Dimensional Polymer Models

In the quantum Efimov effect, identical bosons form infinitely many bound trimer states at the bound dimer dissociation threshold, with their energy spectrum obeying a universal geometrical scaling law. Inspired by the formal correspondence between the possible trajectories of a quantum particle and the possible conformations of a polymer chain, the existence of a triple-stranded DNA bound state when a double-stranded DNA is not stable was recently predicted by modelling three directed polymer chains in low-dimensional lattices, both fractal ( $$d<1$$ ) and euclidean ( $$d=1$$ ). A finite melting temperature for double-stranded DNA requires in $$d\le 2$$ the introduction of a weighting factor penalizing the formation of denaturation bubbles, that is non-base paired portions of DNA. The details of how bubble weighting is defined for a three-chain system were shown to crucially affect the presence of Efimov-like behaviour on a fractal lattice. Here we assess the same dependence on the euclidean $$1+1$$ lattice, by setting up the transfer matrix method for three infinitely long chains confined in a finite size geometry. This allows us to discriminate unambiguously between the absence of Efimov-like behaviour and its presence in a very narrow temperature range, in close correspondence with what was already found on the fractal lattice. When present, however, no evidence is found for triple-stranded bound states other than the ground state at the two-chain melting temperature.