Higher-dimensional Lattices Research Articles

Density-Density Correlation Spectra of Ultracold Bosonic Gas Released from a Deep 1D Optical Lattice.

Density-density correlation analysis is a convenient diagnostic tool to reveal the hidden order in the strongly correlated phases of ultracold atoms. We report on a study of the density-density correlations of ultracold bosonic atoms which were initially prepared in a Mott insulator (MI) state in one-dimensional optical lattices. For the atomic gases released from the deep optical lattice, we extracted the normalized density-density correlation function from the atomic density distributions of freely expanded atomic clouds. Periodic bunching peaks were observed in the density-density correlation spectra, as in the case of higher-dimensional lattices. Treating the bosonic gas within each lattice well as a subcondensate without quantum tunneling, we simulated the post-expansion density distribution along the direction of the 1D lattice, and the calculated density-density correlation spectra agreed with our experimental observations.

Realization of higher-order topological lattices on a quantum computer

Programmable quantum simulators may one day outperform classical computers at certain tasks. But at present, the range of viable applications with noisy intermediate-scale quantum (NISQ) devices remains limited by gate errors and the number of high-quality qubits. Here, we develop an approach that places digital NISQ hardware as a versatile platform for simulating multi-dimensional condensed matter systems. Our method encodes a high-dimensional lattice in terms of many-body interactions on a reduced-dimension model, thereby taking full advantage of the exponentially large Hilbert space of the host quantum system. With circuit optimization and error mitigation techniques, we measured on IBM superconducting quantum processors the topological state dynamics and protected mid-gap spectra of higher-order topological lattices, in up to four dimensions, with high accuracy. Our projected resource requirements scale favorably with system size and lattice dimensionality compared to classical computation, suggesting a possible route to useful quantum advantage in the longer term.

Quench dynamics in lattices above one dimension: The free fermionic case

We begin a systematic investigation of quench dynamics in higher-dimensional lattice systems considering the case of noninteracting fermions with conserved particle number. We prepare the system in a translational-invariant nonequilibrium initial state, the simplest example being a classical configuration with fermions at fixed positions on the lattice, and let it evolve in time. We characterize the system's dynamics by measuring the entanglement between a finite connected region and its complement. We observe the transmutation of entanglement entropy into thermodynamic entropy and investigate how this process depends on the shape and orientation of the region with respect to the underlying lattice. Interestingly, we find that irregular regions display a distinctive multislope entanglement growth, while the dependence on the orientation angle is generically fairly weak. This is particularly true for regions with a large (discrete) rotational symmetry group. The main tool of our analysis is the celebrated quasiparticle picture of Calabrese and Cardy, which we generalize to describe the case at hand. Specifically, we show that for generic initial configurations (even when restricting to classical ones) one has to allow for the production of multiplets involving n>2 quasiparticles and carrying nondiagonal correlations. We obtain quantitatively accurate predictions, tested against exact numerics, and propose an efficient Monte Carlo based scheme to evaluate them for arbitrary connected regions of generic higher-dimensional lattices. Published by the American Physical Society 2024

Delocalized nonlinear vibrational modes and discrete breathers in a body centered cubic lattice

Body centered cubic (bcc) lattice with nearest and next-nearest interactions described by the β-FPUT interatomic potential is considered. Exact dynamical solutions in the form of zone-boundary delocalized nonlinear vibrational modes (DNVMs) are analyzed. 31 such solutions are revealed from the analysis of only the symmetry of the bcc lattice. Frequency response of DNVMs for the case of soft- and hard-type anharmonicity is calculated. In the case of hard-type anharmonicity, four DNVMs have frequencies bifurcating from the upper edge of the phonon spectrum and growing with the amplitude. Various quasi-discrete breathers are obtained by superimposing localizing functions upon these DNVMs. Our work demonstrates a practical approach to finding quasi-discrete breathers in higher-dimensional lattices. The obtained spatially localized long-lived vibrational modes inspire the search for various discrete breathers in bcc metals.

Programmable photonic system for quantum simulation in arbitrary topologies

Synthetic dimensions have generated great interest for studying many types of topological, quantum, and many-body physics, and they offer a flexible platform for simulation of interesting physical systems, especially in high dimensions. In this paper, we describe a programmable photonic device capable of emulating the dynamics of a broad class of Hamiltonians in lattices with arbitrary topologies and dimensions. We derive a correspondence between the physics of the device and the Hamiltonians of interest, and we simulate the physics of the device to observe a wide variety of physical phenomena, including chiral states in a Hall ladder, effective gauge potentials, and oscillations in high-dimensional lattices. Our proposed device opens new possibilities for studying topological and many-body physics in near-term experimental platforms.

An Improved Coppersmith Algorithm Based on Block Preprocessing

Since Coppersmith proposed the use of the LLL algorithm to solve univariate modular polynomial equations at EUROCRYPT’96, it has sparked a fervent research interest in lattice analysis among cryptographers. Despite its polynomial-time nature, the LLL algorithm exhibits a high-order polynomial upper bound in terms of theoretical complexity, particularly with longer computation times when applied to high-dimensional lattices. In addressing this issue, we propose an improved algorithm based on block preprocessing, building on the original Coppersmith algorithm and thus providing proof of correctness for this algorithm. This approach effectively reduces the solution time of the algorithm, offering a maximum improvement of 8.1% compared to the original Coppersmith algorithm. Additionally, we demonstrate the compatibility of our algorithm with the rounding algorithm proposed at PKC 2014. The combined utilization of these approaches further enhances the efficiency of our algorithm. The experimental results show that the combined algorithm achieves a maximum improvement of 22.4% in solution time compared to the original Coppersmith algorithm. It also outperforms the standalone rounding algorithm with a maximum improvement of 12.1%. When compared to the improved Coppersmith algorithm based on row common factor extraction, our proposed algorithm demonstrates comparable or even superior performance in certain dimensions. The block preprocessing algorithm in our approach enables independent execution without data exchange, making it suitable for leveraging multi-processing advantages in scenarios involving higher degrees of modular polynomial equations. This offers a new perspective for achieving the parallel computation of the Coppersmith algorithm, facilitating parallel execution and providing valuable insights.

Ordinal motifs in lattices

Lattices are a commonly used structure for the representation and analysis of relational and ontological knowledge. In particular, the analysis of these requires a decomposition of a large and high-dimensional lattice into a set of understandably large parts. With the present work we propose /ordinal motifs/ as analytical units of meaning. We study these ordinal substructures (or standard scales) through order-embeddings and (full) scale-measures of formal contexts from the field of formal concept analysis. We show that the underlying decision problems are NP-complete and provide results on how one can incrementally identify ordinal motifs to save computational effort. Accompanying our theoretical results, we demonstrate how ordinal motifs can be leveraged to achieve textual explanations based on principles from human computer interaction.

Quantitative Green’s function estimates for lattice quasi-periodic Schrödinger operators

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.

Upper Bound for the Coefficients of the Shortest Vector of Random Lattice

This paper shows that upper bounds on the coefficients of the shortest vector of a lattice can be represented using the smallest eigenvalue of the Gram matrix for the lattice, obtains its distribution for high-dimensional random Goldstein-Mayer lattice, and applies it to determine the percentage of zeros of coefficient vector.

Topological states switching and group velocity control in two-dimensional non-reciprocal Hermitian photonic lattice

We proposed a model with non reciprocal coupling coefficients, in which the imaginary parts γ indicate the phase delay or exceed term. The distributions of band structure and the group velocity are both characterized as a function of the coupling. we studied the system’s topological states and group velocity control. The results show that the movement and breaking of Dirac points exist in the energy band of the system. By changing the coupling coefficients, the conversion between any topological states corresponds to different Chern number. Topological edge states exist in topological non-trivial systems that correspond to the two different Chern numbers. Besides, it is also found that both the coupling coefficient and the wave vector can cause the oscillation of the pulse group velocity. At the same time, the topological state can suppress the amplitude of the group velocity profiles. Our findings enrich the theory of light wave manipulation in high-dimensional photonic lattices and provide a novel view for realizing linear localization and group velocity regulation of light waves, which has potential application in high-speed optical communication and quantum information fields.

Traveling wave solutions in a higher dimensional lattice delayed cooperation system with nonlocal diffusion

This paper is concerned with the existence of traveling wave solutions of a higher dimensional lattice delayed cooperation system with nonlocal diffusion. For sufficiently small intraspecific cooperative delays, we construct upper and lower solutions under two different parameters conditions. And then, by using the monotone iterative and Schauder's fixed point theorem, we obtain the existence of traveling wave solutions. The lower bound of the wave speed is in accordance with the properties of linear determined.

Programmable large-scale simulation of bosonic transport in optical synthetic frequency lattices

Photonic simulators using synthetic frequency dimensions have enabled flexible experimental analogues of condensed-matter systems. However, so far, such photonic simulators have been limited in scale, yielding results that suffer from finite-size effects. Here we present an analogue simulator capable of simulating large two-dimensional (2D) and 3D lattices, as well as lattices with non-planar connectivity. Our simulator takes advantage of the broad bandwidth achievable in photonics, allowing our experiment to realize programmable lattices with over 100,000 lattice sites. We showcase the scale of our simulator by demonstrating the extension of bandstructure spectroscopy from 1D to 2D and 3D lattices. We then report the direct observation of time-reversal symmetry-breaking in a triangular lattice in both momentum and real space, as well as site-resolved occupation measurements in a tree-like geometry that serves as a toy model in quantum gravity. Moreover, we demonstrate a method to excite arbitrary multisite states, which we use to study the response of a 2D lattice to both conventional and exotic input states. Our work highlights the scalability and flexibility of optical synthetic frequency dimensions. Future experiments building on our approach will be able to explore non-equilibrium phenomena in high-dimensional lattices and to simulate models with nonlocal higher-order interactions. Analogue photonic simulators have so far suffered from severe finite size effects and limited programmability. Now, a frequency-mode photonic simulator enables the simulation of large-scale models in two and three dimensions.

Rearrangement on lattices with pick-n-swaps: Optimality structures and efficient algorithms

We study a class of rearrangement problems under a novel pick-n-swap prehensile manipulation model, in which a robotic manipulator, capable of carrying an item and making item swaps, is tasked to sort items stored in lattices of variable dimensions in a time-optimal manner. We systematically analyze the intrinsic optimality structure, which is fairly rich and intriguing, under different levels of item distinguishability (fully-labeled, where each item has a unique label, or partially-labeled, where multiple items may be of the same type) and different lattice dimensions. Focusing on the most practical setting of one and two dimensions, we develop low polynomial time cycle-following-based algorithms that optimally perform rearrangements on 1D lattices under both fully- and partially-labeled settings. On the other hand, we show that rearrangement on 2D and higher-dimensional lattices become computationally intractable to optimally solve. Despite their NP-hardness, we prove that efficient cycle-following-based algorithms remain optimal in the asymptotic sense for 2D fully- and partially-labeled settings, in expectation, using the interesting fact that random permutations induce only a small number of cycles. We further improve these algorithms to provide 1.x-optimality when the number of items is small. Simulation studies corroborate the effectiveness of our algorithms. The implementation of the algorithms from the paper can be found at github.com/arc-l/lattice-rearrangement.

The pigeonhole principle and multicolor Ramsey numbers

A standard proof of Schur's Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, the minimum $N$ such that any $r$-coloring of a complete graph on $N$ vertices yields a monochromatic triangle. We explore generalizations and modifications of this result in higher dimensional integer lattices, showing in particular that if $k\geq d+1$, then any $r$-coloring of $\{1,2,\dots,R_r(k)^d-1\}^d$ yields a monochromatic solution to $x_1+\cdots+x_{k-1}=x_k$ with $\{x_1,\dots,x_d\}$ linearly independent, where $R_r(k)$ is the analogous Ramsey number in which triangles are replaced by complete graphs on $k$ vertices. We also obtain computational results and examples in the case $d=2$, $k=3$, and $r\in\{2,3,4\}$.

Local iterative block-diagonalization of gapped Hamiltonians: A new tool in singular perturbation theory

In this paper, the local iterative Lie–Schwinger block-diagonalization method, introduced and developed in our previous work for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation, consisting of bounded interaction potentials of short range multiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms, which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, which serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step.

Design of Beam-Steerable Aperiodic Linear Array Antenna With Improved Peak SLL Using Strip-Projection Method

This article discusses the design of the beam-steerable aperiodic linear array antenna with improved sidelobe level (SLL), using the strip-projection method. The strip projection method uses the area of rotated higher-dimensional lattice and projects it to lower dimensions to generate an aperiodic array. This article describes the modeling and analysis of such an array and its optimization for the desired far-field performance. The aperiodic arrays are also generated using various evolutionary optimization algorithms, namely genetic algorithm (GA), particle swarm optimization (PSO), and Jaya algorithms. The far-field performance of the generated aperiodic arrays is compared with the conventional periodic array. The proposed 21-element aperiodic array is also populated with a X-band electromagnetically coupled patch antenna and phase shifter designed using varying width of the high dielectric material. The aperiodic patch array antenna is also developed and characterized for various beam scan angles. A significant improvement of 5.72 dB in peak SLL is achieved at ±30° beam scan angle.

Unwrapped two-point functions on high-dimensional tori

We study unwrapped two-point functions for the Ising model, the self-avoiding walk (SAW) and a random-length loop-erased random walk on high-dimensional lattices with periodic boundary conditions. While the standard two-point functions of these models have been observed to display an anomalous plateau behaviour, the unwrapped two-point functions are shown to display standard mean-field behaviour. Moreover, we argue that the asymptotic behaviour of these unwrapped two-point functions on the torus can be understood in terms of the standard two-point function of a random-length random walk model on . A precise description is derived for the asymptotic behaviour of the latter. Finally, we consider a natural notion of the Ising walk length, and show numerically that the Ising and SAW walk lengths on high-dimensional tori show the same universal behaviour known for the SAW walk length on the complete graph.

Discrete integrable equations on face-centred cubics: consistency and Lax pairs of corner equations

A new set of discrete integrable equations, called face-centred quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang–Baxter equation. These equations satisfy a new formulation of multidimensional consistency, known as consistency-around-a-face-centred-cube (CAFCC), which requires consistency of an overdetermined system of 14 five-point equations on the face-centred cubic unit cell. In this paper, a new formulation of CAFCC is introduced where so-called type C equations are centred at faces of the face-centred cubic unit cell, whereas previously they were only centred at corners. This allows type C equations to be regarded as independent multidimensionally consistent integrable systems on higher dimensional lattices and is used to establish their Lax pairs.

An introduction to multiscale techniques in the theory of Anderson localization, Part I

These lectures present some basic ideas and techniques in the spectral analysis of lattice Schrödinger operators with disordered potentials. In contrast to the classical Anderson tight binding model, the randomness is also allowed to possess only finitely many degrees of freedom. This refers to dynamically defined potentials, i.e., those given by evaluating a function along an orbit of some ergodic transformation (or of several commuting such transformations on higher-dimensional lattices). Classical localization theorems by Fröhlich–Spencer for large disorders are presented, both for random potentials in all dimensions, as well as even quasi-periodic ones on the line. After providing the needed background on subharmonic functions, we then discuss the Bourgain–Goldstein theorem on localization for quasiperiodic Schrödinger cocycles assuming positive Lyapunov exponents.

Quasiperiodic metamaterials with broadband absorption: Tailoring electromagnetic wave by Penrose tiling

Quasicrystals can enhance wave transmission and boost nonlinear phenomena due to rotational symmetry forbidden in conventional periodic crystals. As a projection of higher-dimensional quasicrystal lattice, quasiperiodic structures represent a novel platform that exhibits robust microwave-absorbing capacity in thin thickness and digs out the influence of structural configurations on electromagnetic wave absorption from local and global aspects. Due to the synergy effect of composites and structure, quasiperiodic metamaterials with Penrose tiling achieve 12.2 GHz in effective absorption bandwidth (reflection loss < −10 dB), from 5.78 GHz to 18.00 GHz with 2 mm thickness. It has obvious advantages in absorbing electromagnetic wave compared with corresponding periodic absorbers according to the characteristic of local anisotropy and higher order translational symmetry in Brillouin zone . By enhancing multiple scattering and nonlinear coupling effects, Penrose tiling provides new insights to design absorbers with outstanding electromagnetic performance and higher flexibility and tunability.