Large Spectral Gap Research Articles

Projective integration methods in the Runge–Kutta framework and the extension to adaptivity in time

Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge–Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge–Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge–Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge–Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.

Random sparse generators of Markovian evolution and their spectral properties.

The evolution of a complex multistate system is often interpreted as a continuous-time Markovian process. To model the relaxation dynamics of such systems, we introduce an ensemble of random sparse matrices which can be used as generators of Markovian evolution. The sparsity is controlled by a parameter φ, which is the number of nonzero elements per row and column in the generator matrix. Thus, a member of the ensemble is characterized by the Laplacian of a directed regular graph with D vertices (number of system states) and 2φD edges with randomly distributed weights. We study the effects of sparsity on the spectrum of the generator. Sparsity is shown to close the large spectral gap that is characteristic of nonsparse random generators. We show that the first moment of the eigenvalue distribution scales as ∼φ, while its variance is ∼sqrt[φ]. By using extreme value theory, we demonstrate how the shape of the spectral edges is determined by the tails of the corresponding weight distributions and clarify the behavior of the spectral gap as a function of D. Finally, we analyze complex spacing ratio statistics of ultrasparse generators, φ=const, and find that starting already at φ⩾2, spectra of the generators exhibit universal properties typical of Ginibre's orthogonal ensemble.

Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes.

Relation between spectra of Narain CFTs and properties of associated boolean functions

Recently, the construction of Narain CFT from a certain class of quantum error correcting codes has been discovered. In particular, the spectral gap of Narain CFT corresponds to the binary distance of the code, not the genuine Hamming distance. In this paper, we show that the binary distance is identical to the so-called EPC distance of the boolean function uniquely associated with the quantum code. Therefore, seeking Narain CFT with large spectral gap can be addressed by getting a boolean function with high EPC distance. Furthermore, this problem can be undertaken by finding lower Peak-to-Average Power ratio (PAR) with respect to the binary truth table of the boolean function. Though this is neither sufficient nor necessary condition for high EPC distance, we construct some examples of relatively high EPC distances referring to the constructions for lower PAR. We also see that codes with high distance are related to induced graphs with low independence numbers.

Accumulation time of diffusion in a two-dimensional singularly perturbed domain

A general problem of current interest is the analysis of diffusion problems in singularly perturbed domains, within which small subdomains are removed from the domain interior and boundary conditions imposed on the resulting holes. One major application is to intracellular diffusion, where the holes could represent organelles or biochemical substrates. In this paper, we use a combination of matched asymptotic analysis and Green’s function methods to calculate the so-called accumulation time for relaxation to steady state. The standard measure of the relaxation rate is in terms of the principal non-zero eigenvalue of the negative Laplacian. However, this global measure does not account for possible differences in the relaxation rate at different spatial locations, is independent of the initial conditions, and relies on the assumption that the eigenvalues have sufficiently large spectral gaps. As previously established for diffusion-based morphogen gradient formation, the accumulation time provides a better measure of the relaxation process.

Subdiffusive wave transport and weak localization transition in three-dimensional stealthy hyperuniform disordered systems

The purpose of this work is to understand the fundamental connection between structural correlations and light localization in three-dimensional (3D) open scattering systems of finite size. We numerically investigate the transport of vector electromagnetic waves scattered by resonant electric dipoles spatially arranged in 3D space by stealthy hyperuniform disordered point patterns. Three-dimensional stealthy hyperuniform disordered systems are engineered with different structural correlation properties determined by their degree of stealthiness $\ensuremath{\chi}$. Such fine control of exotic states of amorphous matter enables the systematic design of optical media that interpolate in a tunable fashion between uncorrelated random structures and crystalline materials. By solving the electromagnetic multiple scattering problem using Green's matrix spectral method, we establish a transport phase diagram that demonstrates a distinctive transition from a diffusive to a weak localization regime beyond a critical scattering density that depends on $\ensuremath{\chi}$. The transition is characterized by studying the Thouless number and the spectral statistics of the scattering resonances. In particular, by tuning the $\ensuremath{\chi}$ parameter, we demonstrate large spectral gaps and suppressed subradiant proximity resonances, facilitating light localization. Moreover, consistently with previous studies, our results show a region of the transport phase diagram where the investigated scattering systems become transparent. Our work provides a systematic description of the transport and weak localization properties of light in stealthy hyperuniform structures and motivates the engineering of photonic systems with enhanced light-matter interactions for applications to both classical and quantum devices.

Factors and loose Hamilton cycles in sparse pseudo‐random hypergraphs

AbstractWe investigate the emergence of subgraphs in sparse pseudo‐random k‐uniform hypergraphs, using the following comparatively weak notion of pseudo‐randomness. A k‐uniform hypergraph H on n vertices is called ‐pseudo‐random if for all (not necessarily disjoint) vertex subsets with we have urn:x-wiley:rsa:media:rsa21052:rsa21052-math-0004For any linear k‐uniform F, we provide a bound on in terms of and F, such that (under natural divisibility assumptions on n) any k‐uniform ‐pseudo‐random n‐vertex hypergraph H with a mild minimum vertex degree condition contains an F‐factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo‐random hypergraphs and, along the way, we also derive conditions which guarantee the appearance of any fixed sized subgraph. All results imply corresponding bounds for stronger notions of hypergraph pseudo‐randomness such as jumbledness or large spectral gap. As a consequence, ‐pseudo‐random k‐graphs as above contain: (i) a perfect matching if and (ii) a loose Hamilton cycle if . This extends the works of Lenz–Mubayi, and Lenz–Mubayi–Mycroft who studied the analogous problems in the dense setting.

Correlation Length in Random MPS and PEPS

Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of this work is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we are particularly interested in is that of having either rapidly decaying or long-range correlations. In a nutshell, our main result is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. We have two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap. In all these statements, our aim is to get a quantitative estimate of the considered quantity (generic correlation length or spectral gap), which has the best possible dependency on the physical and bond dimensions of the random MPS or PEPS.

Effects of network topology on the performance of consensus and distributed learning of SVMs using ADMM.

The Alternating Direction Method of Multipliers (ADMM) is a popular and promising distributed framework for solving large-scale machine learning problems. We consider decentralized consensus-based ADMM in which nodes may only communicate with one-hop neighbors. This may cause slow convergence. We investigate the impact of network topology on the performance of an ADMM-based learning of Support Vector Machine using expander, and mean-degree graphs, and additionally some of the common modern network topologies. In particular, we investigate to which degree the expansion property of the network influences the convergence in terms of iterations, training and communication time. We furthermore suggest which topology is preferable. Additionally, we provide an implementation that makes these theoretical advances easily available. The results show that the performance of decentralized ADMM-based learning of SVMs in terms of convergence is improved using graphs with large spectral gaps, higher and homogeneous degrees.

Learning Graph Similarity With Large Spectral Gap

Learning a good graph similarity matrix in data clustering is very crucial. The goal of clustering is to construct a good graph similarity matrix such that the similarity of points between the same classes is largest, and the similarity of points between different classes is smallest. In this paper, a more efficient subspace segmentation approach to learn a similarity matrix with large spectral gap is proposed. In our model, a robust self-representation coefficient matrix is learned by utilizing the Schatten- ${p}$ norm instead of the conventional rank function. Besides, the fast block-diagonal structure of the coefficient representation matrix is enhanced by learning and optimizing the co-association matrix with the soft label of clustering results simultaneously in a unified framework. The affinity graphs constructed in this paper can clearly reveal the intrinsic structures of the data sets. Extensive experiments on the real data sets demonstrate that our proposed method can perform better than the state-of-the-art methods.

Large spectral gap and impurity-induced states in a two-dimensional Abrikosov vortex

We study the subgap spectrum of a 2D Abrikosov vortex in an s-wave superconductor in the absence and presence of a point impurity. By solving the Eilenberger equations without impurity for two models of the vortex (including a self-consistent one), we find multiple subgap spectral branches. The number of these branches may be arbitrary large provided that the magnetic field screening length is large enough. The quasiclassical spectrum of the vortex has a local gap with a width of the order of the bulk gap and a spatial extent of several coherence lengths. The existence of such gap is the prerequisite for the appearance of discrete impurity-induced states. Within the Gor'kov equations formalism, we find that a single impurity induces up to four discrete quasiparticle states in the vortex. The energies and wavefunctions of the impurity states are calculated for different parameters. We claim that most of the predicted spectral features can be observed in scanning tunnel spectroscopy experiments.

Proposal for noise-free visible-telecom quantum frequency conversion through third-order sum and difference frequency generation.

Quantum frequency conversion (QFC) between the visible and telecom is a key to connect quantum memories in fiber-based quantum networks. Current methods for linking such widely separated frequencies, such as sum/difference frequency generation and four-wave mixing Bragg scattering, are prone to broadband noise generated by the pump laser(s). To address this issue, we propose to use third-order sum/difference frequency generation (TSFG/TDFG) for an upconversion/downconversion QFC interface. In this process, two long wavelength pump photons combine their energy and momentum to mediate frequency conversion across the large spectral gap between the visible and telecom bands, which is particularly beneficial from the noise perspective. We show that waveguide-coupled silicon nitride microring resonators can be designed for efficient QFC between 606 and 1550 nm via a 1990 nm pump through TSFG/TDFG. We simulate the device dispersion and coupling, and from the simulated parameters, estimate that the frequency conversion can be efficient (${\gt}80 \%$) at 50 mW pump power. Our results suggest that microresonator TSFG/TDFG is promising for compact, scalable, and low-power QFC across large spectral gaps.

Projective integration schemes for hyperbolic moment equations

In this paper, we apply projective integration methods to hyperbolic moment models of the Boltzmann equation and the BGK equation, and investigate the numerical properties of the resulting scheme. Projective integration is an explicit scheme that is tailored to problems with large spectral gaps between slow and (one or many) fast eigenvalue clusters of the model. The spectral analysis of a linearized moment model clearly shows spectral gaps and reveals the multi-scale nature of the model for which projective integration is a matching choice. The combination of the non-intrusive projective integration method with moment models allows for accurate, but efficient simulations with significant speedup, as demonstrated using several 1D and 2D test cases with different collision terms, collision frequencies and relaxation times.

High Order Random Walks: Beyond Spectral Gap

We study high order random walks in high dimensional expanders; namely, in complexes which are local spectral expanders. Recent works have studied the spectrum of high order walks and deduced fast mixing. However, the spectral gap of high order walks is inherently small, due to natural obstructions (called coboundaries) that do not happen for walks on expander graphs. In this work we go beyond spectral gap, and relate the shrinkage of a k-cochain by the walk operator, to its structure under the assumption of local spectral expansion. A simplicial complex is called a one-sided local spectral expander, if its links have large spectral gaps and a two-sided local spectral expander if its links have large two-sided spectral gaps. We show two Decomposition Theorems (one per one-sided/two-sided local spectral assumption): For every k-cochain ϕ defined on an n-dimensional local spectral expander, there exists a decomposition of ϕ into “orthogonal” parts that are, roughly speaking, the “projections” on the j-dimensional cochains for 0 ≤ j ≤ k. The random walk shrinks each of these parts by a factor of $$\frac{k+1-j}{k+2}$$ plus an error term that depends on the spectral expansion. When assuming one-sided local spectral gap, our Decomposition Theorem yields an optimal mixing for the high order random walk operator. Namely, negative eigenvalues of the links do not matter! This improves over [5] that assumed two-sided spectral gap in the links to get optimal mixing. This improvement is crucial in a recent breakthrough [1] proving a conjecture of Mihail and Vazirani. Additionally, we get an optimal mixing for high order random walks on Ramanujan complexes (whose links are one-sided local spectral expanders). When assuming two-sided local spectral gap, our Decomposition Theorem allows us to describe the whole spectrum of the random walk operator (up to an error term that is determined by the spectral gap) and give an explicit orthogonal decomposition of the spaces of cochains that approximates the decomposition to eigenspaces of the random walk operator.

Developing a ratiometric two-photon probe with baseline resolved emissions by through band energy transfer strategy: Tracking mitochondrial SO2 during neuroinflammation

Sulfur dioxide (SO2) with the largest quantity and widest distribution in the atmosphere is closely related to many nervous system diseases via mitochondria respiration. It is of great significance to monitor this gaseous molecule during various physiological and pathological processes, but currently the task still remains challenging due to the lack of reliable tools. Through-bond energy transfer (TBET) is a relatively new strategy to fabricate ratiometric fluorescent probes, which does not need spectral overlap between the energy donor and acceptor while provides high energy-transfer efficiency. It offers strong dual fluorescence emission peaks as well as large wavelength differences between the two peaks, which increases the bioimaging resolution and reliability. Herein, we developed a TBET-based ratiometric probe (TBET-SO2) with a series of superior properties for in vivo SO2 imaging. Excited by near-infrared pulsed laser (810 nm), the probe undergoes TBET and produces far-red emission (611 nm). It achieved significant energy-transfer efficiency (90.5%) and large spectral gap between two peaks (△λ = 118 nm). Upon reacting with SO2, TBET-SO2 showed ~30-fold enhancement of ratiometric signal contributed by the baseline resolved emissions. A detection limit of as low as 0.09 μM was obtained. Furthermore, TBET-SO2 was successfully applied for visualizing the mitochondrial SO2 in living cells and mice brain tissue during the neuroinflammation process induced by SO2 pollution.

Large spectral gap induced by small delay and its application to reduction

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

Mixing time and eigenvalues of the abelian sandpile Markov chain

The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph G G . By viewing this chain as a (nonreversible) random walk on an abelian group, we give a formula for its eigenvalues and eigenvectors in terms of “multiplicative harmonic functions” on the vertices of G G . We show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest noninteger vector in the dual Laplacian lattice, while the mixing time is at most a constant times the smoothing parameter of the Laplacian lattice. We find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on G G : If the latter has a sufficiently large spectral gap, then the former has a small gap! In the case where G G is the complete graph on n n vertices, we show that the sandpile chain exhibits cutoff at time 1 4 π 2 n 3 log ⁡ n \frac {1}{4\pi ^{2}}n^{3}\log n .

Expansion of coset graphs of PSL2(Fp)

Let $G$ be a finite group and let $H_1,H_2<G$ be two subgroups. In this paper, we are concerned with the bipartite graph whose vertices are $G/H_1\cup G/H_2$ and a coset $g_1H_1$ is connected with another coset $g_2H_2$ if and only if $g_1H_1\cap g_2 H_2\neq\varnothing$. The main result of the paper establishes the existence of such graphs with large girth and large spectral gap. Lubotzky, Manning and Wilton use such graphs to construct certain infinite groups of interest in geometric group theory.

Efficient telecom-to-visible spectral translation through ultra-low power nonlinear nanophotonics.

The ability to spectrally translate lightwave signals in a compact, low-power platform is at the heart of the promise of nonlinear nanophotonic technologies. For example, a device to link the telecommunications band with visible and short near-infrared wavelengths can enable a connection between high-performance chip-integrated lasers based on scalable nanofabrication technology with atomic systems used for time and frequency metrology. While second-order nonlinear () systems are the natural approach for bridging such large spectral gaps, here we show that third-order nonlinear () systems, despite their typically much weaker nonlinear response, can realize spectral translation with unprecedented performance. By combining resonant enhancement with nanophotonic mode engineering in a silicon nitride microring resonator, we demonstrate efficient spectral translation of a continuous-wave signal from the telecom band (≈ 1550 nm) to the visible band (≈ 650 nm) through cavity-enhanced four-wave mixing. We achieve such translation over a wide spectral range >250 THz with a translation efficiency of (30.1 ± 2.8) % and using an ultra-low pump power of (329 ± 13) μW. The translation efficiency projects to (274 ± 28) % at 1 mW and is more than an order of magnitude larger than what has been achieved in current nanophotonic devices.

Computing the density of states for optical spectra of molecules by low-rank and QTT tensor approximation

In this paper, we introduce a new interpolation scheme to approximate the density of states (DOS) for a class of rank-structured matrices with application to the Tamm–Dancoff approximation (TDA) of the Bethe–Salpeter equation (BSE). The presented approach for approximating the DOS is based on two main techniques. First, we propose an economical method for calculating the traces of parametric matrix resolvents at interpolation points by taking advantage of the block-diagonal plus low-rank matrix structure described in [6,3] for the BSE/TDA problem. This allows us to overcome the computational difficulties of the traditional schemes since we avoid the construction of the matrix inverse and hence the need of stochastic sampling. Second, we show that a regularized or smoothed DOS discretized on a fine grid of size N can be accurately represented in a low rank quantized tensor train (QTT) format that can be determined through a least squares fitting procedure. The QTT tensor provides good approximation properties for strictly oscillating DOS functions with multiple gaps, in contrast to interpolation by problem independent functions like polynomials, trigonometric functions, etc. Moreover, the QTT approximant requires asymptotically much fewer (e.g., O(log⁡N)) functional calls compared with the full grid size N. Numerical tests indicate that the QTT approach yields accurate recovery of DOS associated with problems that contain relatively large spectral gaps. The QTT tensor rank only weakly depends on the size of a molecular system that paves the way for treatment of large-scale spectral problems.