Ratio Of Spacings Research Articles

Level statistics of the one-dimensional dimerized Hubbard model

The statistical properties of level spacings provide valuable insights into the dynamical properties of a many-body quantum systems. We investigate the level statistics of the Fermi–Hubbard model with dimerized hopping amplitude and find that after taking into account translation, reflection, spin and η pairing symmetries to isolate irreducible blocks of the Hamiltonian, the level spacings in the limit of large system sizes follow the distribution expected for hermitian random matrices from the Gaussian orthogonal ensemble. We show this by analyzing the distribution of the ratios of consecutive level spacings in this system, its cumulative distribution and quantify the deviations of the distributions using their mean, standard deviation and skewness.

Coarse-grained molecular dynamics model of AB diblock copolymers composed of blocks with different Lennard-Jones parameters forming lamellar structures

In thermoplastic elastomers, the structure formation of hard and soft domains and the mechanical response to deformation remain vital areas of interest. To realize the coarse-grained molecular dynamics (CGMD) simulations of systems incorporating the difference in glass transition temperature Tg between domains, we studied AB diblock copolymers (di-BCPs) with asymmetric Lennard-Jones (LJ) parameters (the strength coefficient εLJ and cutoff length rc). For instance, hard and soft domains were realized by assigning the cutoff lengths of each segment to rc,(A,A) = 2.5 (high Tg) and rc,(B,B) = 1.5 (low Tg), respectively. To identify how the cutoff length rc,(B,B) corresponds to the strength coefficient ε(B,B) in the phase separation behavior against the fixed LJ parameters for the A block (rc,(A,A) = 2.5 and ε(A,A) = 1.0), we focused on lamellar structures in which the A and B blocks possess equal segment numbers. For the asymmetric cases with rc,(B,B) < rc,(A,A), we estimated the effective ε(B,B)' on the basis of the DA/DB ratio of the partial domain spacings. We confirmed that the scaling relations (D ∼ Δε1/6, D/N0.7 ∼ Δε, and D/N1/2 ∼ ΔεN) hold, where ε(A,A) = 1, Δε = ((1 + ε(B,B)')/2 – ε(A,B)), and N denotes the number of segments per AB di-BCP chain. Therefore, this work will provide valuable insights for guiding applied research on BCPs.

Long-range spectral statistics of the Rosenzweig-Porter model

The Rosenzweig-Porter model is a single-parameter random matrix ensemble that supports an ergodic, fractal, and localized phase. The names of these phases refer to the properties of the (midspectrum) eigenstates. This work focuses on the long-range spectral statistics of the recently introduced unitary equivalent of this model. By numerically studying the Thouless time obtained from the spectral form factor, it is argued that long-range spectral statistics can be used to probe the transition between the ergodic and the fractal phases. The scaling of the Thouless time as a function of the model parameters is found to be similar to the scaling of the spreading width of the eigenstates. Provided that the transition between the fractal and the localized phases can be probed through short-range level statistics, such as the average ratio of consecutive level spacings, this work establishes that spectral statistics are sufficient to probe both transitions present in the phase diagram. Published by the American Physical Society 2024

Field monitoring of river ice processes in the vicinity of ice control structures in the province of Quebec, Canada

Over the past decades, a variety of ice control structures (ICSs) have been designed and built, but to date, there has been no systematic evaluation of the effectiveness of these structures. To achieve this objective, first an understanding of the interaction between different ice processes and the ICSs must be established. For this purpose, a total of four ICSs located in the province of Québec were monitored during the 2021–2022 winter. The results showed that the ice jam holding time could vary from 1.5 to 68.5 h. The release of the jam was mechanically driven when the ratio of release to initiation Froude number was higher than one and was thermally driven when this ratio was lower than one, and the water temperature increased between initiation and release. Also, as the ratio of the total pier spacing to upstream river width increased, the holding time decreased.

Eigenvalue ratio statistics of complex networks: Disorder versus randomness

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios distribution of various model networks, namely, small-world, Erdős-Rényi random, and (dis)assortative random having a diagonal disorder in the corresponding adjacency matrices. Without any diagonal disorder, the eigenvalues ratio distribution of these model networks depict Gaussian orthogonal ensemble (GOE) statistics. Upon adding diagonal disorder, there exists a gradual transition from the GOE to Poisson statistics depending upon the strength of the disorder. The critical disorder (w_{c}) required to procure the Poisson statistics increases with the randomness in the network architecture. We relate w_{c} with the time taken by maximum entropy random walker to reach the steady state. These analyses will be helpful to understand the role of eigenvalues other than the principal one for various network dynamics such as transient behavior.

Ratio of consecutive level spacings as a signature of chaos in nuclear many-body models

Statistical properties of nuclear energy levels and nuclear transitions have long been juxtaposed to the limit of random matrix theory. A novel practical measure of chaos onset, the ratio of consecutive energy spacings, that does not require the unfolding of energy levels, was previously introduced for all three canonical random matrix ensembles, along with the expressions for the regular and chaotic limits. In this study, an expression for this ratio interpolating between the regular and chaotic limits is introduced and applied to the energy levels calculated in the realistic nuclear shell model and in the interacting boson model. The results are consistent with those extracted from the traditional statistics of nearest-neighbor spacings with unfolding; we find the relation between the ways of convergence to the chaotic limit.

Nonuniform random graphs on the plane: A scaling study.

We consider random geometric graphs on the plane characterized by a nonuniform density of vertices. In particular, we introduce a graph model where n vertices are independently distributed in the unit disk with positions, in polar coordinates (l,θ), obeying the probability density functions ρ(l) and ρ(θ). Here we choose ρ(l) as a normal distribution with zero mean and variance σ∈(0,∞) and ρ(θ) as a uniform distribution in the interval θ∈[0,2π). Then, two vertices are connected by an edge if their Euclidean distance is less than or equal to the connection radius ℓ. We characterize the topological properties of this random graph model, which depends on the parameter set (n,σ,ℓ), by the use of the average degree 〈k〉 and the number of nonisolated vertices V_{×}, while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings r and the Shannon entropy S of eigenvectors. First we propose a heuristic expression for 〈k(n,σ,ℓ)〉. Then, we look for the scaling properties of the normalized average measure 〈X[over ¯]〉 (where X stands for V_{×}, r, and S) over graph ensembles. We demonstrate that the scaling parameter of 〈V_{×}[over ¯]〉=〈V_{×}〉/n is indeed 〈k〉, with 〈V_{×}[over ¯]〉≈1-exp(-〈k〉). Meanwhile, the scaling parameter of both 〈r[over ¯]〉 and 〈S[over ¯]〉 is proportional to n^{-γ}〈k〉 with γ≈0.16.

Spectral statistics of a 1D photonic crystal containing an anisotropic graphene-based hyperbolic metamaterial defect layer

A simple photonic crystal is a periodic and regular nanostructure. Breaking the periodicity of a photonic crystal by a defect layer can be an interesting pattern. We study the spectral statistics of eigenfrequency spacings (the so-called level spacing statistics) for TM-polarized waves in the one-dimensional (1D) defective photonic crystal containing graphene-based hyperbolic metamaterial as a defect layer. The localization of light in the proposed structure has been investigated using the transfer matrix method. Moreover, the effects of different parameters, such as the number of graphene cells and the defect layer's optical axis orientation, are discussed. The performance of the ratios of nearest neighbor level spacings has become a Wigner distribution at the localized phases which are introduced by inserting a defect layer. The spectral statistics are found to approach the Wigner distribution for different repeat numbers of graphene layers. In contrast, in the absence of graphene layers, the level spacing statistics follow the Poisson distribution.

Statistical properties of mutualistic-competitive random networks

Mutualistic networks are used to study the structure and processes inherent to mutualistic relationships. In this paper, we introduce a random matrix ensemble (RME) representing the adjacency matrices of mutualistic networks composed by two vertex sets of sizes n and m−n. Our RME depends on three parameters: the network size n, the size of the smaller set m, and the connectivity between the two sets α, where α is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. We focus on the spectral, eigenvector and topological properties of the RME by computing, respectively, the ratio of consecutive eigenvalue spacings r, the Shannon entropy of the eigenvectors S, and the Randić index R. First, within a random matrix theory approach (i.e. a statistical approach), we identify a parameter ξ≡ξ(n,m,α) that scales the average normalized measures <X¯> (with X representing r, S and R). Specifically, we show that (i) ξ∝αn with a weak dependence on m, and (ii) for ξ<1/10 most vertices in the mutualistic network are isolated, while for ξ>10 the network acquires the properties of a complete network, i.e., the transition from isolated vertices to a complete-like behavior occurs in the interval 1/10<ξ<10. Then, we demonstrate that our statistical approach predicts reasonably well the properties of real-world mutualistic networks; that is, the universal curves <X¯> vs. ξ show good correspondence with the properties of real-world networks.

Deviations from Poisson statistics in the spectra of free rectangular thin plates.

The counterintuitive fact that wave chaos appears in the bending spectrum of free rectangular thin plates is presented. After extensive numerical simulations, varying the ratio between the length of its sides, it is shown that (i) frequency levels belonging to different symmetry classes cross each other and (ii) for levels within the same symmetry sector, only avoided crossings appear. The consequence of anticrossings is studied by calculating the distribution of the ratio of consecutive level spacings for each symmetry class. The resulting ratio distributions disagree with the expected Poissonian result. They are then compared with some well-known transition distributions between Poisson and the Gaussian orthogonal random matrix ensemble. It is found that the distribution of the ratio of consecutive level spacings agrees with the prediction of the Rosenzweig-Porter model. Also, the normal-mode vibration amplitudes are found experimentally on aluminum plates, before and after an avoided crossing for symmetrical-symmetrical, symmetrical-antisymmetrical, and antisymmetrical-symmetrical classes. The measured modes show an excellent agreement with our numerical predictions. The expected Poissonian distribution is recovered for the simply supported rectangular plate.

Electronic transport in chaotic mesoscopic cavities: A Kwant and random matrix theory based exploration

We investigate the spectral fluctuations and electronic transport properties of chaotic mesoscopic cavities using Kwant, an open source Python programming language based package. Discretized chaotic billiard systems are used to model these mesoscopic cavities. For the spectral fluctuations, we study the ratio of consecutive eigenvalue spacings, and for the transport properties, we focus on Landauer conductance and shot noise power. We generate an ensemble of scattering matrices in Kwant, with desired number of open channels in the leads attached to the cavity. The results obtained from Kwant simulations, performed without or with magnetic field, are compared with the corresponding random matrix theory predictions for orthogonally and unitarily invariant ensembles. These two cases apply to the scenarios of preserved and broken time-reversal symmetry, respectively. In addition, we explore the orthogonal to unitary crossover statistics by varying the magnetic field and examine its relationship with the random matrix transition parameter.

Replacing wakes with streaks in wind turbine arrays

AbstractWind turbine wakes negatively affect downwind turbines in wind farms reducing their global efficiency. The reduction of wake‐turbine interactions by actuating control on yaw angles and induction factors is an active area of research. In this study, the capability of spanwise‐periodic rows of wind turbines with tilted rotors to reduce negative wake‐turbine interactions is investigated through large‐eddy simulations. It is shown that, by means of rotor tilt, it is possible to replace turbine far wakes with high‐speed streaks where the streamwise velocity exceeds the freestream velocity at hub height. Considering three aligned rows of wind turbines, it is found that the global power extracted from the wind can be increased by tilting rotors of the upwind turbine rows, similarly to what is already known for the case of a single column of aligned turbines. It is further shown that global tilt‐induced power gains can be significantly increased by operating the tilted turbines at higher induction rates. Power gains are further increased for higher ratios of rotor diameters and turbine spacings to the boundary layer height. All these findings are consistent with those of previous studies where streamwise streaks were artificially forced by means of spanwise‐periodic rows of wall‐mounted roughness elements in order to control canonical boundary layers for drag‐reduction applications.

Distribution of the ratio of consecutive level spacings for different symmetries and degrees of chaos.

Theoretical expressions for the distribution of the ratio of consecutive level spacings for quantum systems with transiting dynamics remain unknown. We propose a family of one-parameter distributions P(r)≡P(r;β), where β∈[0,+∞) is a generalized Dyson index, that describes the eigenlevel statistics of a quantum system characterized by different symmetries and degrees of chaos. We show that this crossover strongly depends on the specific properties of each model, and thus the reduction of such a family to a universal formula, albeit desirable, is not possible. We use the information entropy as a criterion to suggest particular ansätze for different transitions, with a negligible associated error in the limits corresponding to standard random ensembles.

Observation of spectral modulation coupled with broadband transverse-mode-locking in an Yb:CALGO frequency-degenerate cavity

A frequency-degenerate cavity (FDC) is the resonator that the ratio of transverse and longitudinal mode frequency spacings is a simple rational number. When an optical resonator is close to the FDC, transverse-mode-locking (TML) takes place with drastic changes of laser mode. We report for the first time, to the best of our knowledge, the multi-frequency emission and spectral modulation effects coupled with TML in FDC. The Yb:CaGdAlO4 (Yb:CALGO) crystal with large gain bandwidth was used as a gain medium in an off-axis-pumped hemispherical FDC for realizing broadband emission. Interestingly, the spectrum can transform from a single smooth packet shape to a multi-peak structure; meanwhile, the transverse pattern accordingly transforms into some exotic wave-packet profiles through controlling off-axis displacement in a special degenerate state.

Ordered level spacing probability densities

Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the and spacings from a given level are introduced. Analytical predictions are derived using a matrix model. The density is generalized to the spacing density, which allows for investigating long-range correlations. For larger the probability density of spacings is well described by a Gaussian. Using these spacings we propose the ratio of the to the as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found.

Higher-order spacing ratios in random matrix theory and complex quantum systems

The distribution of the ratios of nearest neighbor level spacings has become a popular indicator of spectral fluctuations in complex quantum systems like interacting many-body localized and thermalization phases, quantum chaotic systems, and also in atomic and nuclear physics. In contrast to the level spacing distribution, which requires the cumbersome and at times ambiguous unfolding procedure, the ratios of spacings do not require unfolding and are easier to compute. In this work, for the class of Wigner-Dyson random matrices with nearest neighbor spacing ratios $r$ distributed as $P_{\beta}(r)$ for the three ensembles indexed by $\beta=1,2, 4$, their $k-$th order spacing ratio distributions are shown to be identical to $P_{\beta'}(r)$, where $\beta'$, an integer, is a function of $\beta$ and $k$. This result is shown for Gaussian and circular ensembles of random matrix theory and for several physical systems such as spin chains, chaotic billiards, Floquet systems and measured nuclear resonances.

Finite-size scaling with respect to interaction and disorder strength at the many-body localization transition

We present a finite-size scaling for both interaction and disorder strengths in the critical regime of the many-body localization (MBL) transition for a spin-1/2 XXZ spin chain with a random field by studying level statistics. We show how the dynamical transition from the thermal to MBL phase depends on interaction together with disorder by evaluating the ratio of adjacent level spacings, and thus, extend previous studies in which interaction coupling is fixed. We introduce an extra critical exponent in order to describe the nontrivial interaction dependence of the MBL transition. It is characterized by the ratio of the disorder strength to the power of the interaction coupling with respect to the extra critical exponent and not by the simple ratio between them.

Anderson localization in sigma models

In QCD above the chiral restoration temperature there exists an Anderson transition in the fermion spectrum from localized to delocalized modes. We investigate whether the same holds for nonlinear sigma models which share properties like dynamical mass generation and asymptotic freedom with QCD. In particular we study the spectra of fermions coupled to (quenched) CP(N-1) configurations at high temperatures. We compare results in two and three space-time dimensions: in two dimensions the Anderson transition is absent, since all fermion modes are localized, while in three dimensions it is present. Our measurements include a more recent observable characterizing level spacings: the distribution of ratios of consecutive level spacings.

Reply to “Comment on ‘Lattice gluon and ghost propagators and the strong coupling in pureSU(3)Yang-Mills theory: Finite lattice spacing and volume effects’ ”

The quenched gluon and ghost propagator data published in [1] is reanalysed following the suggestion of [2] to resolve the differences between the infrared data of the simulations. Our results confirms that the procedure works well either for the gluon or for the ghost propagator but not for both propagators simultaneously as the observed deviations in the data follow opposite patterns. Definitive conclusions require improving the determination of the (ratios) of lattice spacings. A simple procedure for the relative calibration of the lattice spacing in lattice simulations is suggested.

Many-body localization phase in a spin-driven chiral multiferroic chain

Many-body localization (MBL) is an emergent phase in correlated quantum systems with promis- ing applications, particularly in quantum information. Here, we unveil the existence and analyse this phase in a chiral multiferroic model system. Conventionally, MBL occurrence is traced via level statistics by implementing a standard finite-size scaling procedure. Here, we present an approach based on the full distribution of the ratio of adjacent energy spacings. We find a strong broadening of the histograms of counts of these level spacings directly at the transition point from MBL to the ergodic phase. The broadening signals reliably the transition point without relying on an averaging procedure. The fast convergence of the histograms even for relatively small systems allows moni- toring the MBL dynamics with much less computational effort. Numerical results are presented for a chiral spin chain with a dynamical Dzyaloshinskii Moriya (DM) interaction, an established model to describe the spin excitations in a single phase spin-driven multiferroic system. The multiferroic MBL phase is uncovered and it is shown how to steer it via electric fields.