Level Spacing Distribution Research Articles

Crossover between the Gaussian orthogonal ensemble, the Gaussian unitary ensemble, and Poissonian statistics.

Until now only for specific crossovers between Poissonian statistics (P), the statistics of a Gaussian orthogonal ensemble (GOE), or the statistics of a Gaussian unitary ensemble (GUE) have analytical formulas for the level spacing distribution function been derived within random matrix theory. We investigate arbitrary crossovers in the triangle between all three statistics. To this aim we propose an according formula for the level spacing distribution function depending on two parameters. Comparing the behavior of our formula for the special cases of P→GUE, P→GOE, and GOE→GUE with the results from random matrix theory, we prove that these crossovers are described reasonably. Recent investigations by F. Schweiner etal. [Phys. Rev. E 95, 062205 (2017)2470-004510.1103/PhysRevE.95.062205] have shown that the Hamiltonian of magnetoexcitons in cubic semiconductors can exhibit all three statistics in dependence on the system parameters. Evaluating the numerical results for magnetoexcitons in dependence on the excitation energy and on a parameter connected with the cubic valence band structure and comparing the results with the formula proposed allows us to distinguish between regular and chaotic behavior as well as between existent or broken antiunitary symmetries. Increasing one of the two parameters, transitions between different crossovers, e.g., from the P→GOE to the P→GUE crossover, are observed and discussed.

Observation of giant conductance fluctuations in a protein

Proteins are insulating molecular solids, yet even those containing easily reduced or oxidized centers can have single-molecule electronic conductances that are too large to account for with conventional transport theories. Here, we report the observation of remarkably high electronic conductance states in an electrochemically inactive protein, the ∼200 kD αVβ3 extracellular domain of human integrin. Large current pulses (up to nA) were observed for long durations (many ms, corresponding to many pC of charge transfer) at large gap (>5 nm) distances in an STM when the protein was bound specifically by a small peptide ligand attached to the electrodes. The effect is greatly reduced when a homologous, weakly binding protein (α4β1) is used as a control. In order to overcome the limitations of the STM, the time- and voltage-dependence of the conductance were further explored using a fixed-gap (5 nm) tunneling junction device that was small enough to trap a single protein molecule at any one time. Transitions to a high conductance (∼nS) state were observed, the protein being ‘on’ for times from ms to tenths of a second. The high-conductance states only occur above ∼100 mV applied bias, and thus are not an equilibrium property of the protein. Nanoamp two-level signals indicate the specific capture of a single molecule in an electrode gap functionalized with the ligand. This offers a new approach to label-free electronic detection of single protein molecules. Electronic structure calculations yield a distribution of energy level spacings that is consistent with a recently proposed quantum-critical state for proteins, in which small fluctuations can drive transitions between localized and band-like electronic states.

Weighted random-geometric and random-rectangular graphs: spectral and eigenfunction properties of the adjacency matrix

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of weighted random--geometric and random--rectangular graphs. A random--geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random--rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths $a$ and $1/a$, the connection radius $r$, and the number of vertices $N$. We then study in detail the case $a=1$ which corresponds to weighted RGGs and explore weighted RRGs characterized by $a\sim 1$, i.e.~two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when $a\gg1$. In general we look for the scaling properties of $P(s)$ and $\ell$ as a function of $a$, $r$ and $N$. We find that the ratio $r/N^\gamma$, with $\gamma(a)\approx -1/2$, fixes the properties of both RGGs and RRGs. Moreover, when $a\ge 10$ we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio $r/{\cal C}N^\gamma$, with ${\cal C}\approx a$.

Manifestations of isospin in nearest neighbor spacing distributions for the f-p model space

The strong interactions are charge independent. If we limit ourselves to the strong interactions, we have the isospin [Formula: see text] as a good quantum number. Here, we consider the lack of level repulsion of states of different isospin and how this effect manifests in nearest neighbor spacing (NNS) histograms, which provide a visual and statistical context in which to study distributions of energy level spacings. In particular, we study nucleons in the [Formula: see text] model space for the nucleus [Formula: see text]. We also compare NNS distributions produced in the isospin formalism to distributions produced in the proton–neutron formalism.

Signatures of chaos in the Brillouin zone.

When the classical dynamics of a particle in a finite two-dimensional billiard undergoes a transition to chaos, the quantum dynamics of the particle also shows manifestations of chaos in the form of scarring of wave functions and changes in energy level spacing distributions. If we "tile" an infinite plane with such billiards, we find that the Bloch states on the lattice undergo avoided crossings, energy level spacing statistics change from Poisson-like to Wigner-like, and energy sheets of the Brillouin zone begin to "mix" as the classical dynamics of the billiard changes from regular to chaotic behavior.

Pseudosymmetric random matrices: Semi-Poisson and sub-Wigner statistics

Real nonsymmetric matrices may have either real or complex conjugate eigenvalues. These matrices can be seen to be pseudosymmetric as ηMη^{-1}=M^{t}, where the metric η could be secular (a constant matrix) or depending upon the matrix elements of M. Here we construct ensembles of a large number N of pseudosymmetric n×n (n large) matrices using N[n(n+1)/2≤N≤n^{2}] independent and identically distributed random numbers as their elements. Based on our numerical calculations, we conjecture that for these ensembles the nearest level spacing distributions [NLSDs, p(s)] are sub-Wigner as p_{abc}(s)=ase^{-bs^{c}}(0<c<2) and the distributions of their eigenvalues fit well to D(ε)=A[tanh{(ε+B)/C}-tanh{(ε-B)/C}] (exceptions also discussed). These sub-Wigner NLSDs are encountered in Anderson metal-insulator transition and topological transitions in a Josephson junction. Interestingly, p(s) for c=1 is called semi-Poisson, and we show that it lies close to the form p(s)=0.59sK_{0}(0.45s^{2}) derived for the case of 2×2 pseudosymmetric matrix where the eigenvalues are most aptly conditionally real, E_{1,2}=a±sqrt[b^{2}-c^{2}], which represent characteristic coalescing of eigenvalues in parity-time (PT) -symmetric systems.

Excited states of two-dimensional hydrogen atom in tilted magnetic field: Quantum chaos

The aim of the current work is the research of the influence of a tilted magnetic field direction on the spectrum and the energy level spacing distribution of a two-dimensional (2D) hydrogen atom and of an exciton in GaAs/Al0.33Ga0.67As quantum well. It was discovered that the quantum chaos (QC) is initiated with an increasing angle α between the magnetic field direction and the normal to the atomic plane. It is characterized by the repulsion of levels leading to the eliminating of the shell structure and by changing the spectrum statistical properties. The statement about the initiation of chaos and its dominance over regular motion with increasing angle α is confirmed by the results of our calculations of the classical dynamics presented in this paper. The evolution of the spatial distribution of the square of the absolute value of the wave function at an increasing angle α was observed. The differences of calculated dependencies of energies for various excited states on the tilt angle at a wide range of the magnetic field strength were described.

Interaction effects in a chaotic graphene quantum billiard

We investigate the local electronic structure of a Sinai-like, quadrilateral graphene quantum billiard with zigzag and armchair edges using scanning tunneling microscopy at room temperature. It is revealed that besides the $(\sqrt{3}\times\sqrt{3})R30${\deg} superstructure, which is caused by the intervalley scattering, its overtones also appear in the STM measurements, which are attributed to the Umklapp processes. We point out that these results can be well understood by taking into account the Coulomb interaction in the quantum billiard, accounting for both the measured density of state values and the experimentally observed topography patterns. The analysis of the level-spacing distribution substantiates the experimental findings as well. We also reveal the magnetic properties of our system which should be relevant in future graphene based electronic and spintronic applications.

Spectral Fluctuations in A=32 Nuclei Using the Framework of the Nuclear Shell Model

Chaotic properties of nuclear energy spectra in A=32 nuclei are investigated via the framework of the nuclear shell model. The energies (the main object of this investigation) are calculated through accomplishing shell model calculations employing the OXBASH computer code with the realistic effective interaction of W in the isospin formalism. The A=32 nuclei are supposed to have an inert 16O core with 16 nucleons move in the 1d5/2, 2s1/2 and 1d3/2 orbitals. For full hamiltonian calculations, the spectral fluctuations (i.e., the nearest neighbor level spacing distributions P(S) and the Δ3 statistics) are well characterized by the Gaussian orthogonal ensemble of random matrices. Besides, they show no dependency on the spin J and isospin T. For unperturbed hamiltonian calculations, we find a regular behavior for the distribution of P(S) and an intermediate behavior between the GOE and the Poisson limits for the Δ3 statistics.

Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers

We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors-Keating, and Smilansky, we formulate an analog of the Hardy-Littlewood prime ...

Spectral statistics, finite-size scaling and multifractal analysis of quasiperiodic chain with p-wave pairing

We study the spectral and wavefunction properties of a one-dimensional incommensurate system with p-wave pairing and unveil that the system demonstrates a series of particular properties in its ciritical region. By studying the spectral statistics, we show that the bandwidth distribution and level spacing distribution in the critical region follow inverse power laws, which however break down in the extended and localized regions. By performing a finite-size scaling analysis, we can obtain some critical exponents of the system and find these exponents fulfilling a hyperscaling law in the whole critical region. We also carry out a multifractal analysis on system's wavefuntions by using a box-counting method and unveil the wavefuntions displaying different behaviors in the critical, extended and localized regions.

Absence of localization in disordered two-dimensional electron gas at weak magnetic field and strong spin-orbit coupling

The one-parameter scaling theory of localization predicts that all states in a disordered two-dimensional system with broken time reversal symmetry are localized even in the presence of strong spin-orbit coupling. While at constant strong magnetic fields this paradigm fails (recall the quantum Hall effect), it is believed to hold at weak magnetic fields. Here we explore the nature of quantum states at weak magnetic field and strongly fluctuating spin-orbit coupling, employing highly accurate numerical procedure based on level spacing distribution and transfer matrix technique combined with one parameter finite-size scaling hypothesis. Remarkably, the metallic phase, (known to exist at zero magnetic field), persists also at finite (albeit weak) magnetic fields, and eventually crosses over into a critical phase, which has already been confirmed at high magnetic fields. A schematic phase diagram drawn in the energy-magnetic field plane elucidates the occurrence of localized, metallic and critical phases. In addition, it is shown that nearest-level statistics is determined solely by the symmetry parameter β and follows the Wigner surmise irrespective of whether states are metallic or critical.

Microwave Realization of the Gaussian Symplectic Ensemble.

Following an idea by Joyner etal. [Europhys. Lett. 107, 50004 (2014)], a microwave graph with an antiunitary symmetry T obeying T^{2}=-1 is realized. The Kramers doublets expected for such systems are clearly identified and can be lifted by a perturbation which breaks the antiunitary symmetry. The observed spectral level spacings distribution of the Kramers doublets is in agreement with the predictions from the Gaussian symplectic ensemble expected for chaotic systems with such a symmetry.

Many-body localization in a quantum simulator with programmable random disorder

When a system thermalizes it loses all local memory of its initial conditions. This is a general feature of open systems and is well described by equilibrium statistical mechanics. Even within a closed (or reversible) quantum system, where unitary time evolution retains all information about its initial state, subsystems can still thermalize using the rest of the system as an effective heat bath. Exceptions to quantum thermalization have been predicted and observed, but typically require inherent symmetries or noninteracting particles in the presence of static disorder. The prediction of many-body localization (MBL), in which disordered quantum systems can fail to thermalize in spite of strong interactions and high excitation energy, was therefore surprising and has attracted considerable theoretical attention. Here we experimentally generate MBL states by applying an Ising Hamiltonian with long-range interactions and programmably random disorder to ten spins initialized far from equilibrium. We observe the essential signatures of MBL: memory retention of the initial state, a Poissonian distribution of energy level spacings, and entanglement growth in the system at long times. Our platform can be scaled to higher numbers of spins, where detailed modeling of MBL becomes impossible due to the complexity of representing such entangled quantum states. Moreover, the high degree of control in our experiment may guide the use of MBL states as potential quantum memories in naturally disordered quantum systems.

Inner structure of vehicular ensembles and random matrix theory

We introduce a special class of random matrices (DUE) whose spectral statistics corresponds to statistics of microscopical quantities detected in vehicular flows. Comparing the level spacing distribution (for ordered eigenvalues in unfolded spectra of DUE matrices) with the time-clearance distribution extracted from various areas of the flux-density diagram (evaluated from original traffic data measured on Czech expressways with high occupancies) we demonstrate that the set of classical systems showing an universality associated with Random Matrix Ensembles can be extended by traffic systems.

Level and width statistics of the open many-body systems

The level and width statistics of the two kinds of the random matrix models coupled to the continuum are analyzed. In the first model, the gaussian orthogonal ensemble with random couplings to the continuum, not only the width statistics deviate from the Porter-Thomas distribution due to the super-radiant mechanism, but also the distribution of the nearest neighbor level spacings shows deviation from the Wigner one simultaneously. In the second model, the two body random ensemble with correlated couplings to the continuum, the correlation between the target and the compound states leads to the global energy dependence of the widths. Within the narrow energy interval where states with widths deviating from the global energy dependence lie, the distributions behave similar way with the case of the random couplings. Namely, the deviation of statistics of the nearest neighbor level spacings from the Wigner distribution and the deviation of the width statistics from the Porter-Thomas distribution take place simultaneously within the models we investigated.

Many-Body Localization in the Presence of a Single-Particle Mobility Edge.

In one dimension, noninteracting particles can undergo a localization-delocalization transition in a quasiperiodic potential. Recent studies have suggested that this transition transforms into a many-body localization (MBL) transition upon the introduction of interactions. It has also been shown that mobility edges can appear in the single particle spectrum for certain types of quasiperiodic potentials. Here, we investigate the effect of interactions in two models with such mobility edges. Employing the technique of exact diagonalization for finite-sized systems, we calculate the level spacing distribution, time evolution of entanglement entropy, optical conductivity, and return probability to detect MBL. We find that MBL does indeed occur in one of the two models we study, but the entanglement appears to grow faster than logarithmically with time unlike in other MBL systems.

Pairing effect and misleading regularity

We study the nearest neighbor spacing distribution of energy levels of even–even nuclei classified according to their reduced electric quadrupole transition probability B(E2)↑ using the available experimental data. We compare between Brody, and Abul-Magd distributions that extract the degree of chaoticity within nuclear dynamics. The results show that Abul-Magd parameter f can represents the chaotic behavior in more acceptable way than Brody, especially if a statistically significant study is desired. A smooth transition from chaos to order is observed as B(E2)↑ increases. An apparent regularity was located at the second interval, namely: at 0.05≤B(E2)<0.1 in e2b2 units, and at 10≤B(E2)<15 in Weisskopf unit. Finally, the chaotic behavior parameterized in terms of B(E2)↑ does not depend on the unit used.

Quench dynamics of dipolar fermions in a one-dimensional harmonic trap

We study a system of few fermions in a one-dimensional harmonic trap, and focus on the case of dipolar majority particles in contact with a single impurity. The impurity is used both for quenching the system, and for tracking the system evolution after the quench. Employing exact diagonalization, we investigate relaxation and thermalization properties. In the absence of dipolar interactions, the system is near integrability, and the dynamics remains oscillatory even on long time scales. On the other hand, repulsive as well as attractive dipolar interactions lead to quick relaxation to the diagonal ensemble average which is significantly different from corresponding thermal averages. A Wigner-shaped level spacing distribution indicates level repulsion and thus chaotic dynamical behavior due to the presence of dipolar interactions.

Entanglement spectrum and block eigenvalue spacing distribution of correlated electron states

Entanglement spectrum of finite-size correlated electron systems are investigated using the Gutzwiller projection technique. The product of largest eigenvalue and rank of the block reduced density matrix, which is a measure of distance of the state from the maximally entangled state of the corresponding rank, is seen to characterise the insulator to metal crossover in the state. The fraction of distinct eigenvalues exhibits a ‘chaotic’ behaviour in the crossover region, and it shows a ‘integrable’ behaviour at both insulating and metallic ends. The integrated entanglement spectrum obeys conformal field theory (CFT) prediction at the metal and insulator ends, but shows a noticeable deviation from CFT prediction in the crossover regime, thus it can also track a metal–insulator crossover. A modification of the CFT result for the entanglement spectrum for finite size is proposed which holds in the crossover regime also. The adjacent level spacing distribution of unfolded non-zero eigenvalues for intermediate values of Gutzwiller projection parameter g is the same as that of an ensemble of random matrices obtained by replacing each block of reduced density matrix by a random real symmetric Toeplitz matrix. It is strongly peaked at zero, with an exponential tail proportional to e−(n/R)s, where s is the adjacent level spacing, n is number of distinct eigenvalues and R is the rank of the reduced density matrix.