Nearest-neighbor Level Spacing Distribution Research Articles

Linear level repulsions near exceptional points of non-Hermitian systems

The nearest-neighbor level-spacing distributions are a fundamental quantity of disordered systems and are classified into different universality classes. They are the Wigner-Dyson and the Poisson functions for extended and localized states in Hermitian systems, respectively. The distributions follow the Ginibre functions for the non-Hermitian systems whose eigenvalues are complex and away from exceptional points (EPs). However, the level-spacing distributions of disordered non-Hermitian systems near EPs are still unknown, and a corresponding random matrix theory is absent. Here, we show another class of universal level-spacing distributions in the vicinity of EPs of non-Hermitian Hamiltonians. Two distribution functions, ${P}_{\text{SP}}(s)$ for the symmetry-preserved phase and ${P}_{\text{SB}}(s)$ for the symmetry-broken phase, are needed to describe the nearest-neighbor level-spacing distributions near EPs. Surprisingly, both ${P}_{\text{SP}}(s)$ and ${P}_{\text{SB}}(s)$ are proportional to $s$ for small $s$, or linear level repulsions, in contrast to cubic level repulsions of the Ginibre ensembles. For disordered non-Hermitian tight-binding Hamiltonians, ${P}_{\text{SP}}(s)$ and ${P}_{\text{SB}}(s)$ can be well described by a surmise ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}}_{\text{ep}}(s)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{1}sexp[\ensuremath{-}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}}_{2}{s}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}}]$ in the thermodynamic limit (infinite systems) with a constant $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}$ that depends on the localization nature of states at EPs rather than the dimensionality of non-Hermitian systems and the order of EPs.

Spectral fluctuations in the interacting boson model* *Supported by National Natural Science Foundation of China (11875158, 11875171)

The energy dependence of the spectral fluctuations in the interacting boson model (IBM) and its connections to the mean-field structures are analyzed by adopting two statistical measures: the nearest neighbor level spacing distribution measuring the chaoticity (regularity) in energy spectra and the statistics of Dyson and Metha measuring the spectral rigidity. Specifically, the statistical results as functions of the energy cutoff are determined for different dynamical scenarios, including the U(5)-SU(3) and SU(3)-O(6) transitions as well as those near the AW arc of regularity. We observe that most of the changes in spectral fluctuations are triggered near the stationary points of the classical potential, particularly for cases in the deformed region of the IBM phase diagram. Thus, the results justify the stationary point effects from the perspective of statistics. In addition, the approximate degeneracies in the spectrum on the AW arc is also revealed from the statistical calculations.

Statistical analysis of the excited-state quantum phase transitions in the interacting boson model

The spectral fluctuations and transition intensity fluctuations in the excited-state quantum phase transitions (ESQPTs) have been investigated within the framework of the interacting boson model (IBM) by adopting three statistical measures, including the nearest neighbor level spacing distribution P(S) measuring the chaoticity (regularity) in energy spectra, the Δ3(L) statistics of Dyson and Mehta measuring the spectral rigidity and the intensity distribution P(y) measuring the chaoticity (regularity) in B(E0) transitions. The results indicate that that the ESQPT as a function of the excitation energy may occur as a transition from regular (or semiregular) to highly chaotic if only the associated whole spectrum is chaotic, which fits most of the deformed situations in the IBM including those in the U(5)–SU(3) and SU(3)–O(6) transitional regions. Otherwise, the ESQPT will appear as a transition from regular (or semiregular) to regular such as the cases in the U(5)–O(6) transitional region or those on the ‘Alhassid–Whelan arc’, which represents a nearly regular parameter region connecting the U(5) and SU(3) limits in the IBM.

Spacing ratio characterization of the spectra of directed random networks.

Previous literature on random matrix and network science has traditionally employed measures derived from nearest-neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices. This approach, however, depends crucially on eigenvalue unfolding procedures, which in many situations represent a major hindrance due to constraints in the calculation, especially in the case of complex spectra. Here we study the spectra of directed networks using the recently introduced ratios between nearest and next-to-nearest eigenvalue spacing, thus circumventing the shortcomings imposed by spectral unfolding. Specifically, we characterize the eigenvalue statistics of directed Erdős-Rényi (ER) random networks by means of two adjacency matrix representations, namely, (1) weighted non-Hermitian random matrices and (2) a transformation on non-Hermitian adjacency matrices which produces weighted Hermitian matrices. For both representations, we find that the distribution of spacing ratios becomes universal for a fixed average degree, in accordance with undirected random networks. Furthermore, by calculating the average spacing ratio as a function of the average degree, we show that the spectral statistics of directed ER random networks undergoes a transition from Poisson to Ginibre statistics for model 1 and from Poisson to Gaussian unitary ensemble statistics for model 2. Eigenvector delocalization effects of directed networks are also discussed.

Spectral fluctuations in 24Mg nucleus using the framework of the nuclear shell model

Random matrix theory is used to study the chaotic properties in nuclear energy spectrum of the 24Mg nucleus. The excitation energies (which are the main object of this study) are obtained via performing shell model calculations using the OXBASH computer code together with an effective interaction of Wildenthal (W) in the isospin formalism. The 24Mg nucleus is assumed to have an inert 16O core with 8 nucleons (4protons and 4neutrons) move in the 1d5/2, 2s1/2 and 1d3/2 orbitals. The spectral fluctuations are studied by two statistical measures: the nearest neighbor level spacing distribution P(s) and the Dyson-Mehta statistics (∆3 statistics). For calculations with the full diagonalization of the Hamiltonian, the spectral fluctuations are found to be in agreement with the Gaussian orthogonal ensemble of random matrices.

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.

Statistical properties of spectral fluctuations ofNinteracting bosons in a harmonic trap

Spectral fluctuations of a system of N weakly interacting bosons in an isotropic harmonic trap are studied, with the focus on the deviations from Poisson spectral statistics, typical of a quantum integrable systems. We have utilized the ideas formulated by Makino et al. [Phys. Rev. E 67, 066205 (2003)1063-651X10.1103/PhysRevE.67.066205] who have extended the approach of Berry and Robnik [J. Phys. A 17, 2413 (1984)JPHAC50305-447010.1088/0305-4470/17/12/013]. Earlier investigations of the Berry conjecture [Proc. R. Soc. London, Ser. A 356, 375 (1977)1364-502110.1098/rspa.1977.0140] of Poisson spectral statistics mainly considered quantum systems whose classical counterparts are integrable. However, the system of N weakly interacting bosons in the external trap has no classical counterpart. Also, it is a realistic and experimentally achievable system with close relation to Bose-Einstein condensation. Thus, a stringent analysis of the applicability of the Berry conjecture to this kind of systems is indeed required. We observe that for small boson number, the system is close to integrability and the nearest-neighbor level spacing distribution and the level number variance exhibit deviations from Poisson statistics similar to those of rational rectangular billiards.

Chaos in Jahn–Teller Rattling

We unveil chaotic behavior hidden in the energy spectrum of a Jahn-Teller ion vibrating in a cubic anharmonic potential as a typical model for rattling in cage-structure materials. When we evaluate the nearest-neighbor level-spacing distribution $P(s)$ of eigenenergies of the present oscillator system, we observe the transition of $P(s)$ from the Poisson to the Wigner distribution with the increase of cubic anharmonicity, showing the occurrence of chaos in the anharmonic Jahn-Teller vibration. The energy scale of the chaotic region is specified from the analysis of $P(s)$ and we discuss a possible way to observe chaotic behavior in the experiment of specific heat. It is an intriguing possibility that chaos in nonlinear physics could be detected by a standard experiment in condensed matter physics.

Anomalous quantum heat transport in a one-dimensional harmonic chain with random couplings

We investigate quantum heat transport in a one-dimensional harmonic system with random couplings. In the presence of randomness, phonon modes may normally be classified as ballistic, diffusive or localized. We show that these modes can roughly be characterized by the local nearest-neighbor level spacing distribution, similarly to their electronic counterparts. We also show that the thermal conductance Gth through the system decays rapidly with the system size (Gth ∼ L−α). The exponent α strongly depends on the system size and can change from α < 1 to α > 1 with increasing system size, indicating that the system undergoes a transition from a heat conductor to a heat insulator. This result could be useful in thermal control of low-dimensional systems.

Vibrational Energies of LiH2+ and LiD2+ in the Ã1Σ+ Electronic State

In connection with the recent study of the ground electronic state of the LiH2(+) molecular ion (Kraemer, W. P.; Spirko, V. Chem. Phys. 2006, 330, 190), the adiabatic three-dimensional double-minimum potential enery surface of the first excited electronic state was evaluated, including its two lowest atom-diatom dissociation channels as well as the three-atom complete fragmentation asymptote. Applying the Sutcliffe-Tennyson Hamiltonian for triatomic molecules, the levels of all bound vibrational states and the levels of the states localized in the two energy minimum regions were separately determined. The validity of statistical methods such as the density of states approach and the nearest-neighbor level spacing distribution (NNSD) was tested for the light LiH2(+) ion. Special effort was put into investigating possible effects of a tunnelling motion across the proton-transfer barrier on the vibrational level pattern using the NNSD approach.

Statistical properties of the energies and electric dipole moments of the bound vibrational states of the system HCO +/COH +

From the ab initio calculated three-dimensional adiabatic double-minimum potential energy surface of the HCO +/COH + system and the corresponding dipole moment surface the energies of all bound vibrational states and their effective dipole moments are determined applying the Suttcliffe–Tennyson Hamiltonian for triatomic molecules. The energy and dipole data are analysed in terms of statistical methods such as the density of states approach and the nearest-neighbor level spacing distribution (NNSD). Special effort is put into investigating the effect of the tunnelling motion across the isomerization barrier on the NNSD representations.

Localization–delocalization transition in two-dimensional system with correlated disorder

We study the localization properties of the two-dimensional (2D) Anderson model with diagonal correlated disorder named visible vw and non-visible nw sites. Using the exact diagonalization of Anderson Hamiltonian, investigating the behavior of the density of states (DOS), we found that states in the center of the band E = 0 show critical behavior. The energy levels statistics (ELS) is also examined, we found a crossover of the nearest-neighbor level spacing distribution P ( s ) from Wigner Surmise distribution at small correlated disorder vw and nw (indicating a metallic behavior) to the Poisson distribution at large correlated disorder vw and nw characteristic for localized states. In addition, an analysis of width M dependence of the reduced localization length λ / M is obtained by the transfer-matrix method (TMM). For all investigated strengths of correlated disorder at E = 0 and E =+ 2 , the reduced localization length λ / M always decreases with increasing width. We found only delocalized states in suggested energy and disorder range.

Exploring remnants of invariants buried in a deep potential well in chemical reactions

We revisit the concept of "remnant of invariant manifolds" originally discussed by Shirts and Reinhardt in a two degrees of freedom Henon-Heiles system [J. Chem. Phys. 77, 5204 (1982)]. This is regarded as the remnants of a destroyed invariant manifold that can dominate the transport in phase space even at high energy regions where most of all tori vanish. We present a novel technique to extract such remnants of invariants from a sea of chaos in highly nonlinear coupled molecular systems in terms of the canonical perturbation theory based on Lie transforms. As an illustrative example we demonstrate in HCN isomerization reaction that the conventional procedure based on a finite order truncation of the coordinate transformation prevent us from detecting remnants of invariants. However, our technique correctly captures the underlying remnants of invariants that shed light on the energetics of chemical reaction, that is, how the reactive mode acquires (releases) energy from (to) the other vibrational mode in order to overcome the potential barrier (to be trapped in the potential well). We also found the qualitative difference between the two potential wells, HCN and CNH, which coincides with the nearest neighbor level spacing distribution of the vibrational quantum states within the wells.

Nonballistic heat conduction in an integrable random-exchange Ising chain studied with quantum master equations

We numerically investigate the heat conduction in a random-exchange Ising spin chain by using the quantum master equation. The chain is subject to a uniform transverse field $h$, while the exchange couplings ${{Q}_{n}}$ between the nearest neighbor spins are random; the largest size we simulate is up to 10. This model is integrable; i.e., the nearest neighbor level spacing distribution is Poissonian. However, we find clear evidence of nonballistic transport. In the small coupling regime $({Q}_{n}\ensuremath{\ll}h)$, an energy and/or temperature gradient in the bulk of the system is observed and the energy current appears to be proportional to the inverse of the system size. Moreover, we find that in the low and high temperature regimes, the thermal conductivity $\ensuremath{\kappa}$ and the specific heat ${C}_{v}$ have the same dependence on temperature. The large coupling case $({Q}_{n}\ensuremath{\sim}h)$ is also discussed.

Energies and Electric Dipole Moments of the Bound Vibrational States of HN2+ and DN2+

The adiabatic three-dimensional potential energy surface and the corresponding dipole moment surface describing the ground electronic state of HN2+ (Χ1Σ+) are calculated at different levels of ab initio theory. The calculations cover the entire bound part of the potential up to its lowest dissociation channel including the isomerization barrier. Energies of all bound vibrational and low-lying ro-vibrational levels are determined in a fully variational procedure using the Suttcliffe-Tennyson Hamiltonian for triatomic molecules. They are in close agreement with the available experimental numbers. From the dipole moment function effective dipoles and transition moments are obtained for all the calculated vibrational and ro-vibrational states. Statistical tools such as the density of states or the nearest-neighbor level spacing distribution (NNSD) are applied to describe and analyse general patterns and characteristics of the energy and dipole results calculated for the massively large number of states of the strongly bound HN2+ ion and its deuterated isotopomer.

Low-lying Dirac eigenmodes and monopoles in 4D compact QED

We study the properties of low-lying Dirac modes in quenched compact QED at $\beta =1.01$, employing $12^3\times N_t$ ($N_t =4,6,8,10,12$) lattices and the overlap formalism for the fermion action. We pay attention to the spatial distributions of low-lying Dirac modes below and above the ``phase transition temperature'' $T_c$. Near-zero modes are found to have universal anti-correlations with monopole currents, and are found to lose their temporal structures above $T_c$ exhibiting stronger spatial localization properties. We also study the nearest-neighbor level spacing distribution of Dirac eigenvalues and find a Wigner-Poisson transition.

Regularity in highly excited vibrational dynamics of NOCl (X 1A′): Quantum mechanical calculations on a new potential energy surface

A new potential energy surface (PES) of NOCl in the ground electronic state (X 1A′) is constructed based on multireference configuration interaction calculations. Almost all the vibrational eigenstates up to the dissociation threshold are obtained using filter diagonalization method with the present PES. The vibrational excitation energies are in good agreement with the experimental values. The nearest neighbor level spacing distribution shows that the vibrational structure is regarded as regular up to the dissociation threshold. The distribution of radiative coupling elements also supports the regularity. It is further found out that the NO stretching mode is strongly decoupled from the other modes. Taking advantage of this result, the two-dimensional (2D) analyses are carried out with the NO distance fixed. The 2D wave functions have clear nodal patterns even near the dissociation threshold, indicating that the 2D quantum dynamics is regular as in the three-dimensional case. Comparing the 2D classical dynamics with the 2D wave functions, quasiperiodic trajectories closely related to the corresponding quantum dynamics are extracted. Although it is generally believed that the phase space becomes fully chaotic with approaching to the dissociation threshold, two stable islands of regular motion still remain even at this energy region.

Bound and continuum vibrational states of the bifluoride anion

The energies of the bound vibrational states and energy density spectra of the continuum vibrational states of FHF − are calculated, ‘exactly’ and ‘adiabatically’, using a new ab initio (CCSD(T)) potential energy surface. Statistical properties of the bound states are probed in terms of the density of states and nearest neighbor level spacing distributions (NNSD). Importantly, the approximate ‘adiabatic’ densities coincide nearly quantitatively with their ‘exact’ counterparts. A quantitative fitting of the NNSDs is achieved with a new empirical modification of the Wigner distribution.

Spectral statistics of a spin- [formula omitted] particle in coupled quartic oscillator potentials

The quantum spectral statistical properties, i.e. the nearest neighbor level spacing distribution (NNSD) and the mode fluctuation distribution of a spin- 1 2 particle in three-dimensional coupled quartic oscillator potentials are numerically studied. Introducing the coupling terms, the system can be continuously transformed from an integrable to chaotic one. With use of three parameter sets, three kinds of the Gaussian ensembles can be realized. The Brody-like NNSD interpolation formula between the Poisson distribution and that of each ensemble is found to work well for the numerically obtained NNSD.

Transition of Spectral Statistics of a Spin-1/2 Coupled Quartic Oscillators from GSE to Integrable Regime

A large amount of work has been done in the past decades to understand quantum chaos. Most work has been concentrated on a spinless particle in two dimensions, which provides the simplest and the most fundamental knowledge of quantum chaotic systems (see Ref.1, and references therein). In this work, we study the system of a spin-1/2 particle in three dimensions, which is also essential for physics. By choosing parameters, our system has three kinds of chaotic limits, which correspond to the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble(GUE) and the Gaussian Symplectic Ensemble(GSE). However, the intermediate regime between the integrable and the GSE (or the GUE) class has been scarcely studied. In addition to the nearest neighbor level spacing distribution(NNSD), We also adopt the mode fluctuation distribution(MFD), which was introduced for the signature of chaoticity. Half-integer spin and three dimensions lead us to the chaotic system that is not the GOE class. In this work, spectral statistics of the GOE, the GSE, the integrable classes and their intermediate ones are studied.