Discrete symmetries and spectral statistics

Keating, Jon P; Robbins, Jonathan M

doi:10.1088/0305-4470/30/7/006

Cited by 40 publications

(45 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To understand this effect we investigate the structure of the Hamiltonian matrix of our systems and explain the previous results in terms of the underlying symmetries of the problem. At the end, we explain the anomalous GUE statistics in the light of a more general analysis 28,29 . From the Eqs.…”

Section: Discussionmentioning

confidence: 99%

“…When spin is not involved, it is expected that the spectrum statistics of a chaotic system is similar to that obtained from the GOE (GUE) if it is (is not) time reversal invariant (TRI). However, there are exceptions to this rule, consisting of a special class of TRI systems with point group irreducible representations, which does exhibit the GUE statistics 28,29 …”

Section: Energy Level Statisticsmentioning

confidence: 99%

See 1 more Smart Citation

Quantum chaos in nanoelectromechanical systems

2006

View full text Add to dashboard Cite

We present a theoretical study of the electron-phonon coupling in suspended nanoelectromechanical systems (NEMS) and investigate the resulting quantum chaotic behavior. The phonons are associated with the vibrational modes of a suspended rectangular dielectric plate, with free or clamped boundary conditions, whereas the electrons are confined to a large quantum dot (QD) on the plate's surface. The deformation potential and piezoelectric interactions are considered. By performing standard energy-level statistics we demonstrate that the spectral fluctuations exhibit the same distributions as those of the Gaussian Orthogonal Ensemble (GOE) or the Gaussian Unitary Ensemble (GUE), therefore evidencing the emergence of quantum chaos. That is verified for a large range of material and geometry parameters. In particular, the GUE statistics occurs only in the case of a circular QD. It represents an anomalous phenomenon, previously reported for just a small number of systems, since the problem is time-reversal invariant. The obtained results are explained through a detailed analysis of the Hamiltonian matrix structure.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Energy Level Statisticsmentioning

confidence: 99%

Quantum chaos in nanoelectromechanical systems

2006

View full text Add to dashboard Cite

show abstract

“…However, there are exceptions to this rule, which consist of the special class of time-reversal invariant systems with point group irreducible representations that can exhibit the GUE statistics [14,15]. The family of systems presently known to show this phenomenon is formed by billiards having threefold symmetry, which have been experimentally implemented [16,17] in classical microwave cavities.…”

mentioning

confidence: 99%

Anomalous quantum chaotic behaviour in suspended electromechanical nanostructures

Rego¹,

Gusso²,

Luz³

2005

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

It is predicted that for sufficiently strong electron-phonon coupling an anomalous quantum chaotic behavior develops in certain types of suspended electro-mechanical nanostructures, here represented by a thin cylindrical quantum dot (billiard) on a suspended rectangular dielectric plate. The deformation potential and piezoelectric interactions are considered. As a result of the electron-phonon coupling between the two systems the spectral statistics of the electro-mechanic eigenenergies exhibit an anomalous behavior. If the center of the quantum dot is located at one of the symmetry axes of the rectangular plate, the energy level distributions correspond to the Gaussian Orthogonal Ensemble (GOE), otherwise they belong to the Gaussian Unitary Ensemble (GUE), even though the system is time-reversal invariant. The possibility of engineering devices at the nano and micro scales has openned a great avenue for testing fundamental aspects of quantum theory, otherwise difficult to probe in natural atomic size systems. In particular, mesoscopic structures have played an important role in the experimental study of quantum chaos [1], mainly through the investigation of the transport properties of quantum dots [2] and quantum well structures [3] in the presence of magnetic field. However, some hard to control characteristics of such structures can prevent the full observation of quantum chaotic behavior. For instance, the incoherent influence of the bulk on the electronic dynamics hinders the observation of the so called eigenstate scars [4] in quantum corrals [5]. Also, Random Matrix Theory (RMT) predictions to the Coulomb blockade peaks in quantum dots may fail due to coupling to the environment [6]. Alternatively, suspended nanostructures, due to their particular architecture, are ideal candidates for investigating as well as implementing coherent phenomena in semiconductor devices [7,8]. In special, to understand in a controlled way how phonons influence electronic states and affect the system dynamics, possibly leading to a chaotic behavior. Such point is of practical relevance since it bears the question of stability of quantum computers [9,10], whose actual implementation could be prevented by the emergence of chaos [11].A remarkable characteristic of the quantum chaotic systems is the manifestation of universal statistical features that occur irrespective to their physical nature (e.g., the energy spectra of spinless particles). According to RMT the resulting spectra for systems with and without time-reversal invariance (TRI) are typically described by random matrices of the Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE), respectively. This property was conjectured by Bohigas et al.[12] and has been firmly established by theoretical and experimental examination [1,13]. However, there are exceptions to this rule, which consist of the special class of time-reversal invariant systems with point group irreducible representations that can exhibit the GUE statistics [14,15]. The family of systems p...

show abstract

“…When time-reversal invariance is violated, the statistics is described by the Gaussian unitary ensembles (GUE). We note, that in nonrelativistic quantum chaotic systems, a magnetic field induces time-reversal symmetry breaking and changes the level statistics from GOE to GUE [4][5][6][7][8]. Although the above correspondences may be violated for certain nongeneric cases, they are generally expected to hold for typical nonrelativistic quantum systems and even for relativistic quasiparticles in two-dimensional systems governed by the Dirac equation such as graphene flakes, also called graphene billiards [9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 88%

Gaussian orthogonal ensemble statistics in graphene billiards with the shape of classically integrable billiards

et al. 2016

Phys. Rev. E

View full text Add to dashboard Cite

A crucial result in quantum chaos, which has been established for a long time, is that the spectral properties of classically integrable systems generically are described by Poisson statistics whereas those of time-reversal symmetric, classically chaotic systems coincide with those of random matrices from the Gaussian orthogonal ensemble (GOE). Does this result hold for two-dimensional Dirac material systems? To address this fundamental question, we investigate the spectral properties in a representative class of graphene billiards with shapes of classically integrable circular-sector billiards. Naively one may expect to observe Poisson statistics, which is indeed true for energies close to the band edges where the quasiparticle obeys the Schrödinger equation. However, for energies near the Dirac point, where the quasiparticles behave like massless Dirac fermions, Poisson statistics is extremely rare in the sense that it emerges only under quite strict symmetry constraints on the straight boundary parts of the sector. An arbitrarily small amount of imperfection of the boundary results in GOE statistics. This implies that, for circular sector confinements with arbitrary angle, the spectral properties will generically be GOE. These results are corroborated by extensive numerical computation. Furthermore, we provide a physical understanding for our results.

show abstract

Discrete symmetries and spectral statistics

Cited by 40 publications

References 17 publications

Quantum chaos in nanoelectromechanical systems

Quantum chaos in nanoelectromechanical systems

Anomalous quantum chaotic behaviour in suspended electromechanical nanostructures

Gaussian orthogonal ensemble statistics in graphene billiards with the shape of classically integrable billiards

Contact Info

Product

Resources

About