Non-Hermitian β-ensemble with real eigenvalues

Bohigas, O.; Pato, M. P.

doi:10.1063/1.4796167

Cited by 17 publications

(35 citation statements)

References 21 publications

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“…It is also noteworthy that the matrix η given in reference [24] which verifies (1) fluctuates around finite values for all its elements [25]. In particular, when N → ∞, the average goes as η N N → 1 and the variance goes as σ [26,27] A is one for which there exists a hermitian linear operator T such that its domain is the Hilbert space being considered, T is positive definite and bounded and, also, T A = A † T .…”

Section: The Hermitian and The Quasi-hermitian β-Hermite Ensemblesmentioning

confidence: 99%

“…In [24] these matrices were made non-Hermitian by filling the two off-diagonals with different values taken from the same distribution. We denote the new matrices byĤ…”

Section: The Hermitian and The Quasi-hermitian β-Hermite Ensemblesmentioning

confidence: 99%

“…It has then been shown [24] that by explicitly allowing the subdiagonals of the β-ensemble matrices to be sorted independently, an ensemble of non-Hermitian tridiagonal matrices in which all the eigenvalues are real is obtained. This non-Hermitian ensemble was constructed as an effort to provide an ensemble whose matrices could, in principle, model aspects of PT symmetric systems.…”

Section: Introductionmentioning

confidence: 99%

“…In Sect. 2, the results of [24] regarding Hermitian and Quasi-Hermitian β-ensembles are reviewed. In the following Sect.…”

Section: Introductionmentioning

confidence: 99%

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Pseudo-Hermitian $$\beta $$ β -Ensembles with Complex Eigenvalues

Marinello

Pato

2016

Springer Proceedings in Physics

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Pseudo-Hermitian and PT-symmetric matrices have been a topic of interest since the first papers on the subject in the late 90s. In this paper we utilize properties of tridiagonal random matrices described in the framework of generalized β ensembles to explore variations of that ensemble, which cause the eigenvalues to move from the real line into the complex plane in conjugate pairs, while still maintaining the pseudo-Hermitian property.

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Section: The Hermitian and The Quasi-hermitian β-Hermite Ensemblesmentioning

confidence: 99%

“…In [24] these matrices were made non-Hermitian by filling the two off-diagonals with different values taken from the same distribution. We denote the new matrices byĤ…”

Section: The Hermitian and The Quasi-hermitian β-Hermite Ensemblesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In Sect. 2, the results of [24] regarding Hermitian and Quasi-Hermitian β-ensembles are reviewed. In the following Sect.…”

Section: Introductionmentioning

confidence: 99%

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Pseudo-Hermitian $$\beta $$ β -Ensembles with Complex Eigenvalues

Marinello

Pato

2016

Springer Proceedings in Physics

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“…Non-Hermitian random matrix models, on the other hand, whose eigenvalues are in general complex, are widely studied, and have applications ranging from dissipative quantum systems and scattering theory to quantum chromodynamics (see, e.g., [26] and references therein). Several attempts towards defining P Tsymmetric random matrices and identifying universality classes for P T -symmetric systems have been made [27][28][29][30][31][32]. Most of them are restricted to 2 × 2 matrices, due to the lack of a natural parameterisation of larger P T -symmetric matrices.…”

mentioning

confidence: 99%

Random matrix ensembles forPT-symmetric systems

Graefe¹,

Mudute-Ndumbe²,

Taylor³

2015

J. Phys. A: Math. Theor.

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Abstract. Recently much effort has been made towards the introduction of nonHermitian random matrix models respecting P T -symmetry. Here we show that there is a one-to-one correspondence between complex P T -symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. We conjecture that these ensembles represent universality classes for P Tsymmetric matrices. For the case of 2 × 2 matrices we derive analytic expressions for the joint probability distributions of the eigenvalues, the one-level densities and the level spacings in the case of real eigenvalues.In recent years there has been a surge of research interest in P T -symmetric quantum theories, accompanied by a multitude of experimental applications and realisations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. For finite-dimensional systems represented by matrices, P Tsymmetry is equivalent to the reality of the characteristic polynomial [18]. That is, P T -symmetric matrices have either real or complex conjugate eigenvalues. Their eigenvectors are orthogonal with respect to a suitably defined CP T inner product [19]. It has recently been conjectured that P T -symmetry is closely related to split-quaternionic extensions of quantum theory [20,21]. Here we show that splitquaternionic extensions of Hermitian matrices are indeed a natural representation of P T -symmetric matrices. This equivalence allows us to introduce new P T -symmetric random matrix ensembles.In conventional quantum systems, random matrices play an important role due to their ability to describe spectral fluctuations in sufficiently complicated systems [22]. In particular, the famous Bohigas-Giannoni-Schmit conjecture states that the spectral fluctuations of quantum systems with chaotic classical counterparts are similar to those of certain random matrices [23,24]. There are three important universality classes for Hermitian quantum systems, depending on the time-reversal properties of the system, corresponding to the Gaussian orthogonal, unitary, and symplectic ensembles [25]. Non-Hermitian random matrix models, on the other hand, whose eigenvalues are in general complex, are widely studied, and have applications ranging from dissipative quantum systems and scattering theory to quantum chromodynamics (see, e.g., [26] and references therein). Several attempts towards defining P Tsymmetric random matrices and identifying universality classes for P T -symmetric systems have been made [27][28][29][30][31][32]. Most of them are restricted to 2 × 2 matrices, due to the lack of a natural parameterisation of larger P T -symmetric matrices. Here we introduce the split-complex and split-quaternionic versions of the Gaussian unitary arXiv:1505.07810v2 [math-ph]

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The Pseudo-Hermitian Condition

Pato

2024

Pseudo-Hermitian Random Matrices

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Non-Hermitian β-ensemble with real eigenvalues

Cited by 17 publications

References 21 publications

Pseudo-Hermitian $$\beta $$ β -Ensembles with Complex Eigenvalues

Pseudo-Hermitian $$\beta $$ β -Ensembles with Complex Eigenvalues

Random matrix ensembles forPT-symmetric systems

The Pseudo-Hermitian Condition

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