Random matrix ensembles for<i>PT</i>-symmetric systems

Graefe, Eva-Maria; Mudute-Ndumbe, Steve; Taylor, M.

doi:10.1088/1751-8113/48/38/38ft02

Cited by 18 publications

(15 citation statements)

References 44 publications

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“…The spectral statistics of random Hermitian Hamiltonians usually exhibits universal behavior, depending only on symmetries of the system [68]. An interesting question then naturally arises whether the spectral statistics of disordered non-Hermitian Hamiltonians also exhibits universal behavior [69][70][71][72][73][74][75][76][77][78][79][80][81][82][83].…”

Section: Introductionmentioning

confidence: 99%

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

et al. 2020

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We explore the eigenvalue statistics of a non-Hermitian version of the Su-Schrieffer-Heeger model, with imaginary on-site potentials and randomly distributed hopping terms. We find that owing to the structure of the Hamiltonian, eigenvalues can be purely real in a certain range of parameters, even in the absence of parity and time-reversal symmetry. As it turns out, in this case of purely real spectrum, the level statistics is that of the Gaussian orthogonal ensemble. This demonstrates a general feature which we clarify that a non-Hermitian Hamiltonian whose eigenvalues are purely real can be mapped to a Hermitian Hamiltonian which inherits the symmetries of the original Hamiltonian. When the spectrum contains imaginary eigenvalues, we show that the density of states (DOS) vanishes at the origin and diverges at the spectral edges on the imaginary axis. We show that the divergence of the DOS originates from the Dyson singularity in chiral-symmetric one-dimensional Hermitian systems and derive analytically the asymptotes of the DOS which is different from that in Hermitian systems.

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Section: Introductionmentioning

confidence: 99%

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

et al. 2020

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“…whose expectation value in any given state (pure or mixed) is likewise independent of the parameters (ξ, η). Note that the sextet (t, x, y, z, ξ, η) corresponds to the six parameters required for specifying most general PTsymmetric 2 × 2 Hamiltonians [19,20]. The foregoing example illustrates the fact that when a HamiltonianĤ is prescribed according to (13), the only parameters that an experimentalist can adjust in a laboratory are the ones in the triplet (x, y, z), and possibly t in some circumstances, but not the remaining two variables (ξ, η).…”

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confidence: 99%

Consistency of PT-symmetric quantum mechanics

Brody¹

2016

J. Phys. A: Math. Theor.

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In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully consistent with standard quantum mechanics. This follows from the surprising fact that the much-discussed metric operator on Hilbert space is not physically observable. In particular, for closed quantum systems in finite dimensions there is no statistical test that one can perform on the outcomes of measurements to determine whether the Hamiltonian is Hermitian in the conventional sense, or PT-symmetric-the two theories are indistinguishable. Nontrivial physical effects arising as a consequence of PT symmetry are expected to be observed, nevertheless, for open quantum systems with balanced gain and loss.Since the realisation by Bender and Boettcher [1] that complex-non-Hermitian-Hamiltonians admitting space-time reflection (parity and time-reversal) symmetry can possess entirely real eigenvalues, considerable amount of research has been carried out into identifying properties of physical systems described by PTsymmetric Hamiltonians. It was subsequently observed that such Hamiltonians, although not Hermitian, can nevertheless be used to generate unitary time evolutions for the characterisation of closed quantum systems, provided that one works with a Hilbert space equipped with a preferentially selected inner product [2,3]. In fact, the idea of modifying the Hilbert space inner product in quantum mechanics has previously been proposed in [4], but the work of [2, 3] has triggered extensive research into the identification of appropriate inner products-the so-called metric operators-for a wide range of complex PT-symmetric Hamiltonians. Nevertheless, the physical significance of the metric operator has hitherto remained elusive, leading to a variety of controversial claims concerning what might be achievable by altering the inner product in a laboratory experiment.The purpose of the present paper is to unambiguously settle this issue by showing that the degrees of freedom in the Hamiltonian associated with the choice of the Hilbert space inner product are not observable. Putting the matter differently, the lack of Hermiticity of the Hamiltonian, or equivalently the Petermann factor (see, e.g., [5]), is not observable in closed systems-in contrast to open systems. The significance of this result is that not all parametric degrees of freedom in a complex Hamiltonian can be perturbed by an experimentalist in a laboratory. Our finding thus answers the open question raised in [6] regarding the possible physical constraints that prohibit experimentalists modifying the inner product (or equivalently, switching between different PT-symmetric Hamiltonians) in a laboratory. For the same token, the results here invalidate the claim in [7] that local PTsymmetric Hamiltonians violate the no-signalling condition, and the claim in [8] that local PT-symmetric Hamiltonians can be used to increase...

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“…Turning now to the real ones, it is reasonable to consider that they form a kind of an incomplete sequence of levels with a reduced repulsion among them. The theory of randomly incomplete spectra [17] that emerged from the theory of missing levels [18], recently, has attracted much attention [19][20][21][22]. In [17], it was indeed conjectured that a situation in which levels move away from the real axis would be a realization of a randomly thinned spectra.…”

Section: Introductionmentioning

confidence: 99%

Statistical properties of eigenvalues of an ensemble of pseudo-Hermitian Gaussian matrices

Marinello

Pato

2019

Phys. Scr.

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We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate eigenvalues, the real ones show characteristics of an intermediate incomplete spectrum, that is, of a so-called thinned ensemble. On the other hand, the complex ones show repulsion compatible with cubic-order repulsion of non normal matrices for the real matrices, but higher order repulsion for the complex and quaternion matrices.

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Random matrix ensembles forPT-symmetric systems

Cited by 18 publications

References 44 publications

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

Consistency of PT-symmetric quantum mechanics

Statistical properties of eigenvalues of an ensemble of pseudo-Hermitian Gaussian matrices

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