Time-Reversal Symmetry in Non-Hermitian Systems

Sato, Masatoshi; Hasebe, Kazuki; Esaki, Kenta; Kohmoto, Mahito

doi:10.1143/ptp.127.937

Cited by 80 publications

(84 citation statements)

References 41 publications

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“…We now generalize the symmetry classes to the non-Hermitian case, making use of the ideas of Bernard-LeClair symmetry classes 65,[89][90][91]98 . The key difference is that in the case of non-Hermitian systems, the scope of symmetries is significantly expanded; in particular, Hermiticity can now be viewed as a special type of non-Hermitian symmetry (type Q).…”

Section: Non-hermitian Symmetry Classesmentioning

confidence: 99%

Periodic table for topological bands with non-Hermitian symmetries

2019

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Classifications of symmetry-protected topological (SPT) phases provide a framework to systematically understand the physical properties and potential applications of topological systems. While such classifications have been widely explored in the context of Hermitian systems, a complete understanding of the roles of more general non-Hermitian symmetries and their associated classification is still lacking. Here, we derive a periodic table for non-interacting SPTs with general non-Hermitian symmetries. Our analysis reveals novel non-Hermitian topological classes, while also naturally incorporating the entire classification of Hermitian systems as a special case of our scheme. Building on top of these results, we derive two independent generalizations of Kramers theorem to the non-Hermitian setting, which constrain the spectra of the system and lead to new topological invariants. To elucidate the physics behind the periodic table, we provide explicit examples of novel non-Hermitian topological invariants, focusing on the symmetry classes in zero, one and two dimensions with new topological classifications (e.g. Z in 0D, Z2 in 1D, 2D). These results thus provide a framework for the design and engineering of non-Hermitian symmetry-protected topological systems. arXiv:1812.10490v2 [cond-mat.mes-hall]

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Section: Non-hermitian Symmetry Classesmentioning

confidence: 99%

Periodic table for topological bands with non-Hermitian symmetries

2019

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“…Though experimentally challenging at first, it was later discovered that optic and photonic systems offer an ideal platform for the PT -symmetric quantum theory, where non-Hermitian effects of terms are naturally introduced by optical radiation and loss/gain of particles [20,[22][23][24][25]. Studies on PT -symmetric quantum theory not only deepen our understanding on fundamental physics [26][27][28][29], but also lead to novel applications [30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

PT -symmetric non-Hermitian Dirac semimetals

2019

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Three-dimensional Dirac semimetals with Dirac points carrying Z2 monopole charges can be realized in systems preserving the combined symmetry of space-inversion (P) and time-reversal (T ).Here we systematically study PT -symmetric non-Hermitian Dirac semimetals incorporating different kinds of symmetry-preserving non-Hermitian potentials. These potentials render non-Hermitian semimetals to be in either PT -unbroken or PT -broken phases. Interestingly, with the same kind of non-Hermitian potentials, systems with periodic and open boundary conditions may belong to different PT phases unique to non-Hermitian systems. We find that the topological properties of Z2 monopole charges are retained in non-Hermitian Dirac semimetals with PT -unbroken phases, where the bulk-boundary correspondence can be established. However, these systems exhibit features with no counterpart in Hermitian theory, such as the reflection-symmetric non-Hermitian skin effect and Fermi ribbon surface states. arXiv:1907.10417v1 [cond-mat.mes-hall]

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“…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.…”

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confidence: 53%

“…The resulting characteristic polynomial is real, thus the eigenvalues are either real or come in complex conjugate pairs. Further, the eigenvalues of the 2N × 2N representation are doubly degenerate, in analogy to Kramer's degeneracy for quaternionic Hermitian matrices [21,22,24]. The split-quaternionic eigenvectors belonging to distinct eigenvalues are orthogonal with respect to the inner product (4).…”

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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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Time-Reversal Symmetry in Non-Hermitian Systems

Cited by 80 publications

References 41 publications

Periodic table for topological bands with non-Hermitian symmetries

Periodic table for topological bands with non-Hermitian symmetries

PT -symmetric non-Hermitian Dirac semimetals

Random matrix ensembles forPT-symmetric systems

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