Chiral symmetry in non-Hermitian systems: Product rule and Clifford algebra

Rivero, Jose D. H.; Li, Ge

doi:10.1103/physrevb.103.014111

Cited by 21 publications

(8 citation statements)

References 69 publications

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“…We refer to the symmetry defined by Eq. ( 6) as pseudochirality, not just because it involves a similar transformation as in the definition of pseudo-Hermiticity, but also due to the fact that it warrants, as we will show, a symmetric spectrum about the origin of the complex energy plane, similar to the consequence of chiral symmetry in non-Hermitian systems [19,21,22]. Here chiral symmetry in both Hermitian [44] and non-Hermitian systems [38] is defined by {H, Ξ} = 0, and it differs from pseudo-chirality by the absence of the matrix transpose.…”

mentioning

confidence: 73%

“…While its ramification in quantum theories is still under intense investigation, its application in different fields has led to a plethora of findings, ranging from nonlinear dynamics [8], atomic physics [9], photonics [2], acoustics [10], microwave [11], electronics [12], to quantum information science [13]. Subsequent explorations have also revealed other novel non-Hermitian symmetries, including, for example, anti-PT symmetry [14,15], odd-time-reversal PT symmetry [16,17], non-Hermitian particle-hole (NHPH) symmetry [18][19][20][21], and non-Hermitian chiral symmetry [21][22][23]. Benefiting from these findings, different devices such as single-mode lasers [24,25], robust power transfer circuits [12], laser-anti-lasers [26][27][28][29] and on-chip lasers carrying orbital angular momentum [30] have been demonstrated, which also shine light on a new type of light-matter interaction [11].…”

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confidence: 99%

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Pseudo-chirality: a manifestation of Noether's theorem in non-Hermitian systems

Rivero,

2021

Preprint

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Noether's theorem relates constants of motion to the symmetries of the system. Here we investigate a manifestation of Noether's theorem in non-Hermitian systems, where the inner product is defined differently from quantum mechanics. In this framework, a generalized symmetry which we term pseudo-chirality emerges naturally as the counterpart of symmetries defined by a commutation relation in quantum mechanics. Using this observation, we reveal previously unidentified constants of motion in non-Hermitian systems with parity-time and chiral symmetries. We further elaborate the disparate implications of pseudo-chirality induced constant of motion: It signals the pair excitation of a generalized "particle" and the corresponding "hole" but vanishes universally when the pseudo-chiral operator is anti-symmetric. This disparity, when manifested in a non-Hermitian topological lattice with the Landau gauge, depends on whether the lattice size is even or odd. We further discuss previously unidentified symmetries of this non-Hermitian topological system, and we reveal how its constant of motion due to pseudo-chirality can be used as an indicator of whether a pure chiral edge state is excited.

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confidence: 73%

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confidence: 99%

Pseudo-chirality: a manifestation of Noether's theorem in non-Hermitian systems

Rivero,

2021

Preprint

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“…Here we show that the Green's function in a 1D non-Hermitian lattice can have the same property, which further stresses the new perspective our method provides. This 1D lattice features a non-Hermitian flatband [15], enabled by imposing non-Hermitian particle-hole (NHPH) symmetry [43][44][45][46] via staggered gain and loss along a lattice of optical resonators. In the tight-binding Hamiltonian description, it is given by (23) where n labels the lattice sites from 1 to N , g ∈ R is the nearest neighbor coupling, and the alternate positive and negative imaginary on-site detuning γ models gain and loss.…”

Section: New Insightsmentioning

confidence: 99%

Green's function as a defect state in a boundary value problem

Rivero

2021

Phys. Rev. B

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A new perspective of the Green's function in a boundary value problem as the only eigenstate in an auxiliary formulation is introduced. In this treatment, the Green's function can be perceived as a defect state in the presence of a δ-function potential, the height of which depends on the Green's function itself. This approach is illustrated in one-dimensional and two-dimensional Helmholtz equation problems, with an emphasis on systems that are open and have a non-Hermitian potential. We then draw an analogy between the Green's function obtained this way and a chiral edge state circumventing a defect in a topological lattice, which shines light on the local minimum of the Green's function at the source position.

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“…On the other hand, eigenvalues are complex in general in non-Hermitian arrays. While chiral symmetry can still be realized in this case 6 , a more prevailing symmetry that leads to a zero mode in a non-Hermitian lattice is NHPH symmetry, giving rise to ε k = −ε * j . A zero mode ( j = k), which always exists in the case of an odd number of cavities, leads to Re(ε j ) = 0.…”

Section: (B) Inset]mentioning

confidence: 99%

“…Being their own anti-particles and hosting non-Abelian braiding properties, their experimental demonstration is being actively pursued in high-energy physics and condensed matter physics [1][2][3][4][5] . The existence of these zero-energy excitations is warranted by particle-hole symmetry, in the form that the (Hermitian) Hamiltonian anticommutes with an anti-linear operator 6 .…”

Section: Introductionmentioning

confidence: 99%

Direct observation of zero modes in a non-Hermitian nanocavity array

Hentinger,

Hedir,

Garbin

et al. 2021

Preprint

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Zero modes are symmetry protected ones whose energy eigenvalues have zero real parts. In Hermitian arrays, they arise as a consequence of the sublattice symmetry, implying that they are dark modes. In non-Hermitian systems, that naturally emerge in gain/loss optical cavities, particle-hole symmetry prevails instead; the resulting zero modes are no longer dark but feature π/2 phase jumps between adjacent cavities. Here we report on the direct observation of zero modes in a non-Hermitian three coupled photonic crystal nanocavity array containing quantum wells. Unlike the Hermitian counterparts, the non-Hermitian zero modes can only be observed for small sublattice detuning, and they can be identified through far-field imaging and spectral filtering of the photoluminescence at selected pump locations. We explain the zero mode coalescence as a parity-time phase transition for small coupling. These zero modes are robust against coupling disorder, and can be used for laser mode engineering and photonic computing.

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Chiral symmetry in non-Hermitian systems: Product rule and Clifford algebra

Cited by 21 publications

References 69 publications

Pseudo-chirality: a manifestation of Noether's theorem in non-Hermitian systems

Pseudo-chirality: a manifestation of Noether's theorem in non-Hermitian systems

Green's function as a defect state in a boundary value problem

Direct observation of zero modes in a non-Hermitian nanocavity array

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