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Register a new userUniversal distributions from non-Hermitian Perturbation of Zero-Modes
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Hermitian operators with exact zero modes subject to non-Hermitian perturbations are considered. Specific focus is on the average distribution of the initial zero modes of the Hermitian operators. The broadening of these zero modes is found to follow an elliptic Gaussian random matrix ensemble of fixed size, where the symmetry class of the perturbation determines the behaviour of the modes. This distribution follows from a central limit theorem of matrices, and is shown to be robust to deformations of the average.
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We show that a non-Hermitian zero mode can exhibit an unusual behavior at the transition between extended and localized regimes: it displays a linearly decreasing amplitude as a function of space in a weakly coupled non-Hermitian reservoir. Through the discussion of a linear homogeneous recurrence relation, we attribute this phenomenon to the underlying non-Hermitian particle-hole symmetry and the zeroness of its energy eigenvalue. We also show that linear localization bears a strong resemblance to critical damping, even though the latter does not display linear temporal dynamics.
We consider the smallest eigenvalues of perturbed Hermitian operators with zero modes, either topological or system specific. To leading order for small generic perturbation we show that the corresponding eigenvalues broaden to a Gaussian random matrix ensemble of size $ utimes u$, where $ u$ is the number of zero modes. This observation unifies and extends a number of results within chiral random matrix theory and effective field theory and clarifies under which conditions they apply. The scaling of the former zero modes with the volume differs from the eigenvalues in the bulk, which we propose as an indicator to identify them in experiments. These results hold for all ten symmetric spaces in the Altland-Zirnbauer classification and build on two facts. Firstly, the broadened zero modes decouple from the bulk eigenvalues and secondly, the mixing from eigenstates of the perturbation form a Central Limit Theorem argument for matrices.
In condensed matter physics, non-Abelian statistics for Majorana zero modes (or Majorana Fermions) is very important, really exotic, and completely robust. The race for searching Majorana zero modes and verifying the corresponding non-Abelian statistics becomes an important frontier in condensed matter physics. In this letter, we generalize the Majorana zero modes to non-Hermitian (NH) topological systems that show universal but quite different properties from their Hermitian counterparts. Based on the NH Majorana zero modes, the orthogonal and nonlocal Majorana qubits are well defined. In particular, the non-Abelian statistics for these NH Majorana zero modes become anomalous, which is different from the usual non-Abelian statistics. The usual Ivanovs braiding operator for two Majorana modes is generalized to a non-Hermitian Ivanovs braiding perator. The one-dimensional NH Kitaev model is taken as an example to numerically verify the anomalous non-Abelian statistics for two NH Majorana zero modes. The numerical results are exactly consistent with the theoretical prediction. With the help of braiding these two zero modes, the $pi/8$ gate can be reached and thus universal topological quantum computation becomes possible.
We investigate the number-anomalous of the Majorana zero modes in the non-Hermitian Kitaev chain, whose hopping and superconductor paring strength are both imbalanced. We find that the combination of two imbalanced non-Hermitian terms can induce defective Majorana edge states, which means one of the two localized edge states will disappear due to the non-Hermitian suppression effect. As a result, the conventional bulk-boundary correspondence is broken down. Besides, the defective edge states are mapped to the ground states of non-Hermitian transverse field Ising model, and the global phase diagrams of ferromagnetic-antiferromagnetic crossover for ground states are given. Our work, for the first time, reveal the break of topological robustness for the Majorana zero modes, which predict more novel effects both in topological material and in non-Hermitian physics.
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 ${pi}/2$ 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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