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arXiv:1912.07227 · Physical Review A

Resonant generation of a p -wave Cooper pair in a non-Hermitian Kitaev chain at the exceptional point

X. M. Yang, Z. Song

Published Aug 19, 2020• 13 citations
arXivHTMLPDF

Abstract

We investigate a non-Hermitian extension of a Kitaev chain by considering imaginary $p$-wave pairing amplitudes. The exact solution shows that the phase diagram consists of two phases with real and complex Bogoliubov--de Gennes spectra, associated with the $\mathcal{PT}$-symmetry breaking, which is separated by a hyperbolic exceptional line. The exceptional points (EPs) correspond to a specific Cooper pair state $(1+{c}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{\ensuremath{-}k}^{\ifmmode\dagger\else\textdagger\fi{}})\left|0\right\ensuremath{\rangle}$ with movable $k$ when the parameters vary along the exceptional line. The non-Hermiticity around EP supports resonant generation of such a pair state from the vacuum state $\left|0\right\ensuremath{\rangle}$ of fermions via the critical dynamic process. In addition, we propose a scheme to generate a superconducting state through a dynamic method.

DOI
10.1103/physreva.102.022219

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We investigate a non-Hermitian extension of a Kitaev chain by considering imaginary ppp-wave pairing amplitudes. The exact solution shows that the phase diagram consists of two phases with real and complex Bogoliubov--de Gennes spectra, associated with the PT\mathcal{PT}PT-symmetry breaking, which is separated by a hyperbolic exceptional line. The exceptional points (EPs) correspond to a specific Cooper pair state (1+{c}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{\ensuremath{-}k}^{\ifmmode\dagger\else\textdagger\fi{}})\left|0\right\ensuremath{\rangle} with movable kkk when the parameters vary along the exceptional line. The non-Hermiticity around EP supports resonant generation of such a pair state from the vacuum state \left|0\right\ensuremath{\rangle} of fermions via the critical dynamic process. In addition, we propose a scheme to generate a superconducting state through a dynamic method.
Parity-time (PT)(\mathcal{PT})(PT) symmetric systems are classical, gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues of a PT\mathcal{PT}PT-symmetric Hamiltonian change from real to complex conjugates at a critical value of gain-loss strength that is called the PT\mathcal{PT}PT breaking threshold. Here, we obtain the PT\mathcal{PT}PT threshold for a one-dimensional, finite Kitaev chain---a prototype for a ppp-wave superconductor---in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, on-site potential, and the distance between the gain and loss sites. In addition to a robust, nonlocal threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band structure of the Hermitian Kitaev model. In particular, for an even chain with zero on-site potential, we find a re-entrant PT\mathcal{PT}PT-symmetric phase bounded by second-order EP contours. Our numerical results are supplemented by analytical calculations for small system sizes.
Exact solutions for non-Hermitian quantum many-body systems are rare but may provide valuable insights into the interplay between Hermitian and non-Hermitian components. We report our investigation of a non-Hermitian variant of a ppp-wave Kitaev chain by introducing staggered imbalanced pair creation and annihilation terms. We find that there exist fixed lines in the phase diagram, at which the ground state remains unchanged in the presence of a non-Hermitian term under the periodic boundary condition for a finite system. This allows the constancy of the topological index in the process of varying the balance strength at arbitrary rate, exhibiting the robustness of the topology for the non-Hermitian Kitaev chain under time-dependent perturbations. The underlying mechanism is investigated through the equivalent quantum spin system obtained by the Jordan-Wigner transformation for infinite chain. In addition, the exact solution shows that a resonant non-Hermitian impurity can induce a pair of zero modes in the corresponding Majorana lattice, which asymptotically approach the edge modes in the thermodynamic limit, manifesting the bulk-boundary correspondence. Numerical simulation is performed for the quench dynamics for the systems with slight deviation from the fixed lines to show the stability region in time. This work reveals the interplay between the pair creation and annihilation pairing processes.
A nonequilibrium state shares macroscopic properties, such as conductivity and superconductivity, with a static state when the two states exhibit an identical average value of an observable over a period. We studied the quench dynamics of a Kitaev chain on the basis of two types of order parameters associated with two channels of pairing: local pairing in real space and Bardeen-Cooper-Schrieffer (BCS)-like pairing in momentum space. Based on the exact solution, we found that the two order parameters were identical for the ground state, which indicated a balance between the two pairing channels and would help determine the quantum phase diagram. However, for a nonequilibrium state obtained through time evolution from an initially prepared vacuum state, the two parameters varied, but both parameters could still help determine the phase diagram. In the region of nontrivial topological phases, nonequilibrium states favored BCS-like pairing. In this paper, we present an alternative approach to dynamically generate a superconducting state from a trivial empty state and illuminate the pairing mechanism.
We study a non-Hermitian extension of the Creutz ladder with generic non-reciprocal hopping. By mapping the ladder onto two decoupled non-Hermitian Su--Schrieffer--Heeger (SSH) chains, we uncover a rich structure in parameter space under different boundary conditions. Under periodic boundary conditions, the spectrum admits a fine-tuned line in parameter space with entirely real eigenvalues, while deviations from this line induce a real--complex spectral transition without crossing exceptional points. In contrast, an exact analytical diagonalization under open boundary conditions reveals extended regions in parameter space with purely real or purely imaginary spectra, separated from complex spectral domains by exceptional lines. The intersections of these exceptional lines define triple-junction points where distinct spectral regimes meet, giving rise to a structured phase diagram that is absent under periodic boundary conditions. We further show that flat bands in this system can occur both as Hermitian diabolical points and as non-Hermitian exceptional points, known as exceptional flat bands, where the dynamics is more stringent than in the Hermitian case, leading to distinct spectral and dynamical signatures.
Exceptional point rings and PT\mathcal{P}\mathcal{T}PT-symmetry in the non-Hermitian XY model
Abstract The XY spin chain is a paradigmatic example of a model solved by free fermions, in which the energy eigenspectrum is built from combinations of quasi-energies. In this article, we show that by extending the XY model’s anisotropy parameter λ\lambdaλ λ to complex values, it is possible for two of the quasi-energies to become degenerate. In the non-Hermitian XY model, these quasi-energy degeneracies give rise to exceptional points (EPs) where eigenvalues and their corresponding eigenvectors coalesce. The distinct λ\lambdaλ λ values at which EPs appear form concentric rings in the complex plane which are shown in the infinite system size limit to converge to the unit circle coinciding with the boundary between distinct topological phases. The non-Hermitian model is also seen to possess a line of broken PT\mathcal{P}\mathcal{T}PT P T symmetry along the pure imaginary λ\lambdaλ λ -axis. For finite systems, there are four EP values on this broken PT\mathcal{P}\mathcal{T}PT P T -symmetric line if the system size is a multiple of 4.
PT\mathcal{PT}PT-Symmetric Spin--Boson Model with a Continuous Bosonic Spectrum: Exceptional Points and Dynamics
This work studies a PT\mathcal{PT}PT-symmetric non-Hermitian spin--boson model, consisting of a non-Hermitian two-level system coupled to a continuous bosonic bath. The static properties of the system are analyzed through a projection method derived from the displacement operator. We find that only a single exceptional point (EP) emerges, in contrast to non-Hermitian spin--boson models with finite modes, which typically exhibit multiple EPs. Notably, only a single real eigenvalue is found before the EP, which differs markedly from typical non-Hermitian systems where a pair of real eigenvalues precedes the EP. The time evolution of observables is further investigated via the Dirac--Frenkel time-dependent variational principle. Compared to its Hermitian counterpart, the non-Hermitian model exhibits distinct dynamical signatures, most notably the emergence of oscillations with periodic amplified amplitude. In the PT\mathcal{PT}PT-unbroken phase, the system exhibits sustained oscillatory dynamics with suppressed decoherence, whereas in the PT\mathcal{PT}PT-broken phase, additional dissipative channels accelerate decoherence and drive rapid convergence toward a stable steady state. These results shed light on how PT\mathcal{PT}PT symmetry protects coherent light--matter interactions in non-Hermitian quantum systems.