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arXiv:1804.08336 · Physical Review A

Simple harmonic oscillation in a non-Hermitian Su-Schrieffer-Heeger chain at the exceptional point

K. L. Zhang, P. Wang, Gang Zhang, Z. Song

Published Aug 21, 2018• 23 citations
arXivHTMLPDF

Abstract

The balance of gain and loss in an open system may maintain certain Hermitian dynamical behaviors which can be hardly observed in a popular Hermitian system. In this paper, we systematically study a 1D $\mathcal{PT}$-symmetry non-Hermitian Su-Schrieffer-Heeger model with open boundary condition based on an exact approximate solution. We show that the long-wavelength standing-wave modes can be achieved within the linear dispersion region when the system is tuned at the exceptional point. The whole Hilbert space can be decomposed into two quasi-Hermitian subspaces, which are consisted of positive and negative energy levels, respectively. Within each subspace, the system maintains all the features of a Hermitian system. We construct a coherentlike state in a subspace and find that it exhibits perfect simple harmonic motion (SHM). In contrast to a canonical coherent state, the shape of the wave packet deforms periodically rather than entirely by translation, and the amplitude of the SHM is not determined by the initial condition but by the shape of the wave packet. Our result indicates that novel Hermitian dynamics can be realized by a non-Hermitian system.

DOI
10.1103/physreva.98.022128

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The balance of gain and loss in an open system may maintain certain Hermitian dynamical behaviors which can be hardly observed in a popular Hermitian system. In this paper, we systematically study a 1D PT\mathcal{PT}PT-symmetry non-Hermitian Su-Schrieffer-Heeger model with open boundary condition based on an exact approximate solution. We show that the long-wavelength standing-wave modes can be achieved within the linear dispersion region when the system is tuned at the exceptional point. The whole Hilbert space can be decomposed into two quasi-Hermitian subspaces, which are consisted of positive and negative energy levels, respectively. Within each subspace, the system maintains all the features of a Hermitian system. We construct a coherentlike state in a subspace and find that it exhibits perfect simple harmonic motion (SHM). In contrast to a canonical coherent state, the shape of the wave packet deforms periodically rather than entirely by translation, and the amplitude of the SHM is not determined by the initial condition but by the shape of the wave packet. Our result indicates that novel Hermitian dynamics can be realized by a non-Hermitian system.
Non-Hermitian systems exhibit many peculiar dynamic behaviors that never showed up in Hermitian systems. The existence of spectral singularity for a non-Hermitian scattering center provides a ``lasing mechanism'' in the context of quantum mechanics. In this paper, we investigate the dynamics of the non-Hermitian Su-Schrieffer-Heeger model around an exceptional point, based on an analysis of parity-time and chiral-time symmetries of the Hamiltonian. We show exactly that a finite system near its exceptional point can support EP-related amplification of cyclic states. It becomes a stationary laser solution with linear-power amplification under certain conditions in an infinite limit. In contrast to the SS lasing mechanism for a scattering system, such an SSH chain acts as an active laser medium at threshold, within which a stationary particle emission can be fired anywhere, rather than a specific location at the non-Hermitian scattering center only. In addition, some relevant peculiar phenomena arising from interference between wave packets are revealed based on the analytical solutions.
Parity-time symmetry constrains the spectrum of non-Hermitian systems to be either real or come in complex conjugate pairs. The transition between a symmetry-preserving phase with real energies and a symmetry-broken phase with complex energies is marked by exceptional points, one of the hallmarks of non-Hermitian systems. Because of these properties, these systems are widely studied, both theoretically and experimentally. In this work, we investigate the thermodynamic properties of the gain and loss Su-Schrieffer-Heeger model for both bosons and fermions, and establish the existence of an imaginary time crystal phase, an imaginary time analog of a time crystal. This phase occurs when there is a resonance condition between the Matsubara frequencies and the spectrum of the system, making the Green's function of the system oscillate in imaginary time with the Matsubara frequency. We show that this phase appears in the symmetry-broken region. In particular, the topological edge states of this system exhibit oscillations that are present for fermions. Finally, we discuss the applicability of our results for experiments. We examine signatures of these phases in terms of correlation functions in real time and oscillations in thermodynamic potential in inverse temperature β , and explore possible experimental platforms to realize this system.
Fock-state lattice (FSL) offers a powerful quantum simulator for topological phenomena due to the unbounded scalability and ease of implementation. Nevertheless, the unique topological properties induced by its site-dependent coupling have remained elusive, mainly due to the challenge of handling an infinite state space without translational symmetry. Here, we rigorously analyze the topological features of a semi-infinite FSL-based Su-Schrieffer-Heeger (SSH) model, in both Hermitian and non-Hermitian realms, by mapping it to the solvable Jaynes-Cummings model via a unitary displacement transformation. We find a topological zero mode persisting across all parameter regimes, which is more stable than the conventional SSH model. It originates from the bound state at the inherent domain wall under anisotropic conditions. With gain and loss introduced, we predict a non-Hermitian bound effect, i.e., any state overlapping with the bound state will quickly stabilize to the domain wall, with the minimal stabilization time occurring in the vicinity of exceptional point. The parity-time phase transition can be observed by the oscillating-to-steady crossover of dynamics in the subspace orthogonal to the bound state. Furthermore, a concrete experimental proposal based on the trapped-ion setup is provided.
We investigate domain-wall physics in a non-Hermitian Su-Schrieffer-Heeger model featuring the non-Hermitian skin effect and its higher-dimensional extensions, focusing on analytical eigenstate solutions under open boundary conditions. By gluing two Su-Schrieffer-Heeger chains with inverted coupling ratios together, we identify two distinct interface geometries, and derive closed-form quantization conditions for the complex wave number and wave functions using symmetry-based ansatzes. Besides conventional skin modes and topological zero modes, the domain walls generate additional localized states at finite energy that detach from the bulk open-boundary-condition continuum. We validate the analytical results by exact diagonalization, and further generalize the interface construction to a two-dimensional Lieb lattice, where competing nonreciprocities produce tunable funneling regimes towards codimension-one and codimension-two interfaces.
The simple harmonic oscillator has a well-known normalizable, positive energy, bound state spectrum. We show that degenerate with each such positive energy eigenvalue there is a non-normalizable positive energy eigenstate whose eigenfunction is orthogonal to that of the standard energy eigenfunction. This class of states is not built on the vacuum that aaa annihilates, but is instead built on the vacuum that a†aa^{\dagger} aa†a annihilates. These non-normalizable but nonetheless stationary energy eigenstates are accompanied by yet another set of non-normalizable states, states whose wave functions however are not stationary but instead grow linearly in time. With these states not being energy eigenstates, the eigenbasis of the Hamiltonian is incomplete; with the full Hilbert space containing states that are not energy eigenstates. Thus each energy eigenvalue of the harmonic oscillator is an exceptional point at which the Hamiltonian becomes of non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form. Such non-Hermitian structures occur for Hamiltonians that have an antilinear PTPTPT symmetry. As is characteristic of such systems, one can construct a probability conserving inner product that despite the linear in time growth is nonetheless time independent, and not only that, it leads to states with zero norm. In addition, as is again characteristic of PTPTPT symmetry, these non-normalizable states can be made normalizable by a continuation into a so-called Stokes wedge domain in the complex plane. In this domain one has a completely consistent quantum theory, one that lives alongside the standard normalizable energy eigenspectrum sector. This thus not quite so simple harmonic oscillator provides an explicit realization of our general contention that antilinearity is more basic to quantum theory than Hermiticity.
We develop a systematically general theory of one-dimensional (1D) non-Hermitian systems, elaborating on the energy bands, the band degeneracy, and the defectiveness of eigenstates under open boundary conditions. We analyze the band degeneracy and defectiveness of two typical 1D non-Hermitian models. We obtain the unusual presence and absence of the exceptional points in the generalized non-Hermitian Su-Schrieffer-Heeger model under open boundary conditions. Beyond the general theory, we discover that infernal points exist in 1D non-Hermitian systems, where the energy spectra under open boundary conditions converge on some discrete energy values. We analyze two relevant 1D non-Hermitian models with the existence of infernal points. Moreover, we generalize the infernal points to the infernal knots in four-dimensional systems. The general theory and the infernal points of non-Hermitian systems developed in this paper are also valid in Hermitian systems.