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arXiv:1404.3662 · Europhysics Letters (EPL)

Exceptional points and Bloch oscillations in non-Hermitian lattices with unidirectional hopping

Stefano Longhi

Published May 1, 2014• 29 citations
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Abstract

The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open) and for the ring lattice geometries the spectrum is complex, lattice truncation makes the spectrum real. However, an exceptional point of order equal to the number of lattice sites emerges. When a homogeneous dc force is applied to the lattice, in all cases an equally-spaced real Wannier-Stark ladder spectrum is obtained, corresponding to periodic oscillatory dynamics in real space. Possible physical realizations of non-Hermitian lattices with unidirectional hopping are briefly discussed.

DOI
10.1209/0295-5075/106/34001

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The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open) and for the ring lattice geometries the spectrum is complex, lattice truncation makes the spectrum real. However, an exceptional point of order equal to the number of lattice sites emerges. When a homogeneous dc force is applied to the lattice, in all cases an equally-spaced real Wannier-Stark ladder spectrum is obtained, corresponding to periodic oscillatory dynamics in real space. Possible physical realizations of non-Hermitian lattices with unidirectional hopping are briefly discussed.
We consider non-Hermitian dynamics of a quantum particle hopping on a one-dimensional tight-binding lattice made of NNN sites with asymmetric hopping rates induced by a time-periodic oscillating imaginary gauge field. A deeply different behavior is found depending on the lattice topology. While in a linear chain (open boundary conditions) an oscillating field can lead to a complex quasienergy spectrum via a multiple parametric resonance; in a ring topology (Born--von Karman periodic boundary conditions) an entirely real quasienergy spectrum can be found and the dynamics is pseudo-Hermitian. In the large-NNN limit, parametric instability and pseudo-Hermitian dynamics in the two different lattice topologies are physically explained on the basis of a simple picture of wave-packet propagation.
Bloch oscillation (BO), i.e., the oscillatory motion of a quantum particle in a periodic potential, is one of the most striking effects of coherent quantum transport in matter. In the semiclassical picture, it is well known that BOs can be explained owing to the periodic band structure of the crystal and the so-called acceleration theorem: since in the momentum space the particle wave packet drifts with a constant speed without being distorted, in real space the probability distribution of the particle undergoes a periodic motion following a trajectory which exactly reproduces the shape of the lattice band. In non-Hermitian lattices with a complex (i.e., not real) energy band, extension of the semiclassical model is not intuitive. Here we show that the acceleration theorem holds for non-Hermitian lattices with a complex energy band only on average, and that the periodic wave-packet motion of the particle in real space is described by a trajectory in the complex plane, i.e., it generally corresponds to reshaping and breathing of the wave packet in addition to a transverse oscillatory motion. The concept of BOs involving complex trajectories is exemplified by considering two examples of non-Hermitian lattices with a complex band dispersion relation, including the Hatano-Nelson tight-binding Hamiltonian describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential and a tight-binding lattice with imaginary hopping rates.
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.
We analyze the scattering dynamics and spectrum of a quantum particle on a tight-binding lattice subject to a non-Hermitian (purely imaginary) local potential. The reflection, transmission, and absorption coefficients are studied as a function of the strength of this absorbing potential. The system is found to have an exceptional point at a certain strength of the potential. Unusually, all (or nearly all) of the spectrum pairs up into mutually coalescing eigenstate pairs at this exceptional point. At large potential strengths, the absorption coefficient decreases and the effect of the imaginary potential is similar to that of a real potential. We quantify this similarity by utilizing properties of a localized eigenstate.
We investigate the quantum dynamics of a one-dimensional tight-binding lattice driven by a spatially quadratic and time-periodic potential. Both Hermitian (J1=J2J_1 = J_2J1​=J2​) and non-Hermitian (J1≠J2J_1 \neq J_2J1​=J2​) hopping regimes are analyzed. Within the framework of Floquet theory, the time-dependent Hamiltonian is mapped onto an effective static Floquet Hamiltonian, enabling a detailed study of the quasi-energy spectrum as function of the driving frequency ωωω. By applying a gauge transformation, we find that critical frequencies ωcω_cωc​ emerge, at which nearly equidistant quasi-energy ladders appear, as revealed by a pronounced minimum in the normalized variance Δ(ω)Δ(ω)Δ(ω) of the level spacings. This spectral regularity leads to robust periodic revivals and Bloch-like oscillations in the time evolution. Numerical simulations confirm that such coherent oscillations persist even in the non-Hermitian regime, where the periodic driving stabilizes an almost real and uniformly spaced quasi-energy ladder.
Many features of Bloch oscillations in one-dimensional quantum lattices with a static force can be described by quasiclassical considerations for example by means of the acceleration theorem, at least for Hermitian systems. Here the quasiclassical approach is extended to non-Hermitian lattices, which are of increasing interest. The analysis is based on a generalised non-Hermitian phase space dynamics developed recently. Applications to a single-band tight-binding system demonstrate that many features of the quantum dynamics can be understood from this classical description qualitatively and even quantitatively. Two non-Hermitian and PT -symmetric examples are studied, a Hatano–Nelson lattice with real coupling constants and a system with purely imaginary couplings, both for initially localised states in space or in momentum. It is shown that the time-evolution of the norm of the wave packet and the expectation values of position and momentum can be described in a classical picture.