Non-Bloch-Band Collapse and Chiral Zener Tunneling

Abstract

Non-Bloch band theory describes bulk energy spectra and topological invariants in non-Hermitian crystals with open boundaries, where the bulk eigenstates are squeezed toward the edges (skin effect). However, the interplay of non-Bloch band theory, skin effect and coherent Bloch dynamics is so far unexplored. In two-band non-Hermitian lattices, it is shown here that collapse of non-Bloch bands and skin modes deeply changes the Bloch dynamics under an external force. In particular, for resonance forcing non-Bloch band collapse results in Wannier-Stark ladder coalescence and chiral Zener tunneling between the two dispersive Bloch bands.Introduction. Bloch band theory is the fundamental tool to describe electronic states in crystals [1]. Under the action of a weak dc electric field, Bloch theory predicts that electrons undergo an oscillatory motion, the famous Bloch oscillations (BOs) [1], which can be explained from the formation of a Wannier-Stark (WS) ladder energy spectrum [2]. Transitions among different bands occur for stronger fields because of Zener tunneling (ZT) [3]. BOs and ZT are ubiquitous phenomena of coherent wave transport in periodic media and have been observed in a wide variety of physical systems [4][5][6][7][8][9][10][11][12][13][14]. It is remarkable that, after one century from the seminal paper by Felix Bloch [1], there are still treasures to be uncovered within Bloch band theory. For example, Bloch band theory is central in the understanding of topological insulators [15][16][17] and in the description of flat band systems showing unconventional localization, anomalous phases, and strongly correlated states of matter [18][19][20][21][22][23][24][25][26][27][28]. Recently, a great interest is devoted to extend Bloch band theory and topological order to non-Hermitian lattices . In Bloch band theory, an electronic state is defined by the quasi-momentum k, which spans the first Brillouin zone, and is delocalized all along the crystal, regardless periodic boundary conditions (PBC) or open boundary conditions (OBC) are assumed. However, in non-Hermitian crystals something strange happens: energy bands in crystals with OBC are described by non-Bloch bands that deviate from ordinary Bloch bands; bulk eigenstates cease to be delocalized and get squeezed toward the edges (non-Hermitian skin effect); and the bulk-boundary correspondence based on Bloch topological invariants generally fails to correctly predict the existence of topological zero-energy modes [29, 33-39, 45, 56, 59]. Recent seminal works [35,37,45] showed that to correctly describe energy spectra and topological invariants in crystals with OBC one needs to extend Bloch band theory so as the quasi-momentum k becomes complex and varies on a generalized Brillouin zone. Bloch and non-Bloch bands can show different symmetry breaking phase transitions and, interestingly, band flat-

Critical non-Hermitian skin effect

“…In non-Hermitian systems with unbalanced gain and loss, the spectra under periodic boundary conditions (PBCs) and open-boundary conditions (OBCs) can be very different 28 , 29 , 31 , 43 – 45 . Indeed, under OBC, eigenstates due to NHSE can exponentially localize at a boundary, in contrast to Bloch states under PBCs.…”

“…Without couplings ( t 0 = 0), the two chains under OBC respectively yields a Jordan-block Hamiltonian matrix in real space, with the spectrum given by E = ± V . Because the eigenstates of the decoupled chains are exclusively localized at the first or the last site, their GBZs collapse 45 . By contrast, for any t 0 ≠ 0, is irreducible (here from Eq.…”

Untying links through anti-parity-time-symmetric coupling

“…If the non-Hermiticity is = (0, 0, γ ), the topological phase transition and the existence of the edge state are unaltered because of the pseudo-anti-Hermiticity protection [34]; the topological properties of the non-Hermitian system are inherited by the EPs (exceptional rings or exceptional surfaces in 2D or 3D) [60][61][62][63][64][65][66][67]. If the non-Hermiticity is = (0, γ , 0), the non-Hermitian skin effect occurs under open boundary condition [54][55][56][57][58][59][68][69][70][71][72][73][74][75][76][77][78], the non-Hermitian Aharonov-Bohm effect under periodical boundary condition invalidates the conventional bulk-boundary correspondence [54], and the non-Bloch band theory is developed for topological characterization [77][78][79][80][81][82]. Here, the dissipation induced anti-PT -symmetric coupling corresponds to the imaginary part = (γ , 0, 0).…”

Non‐Adiabatic Transitions in Parabolic and Super‐Parabolic PT‐Symmetric Non‐Hermitian Systems in 1D Optical Waveguides

“…[18] When encircling an exceptional point, there is an unconventional level-crossing behavior, accompanied by a phase change of one eigenstate but not of the other. [3,4] A particularly intriguing behavior is that dynamically encircling an exceptional point leads to chiral behaviors, [63] such that the encircling direction of the exceptional point determines the final output state. [64] In this work, we consider the dynamics of a non-Hermitian  -symmetric system which directly goes through an assembly of exceptional points.…”