Topological Phase Transition in non-Hermitian Quasicrystals

Abstract

The discovery of topological phases in non-Hermitian open classical and quantum systems challenges our current understanding of topological order. Non-Hermitian systems exhibit unique features with no counterparts in topological Hermitian models, such as failure of the conventional bulk-boundary correspondence and non-Hermitian skin effect. Advances in the understanding of the topological properties of non-Hermitian lattices with translational invariance have been reported in several recent studies, however little is known about non-Hermitian quasicrystals. Here we disclose topological phases in a quasicrystal with parity-time (PT ) symmetry, described by a non-Hermitian extension of the Aubry-André-Harper model. It is shown that the metal-insulating phase transition, observed at the PT symmetry breaking point, is of topological nature and can be expressed in terms of a winding number. A photonic realization of a non-Hermitian quasicrystal is also suggested.Introduction. The discovery of topological phases of matter has introduced a major twist in condensed matter physics [1, 2] with great impact in other areas of physics, such as photonics, atom optics, acoustics and mechanics [3][4][5][6][7][8]. Topological band theory classifies Hermitian topological systems depending on their dimensionality and symmetries [9, 10]. The bulk topological invariants are uniquely reflected in robust edge states localized at open boundaries. The ability to engineer non-Hermitian Hamiltonians, demonstrated in a series of recent experiments [11][12][13][14][15][16][17], and the related observation of unconventional topological boundary modes sparked a great interest to extend topological band theory to open systems . Striking features are the failure of the conventional bulk-boundary correspondence [27,31,35,[37][38][39], eigenstate condensation [30,31], non-Hermitian skin effect [33,37,38], and the sensitivity of the bulk spectra on boundary conditions [19,36,[39][40][41]. Most of previous studies have concerned with crystals, however little is know about topological properties of non-Hermitian quasicrystals. Quasicrystals (QCs) constitute an intermediate phase between fully periodic lattices and fully disordered media, showing a longrange order but no periodicity [42,43]. A paradigmatic model of a one-dimensional (1D) QC is provided by the Aubry-André-Harper (AAH) Hamiltonian [43][44][45], which is known to show a metal-insulator phase transition [44][45][46]. In the Hermitian case, the AAH Hamiltonian is topologically nontrivial because it can be mapped into a two-dimensional quantum Hall system on a square lattice [47][48][49][50]. A few recent studies have considered some non-Hermitian extensions of the AAH model [51][52][53][54][55][56], mainly with a commensurate potential and with open boundary conditions. Such numerical studies investigated how gain and loss distributions affect edge states and parity-time (PT ) symmetry breaking [51][52][53]55], the Hofstadter butterfly spectrum [52], and the localization properties of eig...

Metal-insulator phase transition in a non-Hermitian Aubry-André-Harper model

“…On the other hand, for real-energy on-site potential disorder a non-Hermitian delocalization transition is observed * stefano.longhi@polimi.it by application of an imaginary gauge field (Hatano-Nelson-Anderson model [23-26, 29, 33, 39, 46, 47, 49]). Other studies focused on several non-Hermitian extensions of diagonal or off-diagonal AAH models [31, 35-37, 40, 41, 43, 44, 47-50], with either commensurate or incommensurate potential, showing the impact of non-Hermiticity terms in the Hamiltonian on edge states and parity-time (PT ) symmetry breaking [35-37, 41, 43], on the Hofstadter butterfly spectrum [36], and on the localization properties of eigenstates [43,44,[47][48][49][50]. Recently, the topological nature of a metal-insulator phase transition found in an incommensurate PT -symmetric AAH model has been revealed [48].…”

“…Other studies focused on several non-Hermitian extensions of diagonal or off-diagonal AAH models [31, 35-37, 40, 41, 43, 44, 47-50], with either commensurate or incommensurate potential, showing the impact of non-Hermiticity terms in the Hamiltonian on edge states and parity-time (PT ) symmetry breaking [35-37, 41, 43], on the Hofstadter butterfly spectrum [36], and on the localization properties of eigenstates [43,44,[47][48][49][50]. Recently, the topological nature of a metal-insulator phase transition found in an incommensurate PT -symmetric AAH model has been revealed [48]. Most of available results on localization and phase transition phenomena in non-Hermitian AAH models are based on numerical simulations, while few analytical and rigorous results are available owing to the extraordinary complexity of the spectral problem.…”

Topological Origin of Non-Hermitian Skin Effects

“…As a result, point-gapped topological phases cannot always be continuously deformed into any Hermitian counterparts; topology for a point gap is intrinsic to non-Hermitian systems in sharp contrast to a line gap. A point gap describes unique non-Hermitian topological phenomena such as localization transitions [1,2,46,52,58] and emergence of exceptional points [21-26, 34, 37, 49, 50, 53, 55].A hallmark of topological phases is the presence of the localized states at the boundaries as a result of nontrivial topology of the bulk [61-63]. Remarkably, non-Hermiticity alters the nature of the bulk-boundary correspondence (BBC) .…”

Anomalous features of non-Hermitian topological states

“…Then more complex models has been introduced and studied in the literature . In a recent paper, it was shown that topological phase can also arise in a non-Hermitian quasicrystal with parity-time symmetry [66]. Note that topological edge states in a non-Hermitian system can have real or complex energy eigenvalues.…”