We show that one-dimensional quasiperiodic optical lattice systems can exhibit edge states and topological phases which are generally believed to appear in two-dimensional systems. When the Fermi energy lies in gaps, the Fermi system on the optical superlattice is a topological insulator characterized by a nonzero topological invariant. The topological nature can be revealed by observing the density profile of a trapped fermion system, which displays plateaus with their positions uniquely determined by the ration of wavelengths of the bichromatic optical lattice. The butterflylike spectrum of the superlattice system can be also determined from the finite-temperature density profiles of the trapped fermion system. This finding opens an alternative avenue to study the topological phases and Hofstadter-like spectrum in one-dimensional optical lattices.
Non-Hermiticity from non-reciprocal hoppings has been shown recently to demonstrate the non-Hermitian skin effect (NHSE) under open boundary conditions (OBCs). Here we study the interplay of this effect and the Anderson localization in a non-reciprocal quasiperiodic lattice, dubbed nonreciprocal Aubry-André model, and a rescaled transition point is exactly proved. The non-reciprocity can induce not only the NHSE, but also the asymmetry in localized states with two Lyapunov exponents for both sides. Meanwhile, this transition is also topological, characterized by a winding number associated with the complex eigenenergies under periodic boundary conditions (PBCs), establishing a bulk-bulk correspondence. This interplay can be realized by an elaborately designed electronic circuit with only linear passive RLC devices instead of elusive non-reciprocal ones, where the transport of a continuous wave undergoes a transition between insulating and amplifying. This initiative scheme can be immediately applied in experiments to other non-reciprocal models, and will definitely inspires the study of interplay of NHSEs and more other quantum/topological phenomena.
We study the competition of disorder and superconductivity for a one-dimensional p-wave superconductor in incommensurate potentials. With the increase in the strength of the incommensurate potential, the system undergoes a transition from a topological superconducting phase to a topologically trivial localized phase. The phase boundary is determined both numerically and analytically from various aspects and the topological superconducting phase is characterized by the presence of Majorana edge fermions in the system with open boundary conditions. We also calculate the topological Z2 invariant of the bulk system and find it can be used to distinguish the different topological phases even for a disordered system.
Nonlinear transmission lines (NLTLs) are nonlinear electronic circuits used for parametric amplification and pulse generation, and it is known that left-handed NLTLs support enhanced harmonic generation while suppressing shock wave formation. We show experimentally that in a left-handed NLTL analogue of the Su-Schrieffer-Heeger (SSH) lattice, harmonic generation is greatly increased by the presence of a topological edge state. Previous studies of nonlinear SSH circuits focused on solitonic behaviours at the fundamental harmonic. Here, we show that a topological edge mode at the first harmonic can produce strong propagating higher-harmonic signals, acting as a nonlocal cross-phase nonlinearity. We find maximum third-harmonic signal intensities five times that of a comparable conventional left-handed NLTL, and a 250-fold intensity contrast between topologically nontrivial and trivial configurations. This work advances the fundamental understanding of nonlinear topological states, and may have applications for compact electronic frequency generators.
Non-Hermitian systems can exhibit exotic topological and localization properties. Here we elucidate the non-Hermitian effects on disordered topological systems by studying a nonreciprocal disordered Su-Schrieffer-Heeger model. We show that the non-Hermiticity can enhance the topological phase against disorders by increasing energy gaps. Moreover, we uncover a topological phase which emerges only under both moderate non-Hermiticity and disorders, and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge modes. Such topological phases induced by the combination of non-Hermiticity and disorders are dubbed non-Hermitian topological Anderson insulators. We also find that the system has unique non-monotonous localization behaviour and the topological transition is accompanied by an Anderson transition.Topological states of matter have been widely explored in condensed-matter materials [1-5] and various engineered systems, which include ultracold atoms [6-8], photonic lattices [9, 10], mechanic systems [11], classic electronic circuits [12][13][14][15], and superconducting circuits [16][17][18][19][20]. One hallmark of topological insulators is the robustness of nontrivial boundary states against certain types of weak disorders, since the topological band gap (topological invariants) preserves under these perturbations [1][2][3]. However, the band gap closes for sufficiently strong disorders and the system becomes trivial as all states are localized according to the Anderson localization [21]. Surprisingly, there is a topological phase driven from a trivial phase by disorders, known as topological Anderson insulator (TAI) [22]. The TAI was first predicted in two-dimensional (2D) quantum wells and then shown to exhibit in a wide range of systems [22][23][24][25][26][27][28][29][30], such as Su-Schrieffer-Heeger (SSH) chains [31]. Recently, the TAI has been observed in one-dimensional (1D) cold atomic wires and 2D photonic waveguide arrays [32,33].
We study the p-wave superconducting wire with a periodically modulated chemical potential and show that the Majorana edge states are robust against the periodic modulation. We find that the critical amplitude of modulated potential, at which the Majorana edge fermions and topological phase disappear, strongly depends on the phase shifts. For some specific values of the phase shift, the critical amplitude tends to infinity. The existence of Majorana edge fermions in the open chain can be characterized by a topological Z2 invariant of the bulk system, which can be applied to determine the phase boundary between the topologically trivial and nontrivial superconducting phases. We also demonstrate the existence of the zero-energy peak in the spectral function of the topological superconducting phase, which is only sensitive to the open boundary condition but robust against the disorder.
We study the emergence and disappearance of defect states in the complex Su-Schrieffer-Heeger (cSSH) model, a non-Hermitian one-dimensional lattice model containing gain and loss on alternating sites. Previous studies of this model have focused on the existence of a non-Hermitian defect state that is localized to the interface between two cSSH domains, and is continuable to the topologically protected defect state of the Hermitian Su-Schrieffer-Heeger (SSH) model. For large gain/loss magnitudes, we find that these defect states can disappear into the continuum, or undergo pairwise spontaneous breaking of a composite sublattice/time-reversal symmetry. The symmetry-breaking transition gives rise to a pair of defect states continuable to non-topologically-protected defect states of the SSH model. We discuss the phase diagram for the defect states, and its implications for non-Hermitian defect states. arXiv:1807.07776v1 [cond-mat.mes-hall]
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