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Register a new userTopological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems
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Motivated by recent experimental realizations of topological edge states in Su-Schrieffer-Heeger (SSH) chains, we theoretically study a ladder system whose legs are comprised of two such chains. We show that the ladder hosts a rich phase diagram and related edge mode structure dictated by choice of inter-chain and intra-chain couplings. Namely, we exhibit three distinct physical regimes: a topological hosting localized zero energy edge modes, a topologically trivial phase having no edge mode structure, and a regime reminiscent of a weak topological insulator having unprotected edge modes resembling a twin-SSH construction. In the topological phase, the SSH ladder system acts as an analog of the Kitaev chain, which is known to support localized Majorana fermion end modes, with the difference that bound states of the SSH ladder having the same spatial wavefunction profiles correspond to Dirac fermion modes. Further, inhomogeneity in the couplings can have a drastic effect on the topological phase diagram of the ladder system. In particular for quasiperiodic variations of the inter-chain coupling, the phase diagram reproduces Hofstadters butterfly pattern. We thus identify the SSH ladder system as a potential candidate for experimental observation of such fractal structure.
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We address the conditions required for a $mathbb{Z}$ topological classification in the most general form of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. Any chirally-symmetric SSH model will possess a conjugated-pseudo-Hermiticity which we show is responsible for a quantized complex Berry phase. Consequently, we provide the first example where the complex Berry phase of a band is used as a quantized invariant to predict the existence of gapless edge modes in a non-Hermitian model. The chirally-broken, $PT$-symmetric model is studied; we suggest an explanation for why the topological invariant is a global property of the Hamiltonian. A geometrical picture is provided by examining eigenvector evolution on the Bloch sphere. We justify our analysis numerically and discuss relevant applications.
In this work, we study the topological phases of the dimerized square lattice in the presence of an external magnetic field. The dimerization pattern in the lattices hopping amplitudes can induce a series of bulk energy gap openings in the Hofstadter spectrum at certain fractional fillings, giving rise to various topological phases. In particular, we show that at $frac{1}{2}$-filling the topological quadrupole insulator phase with a quadrupole moment quantized to $frac{e}{2}$ and associated corner-localized mid-gap states exists in certain parameter regime for all magnetic fluxes. At $frac{1}{4}$ filling, the system can host obstructed atomic limit phases or Chern insulator phases. For those configurations gapped at fillings below $frac{1}{4}$, the system is in Chern insulator phases of various non-vanishing Chern numbers. Across the phase diagram, both bulk-obstructed and boundary-obstructed topological phase transitions exist in this model.
In this paper we study the formation of topological Tamm states at the interface between a semi-infinite one-dimensional photonic-crystal and a metal. We show that when the system is topologically non-trivial there is a single Tamm state in each of the band-gaps, whereas if it is topologically trivial the band-gaps host no Tamm states. We connect the disappearance of the Tamm states with a topological transition from a topologically non-trivial system to a topologically trivial one. This topological transition is driven by the modification of the dielectric functions in the unit cell. Our interpretation is further supported by an exact mapping between the solutions of Maxwells equations and the existence of a tight-binding representation of those solutions. We show that the tight-binding representation of the 1D photonic crystal, based on Maxwells equations, corresponds to a Su-Schrieffer-Heeger-type model (SSH-model) for each set of pairs of bands. Expanding this representation near the band edge we show that the system can be described by a Dirac-like Hamiltonian. It allows one to characterize the topology associated with the solution of Maxwells equations via the winding number. In addition, for the infinite system, we provide an analytical expression for the photonic bands from which the band-gaps can be computed.
We investigate topological and disorder effects in non-Hermitian systems with chiral symmetry. The system under consideration consists in a finite Su-Schrieffer-Heeger chain to which two semi-infinite leads are attached. The system lacks the parity-time and time-reversal symmetries and is appropriate for the study of quantum transport properties. The complex energy spectrum is analyzed in terms of the chain-lead coupling and chiral disorder strength, and shows substantial differences between chains with even and odd number of sites. The mid-gap edge states acquire a finite lifetime and are both of topological origin or generated by a strong coupling to the leads. The disorder induces coalescence of the topological eigenvalues, associated with exceptional points and vanishing of the eigenfunction rigidity. The electron transmission coefficient is approached in the Landauer formalism, and an analytical expression for the transmission in the range of topological states is obtained. Notably, the chiral disorder in this non-Hermitian system induces unitary conductance enhancement in the topological phase.
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