We use multi-band SSH models to demonstrate a prescription for calculating the correct Zak phase and winding number of multi-band systems. We verify our prescription by comparing the resultant winding number of a four-band SSH model with the edge states of a semi-infinite chain which we find by solving the equation of motion. We then also carry out an extensive comparison with the numerical results obtained by solving the matrix eigenvalue problem of finite chains with various number of sites. As a double check of our prescription, we also confirm the bulk edge correspondence in a six-band SSH model. Similar to the usual SSH model, the winding numbers associated to the left and right boundaries in a finite chain may be different if space inversion symmetry is violated in the system. We believe the prescription we propose here may also be applied to other 1D multi-band systems.
Bulk-edge correspondence is one of the most distinct properties of topological insulators. In particular, the 1D winding number $\n$ has a one-to-one correspondence to the number of edge states in a chain of topological insulators with boundaries. By properly choosing the unit cells, we carry out numerical calculation to show explicitly in the extended SSH model that the winding numbers corresponding to the left and right unit cells may be used to predict the numbers of edge states on the two boundaries in a finite chain. Moreover, by drawing analogy between the SSH model and QWZ model, we sho…
Non-Hermitian generalizations of the Su-Schrieffer-Heeger (SSH) models with higher periods of the hopping coefficients, called the SSH3 and SSH4 models, are analyzed. The conventional construction of the winding number fails for the Hermitian SSH3 model, but the non-Hermitian generalization leads to a topological system due to a point gap on the complex plane. The non-Hermitian SSH3 model thus has a winding number and exhibits the non-Hermitian skin effect. Moreover, the SSH3 model has two types of localized states and a zero-energy state associated with special symmetries. The total Zak phase…
In this paper, we discuss the topological and transport properties of binary heterostructures of different topological materials. Using the Su-Schrieffer-Heeger model and the method of Green's functions, we calculate and characterize the edge states of these heterostructures. We study the topological phase through an invariant calculated from the Zak phase in order to build phase diagrams. We obtained an analytical result that allows us to find the topological phase diagram for a one-dimensional general chain and for heterostructures from the cross band condition. We also calculate the differe…
We have studied topological edge states in bowtie ladders with various edge truncations. The symmetric bowtie ladder, which comprises two trivial Su-Schrieffer-Heeger (SSH) lattices, exhibits an insulator-metal transition with trivial insulating states. On the other hand, the lattice can be transformed into an extended SSH lattice depending on the edge shapes with non-trivial insulating states in that the winding number is non-zero. The winding numbers are permutationally designated in the phase diagram depending on the choice of unit cell. The topological edge states are affected by the shape…
The Su, Schrieffer and Heeger (SSH) model, describing the soliton excitations in polyacetylene due to the formation of antiphase domain walls (DW) from the alternating bond pattern, has served as a paradigmatic example of one-dimensional (1D) chiral topological insulators. While the SSH model has been realized in photonic and plasmonic systems, there have been limited analogues in three-dimensional (3D) electronic systems, especially regarding the formation of antiphase DWs. Here, we propose that pristine bulk Bi, in which the dimerization of $(111)$ atomic layers renders alternating covalent …
Nonlinearity induced topological properties in nonlinear lattice systems are studied in both momentum space and real space. Experimentally realizable through the Kerr effect on photonic waveguide systems, our working model depicts on-site nonlinearity added to the Su-Schrieffer-Heeger (SSH) model plus a chiral-symmetry breaking term. Under the periodic-boundary condition, two of the nonlinear energy bands approach the energy bands of the chiral-symmetric SSH model as nonlinearity strength increases. Further, we account for a correction to the Zak phase and obtain a general expression for nonli…
The discrete Hamiltonian of Su, Schrieffer and Heeger (SSH) is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on the values $0$ and $1$ labeling the two distinct phases. We display two homotopically equivalent continuum Ham…
We study edge states in AB-stacked bilayer graphene (BLG) ribbon where the Chern number of the corresponding two-dimensional (2D) bulk Hamiltonian is zero. The existence and topological features of edge states when two layers ended with the same or different edge terminations (zigzag, bearded, armchair) are discussed. The edge states (non-dispersive bands near the Fermi level) are states localized at the edge of graphene nanoribbon that only exists in certain range of momentum $k_y$. Their existence near the Fermi level are protected by the chiral symmetry with topology well described by coupl…
A remarkable feature of the trimer Su-Schrieffer-Heeger (SSH3) model is that it supports localized edge states. Although Zak's phase remains quantized for the case of a mirror-symmetric chain, it is known that it fails to take integer values in the absence of this symmetry and thus it cannot play the role of a well-defined bulk invariant in the general case. Attempts to establish a bulk-edge correspondence have been made via Green's functions or through extensions to a synthetic dimension. Here we propose a simple alternative for SSH3, utilizing the previously introduced sublattice Zak's phase…
The topological invariants of band insulators are usually assumed to depend only on the connectivity between orbitals and not on their intra-cell position (orbital embedding), which is a separate piece of information in the tight-binding description. For example, in two dimensions, the orbital embedding is known to change the Berry curvature but not the Chern number. Here, we consider one-dimensional inversion-symmetric insulators classified by a $\mathbb{Z}_2$ topological invariant $\vartheta=0$ or $π$, related to the Zak phase, and show that $\vartheta$ crucially depends on orbital embedding…
The Connection of Topology between Systems with Different Dimensions: 1D Zak Phases to 2D Chern Number, Weyl Point as the Jumping Channel for One Singularity and Nodal Line to Merge All Singularities
We construct microscopical models of one-dimensional non-interacting topological insulators in all of the chiral universality classes. Specifically, we start with a deformation of the Su-Schrieffer-Heeger (SSH) model that breaks time-reversal symmetry, which is in the AIII class. We then couple this model to its time-reversal counterpart in order to build models in the classes BDI, CII, DIII and CI. We find that the $\mathbb{Z}$ topological index (the winding number) in individual chains is defined only up to a sign. This comes from noticing that changing the sign of the chiral symmetry operat…
Topological characterization of non-Hermitian band structures demands more than a straightforward generalization of the Hermitian cases. Even for one-dimensional tight-binding models with nonreciprocal hopping, the appearance of point gaps and the skin effect leads to the breakdown of the usual bulk-boundary correspondence. Luckily, the correspondence can be resurrected by introducing a winding number for the generalized Brillouin zone for systems with even number of bands and chiral symmetry. Here, we analyze the topological phases of a nonreciprocal hopping model on the stub lattice, where o…
Here, we introduce and apply non-Abelian tensor Berry connections to topological phases in multi-band systems. These gauge connections behave as non-Abelian antisymmetric tensor gauge fields in momentum space and naturally generalize Abelian tensor Berry connections and ordinary non-Abelian (vector) Berry connections. We build these novel gauge fields from momentum-space Higgs fields, which emerge from the degenerate band structure of degenerate-band models. Firstly, we show that the conventional topological invariants of two-dimensional topological insulators and three-dimensional Dirac semim…
