Exact edge, bulk, and bound states of finite topological systems

Abstract

Finite topologically non-trivial systems are characterised, among many other unique properties, by the presence of bound states at their physical edges. These topological edge modes can be distinguished from usual Shockley waves energetically, as their energies remain finite and in-gap even when the boundaries of the system represent an effectively infinite and sharp energetic barrier. Theoretically, the existence of topological edge modes can be shown by means of the bulkedge correspondence and topological invariants. On a clean one-dimensional lattice and reducible two-dimensional models, in either the commensurate or semi-infinite case, the edge modes can be essentially obtained analytically, as shown in [Phys. Rev. Lett. 71, 3697 (1993)] and [Phys. Rev. A 89, 023619 (2014)]. In this work, we put forward a method for obtaining the spectrum and wave functions of topological edge modes for arbitrary finite lattices, including the incommensurate case. A small number of parameters are easily determined numerically, with the form of the eigenstates remaining fully analytical. We also obtain the bulk modes in the finite system analytically and their associated eigenenergies, which lie within the infinite-size limit continuum. Our method is general and can be easily applied to obtain the properties of non-topological models and/or extended to include impurities. As an example, we consider a relevant case of an impurity located next to one edge of a one-dimensional system, equivalent to a softened boundary in a separable two-dimensional model. We show that a localised impurity can have a drastic effect on the original topological edge modes of the system. Using the periodic Harper and Hofstadter models to illustrate our method, we find that, on increasing the impurity strength, edge states can enter or exit the continuum, and a trivial Shockley state bound to the impurity may appear. The fate of the topological edge modes in the presence of impurities can be addressed by quenching the impurity strength. We find that at certain critical impurity strengths, the transition probability for a particle initially prepared in an edge mode to decay into the bulk exhibits discontinuities that mark the entry and exit points of edge modes from and into the bulk spectrum.

Joint entanglement of topology and polarization enables error-protected quantum registers

“…We also see that, unlike the topologically trivial case of the previous figure, a stable state accumulates at the boundary, as can be clearly seen in the next figure. The initial positions in parts (a)-(c) are, respectively, 68, 85, and 88. points of topological phase change as considered here (see also [43] for an alternative approach), but something very similar happens at many other types of sudden inhomogeneities, including boundaries between regions governed by nonrelativistic (Schrödinger) and relativistic (Dirac) dynamics [44][45][46].…”

Extended Bloch theorem for topological lattice models with open boundaries

“…We, moreover, note that the approach for finding bulk modes presented in Ref. 39 for systems with open boundary conditions in one direction carries some resemblance to our method in Sec. II B.…”

“…Whereas the focus in Ref. 39 is on timereversal-symmetric models, we observe that the minimal generic requirement for using this method is in fact the presence of a spectral mirror symmetry relating the open boundaries.…”

“…On these lattices, we find n exact solutions, where n is the number of degrees of freedom of the A motifs, which exponentially localize to the boundaries with a completely disappearing amplitude on the B-type motifs and whose eigenenergies correspond to the eigenenergies of the local Hamiltonian on A. Other methods also exist [34][35][36][37][38][39][40][41], which are restricted to finding solutions for models with open boundary conditions in one direction only, while a generalization to higherorder phases is also made in an approximate fashion in Ref. 42.…”

Boundaries of boundaries: A systematic approach to lattice models with solvable boundary states of arbitrary codimension

“…While solutions for first-order topological boundary states have been derived in several specific models [34][35][36][37][38][39][40][41][42][43][44] and several general approaches have been developed to retrieve them [45][46][47][48], there is a surprising lack of methods to find analytical solutions for these boundary wave functions that are straightforward and transparent and can be used to describe modes of any codimension. Such a method is not only of theoretical relevance but also provides practical insight on how to engineer lattices that support these states.…”