Extended Bloch theorem for topological lattice models with open boundaries

Abstract

While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible numerically or in idealized limits. Here we present exact analytic solutions for the boundary states in a number of lattice models of current interest, including nodal-line semimetals on a hyperhoneycomb lattice, spin-orbit coupled graphene, and three-dimensional topological insulators on a diamond lattice, for which no previous exact finite-size solutions are available in the literature. Furthermore, we identify spectral mirror symmetry as the key criterium for analytically obtaining the entire (bulk and boundary) spectrum as well as the concomitant eigenstates, and exemplify this for Chern and Z2 insulators with open boundaries of co-dimension one. In the case of the two-dimensional Lieb lattice, we extend this further and show how to analytically obtain the entire spectrum in the presence of open boundaries in both directions, where it has a clear interpretation in terms of bulk, edge, and corner states. arXiv:1812.03099v2 [cond-mat.mes-hall]

Enhanced localization and protection of topological edge states due to geometric frustration

“…In fact, one can show that this state is related to the square-root topological insulator [8][9][10][11][12][13] and the protecting symmetry is a sublattice chiral-like hidden symmetry. To our knowledge, topological characterization of a weak topological insulator with compact edge states has not been addressed in the literature where geometrically frustrated lattices are studied [14][15][16]. In our paper, we show that these compact localized states reflect a different topological transition point (the atomic limit in our case) and consequently they remain completely localized (in the system boundaries, edge or corner) even at the usual transition point t 1 = t 2 where the remaining edge states converge.…”

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

“…III to illustrate our findings, by dimensional extension it also describes two-and threedimensional systems with each of the sites now corresponding to one-dimensional, periodic chains and twodimensional, periodic planes, respectively [55]. We can thus use our results to find exact solutions for a plethora of different systems, such as for the chiral edge states of Chern insulators and Fermi arcs of Weyl semimetals [55] as well as for the helical edge states of two-dimensional Z 2 insulators and Dirac surface states of three-dimensional strong topological insulators [59] as summarized in Table I.…”

“…In that section, we also solve explicitly for the hinge modes on the pyrochlore lattice. The Lieb lattice is treated explicitly in a forthcoming publication [59], where we solve the complete eigensystem by making use of a symmetry that relates the energies E(k x , k y ) = E(−k x , k y ) = E(k x , −k y ). To construct the lattice in Fig.…”

“…Hinge insulators Strong TI [59], Weyl semimetals [55] TABLE I: Examples of D-dimensional models that host ddimensional boundary modes. Solvable models corresponding to each of these can be constructed using the framework presented in this work.…”

Topological phase transition independent of system non-Hermiticity

“…Non-Hermitian systems [9][10][11] exhibit many intriguing features and applications that not limited to power oscillation [12,13], coherent perfect absorption [14], unidirectionality behaviors [15][16][17][18][19], single-mode laser [20,21], robust energy transfer [22,23], and exceptional point (EP) enhanced sensing [24][25][26][27] due to its nonorthogonal eigenstates and the exotic topology of EPs [28][29][30][31][32][33]. The scope of topological phase of matter has also been extended to non-Hermitian region [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] and stimulates several interesting discussions on PT -symmetric topological interface states [52][53][54][55]<...>…”