Of bulk and boundaries: Generalized transfer matrices for tight-binding models

Abstract

We construct a generalized transfer matrix corresponding to noninteracting tight-binding lattice models, which can subsequently be used to compute the bulk bands as well as the edge states. Crucially, our formalism works even in cases where the hopping matrix is non-invertible. Following Hatsugai [PRL 71, 3697 (1993)], we explicitly construct the energy Riemann surfaces associated with the band structure for a specific class of systems which includes systems like Chern insulator, Dirac semimetal and graphene. The edge states can then be interpreted as non-contractible loops, with the winding number equal to the bulk Chern number. For these systems, the transfer matrix is symplectic, and hence we also describe the windings associated with the edge states on Sp(2, R) and interpret the corresponding winding number as a Maslov index.

Spectral and transmission properties of periodic 1D tight-binding lattices with a generic unit cell: an analysis within the transfer matrix approach

“…In figure 9, we illustrate the DOS of a system with three sites per unit cell, for the parameters used in figure 5, as obtained by equation (32). As a careful comparison of figures 5(b) and 9 suggests, the relative sharpness of the VHS as well as the position of each band's DOS minimum can be expected from the density of points in the eigenspectrum, as N increases.…”

“…Apart from the interest these systems have in themselves, they could also serve as a starting point to gain more understanding about the inclusion of chemical complexity effects in the electronic structure of periodic lattices, as well as of quasi-periodic, fractal and amorphous atomic or molecular sequences. Various quantum chain models, which are studied in the context of low-dimensional systems, have been treated with the transfer matrix method [28][29][30][31][32]. Here, we use the transfer matrix method [33,34] to solve the TB system of equations and determine expressions for its eigenvalues, for both cyclic and fixed boundary conditions, by combining the spectral duality relations [35,36] with the connection of the elements of the powers of a 2×2 unimodular matrix to the Chebyshev polynomials of the second kind [37,38].…”

Oriented propagation of magnetization due to chiral edge modes in Kitaev-type models

“…To obtain the exact solution of the chiral edge mode, we employ a transfer matrix method. Although the method is already established as a standard tool [24][25][26][27] , known exact solutions of chiral edge states 27,28 are still rare so far. Even in the case of the well-known Haldane tight-binding model on the honeycomb lattice [29][30][31][32][33] , the exact solution of the chiral edge state has not yet been obtained.…”

Connecting the dots: Time-reversal symmetric Weyl semimetals with tunable Fermi arcs

“…Using the methods proposed in Ref. 16 to construct a transfer matrix for systems with a noninvertible hopping matrix, we derive (for details, see Appendix A):…”