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

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 noninvertible. Following Hatsugai [Phys. Rev. Lett. 71, 3697 (1993)], we explicitly construct the energy Riemann surfaces associated with the band structure for a specific class of systems which includes systems such as Chern insulator, Dirac semimetal, and graphene. The edge states can then be interpreted as noncontractible 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 $\mathrm{Sp}(2,\mathbb{R})$ and interpret the corresponding winding number as a Maslov index.

Figure 1

(a) A schematic depiction of the recursion relation, with $q$ internal degrees of freedom, range of interaction $R=2$, and Dirichlet boundary conditions at the left edge. We can form blocks (supercells) of such sites with two sites each, so the there is only nearest-neighbor hopping between them. (b) A simplified depiction of the reduced recursion relation, with $\mathbf{\alpha },\mathbf{\beta },\mathbf{\gamma }$ corresponding to the coefficients of $V,W$, and $X$ subspaces (introduced in Sec. 2b), respectively. (c) We club together ${\mathbf{\beta }}_n$ with ${\mathbf{\alpha }}_{n-1}$ to obtain ${\mathrm{\Phi }}_n$, which is translated by one step using the transfer matrix.

Figure 2

Diagrammatic representation of the recursion relations [Eq. (20)] for (a) ${\mathbf{\alpha }}_n$ and (b) ${\mathbf{\beta }}_n$.

Figure 3

The spectrum of Hofstadter model, with the band edges (dark blue) computed using the transfer matrix formalism and the left and right edge state dispersions (dashed and dashed-dotted curves) from the Evans equation (73), overlaid on the spectrum computed using exact diagonalization for a (top) commensurate and (bottom) incommensurate system. Note that in the latter case, the edge states seen in exact diagonalization exactly follow the winding right edge state obtained from the transfer matrix.

Figure 4

The spectrum of Chern insulator for $m=0.8$, with the band edges (dark blue) computed using the transfer matrix formalism and the left and right edge state dispersions (dashed and dashed-dotted curves) from the Evans equation (73), overlaid on the spectrum computed using exact diagonalization for a (top) commensurate and (bottom) incommensurate system. Note that in the latter case, the edge states seen in exact diagonalization exactly follow the winding right edge state obtained from the transfer matrix.

Figure 5

Unfolding the Chern insulator: In (a), we see the Chern insulator in the usual basis, treating the two degrees of freedom as sites. A change of basis in (b) transforms the model to a 1D chain with alternating hopping.

Figure 6

The spectrum of (top) Dirac semimetal, (middle) graphene, and (bottom) kagome semimetal. See Sec. 3e and Table 2 for details.

Figure 7

The schematic for plotting the Riemann sheet corresponding to Chern insulator.

Figure 8

The energy $\varepsilon$-Riemann surface for a Chern insulator, plotted explicitly using Mathematica. The black curve corresponds to an edge state.

Figure 9

The spectrum of Chern insulator for $m=-0.8$, with the band edges (dark blue) computed using the transfer matrix formalism and the left and right edge state dispersions (dashed and dashed-dotted curves) from the Evans equation (73), overlaid on the spectrum computed using exact diagonalization for a (top) commensurate and (bottom) incommensurate system.

Figure 10

The plot of the transfer matrix corresponding to the left edge state for Chern insulator, with $m=+0.8$ (red dashed curve) and $m=-0.8$ (black solid curve) on the $\mathrm{Sp}(2,\mathbb{R})$ manifold, which is homeomorphic to a solid torus. See Figs. 4 and 9, respectively, for the corresponding spectra.

Figure 11

The spectrum of the Hofstadter model for $\varphi =\frac{1}{5}$ (top), and (bottom) the curve of transfer matrices on $\mathrm{Sp}(2,\mathbb{R})$ corresponding to the left edge state in the second gap from the bottom. Note that the curve winds around twice in $\mathrm{Sp}(2,\mathbb{R})$, as expected from the spectrum.

Figure 12

The spectrum of the topological crystalline insulator model due Fu [31], with the parameters $t_1=0.5,t_2=0.25,t_1^{\prime }=1.25,t_2^{\prime }=0.25$, and $t_z^{\prime }=1$ in Eq. (121). The band edges (dark blue) and the left and right edge state dispersions (dashed and dashed-dotted) computed using the transfer matrix formalism, overlaid on the spectrum computed using exact diagonalization equivalent to Fig. 2 of Ref. [31].

Figure 13

Numerically computed estimates of Lyapunov exponents as a function of system length $N$ for Chern insulator on a strip geometry with width $L_y=40$ and parameters $m=1.0,\varepsilon =0$. For large $N$, the estimates converge to the Lyapunov exponents $\left\{{\lambda }_i\right\}$. (a) Open boundaries along the vertical and disorder $W=0.1$ show robust metallic edge modes (red trace) with ${\lambda }_i=0$ in this scale. Also highlighted is an insulating bulk mode (green trace) with ${\lambda }_i\approx 3$. (b) Vertical spatial profiles of the eigenstates for $N={10}^3$ with components $({\alpha }_{N-1},{\beta }_N)$ for the modes highlighted in (a), where the top is the bulk insulating mode (green trace) and the bottom the metallic edge state (red trace) which is strongly localized at the vertical boundaries. Arrows mark the position of these eigenmodes in the Lyapunov spectrum. (c) The same system as in (a) with periodic boundary conditions which shows no metallic edge states. (d) Strongly disordered case $\left(W=5.0\right)$ with open boundaries and absent metallic states.