Generalization of Bloch's theorem for arbitrary boundary conditions: Theory

We present a generalization of Bloch's theorem to finite-range lattice systems of independent fermions, in which translation symmetry is broken solely due to arbitrary boundary conditions, by providing exact, analytic expressions for all energy eigenvalues and eigenstates. Starting with a reordering of the fermionic basis that transforms the single-particle Hamiltonian into a corner-modified banded block-Toeplitz matrix, a key step is a Hamiltonian-dependent bipartition of the lattice, which splits the eigenvalue problem into a system of bulk and boundary equations. The eigensystem inherits most of its solutions from an auxiliary, infinite translation-invariant Hamiltonian that allows for nonunitary representations of translation—hence complex values of crystal momenta with specific localization properties. A reformulation of the boundary equation in terms of a boundary matrix ensures compatibility with the boundary conditions, and determines the allowed energy eigenstates in the form of generalized Bloch states. We show how the boundary matrix quantitatively captures the interplay between bulk and boundary properties, leading to the construction of efficient indicators of bulk-boundary correspondence. Remarkable consequences of our generalized Bloch theorem are the engineering of Hamiltonians that host perfectly localized, robust zero-energy edge modes, and the predicted emergence, for instance, in Kitaev's Majorana chain, of localized excitations whose amplitudes decay in space exponentially with a power-law prefactor. We further show how the theorem may be used to construct numerical and algebraic diagonalization algorithms for the class of Hamiltonians under consideration, and use the proposed bulk-boundary indicator to characterize the topological response of a multiband time-reversal invariant $s$-wave topological superconductor under twisted boundary conditions, showing how a fractional Josephson effect can occur without entailing a fermionic parity switch. Finally, we establish connections to the transfer matrix method and demonstrate, using the paradigmatic Kitaev's chain example, that a defective (nondiagonalizable) transfer matrix signals the presence of solutions with a power-law prefactor.

Figure 1

Example of a $D=2$-dimensional Bravais lattice ${\mathrm{\Lambda }}_2$ terminated along two parallel lines (bordered with pattern). ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ denote the primitive vectors of ${\mathrm{\Lambda }}_2$, and its primitive cell is shaded (in blue). The dotted (black) line connects the points of the surface mesh $\left({\mathrm{\Lambda }}_1\right)$ generated by the primitive vector ${\mathbf{m}}_1$. The primitive cell of the Bravais lattice ${\overline{\mathrm{\Lambda }}}_2$, generated by ${\mathbf{m}}_1$ and the stacking vector $\mathbf{s}$, is also shown (shaded in brown). The original lattice ${\mathrm{\Lambda }}_2$ is obtained by attaching the basis vectors ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$ to each point of ${\overline{\mathrm{\Lambda }}}_2$.

Figure 2

Bulk-boundary separation for a system with two fermionic modes per unit cell, $d=2$, and next-nearest-neighbor hopping, $R=2$. Each (blue) circle stands for a fermionic mode. Thick and thin solid lines indicate two different hopping strengths in the bulk. Since the size of the boundary depends on the range $R$, the boundary comprises the first and last two unit cells of the chain. Dotted lines stand for arbitrary hopping strengths at the boundary.

Figure 3

Flowchart of the numerical diagonalization algorithm. The steps inside the dashed rectangle form the loop for scanning over $\epsilon$. The crucial step is solving the bulk equation, which encompasses steps (4)–(8) as described in the text.

Figure 4

Imaginary part of $detB\left(\epsilon \right)$ for $N=10$ as a function of the dimensionless parameter $\epsilon /2t$. Here, $B\left(\epsilon \right)$ is numerically evaluated from the top expression in Eq. (41), $\epsilon \ne \pm 2t$. Its real part vanishes identically in this range of energies. The impurity potential is $w=0.7>|w_N|$ for the solid blue curve, and $w=0.3<|w_N|$ for the dashed red curve. In the regime $w<w_N$ $\left(w>w_N\right)$, the system hosts zero (two) edge modes, which is reflected in the number of zeros $(N$ and $N-2)$ of the respective curves, in the energy range $-1<\epsilon <1$. In both cases, the crossings through zero at $\epsilon =\pm 1$ do not have associated eigenstates of the Hamiltonian. The origin of such fictitious zeros was discussed in Sec. 4a.

Figure 5

Two variants of the topological comb model. In (a), thin (thick) black lines indicate intra-ladder (diagonal) hopping with strength $t_0$ $\left(t_1\right)$. Red ovals or circles show the support of the zero-energy edge modes. In (b), upon shifting the lower chain by one site to the right, $t_1$ can be interpreted as direct inter-ladder hopping strength. (c) Band structure for the parameter regime $t_1/t_0=0.7$. The (black) dashed line represents zero energy, which lies in the band gap.

Figure 6

Spatial behavior of Majorana wave functions for various parameter regimes of the Kitaev chain under open BCs in the large-$N$ limit. The origin (blue diamond), $\mu =0,\mathrm{\Delta }=0$, corresponds to a metal at half-filling. The region shaded in black pattern is the trivial regime, which does not host Majoranas, and is separated from the nontrivial phase by solid black lines indicating the critical points. The interior of the circle of oscillations [Eq. (50)] (shaded in light blue) hosts Majoranas whose wave function decays with oscillations, whereas the region outside show a behavior similar to overdamped decay of a classical harmonic oscillator. On the circle, the wave function decays exponentially with a power-law prefactor. The “sweet spots” (red dots) host perfectly localized Majorana modes on the edge.

Figure 7

Ring (a) vs bridge (b) configurations of a chain Hamiltonian, $d=1=R$. The solid (black) lines denote nearest-neighbor bulk hopping, whereas the thick (red) line indicates hopping between the left $\left(j=N\right)$ and the right $\left(j=1\right)$ boundary (shaded gray rectangle). The bound states of (a) converge to the ones of (b) in the large-$N$ limit.

Figure 8

(a) Low-lying many-body energy eigenvalues in the Josephson ring configuration, as a function of flux $\varphi$. The energy level $E_1\left(\varphi \right)$ is doubly degenerate. (b) Energy of the bound mode and its antiparticle excitation. The shaded (blue) area denotes the continuum of energy states in the bulk. (c) Comparison of the indicator defined in Ref. [19] (dashed red line) and the generalized indicator of Eq. (54) (solid red line) in the topologically nontrivial phase. (d) Degeneracy of the zero-energy level inferred from the dimension of the kernel of $B_{\infty }(\epsilon =0,\varphi )$. The parameters are $w=0.2,\mu =0,u_{cd}=t=\lambda =1,\mathrm{\Delta }=2,\text{and}N=60$ in (a) and (b).

Figure 9

Four topologically inequivalent one-dimensional manifolds. The classification (Yes $=$ Y, No $=$ N) encompasses compactness and whether the boundary is empty.