Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure

We describe a method for exactly diagonalizing clean $D$-dimensional lattice systems of independent fermions subject to arbitrary boundary conditions in one direction, as well as systems composed of two bulks meeting at a planar interface. The specification of boundary conditions and interfaces can be easily adjusted to describe relaxation, reconstruction, or disorder away from the clean bulk regions of the system. Our diagonalization method builds on the generalized Bloch theorem [A. Alase et al., Phys. Rev. B 96, 195133 (2017)] and the fact that the bulk-boundary separation of the Schrödinger equation is compatible with a partial Fourier transform. Bulk equations admit a rich symmetry analysis that can considerably simplify the structure of energy eigenstates, often allowing a solution in fully analytical form. We illustrate our approach to multicomponent systems by determining the exact Andreev bound states for a simple SNS junction. We then analyze the Creutz ladder model, by way of a conceptual bridge from one to higher dimensions. Upon introducing a new Gaussian duality transformation that maps the Creutz ladder to a system of two Majorana chains, we show how the model provides a first example of a short-range chiral topological insulator that hosts topological zero modes with a power-law profile. Additional applications include the complete analytical diagonalization of graphene ribbons with both zigzag-bearded and armchair boundary conditions, and the analytical determination of the edge modes in a chiral $p+ip$ two-dimensional topological superconductor. Lastly, we revisit the phenomenon of Majorana flat bands and anomalous bulk-boundary correspondence in a two-band gapless $s$-wave topological superconductor. Beside obtaining sharp indicators for the presence of Majorana modes through the use of the boundary matrix, we analyze the equilibrium Josephson response of the system, showing how the presence of Majorana flat bands implies a substantial enhancement in the $4\pi$-periodic supercurrent.

Figure 1

Sketch of a $D=2$ lattice system with nearest-neighbor (NN) hopping subject to arbitrary BCs. The filled and hollow circles together form a Bravais lattice ${\mathrm{\Lambda }}_2$. The two $D=1$ edges of the system are shown by horizontal lines decorated by pattern. The surface mesh ${\mathrm{\Lambda }}_1$ is generated by ${\mathbf{m}}_1$, and consists of all points connected by dashed black lines. ${\mathbf{m}}_1$ and $\mathbf{s}$ generate the Bravais lattice ${\overline{\mathrm{\Lambda }}}_2$, formed only by the filled circles. A primitive cell of ${\overline{\mathrm{\Lambda }}}_2$ (shaded brown) encloses two points of ${\mathrm{\Lambda }}_2$. In this case, assuming there are $d_{\mathrm{int}}$ internal degrees associated to each point of ${\mathrm{\Lambda }}_2$, we get ${\overline{d}}_{\mathrm{int}}=2d_{\mathrm{int}}$. Since, for NN hopping, $R=1$, the operator $W$ that implements the BCs has its support on single-particle states in the boundary region (shaded gray).

Figure 2

(a) Sketch of a $D=2$ crystal with ideal surface. The remaining panels show the same crystal with (b) relaxed, (c) reconstructed, and (d) disordered surface. The unfilled circle in panel (d) shows a surface impurity atom.

Figure 3

Andreev bound states of an SNS junction. The energies of the states are those values of $\epsilon$ (in units of $t$) for which two independent constraints become compatible. These constraints, derived from the boundary matrix, are equations of the form $F_{\pm 1}\left(\epsilon \right)=g_{\pm 1}\left(\epsilon \right)$ and $F_{\pm 1}\left(\epsilon \right)=g_{\mp 1}\left(\epsilon \right)$, see Eqs. (C2) and (C3) of Appendix pp3. The solid (blue) lines are the curves traced by $F_{\pm 1}\left(\epsilon \right)$, the thick black dotted lines are the curves traced by $g_{+1}\left(\epsilon \right)$, and the thin black dotted lines are the curves traced by $g_{-1}\left(\epsilon \right)$. Each intersection corresponds to an Andreev bound state. The top and bottom panels correspond to $N=4\mathsf{L}-1$ sites for the normal dot, with $\mathsf{L}=2$ and $\mathsf{L}=4$, respectively. Other parameters are $t=1,t^{\prime }=0.5$ (a.u.) throughout.

Figure 4

Schematic of the duality between the Creutz ladder (left) and cross-linked Majorana chains (right).

Figure 5

Quantum (zero-temperature) phase diagram of the Creutz ladder at half filling and for $\mathsf{r}=1$. The solid black lines indicate points where the bulk gap closes. The gapped regimes indicated by shaded vertical strips do not host midgap states. Everywhere else, the energy of the edge states is $\epsilon =\mathsf{M}cos\theta$. In the parameter regime indicated by the vertical thick solid (blue) line, the Creutz ladder is dual to the periodic Anderson model, and in the regime indicated by the two dashed black horizontal lines, it is dual to two decoupled, topologically nontrivial Kitaev chains.

Figure 6

Graphene ribbon, periodic or infinite in the horizontal ${\mathbf{m}}_1$ direction. (Left) The ribbon is terminated in the vertical direction by a zigzag edge on the bottom and a “bearded” edge on top. The decoupled $B$ sites at the top are auxiliary degrees of freedom. (Right) The ribbon is terminated by armchair edges. The system has mirror symmetry about the dashed (red) line. In both cases, on-site potentials $v_1$ and $v_2$ are associated with the $A$ and $B$ sublattice, respectively.

Figure 7

The value of the Chern invariant as a function of $\mu$ in the parameter regime $\mathrm{\Delta }=1=t$.

Figure 8

Surface bands for $\mu =1.5$, centered at $k_{\parallel }=0$ (top left), and $\mu =6.5$, centered at $k_{\parallel }=-\pi$ (top right). The shaded (gray) region shows the bulk bands. The electrons on the right edge (dashed red curve) propagate to the right only, and those on the left edge (solid blue curve) to the left only, that is, the surface bands are chiral. The lower panels show the behavior of $z_1$ (solid blue curve) and $z_2$ (dashed red curve) with $k_{\parallel }$. Notice how $z_1$ ($z_2$) enters (exits) the unit circle precisely when the surface bands touch the bulk bands, as marked by vertical solid black lines.

Figure 9

Energy spectrum (blue scatter plot) and degeneracy indicator ${\mathcal{K}}_{k_z}\left(0\right)$ for the zero energy level (red solid line) in the large-$N$ limit for BC1 (top) vs BC2 (bottom) for various values of the SC pairing $\mathrm{\Delta }$. The other parameters are $\mu =0$, $t=\lambda =u_{cd}=1$, $N_x=120,\text{and}{N}_y=30$.

Figure 10

Penetration depth (in units of the lattice constant) of flat-band Majoranas as a function of $k_z$. The parameters are $\mu =0$, $u_{cd}=t=\lambda =1$, and $\mathrm{\Delta }=4$.

Figure 11

Total Josephson current $I\left(\varphi \right)$ (black solid line), $2\pi$-periodic component $I_{2\pi }\left(\varphi \right)$ (blue dotted line), and $4\pi$-periodic component $I_{4\pi }\left(\varphi \right)$ (red dashed line) in units of $2e/\hslash$, as a function of flux $\varphi$. Top (bottom) panels correspond to the gapless (gapped) model of a $D=2s$-wave TSC, whereas left (right) panels correspond to BC1 (BC2), respectively. The parameters used for both models are $\mu =0$, $u_{cd}=t=\lambda =1$, $\mathrm{\Delta }=4$, and $N_x=N_z=60$.