Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions

We present a procedure for exactly diagonalizing finite-range quadratic fermionic Hamiltonians with arbitrary boundary conditions in one of $D$ dimensions, and periodic in the remaining $D-1$. The key is a Hamiltonian-dependent separation of the bulk from the boundary. By combining information from the two, we identify a matrix function that fully characterizes the solutions, and may be used to construct an efficiently computable indicator of bulk-boundary correspondence. As an illustration, we show how our approach correctly describes the zero-energy Majorana modes of a time-reversal-invariant $s$-wave two-band superconductor in a Josephson ring configuration, and predicts that a fractional $4\pi$-periodic Josephson effect can only be observed in phases hosting an odd number of Majorana pairs per boundary.

Figure 1

Quasiparticle energies (solid black lines) and BB indicator $\mathcal{D}$ [Eq. (8)] (dashed red line) vs flux $\varphi$ in a phase supporting one (top), two (middle), and zero (bottom) pairs of Majorana modes per edge. The grey shaded regions indicate bulk quasiparticle energies. Insets: many-body ground state energy $E(\varphi )$. With reference to [24], the parameter values are $t=\lambda =\mathrm{\Delta }=1$, $\mu =0$, $w=0.2$, $u_{cd}=2$ (top), $u_{cd}=0.6$ (middle), $u_{cd}=3.7$ (bottom). In all calculations, lattice size $L=60$ is used.