Pfaffian-based topological invariants for one dimensional semiconductor-superconductor heterostructures

Abstract

We review the Pfaffian-based $\mathbb{Z}_2$ topological invariants in one dimensional semiconductor-superconductor (SM-SC) nanowire heterostructures and clarify their validity in finite and disordered systems. For the clean nanowire, the product of the Pfaffians of the Hamiltonian at particle-hole symmetric momenta $k=0,π$ changes sign at the topological phase transition defined by the bulk gap closing, leading to the definition of $\mathbb{Z}_2$ Kitaev invariant also known as Majorana number. We show that this momentum-space invariant is equivalent to a real space construction based on twisted boundary conditions, in which the sign of the product of the Pfaffians of the Hamiltonian under periodic and anti-periodic boundary conditions defines the $\mathbb{Z}_2$ index. By introducing a superlattice description of periodically repeated disorder, we demonstrate that the real space Pfaffian invariant defined as the sign of the Pfaffians of the Hamiltonian with periodic and anti-periodic boundary conditions, remains a well defined invariant even in the absence of microscopic translational symmetry. Within this framework, it is also equivalent to the recently defined periodic disorder invariant (PDI), which constitutes an integer valued ($\mathbb{Z}$) topological invariant in the presence of chiral symmetry. Finally, we prove that the sign of the Pfaffian of a quadratic Hamiltonian equals the fermion parity of its ground state, establishing a direct physical interpretation of the invariant, in terms of sign of the product of the ground state fermion parity with periodic and anti-periodic boundary conditions. Numerical results confirm the correspondence between sign of the Pfaffian reversals, flux-induced level crossings, and ground-state parity switching in clean and disordered nanowires.

Figures (6)

• Figure 1: Schematic illustration of the superlattice formulation and indexing convention used throughout the manuscript. The global site index $j\in[0,L-1]$ is decomposed into a supercell index $n\in[0,N_c-1]$ and a local index $a\in[0,l-1]$, with $j = nl + a$, where $l$ denotes the number of microscopic sites per supercell.
• Figure 2: Invariant maps for a clean $3\,\,\mu$m long wire. The left panel shows the real space twisted-boundary Pfaffian invariant (refer to Sec. \ref{['sec:relation_bw_ms_rs']}) and the right panel shows the corresponding PDI (refer to Sec. \ref{['sec:PDI']}). The black regions indicate topologically trivial region ($Tr$) where the Pfaffian invariant has a positive sign while PDI has a value of 0 and the white regions are the topologically non-trivial ($Tp$) regimes where the Pfaffian invariant has a negative sign and PDI is 1.
• Figure 3: Flux dependent energy spectrum for a clean $3\,\,\mu$m long wire. The left panel shows the variation of 2 lowest BdG energy eigenvalues for the trivial region point ($\mu=1.0$ meV,$\Gamma=0.5$ meV) and the right panel shows the behavior of the 2 energy curves for topological regime ($\mu=0.0$ meV,$\Gamma=0.5$ meV). The blue curve corresponds to odd parity while the red curve corresponds to even parity.
• Figure 4: Top row: Real space twsited-boundary Pfaffian invariant (refer to Sec. \ref{['sec:relation_bw_ms_rs']}) maps for varying chemical potential $\mu$ on y-axis and Zeeman field $\Gamma$ on x-axis for increasing disorder amplitude $V_0=0.05, 0.5, 1.0$ and $1.5$ meV. Bottom row: Corresponding PDI maps (refer to Sec. \ref{['sec:PDI']}) for the same set of parameters.
• Figure 5: Fermion parity switch indicator $\mathcal{F}$ (refer to Eq. \ref{['eq:FPSI']}) as a function of Zeeman field for different chemical potential values, $\mu=[0.0,1.0,2.3]$ meV for increasing disorder in the nanowire. The top panel corresponds to weak disorder limit $V_0=0.05$ meV and moving down towards the bottom panel representing the high disorder limit $V_0=1.5$ meV. The ribbon cuts in each panel show the value of the fermion parity switch indicator as black corresponding to $\mathcal{F}=0$ (no fermion parity switch) and white corresponding to $\mathcal{F}=1$ (fermion parity switch taking place).