Unwinding of a one-dimensional topological superconductor

We show that a topological superconductor made of four chains of superconducting spinless fermions characterized by four Majorana edge states can adiabatically be deformed into a trivial band insulator. To unwind this time-reversal invariant topological superconductor, interactions to spinful fermions are switched on along an adiabatic path. Thereby, we couple modes which belong to two different representations of the time-reversal symmetry operator $\mathcal{T}$ with ${\mathcal{T}}^2=1$ and ${\mathcal{T}}^2=-1$, respectively. This observation can be understood by investigating how the relevant symmetries act on the entanglement spectrum giving rise to four instead of eight different topological phases with Majorana edge modes. We also show that a simple level crossing of doubly and singly degenerate states occurs in the entanglement spectrum on deforming the quantum state.

Figure 1

Schematic sketch of the adiabatic transformation from a trivial insulator (left, $\lambda =0$) to a superconducting state characterized by four Majorana edge states on each side of the open chain (right, $\lambda =2$). During this transformation the coupling to spinful fermions (upper line) is switched on. At $\lambda =1$ a pseudospin made from four Majoranas forms a spin-singlet with the spin of an electron.

Figure 2

Lower part of the energy spectrum of $H$ as a function of $\lambda$ for $1⩽\lambda ⩽2$ (lower axes labels). The spectra for $0⩽\lambda ⩽1$ are identical (see upper axes labels) as $E\left(\lambda \right)=E(2-\lambda )$. The ground state is unique and the gap is finite.

Figure 3

Entanglement spectrum as a function of $\lambda$ for $1⩽\lambda ⩽2$ for an infinite system cut into two parts. It evolves from an unentangled state for $\lambda ⩽1$ ($E=\ln 1=0$) towards a fourfold degenerate level ($E=\ln 4$) at $\lambda =2$ reflecting four Majorana edge states. The qualitatively different entanglement spectra are connected by a simple level crossing. (Inset) The edge spectrum of an open chain also shows a simple level crossing of a unique ground state and a doubly degenerate state which merges with two other states to form a zero-energy fourfold degenerate state at $\lambda =2$.