Number-conserving interacting fermion models with exact topological superconducting ground states

We present a method to construct number-conserving Hamiltonians whose ground states exactly reproduce an arbitrarily chosen BCS-type mean-field state. Such parent Hamiltonians can be constructed not only for the usual $s$-wave BCS state, but also for more exotic states of this form, including the ground states of Kitaev wires and two-dimensional topological superconductors. This method leads to infinite families of locally interacting fermion models with exact topological superconducting ground states. After explaining the general technique, we apply this method to construct two specific classes of models. The first one is a one-dimensional double wire lattice model with Majorana-like degenerate ground states. The second one is a two-dimensional $p_x+i{p}_y$ superconducting model, where we also obtain analytic expressions for topologically degenerate ground states in the presence of vortices. Our models may provide a deeper conceptual understanding of how Majorana zero modes could emerge in condensed matter systems, as well as inspire novel routes to realize them in experiment.

Figure 1

Parameter region of the double wire system. The constraint $\alpha +\beta +\gamma =0$ restricts the parameter space to the 2D plane drawn in this figure, where the coordinate $(\alpha ,\beta ,\gamma )$ of an arbitrary point $P$ is given by the projections from $P$ to the $\alpha ,\beta ,\gamma$ axes. The conditions $\alpha <0,\beta <t,\gamma <t$, which guarantee $\widehat{H}$ to be positive-definite, give a triangle region (shaded area) in this plane.

Figure 2

Topological superconducting phase in an unbounded 2D plane with $2M$ vortices located at ${\eta }_1,{\eta }_2,...,{\eta }_{2M}$. Here we show the $2M=4$ case. In our gauge convention fermion fields acquire a minus sign on going around each vortex.

Figure 3

Diagrammatic representation of interaction terms in the double wire parent Hamiltonian in Eq. (8). Double arrows represent pair hopping while wavy lines represent interactions.