Exactly solvable spin chain models corresponding to BDI class of topological superconductors

We present an exactly solvable extension of the quantum XY chain with longer range multi-spin interactions. Topological phase transitions of the model are classified in terms of the number of Majorana zero modes, n(M) which are in turn related to an integer winding number, n(W). The present class of exactly solvable models belong to the BDI class in the Altland-Zirnbauer classification of topological superconductors. We show that time reversal symmetry of the spin variables translates into a sliding particle-hole (PH) transformation in the language of Jordan-Wigner fermions – a PH transformation followed by a π shift in the wave vector which we call it the πPH. Presence of πPH symmetry restricts the n(W) (n(M)) of time-reversal symmetric extensions of XY to odd (even) integers. The πPH operator may serve in further detailed classification of topological superconductors in higher dimensions as well.