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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 which are in turn related to an integer winding number. We further find a general relation between the winding number and the number of Majorana fermions at the ends of an open chain. The present class of exactly solvable models belong to the BDI class in the Altland-Zirnbauer classification (A. Altland, M. R. Zirnbauer, Phys. Rev. B (1997) 55 1142) of topological superconductors. We show that time reversal (TR) symmetry of the spin variables translates into a peculiar particle-hole transformation in the language of Jordan-Wigner (JW) fermions that is accompanied by a π shift in the wave vector ($π$PH). Presence of $π$PH symmetry restricts the winding number of TR symmetric extensions of XY to odd integers. The πPH operator may serve in further detailed classification of topological superconductors.