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We present an exactly solvable extension of the quantum XY chain with longerrange multi-spin interactions. Topological phase transitions of the model areclassified in terms of the number of Majorana zero modes which are in turnrelated to an integer winding number. We further find a general relationbetween the winding number and the number of Majorana fermions at the ends ofan open chain. The present class of exactly solvable models belong to the BDIclass in the Altland-Zirnbauer classification (A. Altland, M. R. Zirnbauer,Phys. Rev. B (1997) 55 1142) of topological superconductors. We show that timereversal (TR) symmetry of the spin variables translates into a peculiarparticle-hole transformation in the language of Jordan-Wigner (JW) fermionsthat is accompanied by a {\pi} shift in the wave vector ($\pi$PH). Presence of$\pi$PH symmetry restricts the winding number of TR symmetric extensions of XYto odd integers. The {\pi}PH operator may serve in further detailedclassification of topological superconductors.

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