  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 {\pi} shift in the wave vector ($\pi$PH). Presence of
$\pi$PH symmetry restricts the winding number of TR symmetric extensions of XY
to odd integers. The {\pi}PH operator may serve in further detailed
classification of topological superconductors.

Strongly Correlated Electrons;Superconductivity