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

Abstract

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 ($\pi$PH). Presence of $\pi$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.

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References in corpus (5)

• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological quantum matter with symmetries
• Introduction to topological superconductivity and Majorana fermions

Cited by in corpus (5)

• Resilience of Majorana Fermions in the face of Disorder
• Coexistence of Strong and Weak Majorana Zero Modes in Anisotropic XY Spin Chain with Second-Neighbor Interaction
• Decoherence and relaxation of topological states in extended quantum Ising models
• Exact phase diagram and topological phase transitions of the XYZ spin chain
• Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii-Moriya Interaction