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arXiv:2507.10509

Topological phases and Edge states in an exactly solvable Gamma matrix model

Akhil Pravin Furtado, Kusum Dhochak

Published Jul 14, 2025• 0 citations
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Abstract

We study the phases of an exactly solvable one dimensional model with $4-$dimensional $Γ-$matrix degrees of freedom on each site. The $Γ-$matrix model has a large set of competing interactions and displays a rich phase diagram with critical lines and multi-critical points. We work with the model with certain $Z_2$ symmetries and identify the allowed symmetry protected topological phases using the winding number as the topological invariant. The model belongs to the CII-class of the $10-$fold classification and allows for integer values of the winding number. We confirm that the system also hosts localized zero energy Majorana edge modes, consistent with the integer value of the winding number of the corresponding phase. We further study scaling and universality behaviour of the various topological phase transitions.

DOI
10.48550/arxiv.2507.10509

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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 (π\piπPH). Presence of π\piπPH 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.