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, nM which are in turn related to an integer winding number, nW. The present class of exactly solvable models belong to the BDI class in the Altland-Zirnbauer classification of topological superconductors. We show that time reversal symmetry of the spin variables translates into a sliding particle-hole (PH) transformation in the language of Jordan-Wigner fermions – a PH transformation followed by a π shift in the wave vector which we call it the πPH. Presence of πPH symmetry restricts the nW (nM) of time-reversal symmetric extensions of XY to odd (even) integers. The πPH operator may serve in further detailed classification of topological superconductors in higher dimensions as well.
Highlights
We present an exactly solvable extension of the quantum XY chain with longer range multi-spin interactions
We introduce a class of exactly solvable 1D spin chains with specific type of interactions, which incorporate the topological properties leading to presence of an arbitrary number of Majorana fermion (MF) end-modes depending on the range of the interactions
In this letter we classify generalizations of the XY model with arbitrary n-spin interactions in terms of a πPH symmetry that is a PH transformation followed by a sign alternation in one sublattice
Summary
To proceed further let us first elucidate the meaning of πPH in the language of MFs: If we represent the JW “electron” and “hole” operators as cj† = aj + ibj and h j = aj′ − ibj′ and if we search for MFs with vanishing b, b′component, every zero mode solution (A1, 0, A2, 0, ...) is mapped by πPH to a partner MF (A1′, 0, A2′, 0, ...) with Aj′ = −(−1)j Aj. For even values of r where odd number of spin variables are added to the XY model, consider any point in the phase diagram (see left panel of Fig. 3) with given number nM of MFs. Obviously the generalized r + 1-spin interaction breaks TR symmetry.
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