Exact ground state of the sine-square deformed XY spin chain

TL;DR

This paper proves that the ground state of the sine-square deformed XY spin chain with open boundaries is identical to that of a uniform XY chain with periodic boundaries, confirming a previous numerical conjecture.

Contribution

It provides an exact analytical proof that the ground state of the sine-square deformed XY chain matches that of the uniform periodic chain, resolving a longstanding conjecture.

Findings

Ground state of the sine-square deformed XY chain is identical to the uniform periodic chain.

The model can be mapped to a free fermion system with site-dependent parameters.

The proof confirms a conjecture based on numerical evidence.

Abstract

We study the sine-square deformed quantum XY chain with open boundary conditions, in which the interaction strength at the position $x$ in the chain of length $L$ is proportional to the function $f_x = \sin^2 [\pi/L (x-1/2)]$. The model can be mapped onto a free spinless fermion model with site-dependent hopping amplitudes and on-site potentials via the Jordan-Wigner transformation. Although the single-particle eigenstates of this system cannot be obtained in closed form, it is shown that the many-body ground state is identical to that of the uniform XY chain with periodic boundary conditions. This proves a conjecture of Hikihara and Nishino [Hikihara T and Nishino T 2011 {\it Phys. Rev. B} \textbf{83} 060414(R)] based on numerical evidence.