Painlevé Transcendent Describes Quantum Correlation Function of the XXZ Antiferromagnet away from the free-fermion point

Abstract

We consider quantum correlation functions of the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg XXZ spin chain in a magnetic field. We show that for a magnetic field close to the critical field $h_c$ (for the critical magnetic field the ground state is ferromagnetic) certain correlation functions can be expressed in terms of the solution of the Painlevé V transcendent. This establishes a relation between solutions of Painlevé differential equations and quantum correlation functions in models of {\sl interacting} fermions. Painlevé transcendents were known to describe correlation functions in models with free fermionic spectra.