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Painlevé transcendent describes quantum correlation function of the XXZ antiferromagnet away from the free-fermion point. (English) Zbl 0904.34010
Summary: The authors consider quantum correlation functions of the antiferromagnetic spin-${1\over 2}$ Heisenberg XXZ spin chain in a magnetic field. 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 a solution of the Painlevé V transcendent. This establishes a relation between solutions to Painlevé differential equations and quantum correlation functions in models of interacting fermions. Painlevé transcendents are known to describe correlation functions in models with free fermionic spectra.
##### MSC:
 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 82D40 Magnetic materials (statistical mechanics) 34M55 Painlevé and other special equations; classification, hierarchies
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