Density matrices for finite segments of Heisenberg chains of arbitrary length

We derive a multiple integral representing the ground state density matrix of a segment of length $m$ of the XXZ spin chain on $L$ lattice sites, which depends on $L$ only parametrically. This allows us to treat chains of arbitrary finite length. Specializing to the isotropic limit of the XXX chain we show for small $m$ that the multiple integrals factorize. We conjecture that this property holds for arbitrary $m$ and suggest an exponential formula for the density matrix which involves only a double Cauchy type integral in the exponent. We demonstrate the efficiency of our formula by computing the next-to-nearest neighbour $zz$-correlation function for chain lengths ranging from two to macroscopic numbers.