Density matrices for finite segments of Heisenberg chains of arbitrary length

TL;DR

This paper derives a new integral representation for the ground state density matrix of finite segments in the Heisenberg XXZ chain, enabling analysis of chains of arbitrary length and revealing factorization properties in the isotropic limit.

Contribution

It introduces a parametric multiple integral formula for the density matrix of finite Heisenberg chains, generalizing previous results and proposing an exponential form for the density matrix.

Findings

Integral representation depends only parametrically on chain length

Factorization observed in the isotropic XXX limit for small segments

Efficient computation of next-to-nearest neighbor correlations

Abstract

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.