Factorization of multiple integrals representing the density matrix of a finite segment of the Heisenberg spin chain

TL;DR

This paper develops a new factorization approach for the density matrix of finite segments of the Heisenberg spin chain, simplifying calculations of correlation functions and proposing a conjecture for a general exponential formula.

Contribution

It introduces a finite-sum decomposition of multiple integrals for the density matrix of short segments, leading to more efficient computations and a conjectured exponential formula for arbitrary segment length.

Findings

Finite-sum decomposition for segments of length 2 and 3

New numerically efficient expressions for two-point functions

Conjecture of an exponential density matrix formula for arbitrary length and zero magnetic field

Abstract

We consider the inhomogeneous generalization of the density matrix of a finite segment of length $m$ of the antiferromagnetic Heisenberg chain. It is a function of the temperature $T$ and the external magnetic field $h$, and further depends on $m$ `spectral parameters' $\xi_j$. For short segments of length 2 and 3 we decompose the known multiple integrals for the elements of the density matrix into finite sums over products of single integrals. This provides new numerically efficient expressions for the two-point functions of the infinite Heisenberg chain at short distances. It further leads us to conjecture an exponential formula for the density matrix involving only a double Cauchy-type integral in the exponent. We expect this formula to hold for arbitrary $m$ and $T$ but zero magnetic field.