Reduced density matrix and entanglement entropy of permutationally invariant quantum many-body systems

TL;DR

This paper analyzes the reduced density matrix and entanglement entropy in permutationally invariant quantum many-body systems, providing analytical expressions and exploring temperature effects and phase transitions.

Contribution

It introduces analytical formulas for the RDM and entanglement measures in permutationally symmetric states, including the Heisenberg model, and examines their temperature dependence.

Findings

RDM acquires block diagonal form respecting symmetry

Analytical expressions for RDM elements and spectrum

Entanglement entropy varies with temperature and phase transitions

Abstract

In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin $1/2$ Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number $k$ fixing the polarization in the subsystem conservation of $S_{z}$ and with respect to the irreducible representations of the $\mathbf{S_{n}}$ group. Analytical expression for the RDM elements and for the RDM spectrum are derived for states of arbitrary permutational symmetry and for arbitrary polarizations. The temperature dependence and scaling of the VNE across a finite temperature phase transition is discussed and the RDM moments and the R\'{e}nyi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.