Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2011-12-29
Physics
Condensed Matter
Statistical Mechanics
Festschrift in honor of the 60th birthday of Professor Vladimir Korepin (11 pages, 5 figures)
Scientific paper
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.
Popkov Vladislav
Salerno Mario
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