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arXiv:0804.1061 · Europhysics Letters (EPL)

Scaling of the von Neumann entropy across a finite-temperature phase transition

V. Popkov, Mario Salerno

Published Oct 28, 2008• 7 citations
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Abstract

The spectrum of the reduced density matrix and the temperature dependence of the von Neumann entropy (VNE) are analytically obtained for a system of hard-core bosons on a complete graph which exhibits a phase transition to a Bose-Einstein condensate at T = T c . It is demonstrated that the VNE undergoes a crossover from purely logarithmic at T =0 to purely linear in block size n behaviour for T ⩾ T c . For intermediate temperatures, VNE is a sum of two contributions which are interpreted as the classical (Gibbs) and the quantum (due to entanglement) parts of the von Neumann entropy.

DOI
10.1209/0295-5075/84/30007

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The spectrum of the reduced density matrix and the temperature dependence of the von Neumann entropy (VNE) are analytically obtained for a system of hard-core bosons on a complete graph which exhibits a phase transition to a Bose-Einstein condensate at T = T c . It is demonstrated that the VNE undergoes a crossover from purely logarithmic at T =0 to purely linear in block size n behaviour for T ⩾ T c . For intermediate temperatures, VNE is a sum of two contributions which are interpreted as the classical (Gibbs) and the quantum (due to entanglement) parts of the von Neumann entropy.
Spectral properties of the reduced density matrix (RDM) of permutational invariant quantum many-body systems are investigated. The RDM block diagonalization which accounts for all symmetries of the Hamiltonian is achieved. The analytical expression of the RDM spectrum is provided for arbitrary parameters and rigorously proved in the thermodynamical limit. The existence of several sum rules and recurrence relations among RDM eigenvalues is also demonstrated and the distribution function of RDM eigenvalues (including degeneracies) characterized. In particular, we prove that the distribution function approaches a two-dimensional Gaussian in the limit of large subsystem sizes n>>1. As a physical application we discuss the von Neumann entropy (VNE) of a block of size n for a system of hard-core bosons on a complete graph, as a function of n and of the temperature T. The occurrence of a crossover of VNE from purely logarithmic behavior at T=0 to a purely linear behavior in n for T≥Tc, is demonstrated.
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 Sz and with respect to the irreducible representations of the Sn 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 Rényi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.
It is shown that the von Neumann entropy, a measure of quantum entanglement, does have its classical counterpart in thermodynamic systems, which we call partial entropy. Close to the critical temperature, the partial entropy shows perfect finite-size scaling behavior even for quite small system sizes. This provides a powerful tool to quantify finite-temperature phase transitions as demonstrated on the classical Ising model on a square lattice and the ferromagnetic Heisenberg model on a cubic lattice.
The critical behavior of entanglement near a quantum phase transition has been studied intensely over the past few years. Few-body quantum systems show critical behavior near the ionization point. We investigate the scaling properties of the von Neumann entropy for an atomiclike system near the ionization threshold. Using finite size scaling methods we calculate the critical charge and the critical exponent associated to the von Neumann entropy. The parallelism between the behavior of entanglement near a quantum phase transition and the behavior of the von Neumann entropy in a critical few-body quantum system is analyzed.
Abstract We propose a model for hard-core bosons in a lattice which allows to achieve the optimal occupation number predicted by Tennie et al (2017 Phys. Rev. B 96 064502) for a finite number of sites. The model is based on an extension of the Hamiltonian of the so-called Hubbard star, whose quantum properties are studied by means of quantum information descriptors such as the von Neumann entropy and the mutual information. These metrics are analyzed as a function of the one- and two-particle reduced density matrices, allowing to explore the relationship between condensation and entanglement by means of a control parameter that, under a given limit, connects our findings with previous results. All developments comprised in this article have been derived by analytical methods.