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arXiv:2105.13269 · Physical Review A

Entanglement properties and ground-state statistics of free bosons

Luca Dell’Anna

Published Mar 4, 2022• 6 citations
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Abstract

We calculate analytically the entanglement and R\'enyi entropies, the negativity, and the mutual information together with all the density and many-particle correlation functions for free bosons on a lattice in the ground state, for both homogeneous and inhomogeneous systems. We show that all those quantities can be derived from a multinomial form of the reduced density matrix in the configuration space whose diagonal elements dictate the statistics of the particle distribution, while the off-diagonal coherence terms control the quantum fluctuations. We provide by this analysis a unified approach based on a reduced density matrix technique useful to calculate both the entanglement properties and an infinite number of correlation functions.

DOI
10.1103/physreva.105.032412

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We calculate analytically the entanglement and R'enyi entropies, the negativity, and the mutual information together with all the density and many-particle correlation functions for free bosons on a lattice in the ground state, for both homogeneous and inhomogeneous systems. We show that all those quantities can be derived from a multinomial form of the reduced density matrix in the configuration space whose diagonal elements dictate the statistics of the particle distribution, while the off-diagonal coherence terms control the quantum fluctuations. We provide by this analysis a unified approach based on a reduced density matrix technique useful to calculate both the entanglement properties and an infinite number of correlation functions.
We calculate analytically the entanglement and Renyi entropies, the negativity and the mutual information together with all the density and many-particle correlation functions for free bosons on a lattice in the ground state. We show that those quantities can be derived from a multinomial form of the reduced density matrix in the configuration space whose diagonal elements dictate the statistics of the particle distribution, while the off-diagonal coherence terms control the quantum fluctuations. We clarify by this analysis how to reconcile the logarithmic behavior of the entanglement entropy with the volume law of the particle number fluctuations.
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We review the properties of reduced density matrices for free fermionic or bosonic many-particle systems in their ground state. Their basic feature is that they have a thermal form and thus lead to a quasi-thermodynamic problem with a certain free-particle Hamiltonian. We discuss the derivation of this result, the character of the Hamiltonian and its eigenstates, the single-particle spectra and the full spectra, the resulting entanglement and in particular the entanglement entropy. This is done for various one- and two-dimensional situations, including also the evolution after global or local quenches.
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Universal scaling terms occurring in R\'enyi entanglement entropies have the potential to bring new understanding to quantum critical points in free and interacting systems. Quantitative comparisons between analytical continuum theories and numerical calculations on lattice models play a crucial role in advancing such studies. In this paper, we exactly calculate the universal two-cylinder shape dependence of entanglement entropies for free bosons on finite-size square lattices, and compare to approximate functions derived in the continuum using several different Ans\"atze. Although none of these Ans\"atze are exact in the thermodynamic limit, we find that numerical fits are in good agreement with continuum functions derived using the anti--de Sitter/conformal field theory correspondence, an extensive mutual information model, and a quantum Lifshitz model. We use fits of our lattice data to these functions to calculate universal scalars defined in the thin-cylinder limit, and compare to values previously obtained for the free boson field theory in the continuum.