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arXiv:1002.3306 · Fortschritte der Physik

Free fermions violate the area law for entanglement entropy

Robert C. Helling, W. Spitzer

Published May 17, 2010• 2 citations
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Abstract

Abstract We show that the entanglement entropy associated to a region grows faster than the area of its boundary surface. This is done by proving a special case of a conjecture due to Widom that yields a surprisingly simple expression for the leading behaviour of the entanglement entropy.
DOI
10.1002/prop.201000029

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We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and next-to-leading term of a semi-classical expansion of the trace of the square of certain integral operators on the Hilbert space L2(Rd)L^2(\R^d)L2(Rd). As already observed by Gioev and Klich, this implies that the bi-partite entanglement entropy of the free Fermi gas in its ground state grows at least as fast as the surface area of the spatially bounded part times a logarithmic enhancement.
We study the entanglement entropy of connected bipartitions in free-fermion gases of N particles in arbitrary dimension d . We show that the von Neumann and Rényi entanglement entropies grow asymptotically as N ( d −1)/ d ln N , with a prefactor that is analytically computed using the Widom conjecture both for periodic and open boundary conditions. The logarithmic correction to the power-law behavior is related to the area-law violation in lattice free fermions. These asymptotic large- N behaviors are checked against exact numerical calculations for N -particle systems.
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We prove a logarithmically enhanced area law for all Rényi entanglement entropies of the ground state of a free gas of relativistic Dirac fermions. Such asymptotics occur in any dimension if the modulus of the Fermi energy is larger than the mass of the particles and in the massless case at Fermi energy zero in one space dimension. In all other cases of mass, Fermi energy and dimension, the entanglement entropy grows no faster than the area of the involved spatial region. The result is established for a general class of test functions which includes the ones corresponding to Rényi entropies and relies on a recently proved extension of the Widom-Sobolev formula to matrix-valued symbols by the authors.