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.
Keywords
Dirac gas, Enhanced area law, Widom–Sobolev formula,
matrix-valued pseudodifferential operators
