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Enhanced area law in the Widom–Sobolev formula for the free Dirac operator in arbitrary dimension

Leon Bollmann and Peter Müller
Vol. 7 (2025), No. 3, 595–613
DOI: 10.2140/paa.2025.7.595
Abstract

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.

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Keywords
Dirac gas, Enhanced area law, Widom–Sobolev formula, matrix-valued pseudodifferential operators
Mathematical Subject Classification
Primary: 47B35, 47G30, 81P40
Milestones
Received: 28 May 2024
Revised: 24 January 2025
Accepted: 3 May 2025
Published: 18 June 2025
Authors
Leon Bollmann
Mathematisches Institut
Ludwig-Maximilians-Universität München
München
Germany
Peter Müller
Mathematisches Institut
Ludwig-Maximilians-Universität München
München
Germany
© 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers).