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Modular Hamiltonians for the massless Dirac field in the presence of a defect
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Modular Hamiltonians for the massless Dirac field in the presence of a boundary

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  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 22 March 2021

Modular Hamiltonians for the massless Dirac field in the presence of a defect

  • Mihail Mintchev1 &
  • Erik Tonni2 

Journal of High Energy Physics volume 2021, Article number: 205 (2021) Cite this article

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A preprint version of the article is available at arXiv.

Abstract

We study the massless Dirac field on the line in the presence of a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission. Considering this system in its ground state, we derive the modular Hamiltonians of the subregion given by the union of two disjoint equal intervals at the same distance from the defect. The absence of energy dissipation at the defect implies the existence of two phases, where either the vector or the axial symmetry is preserved. Besides a local term, the densities of the modular Hamiltonians contain also a sum of scattering dependent bi-local terms, which involve two conjugate points generated by the reflection and the transmission. The modular flows of each component of the Dirac field mix the trajectory passing through a given initial point with the ones passing through its reflected and transmitted conjugate points. We derive the two-point correlation functions along the modular flows in both phases and show that they satisfy the Kubo-Martin-Schwinger condition. The entanglement entropies are also computed, finding that they do not depend on the scattering matrix.

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Pisa and INFN Sezione di Pisa, largo Bruno Pontecorvo 3, 56127, Pisa, Italy

    Mihail Mintchev

  2. SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136, Trieste, Italy

    Erik Tonni

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  1. Mihail Mintchev
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  2. Erik Tonni
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Correspondence to Erik Tonni.

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Mintchev, M., Tonni, E. Modular Hamiltonians for the massless Dirac field in the presence of a defect. J. High Energ. Phys. 2021, 205 (2021). https://doi.org/10.1007/JHEP03(2021)205

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  • Received: 16 December 2020

  • Accepted: 31 January 2021

  • Published: 22 March 2021

  • DOI: https://doi.org/10.1007/JHEP03(2021)205

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Keywords

  • Conformal Field Theory
  • Field Theories in Lower Dimensions
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