Entanglement spectrum in general free fermionic systems

Eldad Bettelheim, Aditya Banerjee, Martin B. Plenio, Susana F. Huelga

Research output: Contribution to journalArticlepeer-review


The statistical mechanics characterization of finite subsystems embedded in an infinite system is a fundamental question of quantum physics. Nevertheless, a full closed form for all required entropic measures does not exist in the general case, even for free systems, when the finite system in question is composed of several disjoint intervals. Here we develop a mathematical framework based on the Riemann-Hilbert approach to treat this problem in the one-dimensional case where the finite system is composed of two disjoint intervals and in the thermodynamic limit (both intervals and the space between them contains an infinite number of lattice sites and the result is given as a thermodynamic expansion). To demonstrate the usefulness of our method, we compute the change in the entanglement and negativity spectra namely the spectrum of eigenvalues of the reduced density matrix with or without time reversal of one of the intervals. We do this in the case that the distance between the intervals is much larger than their size. The method we use can be easily applied to compute any power in an expansion in the ratio of the distance between the intervals to their size. We expect these results to provide the necessary mathematical apparatus to address relevant questions in concrete physical scenarios, namely the structure and extent of quantum correlations in fermionic systems subject to local environments.

Original languageEnglish
Article number135001
JournalJournal of Physics A: Mathematical and Theoretical
Issue number13
StatePublished - 1 Apr 2022


  • Fisher-Hartwig theorem
  • Riemann-Hilbert problem
  • entanglement
  • entanglement entropy
  • free fermions
  • logarithmic negativity

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)


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