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Cited by 3,275Quantum many-body systemsBlack Holes and Theoretical PhysicsTheoretical and Computational Physics

Entanglement entropy and quantum field theory

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Pasquale Calabrese, John Cardy·2004·Journal of Statistical Mechanics: Theory and Experiment
DOI 10.1088/1742-5468/2004/06/p06002
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Abstract

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A of a subsystem A . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c , we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

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Crossref · 10.1088/1742-5468/2004/06/p06002

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  • The ingoing modes are in a thermal state at the same inverse temperature β as the black hole, and therefore satisfy dS_in/dv = c_evap π/(6β).

    “The ingoing modes are in a thermal state at the same inverse temperature β as the black hole.They therefore satisfy [57, 58]”

    From Entanglement wedge reconstruction and the information paradox (2020) · cites this paper as [57]

    Equation (3.9) and the discussion following it give an extensive entropy for a finite-temperature CFT, with density πc/(3β). The paper does not address black holes or ingoing modes, establish their temperature, or state the claimed rate dS_in/dv = c_evapπ/(6β). (LLM, conf=0.83)

    Experimental — accuracy not yet measured

  • The bulk entropy of outgoing radiation increases at rate c_evap π/(6β) with respect to v_rad.

    “∂S bulk ∂v rad = c evap π 6β . (3.16)”

    From Entanglement wedge reconstruction and the information paradox (2020) · cites this paper as [57]

    Equation (3.9) and its discussion in the finite-temperature CFT section report entropy growing linearly with subsystem length at large lengths, with coefficient πc/(3β). The paper does not identify that subsystem with outgoing radiation, define v_rad or c_evap, or establish the claimed rate πc_evap/(6β) with respect to v_rad. (LLM, conf=0.82)

    Experimental — accuracy not yet measured

  • In a finite system, there is a crossover between the adiabatic and excited regimes, where in the adiabatic regime the entanglement entropy at the critical point is S ∝ log(N).

    “Focusing on a finite system 31 , there is a crossover between those two regimes, where, in the adiabatic regime, the entanglement entropy at the critical point is S ∝ log(N ) 118 .”

    From Beyond-classical computation in quantum simulation (2025) · cites this paper as [118]

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Cites

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Research works printed but not linked: 17

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Authors
Calabrese, P., & Cardy, J.
Affiliations
University of Oxford (GB)
ORCID
Venue
Journal of Statistical Mechanics: Theory and Experiment
Publisher
IOP Publishing
Published
2004-06-01
Volume
2004
Issue
06
Pages
P06002
ISSN
1742-5468 (ISSN-L)
DOI
10.1088/1742-5468/2004/06/p06002
OpenAlex
W2069840277
Cited by
3,275
References
2
Citations by year
2026: 169 · 2025: 260 · 2024: 232 · 2023: 203 · 2022: 263 · 2021: 234 · 2020: 235 · 2019: 188 · 2018: 154 · 2017: 181 · 2016: 194 · 2015: 165 · 2014: 164 · 2013: 122 · 2012: 107
FWCI
18.41
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of Statistical Mechanics Theory and Experiment (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CINECA IRIS Institutial research information system (University of Pisa) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • SDL (submittedVersion)
  • arXiv (Cornell University)
Funders
Engineering and Physical Sciences Research Council (GR/R83712/01)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
Keywords
entanglement entropy, relativistic quantum field theory, von Neumann entropy, reduced density matrix, conformal field theory, central charge, finite temperature, disjoint intervals, integrable lattice models, Ising model, XXZ model, corner transfer matrix, quantum phase transitions, scaling ansatz, correlation length

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