Abstract
In this paper we present the detailed calculation of a new modular Hamiltonian, namely that of a chiral fermion on a circle at non-zero temperature. We provide explicit results for an arbitrary collection of intervals, which we discuss at length by checking against known results in different limits and by computing the associated modular flow. We also compute the entanglement entropy, and we obtain a simple expression for it which appears to be more manageable than those already existing in the literature.
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H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys.A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev.D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
H. Casini, M. Huerta, R.C. Myers and A. Yale, Mutual information and the F-theorem, JHEP10 (2015) 003 [arXiv:1506.06195] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Markov property of the conformal field theory vacuum and the a theorem, Phys. Rev. Lett.118 (2017) 261602 [arXiv:1704.01870] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for deformed half-spaces and the averaged null energy condition, JHEP09 (2016) 038 [arXiv:1605.08072] [INSPIRE].
D.D. Blanco and H. Casini, Localization of negative energy and the Bekenstein bound, Phys. Rev. Lett.111 (2013) 221601 [arXiv:1309.1121] [INSPIRE].
D. Blanco, H. Casini, M. Leston and F. Rosso, Modular energy inequalities from relative entropy, JHEP01 (2018) 154 [arXiv:1711.04816] [INSPIRE].
H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav.25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].
T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, JHEP03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from entanglement ‘thermodynamics’, JHEP04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
D. Blanco, M. Leston and G. Pérez-Nadal, Gravity from entanglement for boundary subregions, JHEP18 (2018) 130 [arXiv:1803.01874] [INSPIRE].
B. Swingle and M. Van Raamsdonk, Universality of gravity from entanglement, arXiv:1405.2933 [INSPIRE].
J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys.17 (1976) 303 [INSPIRE].
W.G. Unruh, Notes on black hole evaporation, Phys. Rev.D 14 (1976) 870 [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
J. Cardy and E. Tonni, Entanglement Hamiltonians in two-dimensional conformal field theory, J. Stat. Mech.1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
T. Hartman and N. Afkhami-Jeddi, Speed limits for entanglement, arXiv:1512.02695 [INSPIRE].
H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav.26 (2009) 185005 [arXiv:0903.5284] [INSPIRE].
G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement temperature and entanglement entropy of excited states, JHEP12 (2013) 020 [arXiv:1305.3291] [INSPIRE].
R.E. Arias, H. Casini, M. Huerta and D. Pontello, Entropy and modular Hamiltonian for a free chiral scalar in two intervals, Phys. Rev.D 98 (2018) 125008 [arXiv:1809.00026] [INSPIRE].
D. Blanco and G. Pérez-Nadal, Modular Hamiltonian of a chiral fermion on the torus, Phys. Rev.D 100 (2019) 025003 [arXiv:1905.05210] [INSPIRE].
P. Fries and I.A. Reyes, The entanglement spectrum of chiral fermions on the torus, arXiv:1905.05768 [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys.A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
T. Azeyanagi, T. Nishioka and T. Takayanagi, Near extremal black hole entropy as entanglement entropy via AdS 2/CFT 1, Phys. Rev.D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].
C.P. Herzog and T. Nishioka, Entanglement entropy of a massive fermion on a torus, JHEP03 (2013) 077 [arXiv:1301.0336] [INSPIRE].
I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys.A 36 (2003) L205.
K. Chandrasekharan, Elliptic functions, Springer, Berlin Heidelberg, Germany (2012).
G. Pastras, Four lectures on Weierstrass elliptic function and applications in classical and quantum mechanics, arXiv:1706.07371 [INSPIRE].
T. Nishioka and A. Yarom, Anomalies and entanglement entropy, JHEP03 (2016) 077 [arXiv:1509.04288] [INSPIRE].
N. Iqbal and A.C. Wall, Anomalies of the entanglement entropy in chiral theories, JHEP10 (2016) 111 [arXiv:1509.04325] [INSPIRE].
T.L. Hughes, R.G. Leigh, O. Parrikar and S.T. Ramamurthy, Entanglement entropy and anomaly inflow, Phys. Rev.D 93 (2016) 065059 [arXiv:1509.04969] [INSPIRE].
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Blanco, D., Garbarz, A. & Pérez-Nadal, G. Entanglement of a chiral fermion on the torus. J. High Energ. Phys. 2019, 76 (2019). https://doi.org/10.1007/JHEP09(2019)076
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DOI: https://doi.org/10.1007/JHEP09(2019)076