Abstract
We study the entanglement entropies of an interval adjacent to the boundary of the half line for the free fermionic spinless Schrödinger field theory at finite density and zero temperature, with either Neumann or Dirichlet boundary conditions. They are finite functions of the dimensionless parameter given by the product of the Fermi momentum and the length of the interval. The entanglement entropy displays an oscillatory behaviour, differently from the case of the interval on the whole line. This behaviour is related to the Friedel oscillations of the mean particle density on the half line at the entangling point. We find analytic expressions for the expansions of the entanglement entropies in the regimes of small and large values of the dimensionless parameter. They display a remarkable agreement with the curves obtained numerically. The analysis is extended to a family of free fermionic Lifshitz models labelled by their integer Lifshitz exponent, whose parity determines the properties of the entanglement entropies. The cumulants of the local charge operator and the Schatten norms of the underlying kernels are also explored.
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P. Calabrese, J. Cardy and B. Doyon, Entanglement entropy in extended quantum systems, J. Phys. A 42 (2009) 500301.
I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42 (2009) 504003 [arXiv:0906.1663].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
J. Eisert, M. Cramer and M.B. Plenio, Area laws for the entanglement entropy — a review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Lect. Notes Phys. 931 (2017) 1 [arXiv:1609.01287] [INSPIRE].
M. Headrick, Lectures on entanglement entropy in field theory and holography, arXiv:1907.08126 [INSPIRE].
E. Tonni, An introduction to entanglement measures in conformal field theories and AdS/CFT, Springer Proc. Phys. 239 (2020) 69 [INSPIRE].
I. Peschel and J. Zhao, On single-copy entanglement, J. Stat. Mech. 2005 (2005) P11002 [quant-ph/0509002].
J. Eisert and M. Cramer, Single-copy entanglement in critical quantum spin chains, Phys. Rev. A 72 (2005) 042112 [quant-ph/0506250].
R. Orús, J.I. Latorre, J. Eisert and M. Cramer, Half the entanglement in critical systems is distillable from a single specimen, Phys. Rev. A 73 (2006) 060303 [quant-ph/0509023] [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
C.G. Callan, Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402 [hep-th/0312197] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140 [arXiv:1607.00390] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, Irreversibility in quantum field theories with boundaries, JHEP 04 (2019) 166 [arXiv:1812.08183] [INSPIRE].
J.A. Hertz, Quantum critical phenomena, Phys. Rev. B 14 (1976) 1165 [INSPIRE].
U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972) 802 [INSPIRE].
C.R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D 5 (1972) 377 [INSPIRE].
M. Henkel, Schrödinger invariance in strongly anisotropic critical systems, J. Statist. Phys. 75 (1994) 1023 [hep-th/9310081] [INSPIRE].
Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].
E. Ardonne, P. Fendley and E. Fradkin, Topological order and conformal quantum critical points, Annals Phys. 310 (2004) 493 [cond-mat/0311466] [INSPIRE].
D. Gioev and I. Klich, Entanglement entropy of fermions in any dimension and the Widom conjecture, Phys. Rev. Lett. 96 (2006) 100503 [quant-ph/0504151] [INSPIRE].
M.M. Wolf, Violation of the entropic area law for fermions, Phys. Rev. Lett. 96 (2006) 010404 [quant-ph/0503219] [INSPIRE].
E. Fradkin and J.E. Moore, Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum, Phys. Rev. Lett. 97 (2006) 050404 [cond-mat/0605683] [INSPIRE].
B. Hsu, M. Mulligan, E. Fradkin and E.-A. Kim, Universal entanglement entropy in 2D conformal quantum critical points, Phys. Rev. B 79 (2009) 115421 [arXiv:0812.0203] [INSPIRE].
E. Fradkin, Scaling of entanglement entropy at 2D quantum Lifshitz fixed points and topological fluids, J. Phys. A 42 (2009) 504011 [arXiv:0906.1569] [INSPIRE].
S.N. Solodukhin, Entanglement entropy in non-relativistic field theories, JHEP 04 (2010) 101 [arXiv:0909.0277] [INSPIRE].
H. Leschke, A.V. Sobolev and W. Spitzer, Scaling of Rényi entanglement entropies of the free Fermi-gas ground state: a rigorous proof, Phys. Rev. Lett. 112 (2014) 160403 [arXiv:1312.6828].
V. Keranen, W. Sybesma, P. Szepietowski and L. Thorlacius, Correlation functions in theories with Lifshitz scaling, JHEP 05 (2017) 033 [arXiv:1611.09371] [INSPIRE].
T. Alho, V.G.M. Puletti, R. Pourhasan and L. Thorlacius, Monopole correlation functions and holographic phases of matter in 2 + 1 dimensions, Phys. Rev. D 94 (2016) 106012 [arXiv:1607.04059] [INSPIRE].
M.R. Mohammadi Mozaffar and A. Mollabashi, Entanglement in Lifshitz-type quantum field theories, JHEP 07 (2017) 120 [arXiv:1705.00483] [INSPIRE].
M.R.M. Mozaffar and A. Mollabashi, Time scaling of entanglement in integrable scale-invariant theories, Phys. Rev. Res. 4 (2022) L022010 [arXiv:2106.14700] [INSPIRE].
A. Sobolev, Quasi-classical asymptotics for pseudodifferential operators with discontinuous symbols: Widom’s conjecture, Funct. Anal. Appl. 44 (2010) 313 [arXiv:1004.2576].
G. Benfatto and G. Gallavotti, Renormalization group, Princeton University Press (1995).
S. Sachdev, Quantum phase transitions, second edition, Cambridge University Press (2011).
M. Mintchev, D. Pontello, A. Sartori and E. Tonni, Entanglement entropies of an interval in the free Schrödinger field theory at finite density, JHEP 07 (2022) 120 [arXiv:2201.04522] [INSPIRE].
S. Pal and B. Grinstein, Heat kernel and Weyl anomaly of Schrödinger invariant theory, Phys. Rev. D 96 (2017) 125001 [arXiv:1703.02987] [INSPIRE].
I. Hason, Triviality of entanglement entropy in the Galilean vacuum, Phys. Lett. B 780 (2018) 149 [arXiv:1708.08303] [INSPIRE].
D. Hartmann, K. Kavanagh and S. Vandoren, Entanglement entropy with Lifshitz fermions, SciPost Phys. 11 (2021) 031 [arXiv:2104.10913] [INSPIRE].
J. Friedel, XIV. The distribution of electrons round impurities in monovalent metals, London Edinburgh Dublin Phil. Mag. J. Sci. 43 (1952) 153.
T. Giamarchi, Quantum physics in one dimension, Oxford University Press (2003).
A. Imambekov, T.L. Schmidt and L.I. Glazman, One-dimensional quantum liquids: beyond the Luttinger liquid paradigm, Rev. Mod. Phys. 84 (2012) 1253 [arXiv:1110.1374].
D. Slepian and H.O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty — I, Bell Syst. Tech. J. 40 (1961) 43.
H.J. Landau and H.O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty — II, Bell Syst. Tech. J. 40 (1961) 65.
H.J. Landau and H.O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty — III: the dimension of the space of essentially time- and band-limited signals, Bell Syst. Tech. J. 41 (1962) 1295.
D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty — IV: extensions to many dimensions; generalized prolate spheroidal functions, Bell Syst. Tech. J. 43 (1964) 3009.
D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25 (1983) 379.
A. Osipov, V. Rokhlin and H. Xiao, Prolate spheroidal wave functions of order zero, Springer (2013).
V. Eisler and I. Peschel, Free-fermion entanglement and spheroidal functions, J. Stat. Mech. 2013 (2013) P04028 [arXiv:1302.2239].
B.-Q. Jin and V.E. Korepin, Quantum spin chain, Toeplitz determinants and the Fisher-Hartwig conjecture, J. Statist. Phys. 116 (2004) 79 [quant-ph/0304108].
J. Keating and F. Mezzadri, Random matrix theory and entanglement in quantum spin chains, Commun. Math. Phys. 252 (2004) 543 [quant-ph/0407047].
P. Calabrese, M. Campostrini, F. Essler and B. Nienhuis, Parity effects in the scaling of block entanglement in gapless spin chains, Phys. Rev. Lett. 104 (2010) 095701 [arXiv:0911.4660] [INSPIRE].
P. Calabrese and F.H.L. Essler, Universal corrections to scaling for block entanglement in spin-1/2 XX chains, J. Stat. Mech. 2010 (2010) P08029 [arXiv:1006.3420].
C.A. Tracy and H. Widom, Level spacing distributions and the Bessel kernel, Commun. Math. Phys. 161 (1994) 289 [hep-th/9304063] [INSPIRE].
O. Gamayun, N. Iorgov and O. Lisovyy, How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys. A 46 (2013) 335203 [arXiv:1302.1832] [INSPIRE].
G. Bonelli, O. Lisovyy, K. Maruyoshi, A. Sciarappa and A. Tanzini, On Painlevé/gauge theory correspondence, Lett. Math. Phys. 107 (2017) pages 2359 [arXiv:1612.06235] [INSPIRE].
T. Bothner, A. Its and A. Prokhorov, On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential, Adv. Math. 345 (2019) 483.
O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP 10 (2012) 038 [Erratum ibid. 10 (2012) 183] [arXiv:1207.0787] [INSPIRE].
E.L. Basor and T. Ehrhardt, Asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices, Math. Nach. 228 (2001) 5 [math.FA/9809088].
E.L. Basor and T. Ehrhardt, Asymptotic formulas for the determinants of symmetric Toeplitz plus Hankel matrices, in Toeplitz matrices and singular integral equations: the Bernd Silbermann anniversary volume, A. Böttcher, I. Gohberg and P. Junghanns eds., Birkhäuser (2002), p. 61.
E.L. Basor and T. Ehrhardt, Determinant computations for some classes of Toeplitz-Hankel matrices, Operat. Matr. (2008) 167 [arXiv:0804.3073].
P. Deift, A. Its and I. Krasovsky, Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. Math. 174 (2011) 1243 [arXiv:0905.0443].
M. Fagotti and P. Calabrese, Universal parity effects in the entanglement entropy of XX chains with open boundary conditions, J. Stat. Mech. 1101 (2011) P01017 [arXiv:1010.5796] [INSPIRE].
I. Klich and L. Levitov, Quantum noise as an entanglement meter, Phys. Rev. Lett. 102 (2009) 100502 [arXiv:0804.1377] [INSPIRE].
H.F. Song, C. Flindt, S. Rachel, I. Klich and K.L. Hur, Entanglement entropy from charge statistics: exact relations for noninteracting many-body systems, Phys. Rev. B 83 (2011) 161408 [arXiv:1008.5191].
H.F. Song, S. Rachel, C. Flindt, I. Klich, N. Laflorencie and K. Le Hur, Bipartite fluctuations as a probe of many-body entanglement, Phys. Rev. B 85 (2012) 035409 [arXiv:1109.1001] [INSPIRE].
R. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, self-adjointness, Academic Press (1975).
M. Mintchev, L. Santoni and P. Sorba, Quantum transport in presence of bound states — noise power, Annalen Phys. 529 (2017) 1600274 [arXiv:1609.05427] [INSPIRE].
O. Bratteli and D.W. Robinson, Operator algebras and quantum statistical mechanics 2: equilibrium states. Models in quantum statistical mechanics, Springer (1996).
O. Bratteli and D. Robinson, Operator algebras and quantum statistical mechanics 1: C∗- and W∗-algebras. Symmetry groups. Decomposition of states, Springer (1987).
NIST digital library of mathematical functions, release 1.1.5, http://dlmf.nist.gov/, 15 March 2022.
R.T. Powers and E. Stormer, Free states of the canonical anticommutation relations, Commun. Math. Phys. 16 (1970) 1 [INSPIRE].
P.M. Morse and H. Feshbach, Methods of theoretical physics, McGraw-Hill (1953).
C. Flammer, Spheroidal wave functions, Stanford University Press (1957).
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover (1964).
I.C. Moore and M. Cada, Prolate spheroidal wave functions, an introduction to the slepian series and its properties, Appl. Comput. Harmon. Anal. 16 (2004) 208.
H.J. Landau, The eigenvalue behavior of certain convolution equations, Trans. Amer. Math. Soc. 115 (1965) 242.
A. Bonami, P. Jaming and A. Karoui, Non-asymptotic behavior of the spectrum of the sinc-kernel operator and related applications, J. Math. Phys. 62 (2021) 033511 [arXiv:1804.01257].
N.S. Witte, Gap probabilities for double intervals in hermitian random matrix ensembles as τ-functions — spectrum singularity case, Lett. Math. Phys. 68 (2004) 139 [math-ph/0307063].
M. Gaudin, Sur la loi limite de l’espacement des valeurs propres d’une matrice aléatoire (in French), Nucl. Phys. 25 (1961) 447.
F.J. Dyson, Fredholm determinants and inverse scattering problems, Commun. Math. Phys. 47 (1976) 171 [INSPIRE].
M. Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982) 1137.
P. Forrester, Log-gases and random matrices, Princeton University Press (2010).
M. Mintchev and E. Tonni, Modular Hamiltonians for the massless Dirac field in the presence of a boundary, JHEP 03 (2021) 204 [arXiv:2012.00703] [INSPIRE].
N.I. Muskhelishvili, Singular integral equations: boundary problems of functions theory and their applications to mathematical physics, Springer (1977).
H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav. 26 (2009) 185005 [arXiv:0903.5284] [INSPIRE].
R. Arias, D. Blanco, H. Casini and M. Huerta, Local temperatures and local terms in modular Hamiltonians, Phys. Rev. D 95 (2017) 065005 [arXiv:1611.08517] [INSPIRE].
D. Slepian, Some asymptotic expansions for prolate spheroidal wave functions, J. Math. Phys. 44 (1965) 99.
T. Ehrhardt, Dyson’s constants in the asymptotics of the determinants of Wiener-Hopf-Hankel operators with the sine kernel, Commun. Math. Phys. 272 (2007) 683 [math.FA/0605003].
R. Süsstrunk and D.A. Ivanov, Free fermions on a line: asymptotics of the entanglement entropy and entanglement spectrum from full counting statistics, EPL 100 (2012) 60009 [arXiv:1208.5845].
D.A. Ivanov, A.G. Abanov and V.V. Cheianov, Counting free fermions on a line: a Fisher-Hartwig asymptotic expansion for the Toeplitz determinant in the double-scaling limit, J. Phys. A 46 (2013) 085003 [arXiv:1112.2530] [INSPIRE].
H. Leschke, A.V. Sobolev and W. Spitzer, Large-scale behaviour of local and entanglement entropy of the free Fermi gas at any temperature, J. Phys. A 49 (2016) 30LT04 [arXiv:1501.03412].
H. Leschke, A.V. Sobolev and W. Spitzer, Trace formulas for Wiener-Hopf operators with applications to entropies of free fermionic equilibrium states, J. Funct. Anal. 273 (2017) 1049 [arXiv:1605.04429].
H. Leschke, A.V. Sobolev and W. Spitzer, Rényi entropies of the free Fermi gas in multi-dimensional space at high temperature, arXiv:2201.11087.
K. Sakai and Y. Satoh, Entanglement through conformal interfaces, JHEP 12 (2008) 001 [arXiv:0809.4548] [INSPIRE].
V. Eisler and I. Peschel, Entanglement in fermionic chains with interface defects, Annalen Phys. 522 (2010) 679 [arXiv:1005.2144].
P. Calabrese, M. Mintchev and E. Vicari, Entanglement entropy of quantum wire junctions, J. Phys. A 45 (2012) 105206 [arXiv:1110.5713] [INSPIRE].
I. Peschel and V. Eisler, Exact results for the entanglement across defects in critical chains, J. Phys. A 45 (2012) 155301 [arXiv:1201.4104].
M. Mintchev and E. Tonni, Modular Hamiltonians for the massless Dirac field in the presence of a defect, JHEP 03 (2021) 205 [arXiv:2012.01366] [INSPIRE].
J. Kruthoff, R. Mahajan and C. Murdia, Free fermion entanglement with a semitransparent interface: the effect of graybody factors on entanglement islands, SciPost Phys. 11 (2021) 063 [arXiv:2106.10287] [INSPIRE].
L. Capizzi, S. Murciano and P. Calabrese, Rényi entropy and negativity for massless Dirac fermions at conformal interfaces and junctions, JHEP 08 (2022) 171 [arXiv:2205.04722] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 0911 (2009) P11001 [arXiv:0905.2069] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
A. Coser, L. Tagliacozzo and E. Tonni, On Rényi entropies of disjoint intervals in conformal field theory, J. Stat. Mech. 1401 (2014) P01008 [arXiv:1309.2189] [INSPIRE].
A. Coser, E. Tonni and P. Calabrese, Spin structures and entanglement of two disjoint intervals in conformal field theories, J. Stat. Mech. 1605 (2016) 053109 [arXiv:1511.08328] [INSPIRE].
C. De Nobili, A. Coser and E. Tonni, Entanglement entropy and negativity of disjoint intervals in CFT: some numerical extrapolations, J. Stat. Mech. 1506 (2015) P06021 [arXiv:1501.04311] [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].
T. Grava, A.P. Kels and E. Tonni, Entanglement of two disjoint intervals in conformal field theory and the 2D Coulomb gas on a lattice, Phys. Rev. Lett. 127 (2021) 141605 [arXiv:2104.06994] [INSPIRE].
H. Casini, C.D. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 0507 (2005) P07007 [cond-mat/0505563] [INSPIRE].
H. Casini and M. Huerta, Entanglement and alpha entropies for a massive scalar field in two dimensions, J. Stat. Mech. 0512 (2005) P12012 [cond-mat/0511014] [INSPIRE].
L. Daguerre, R. Medina, M. Solis and G. Torroba, Aspects of quantum information in finite density field theory, JHEP 03 (2021) 079 [arXiv:2011.01252] [INSPIRE].
G. Gallavotti, The Luttinger model: its role in the RG-theory of one dimensional many body Fermi systems, J. Statist. Phys. 103 (2001) 459 [cond-mat/0008090].
G. Gentile and V. Mastropietro, Renormalization group for one-dimensional fermions: a review on mathematical results, Phys. Rept. 352 (2001) 273 [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger invariance from Lifshitz isometries in holography and field theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Field theory on Newton-Cartan backgrounds and symmetries of the Lifshitz vacuum, JHEP 08 (2015) 006 [arXiv:1502.00228] [INSPIRE].
J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].
H.R. Afshar, E.A. Bergshoeff, A. Mehra, P. Parekh and B. Rollier, A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities, JHEP 04 (2016) 145 [arXiv:1512.06277] [INSPIRE].
P. Forrester, The spectrum edge of random matrix ensembles, Nucl. Phys. B 402 (1993) 709.
K. Okamoto, Studies on the Painlevé equations: IV. Third Painlevé equation PIII, Funk. Ekvac. 30 (1987) 305.
D. Dai, P.J. Forrester and S.-X. Xu, Applications in random matrix theory of a PIII’ τ-function sequence from Okamoto’s Hamiltonian formulation, Random Matr. 11 (2021) 2250014 [arXiv:1909.07634].
O. Lisovyy, H. Nagoya and J. Roussillon, Irregular conformal blocks and connection formulae for Painlevé V functions, J. Math. Phys. 59 (2018) 091409 [arXiv:1806.08344] [INSPIRE].
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Mintchev, M., Pontello, D. & Tonni, E. Entanglement entropies of an interval in the free Schrödinger field theory on the half line. J. High Energ. Phys. 2022, 90 (2022). https://doi.org/10.1007/JHEP09(2022)090
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DOI: https://doi.org/10.1007/JHEP09(2022)090