Abstract
In two recent works, we studied the symmetry resolved Rényi entropies of quasi-particle excited states in quantum field theory. We found that the entropies display many model-independent features which we discussed and analytically characterised. In this paper we extend this line of investigation by providing analytical and numerical evidence that a similar universal behavior arises for the symmetry resolved negativity. In particular, we compute the ratio of charged moments of the partially transposed reduced density matrix as an expectation value of twist operators. These are “fused” versions of the more traditionally used branch point twist fields and were introduced in a previous work. The use of twist operators allows us to perform the computation in an arbitrary number of spacial dimensions. We show that, in the large-volume limit, only the commutation relations between the twist operators and local fields matter, and computations reduce to a purely combinatorial problem. We address some specific issues regarding fermionic excitations, whose treatment requires the notion of partial time-reversal transformation, and we discuss the differences and analogies with their bosonic counterpart. We find that although the operation of partial transposition requires a redefinition for fermionic theories, the ratio of the negativity moments between an excited state and the ground state is universal and identical for fermions and bosons as well as for a large variety of very different states, ranging from simple qubit states to the excited states of free quantum field theories. Our predictions are tested numerically on a 1D Fermi chain.
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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].
B.-Q. Jin and V.E. Korepin, Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture, J. Stat. Phys. 116 (2004) 79 [quant-ph/0304108].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (2003) 227902 [quant-ph/0211074] [INSPIRE].
J.I. Latorre, E. Rico and G. Vidal, Ground state entanglement in quantum spin chains, Quant. Inf. Comput. 4 (2004) 48 [quant-ph/0304098] [INSPIRE].
J.I. Latorre, C.A. Lutken, E. Rico and G. Vidal, Fine grained entanglement loss along renormalization group flows, Phys. Rev. A 71 (2005) 034301 [quant-ph/0404120] [INSPIRE].
C.H. Bennett, H.J. Bernstein, S. Popescu and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53 (1996) 2046 [quant-ph/9511030] [INSPIRE].
J.L. Cardy, O.A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [INSPIRE].
K. Audenaert, J. Eisert, M.B. Plenio and R.F. Werner, Entanglement Properties of the Harmonic Chain, Phys. Rev. A 66 (2002) 042327 [quant-ph/0205025] [INSPIRE].
K. Zyczkowski, P. Horodecki, A. Sanpera and M. Lewenstein, On the volume of the set of mixed entangled states, Phys. Rev. A 58 (1998) 883 [quant-ph/9804024] [INSPIRE].
J. Eisert and M.B. Plenio, A Comparison of entanglement measures, J. Mod. Opt. 46 (1999) 145 [quant-ph/9807034] [INSPIRE].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
M.B. Plenio, Logarithmic Negativity: A Full Entanglement Monotone That is not Convex, Phys. Rev. Lett. 95 (2005) 090503 [quant-ph/0505071] [INSPIRE].
M.B. Plenio, Erratum: Logarithmic negativity: A full entanglement monotone that is not convex, Phys. Rev. Lett. 95 (2005) 119902.
J. Eisert, Entanglement in quantum information theory, quant-ph/0610253.
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: A field theoretical approach, J. Stat. Mech. 1302 (2013) P02008 [arXiv:1210.5359] [INSPIRE].
V. Eisler and Z. Zimborás, Entanglement negativity in two-dimensional free lattice models, Phys. Rev. B 93 (2016) 115148 [arXiv:1511.08819].
H. Shapourian, K. Shiozaki and S. Ryu, Partial time-reversal transformation and entanglement negativity in fermionic systems, Phys. Rev. B 95 (2017) 165101 [arXiv:1611.07536] [INSPIRE].
H. Shapourian and S. Ryu, Entanglement negativity of fermions: monotonicity, separability criterion, and classification of few-mode states, Phys. Rev. A 99 (2019) 022310 [arXiv:1804.08637] [INSPIRE].
M. Goldstein and E. Sela, Symmetry-resolved entanglement in many-body systems, Phys. Rev. Lett. 120 (2018) 200602 [arXiv:1711.09418] [INSPIRE].
J.C. Xavier, F.C. Alcaraz and G. Sierra, Equipartition of the entanglement entropy, Phys. Rev. B 98 (2018) 041106 [arXiv:1804.06357] [INSPIRE].
E. Cornfeld, M. Goldstein and E. Sela, Imbalance entanglement: Symmetry decomposition of negativity, Phys. Rev. A 98 (2018) 032302 [arXiv:1804.00632] [INSPIRE].
S. Murciano, R. Bonsignori and P. Calabrese, Symmetry decomposition of negativity of massless free fermions, SciPost Phys. 10 (2021) 111 [arXiv:2102.10054] [INSPIRE].
H.-H. Chen, Charged Rényi negativity of massless free bosons, JHEP 02 (2022) 117 [arXiv:2111.11028] [INSPIRE].
H. Gaur and U.A. Yajnik, Charge imbalance resolved Rényi negativity for free compact boson: Two disjoint interval case, JHEP 02 (2023) 118 [arXiv:2210.06743] [INSPIRE].
A. Neven et al., Symmetry-resolved entanglement detection using partial transpose moments, npj Quantum Inf. 7 (2021) 152 [arXiv:2103.07443] [INSPIRE].
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement Content of Quasiparticle Excitations, Phys. Rev. Lett. 121 (2018) 170602 [arXiv:1805.04948] [INSPIRE].
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement content of quantum particle excitations. Part I. Free field theory, JHEP 10 (2018) 039 [arXiv:1806.03247] [INSPIRE].
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement content of quantum particle excitations. Part II. Disconnected regions and logarithmic negativity , JHEP 11 (2019) 058 [arXiv:1904.01035] [INSPIRE].
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement Content of Quantum Particle Excitations III. Graph Partition Functions, J. Math. Phys. 60 (2019) 082301 [arXiv:1904.02615] [INSPIRE].
L. Capizzi et al., Symmetry resolved entanglement of excited states in quantum field theory. Part I. Free theories, twist fields and qubits, JHEP 12 (2022) 127 [arXiv:2203.12556] [INSPIRE].
L. Capizzi et al., Symmetry resolved entanglement of excited states in quantum field theory. Part II. Numerics, interacting theories and higher dimensions, JHEP 12 (2022) 128 [arXiv:2206.12223] [INSPIRE].
D.X. Horváth and P. Calabrese, Symmetry resolved entanglement in integrable field theories via form factor bootstrap, JHEP 11 (2020) 131 [arXiv:2008.08553] [INSPIRE].
V.G. Knizhnik, Analytic Fields on Riemann Surfaces. 2, Commun. Math. Phys. 112 (1987) 567 [INSPIRE].
L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The Conformal Field Theory of Orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].
P. Bouwknegt, Coset construction for winding subalgebras and applications, q-alg/9610013.
L. Borisov, M.B. Halpern and C. Schweigert, Systematic approach to cyclic orbifolds, Int. J. Mod. Phys. A 13 (1998) 125 [hep-th/9701061] [INSPIRE].
O.A. Castro-Alvaredo, B. Doyon and E. Levi, Arguments towards a c-theorem from branch-point twist fields, J. Phys. A 44 (2011) 492003 [arXiv:1107.4280] [INSPIRE].
E. Levi, Composite branch-point twist fields in the Ising model and their expectation values, J. Phys. A 45 (2012) 275401 [arXiv:1204.1192] [INSPIRE].
D. Bianchini et al., Entanglement Entropy of Non Unitary Conformal Field Theory, J. Phys. A 48 (2015) 04FT01 [arXiv:1405.2804] [INSPIRE].
D. Bianchini and O.A. Castro-Alvaredo, Branch Point Twist Field Correlators in the Massive Free Boson Theory, Nucl. Phys. B 913 (2016) 879 [arXiv:1607.05656] [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].
V. Eisler and Z. Zimborás, On the partial transpose of fermionic Gaussian states, New J. Phys. 17 (2015) 053048 [arXiv:1502.01369].
J. Eisert, V. Eisler and Z. Zimborás, Entanglement negativity bounds for fermionic Gaussian states, Phys. Rev. B 97 (2018) 165123 [arXiv:1611.08007] [INSPIRE].
L. Capizzi, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement entropy of excited states in a CFT, J. Stat. Mech. 2007 (2020) 073101 [arXiv:2003.04670] [INSPIRE].
J. Zhang and M.A. Rajabpour, Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part I. Single-particle states, JHEP 12 (2020) 160 [arXiv:2009.00719] [INSPIRE].
J. Zhang and M.A. Rajabpour, Universal Rényi entanglement entropy of quasiparticle excitations, EPL 135 (2021) 60001 [arXiv:2010.13973] [INSPIRE].
J. Zhang and M.A. Rajabpour, Corrections to universal Rényi entropy in quasiparticle excited states of quantum chains, J. Stat. Mech. 2109 (2021) 093101 [arXiv:2010.16348] [INSPIRE].
J. Zhang and M.A. Rajabpour, Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part II. Multi-particle states, JHEP 08 (2021) 106 [arXiv:2011.11006] [INSPIRE].
J. Zhang and M.A. Rajabpour, Entanglement of magnon excitations in spin chains, JHEP 02 (2022) 072 [arXiv:2109.12826] [INSPIRE].
J. Zhang and M.A. Rajabpour, Subsystem distances between quasiparticle excited states, JHEP 07 (2022) 119 [arXiv:2202.11448] [INSPIRE].
G. Mussardo and J. Viti, ħ →0 limit of the entanglement entropy, Phys. Rev. A 105 (2022) 032404 [arXiv:2112.06840] [INSPIRE].
H. Shapourian, R.S.K. Mong and S. Ryu, Anyonic Partial Transpose I: Quantum Information Aspects, arXiv:2012.02222 [INSPIRE].
Y.-H. Lin, M. Okada, S. Seifnashri and Y. Tachikawa, Asymptotic density of states in 2d CFTs with non-invertible symmetries, JHEP 03 (2023) 094 [arXiv:2208.05495] [INSPIRE].
F.C. Alcaraz, M.I. Berganza and G. Sierra, Entanglement of low-energy excitations in Conformal Field Theory, Phys. Rev. Lett. 106 (2011) 201601 [arXiv:1101.2881] [INSPIRE].
G. Parez, R. Bonsignori and P. Calabrese, Dynamics of charge-imbalance-resolved entanglement negativity after a quench in a free-fermion model, J. Stat. Mech. 2205 (2022) 053103 [arXiv:2202.05309] [INSPIRE].
H.-H. Chen, Dynamics of charge imbalance resolved negativity after a global quench in free scalar field theory, JHEP 08 (2022) 146 [Erratum ibid. 10 (2022) 157] [arXiv:2205.09532] [INSPIRE].
J.J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press (2005).
G.E. Bergum and V.E. Hoggatt Jr, Irreducibility of Lucas and generalized Lucas polynomials, The Fibonacci Quarterly 12 (1974) 95.
P.E. Ricci, Generalized Lucas polynomials and Fibonacci polynomials, Riv. Mat. Univ. Parma 4 (1995) 137.
Acknowledgments
The authors are grateful to Cecilia De Fazio and Lucía Sanz-Santamaría for their involvement in Part I and Part II of this project and for many useful discussions that have contributed to the present work. Luca Capizzi thanks ERC for support under Consolidator grant number 771536 (NEMO) and the Department of Mathematics of City, University of London for hospitality during a two month visit (October-November 2022) when most of this work was completed. Michele Mazzoni is grateful for funding under the EPSRC Mathematical Sciences Doctoral Training Partnership EP/W524104/1. Olalla A. Castro-Alvaredo thanks EPSRC for financial support under Small Grant EP/W007045/1.
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Capizzi, L., Mazzoni, M. & Castro-Alvaredo, O.A. Symmetry resolved entanglement of excited states in quantum field theory. Part III. Bosonic and fermionic negativity. J. High Energ. Phys. 2023, 74 (2023). https://doi.org/10.1007/JHEP06(2023)074
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DOI: https://doi.org/10.1007/JHEP06(2023)074