Abstract
The entanglement asymmetry is an information based observable that quantifies the degree of symmetry breaking in a region of an extended quantum system. We investigate this measure in the ground state of one dimensional critical systems described by a CFT. Employing the correspondence between global symmetries and defects, the analysis of the entanglement asymmetry can be formulated in terms of partition functions on Riemann surfaces with multiple non-topological defect lines inserted at their branch cuts. For large subsystems, these partition functions are determined by the scaling dimension of the defects. This leads to our first main observation: at criticality, the entanglement asymmetry acquires a subleading contribution scaling as log ℓ/ℓ for large subsystem length ℓ. Then, as an illustrative example, we consider the XY spin chain, which has a critical line described by the massless Majorana fermion theory and explicitly breaks the U(1) symmetry associated with rotations about the z-axis. In this situation the corresponding defect is marginal. Leveraging conformal invariance, we relate the scaling dimension of these defects to the ground state energy of the massless Majorana fermion on a circle with equally-spaced point defects. We exploit this mapping to derive our second main result: the exact expression for the scaling dimension associated with n defects of arbitrary strengths. Our result generalizes a known formula for the n = 1 case derived in several previous works. We then use this exact scaling dimension to derive our third main result: the exact prefactor of the log ℓ/ℓ term in the asymmetry of the critical XY chain.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. Ares, S. Murciano and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nature Commun. 14 (2023) 2036 [arXiv:2207.14693] [INSPIRE].
L.K. Joshi et al., Observing the quantum Mpemba effect in quantum simulations, arXiv:2401.04270 [INSPIRE].
C. Rylands et al., Microscopic origin of the quantum Mpemba effect in integrable systems, arXiv:2310.04419 [INSPIRE].
S. Murciano, F. Ares, I. Klich and P. Calabrese, Entanglement asymmetry and quantum Mpemba effect in the XY spin chain, J. Stat. Mech. 2401 (2024) 013103 [arXiv:2310.07513] [INSPIRE].
B. Bertini et al., Dynamics of charge fluctuations from asymmetric initial states, arXiv:2306.12404 [INSPIRE].
F. Ferro, F. Ares and P. Calabrese, Non-equilibrium entanglement asymmetry for discrete groups: the example of the XY spin chain, J. Stat. Mech. 2402 (2024) 023101 [arXiv:2307.06902] [INSPIRE].
F. Ares, S. Murciano, E. Vernier and P. Calabrese, Lack of symmetry restoration after a quantum quench: an entanglement asymmetry study, SciPost Phys. 15 (2023) 089 [arXiv:2302.03330] [INSPIRE].
B.J.J. Khor et al., Confinement and kink entanglement asymmetry on a quantum Ising chain, arXiv:2312.08601 [INSPIRE].
F. Ares, S. Murciano, L. Piroli and P. Calabrese, An entanglement asymmetry study of black hole radiation, arXiv:2311.12683 [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].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
L. Capizzi and V. Vitale, A universal formula for the entanglement asymmetry of matrix product states, arXiv:2310.01962 [INSPIRE].
L. Capizzi and M. Mazzoni, Entanglement asymmetry in the ordered phase of many-body systems: the Ising field theory, JHEP 12 (2023) 144 [arXiv:2307.12127] [INSPIRE].
M. Chen and H.-H. Chen, Rényi entanglement asymmetry in (1 + 1)-dimensional conformal field theories, Phys. Rev. D 109 (2024) 065009 [arXiv:2310.15480] [INSPIRE].
J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93 (2004) 070601 [cond-mat/0404051] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized global symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
M. Henkel and A. Patkós, Conformal invariance and line defects in the two-dimensional Ising model, J. Phys. A 21 (1988) L231.
M. Henkel, A. Patkós and M. Schlottmann, The Ising quantum chain with defects. 1. The exact solution, Nucl. Phys. B 314 (1989) 609 [INSPIRE].
L. Turban, Conformal invariance and linear defects in the two-dimensional Ising model, J. Phys. A 18 (1985) L325.
F. Iglói, I. Peschel and L. Turban, Inhomogeneous systems with unusual critical behaviour, Adv. Phys. 42 (1993) 683.
M. Oshikawa and I. Affleck, Defect lines in the Ising model and boundary states on orbifolds, Phys. Rev. Lett. 77 (1996) 2604 [hep-th/9606177] [INSPIRE].
M. Oshikawa and I. Affleck, Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line, Nucl. Phys. B 495 (1997) 533 [cond-mat/9612187] [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
T. Quella, I. Runkel and G.M.T. Watts, Reflection and transmission for conformal defects, JHEP 04 (2007) 095 [hep-th/0611296] [INSPIRE].
J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].
P. Fendley, M.P.A. Fisher and C. Nayak, Boundary conformal field theory and tunneling of edge quasiparticles in non-Abelian topological states, Annals Phys. 324 (2009) 1547 [arXiv:0902.0998] [INSPIRE].
C. Bachas, I. Brunner and D. Roggenkamp, Fusion of critical defect lines in the 2D Ising model, J. Stat. Mech. 1308 (2013) P08008 [arXiv:1303.3616] [INSPIRE].
D. Aasen, R.S.K. Mong and P. Fendley, Topological defects on the lattice I: the Ising model, J. Phys. A 49 (2016) 354001 [arXiv:1601.07185] [INSPIRE].
D. Aasen, P. Fendley and R.S.K. Mong, Topological defects on the lattice: dualities and degeneracies, arXiv:2008.08598 [INSPIRE].
S. Murciano et al., Measurement-altered Ising quantum criticality, Phys. Rev. X 13 (2023) 041042 [arXiv:2302.04325] [INSPIRE].
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.
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.
E.M. Brehm and I. Brunner, Entanglement entropy through conformal interfaces in the 2D Ising model, JHEP 09 (2015) 080 [arXiv:1505.02647] [INSPIRE].
M. Gutperle and J.D. Miller, Entanglement entropy at CFT junctions, Phys. Rev. D 95 (2017) 106008 [arXiv:1701.08856] [INSPIRE].
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].
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].
L. Capizzi, S. Murciano and P. Calabrese, Full counting statistics and symmetry resolved entanglement for free conformal theories with interface defects, J. Stat. Mech. 2307 (2023) 073102 [arXiv:2302.08209] [INSPIRE].
L. Capizzi and A. Rotaru, Thermal entanglement in conformal junctions, arXiv:2312.08275 [INSPIRE].
Z. Yang, D. Mao and C.-M. Jian, Entanglement in a one-dimensional critical state after measurements, Phys. Rev. B 108 (2023) 165120 [arXiv:2301.08255] [INSPIRE].
Z. Weinstein, R. Sajith, E. Altman and S.J. Garratt, Nonlocality and entanglement in measured critical quantum Ising chains, Phys. Rev. B 107 (2023) 245132 [arXiv:2301.08268] [INSPIRE].
J.Y. Lee, C.-M. Jian and C. Xu, Quantum criticality under decoherence or weak measurement, PRX Quantum 4 (2023) 030317 [arXiv:2301.05238] [INSPIRE].
Y. Ashida, S. Furukawa and M. Oshikawa, System-environment entanglement phase transitions, arXiv:2311.16343 [INSPIRE].
P. Ruggiero, P. Calabrese, T. Giamarchi and L. Foini, Electrostatic solution of massless quenches in Luttinger liquids, SciPost Phys. 13 (2022) 111 [arXiv:2203.06740] [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].
R. Bonsignori, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement in free fermionic systems, J. Phys. A 52 (2019) 475302 [arXiv:1907.02084] [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, G. Di Giulio and P. Calabrese, Entanglement and symmetry resolution in two dimensional free quantum field theories, JHEP 08 (2020) 073 [arXiv:2006.09069] [INSPIRE].
P. Calabrese, J. Dubail and S. Murciano, Symmetry-resolved entanglement entropy in Wess-Zumino-Witten models, JHEP 10 (2021) 067 [arXiv:2106.15946] [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].
H.-H. Chen, Symmetry decomposition of relative entropies in conformal field theory, JHEP 07 (2021) 084 [arXiv:2104.03102] [INSPIRE].
L. Capizzi and P. Calabrese, Symmetry resolved relative entropies and distances in conformal field theory, JHEP 10 (2021) 195 [arXiv:2105.08596] [INSPIRE].
R. Bonsignori and P. Calabrese, Boundary effects on symmetry resolved entanglement, J. Phys. A 54 (2021) 015005 [arXiv:2009.08508] [INSPIRE].
B. Estienne, Y. Ikhlef and A. Morin-Duchesne, Finite-size corrections in critical symmetry-resolved entanglement, SciPost Phys. 10 (2021) 054 [arXiv:2010.10515] [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].
A. Milekhin and A. Tajdini, Charge fluctuation entropy of Hawking radiation: a replica-free way to find large entropy, SciPost Phys. 14 (2023) 172 [arXiv:2109.03841] [INSPIRE].
F. Ares, P. Calabrese, G. Di Giulio and S. Murciano, Multi-charged moments of two intervals in conformal field theory, JHEP 09 (2022) 051 [arXiv:2206.01534] [INSPIRE].
H.-H. Chen, Charged Rényi negativity of massless free bosons, JHEP 02 (2022) 117 [arXiv:2111.11028] [INSPIRE].
M. Ghasemi, Universal thermal corrections to symmetry-resolved entanglement entropy and full counting statistics, JHEP 05 (2023) 209 [arXiv:2203.06708] [INSPIRE].
M. Fossati, F. Ares and P. Calabrese, Symmetry-resolved entanglement in critical non-Hermitian systems, Phys. Rev. B 107 (2023) 205153 [arXiv:2303.05232] [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].
G. Di Giulio et al., On the boundary conformal field theory approach to symmetry-resolved entanglement, SciPost Phys. Core 6 (2023) 049 [arXiv:2212.09767] [INSPIRE].
A. Foligno, S. Murciano and P. Calabrese, Entanglement resolution of free Dirac fermions on a torus, JHEP 03 (2023) 096 [arXiv:2212.07261] [INSPIRE].
C. Northe, Entanglement resolution with respect to conformal symmetry, Phys. Rev. Lett. 131 (2023) 151601 [arXiv:2303.07724] [INSPIRE].
Y. Kusuki, S. Murciano, H. Ooguri and S. Pal, Symmetry-resolved entanglement entropy, spectra & boundary conformal field theory, JHEP 11 (2023) 216 [arXiv:2309.03287] [INSPIRE].
A. Bruno, F. Ares, S. Murciano and P. Calabrese, Symmetry resolution of the computable cross-norm negativity of two disjoint intervals in the massless Dirac field theory, JHEP 02 (2024) 009 [arXiv:2312.02926] [INSPIRE].
L. Bhardwaj et al., Lectures on generalized symmetries, Phys. Rept. 1051 (2024) 1 [arXiv:2307.07547] [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information: 10th anniversary edition, Cambridge University Press, Cambridge, U.K. (2010) [https://doi.org/10.1017/cbo9780511976667] [INSPIRE].
Z. Ma, C. Han, Y. Meir and E. Sela, Symmetric inseparability and number entanglement in charge-conserving mixed states, Phys. Rev. A 105 (2022) 042416 [arXiv:2110.09388] [INSPIRE].
J.L. Cardy and I. Peschel, Finite size dependence of the free energy in two-dimensional critical systems, Nucl. Phys. B 300 (1988) 377 [INSPIRE].
J.-M. Stéphan and J. Dubail, Logarithmic corrections to the free energy from sharp corners with angle 2π, J. Stat. Mech. 1309 (2013) P09002 [arXiv:1303.3633] [INSPIRE].
J.-M. Stéphan, Emptiness formation probability, Toeplitz determinants, and conformal field theory, J. Stat. Mech. 2014 (2014) P05010 [arXiv:1303.5499].
A. Lamacraft and P. Fendley, Order parameter statistics in the critical quantum Ising chain, Phys. Rev. Lett. 100 (2008) 165706 [arXiv:0802.1246] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer-Verlag, New York, NY, U.S.A. (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
R.W. Cherng and E. Demler, Quantum noise analysis of spin systems realized with cold atoms, New J. Phys. 9 (2007) 7.
S. Groha, F. Essler and P. Calabrese, Full counting statistics in the transverse field Ising chain, SciPost Phys. 4 (2018) 043.
D.A. Ivanov and A.G. Abanov, Characterizing correlations with full counting statistics: classical Ising and quantum XY spin chains, Phys. Rev. E 87 (2013) 022114.
F. Ares, M.A. Rajabpour and J. Viti, Exact full counting statistics for the staggered magnetization and the domain walls in the XY spin chain, Phys. Rev. E 103 (2021) 042107 [arXiv:2012.14012] [INSPIRE].
C.-M. Chang et al., Topological defect lines and renormalization group flows in two dimensions, JHEP 01 (2019) 026 [arXiv:1802.04445] [INSPIRE].
L. Bhardwaj and Y. Tachikawa, On finite symmetries and their gauging in two dimensions, JHEP 03 (2018) 189 [arXiv:1704.02330] [INSPIRE].
E.H. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals Phys. 16 (1961) 407 [INSPIRE].
I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A 36 (2003) L205 [cond-mat/0212631] [INSPIRE].
Acknowledgments
We are grateful to L. Capizzi, A. Foligno, D. X Horváth, S. Murciano, F. Rottoli, H. Saleur, and J. M. Stéphan for fruitful discussions. PC, FA and MF acknowledge support from the European Research Council (ERC) under Consolidator Grant NEMO No. 771536. JD acknowledges support from ‘Lorraine Université d’Excellence’ program and from Agence Nationale de la Recherche through ANR-22- CE30-0004-01 project ‘UNIOPEN’.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2402.03446
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Fossati, M., Ares, F., Dubail, J. et al. Entanglement asymmetry in CFT and its relation to non-topological defects. J. High Energ. Phys. 2024, 59 (2024). https://doi.org/10.1007/JHEP05(2024)059
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2024)059