Abstract
We consider the ground state of a one-dimensional critical quantum system carrying a global symmetry in the bulk, which is explicitly broken by its boundary conditions. We probe the system via a string-order parameter, showing how it detects the symmetry breaking pattern. We give a precise characterization of the mechanism depicted above in Boundary CFT, and we find a general logarithmic scaling for the order parameter. As a first example we analyze the breaking of a U(1) symmetry for complex free theories induced by a boundary pairing term. Moreover, we give predictions for the breaking of U(N) in free theories, arising from a boundary mixing. We test our predictions with numerical calculations for some lattice realizations of free fermionic system with boundary symmetry breaking, finding a good agreement.
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References
R. Shankar, Quantum Field Theory and Condensed Matter, Cambridge University Press (2017) [https://doi.org/10.1017/9781139044349].
J.F. Annett, Superconductivity, superfluids and condensates, Oxford University Press (2004) [ISBN: 9780198507550].
S. Weinberg, The Quantum theory of fields. Volume 1: Foundations, Cambridge University Press (2005) [https://doi.org/10.1017/CBO9781139644167] [INSPIRE].
J. Zinn-Justin, Phase Transitions and Renormalization Group, Oxford University Press (2007) [https://doi.org/10.1093/acprof:oso/9780199227198.001.0001].
S. Sachdev, Quantum Phase Transitions, Cambridge University Press (2011) [https://doi.org/10.1017/cbo9780511973765].
J. Kondo, Resistance Minimum in Dilute Magnetic Alloys, Prog. Theor. Phys. 32 (1964) 37 [INSPIRE].
I. Affleck, Conformal field theory approach to the Kondo effect, Acta Phys. Polon. B 26 (1995) 1869 [cond-mat/9512099] [INSPIRE].
J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997) [https://doi.org/10.1007/978-1-4612-2256-9].
T. Kennedy and H. Tasaki, Hidden symmetry breaking and the Haldane phase in S = 1 quantum spin chains, Commun. Math. Phys. 147 (1992) 431.
T. Kennedy and H. Tasaki, Hidden Z2 × Z2 symmetry breaking in Haldane-gap antiferromagnets, Phys. Rev. B 45 (1992) 304 [INSPIRE].
A.-L. Cauchy, Sur les formules qui résultent de l’emploi du signe > ou<, et sur les moyennes entre plusieurs quantités, in Cours d’analyse de l’École Royale Polytechnique, Cambridge Library Collection — Mathematics, p. 438–459, Cambridge University Press (2009) [https://doi.org/10.1017/CBO9780511693328.017].
K. Sakai and Y. Satoh, Entanglement through conformal interfaces, JHEP 12 (2008) 001 [arXiv:0809.4548] [INSPIRE].
E.M. Brehm and I. Brunner, Entanglement entropy through conformal interfaces in the 2D Ising model, JHEP 09 (2015) 080 [arXiv:1505.02647] [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, Rényi entropy and negativity for massless complex boson at conformal interfaces and junctions, JHEP 11 (2022) 105 [arXiv:2208.14118] [INSPIRE].
L. Capizzi, S. Murciano and P. Calabrese, Full counting statistics and symmetry resolved entanglement for free conformal theories with interface defects, arXiv:2302.08209 [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].
C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
K. Ohmori and Y. Tachikawa, Physics at the entangling surface, J. Stat. Mech. 1504 (2015) P04010 [arXiv:1406.4167] [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].
C. Bachas, J. de Boer, R. Dijkgraaf and H. Ooguri, Permeable conformal walls and holography, JHEP 06 (2002) 027 [hep-th/0111210] [INSPIRE].
B. Bellazzini and M. Mintchev, Quantum Fields on Star Graphs, J. Phys. A 39 (2006) 11101 [hep-th/0605036] [INSPIRE].
I. Peschel and V. Eisler, Exact results for the entanglement across defects in critical chains, J. Phys. A 45 (2012) 155301.
V. Eisler and I. Peschel, On entanglement evolution across defects in critical chains, EPL (Europhysics Letters) 99 (2012) 20001.
L. Capizzi and V. Eisler, Entanglement evolution after a global quench across a conformal defect, SciPost Phys. 14 (2023) 070 [arXiv:2209.03297] [INSPIRE].
L. Capizzi and V. Eisler, Zero-mode entanglement across a conformal defect, arXiv:2303.10425 [INSPIRE].
I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42 (2009) 504003.
M. Bortz, J. Sato and M. Shiroishi, String correlation functions of the spin-1/2 Heisenberg XXZ chain, J. Phys. A 40 (2007) 4253 [cond-mat/0612348] [INSPIRE].
A.G. Abanov, D.A. Ivanov and Y. Qian, Quantum fluctuations of one-dimensional free fermions and Fisher-Hartwig formula for Toeplitz determinants, J. Phys. A 44 (2011) 485001 [arXiv:1108.1355] [INSPIRE].
I.V. Cherednik, Factorizing Particles on a Half Line and Root Systems, Theor. Math. Phys. 61 (1984) 977 [INSPIRE].
E.K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys. A 21 (1988) 2375 [INSPIRE].
S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841 [Erratum ibid. 9 (1994) 4353] [hep-th/9306002] [INSPIRE].
A. LeClair, G. Mussardo, H. Saleur and S. Skorik, Boundary energy and boundary states in integrable quantum field theories, Nucl. Phys. B 453 (1995) 581 [hep-th/9503227] [INSPIRE].
G. Delfino, G. Mussardo and P. Simonetti, Statistical models with a line of defect, Phys. Lett. B 328 (1994) 123 [hep-th/9403049] [INSPIRE].
G. Delfino, G. Mussardo and P. Simonetti, Scattering theory and correlation functions in statistical models with a line of defect, Nucl. Phys. B 432 (1994) 518 [hep-th/9409076] [INSPIRE].
R. Konik, A. LeClair and G. Mussardo, On Ising correlation functions with boundary magnetic field, Int. J. Mod. Phys. A 11 (1996) 2765 [hep-th/9508099] [INSPIRE].
E. Corrigan, Integrable models with boundaries and defects, in the proceedings of the UK-Japan Winter School on Geometry and Analysis Towards Quantum Theory, Durham U.K., January 6–9 (2004) [math-ph/0411043] [INSPIRE].
Z. Bajnok, L. Palla and G. Takács, On the boundary form-factor program, Nucl. Phys. B 750 (2006) 179 [hep-th/0603171] [INSPIRE].
O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive QFT with a boundary: The Ising model, J. Statist. Phys. 134 (2009) 105 [arXiv:0810.0219] [INSPIRE].
A. Fring and R. Koberle, Boundary bound states in affine Toda field theory, Int. J. Mod. Phys. A 10 (1995) 739 [hep-th/9404188] [INSPIRE].
V. Riva, Boundary bootstrap principle in two-dimensional integrable quantum field theories, Nucl. Phys. B 604 (2001) 511 [hep-th/0102163] [INSPIRE].
E. Corrigan, P.E. Dorey, R.H. Rietdijk and R. Sasaki, Affine Toda field theory on a half line, Phys. Lett. B 333 (1994) 83 [hep-th/9404108] [INSPIRE].
E. Corrigan and C. Zambon, A New class of integrable defects, J. Phys. A 42 (2009) 475203 [arXiv:0908.3126] [INSPIRE].
F. Ares, S. Murciano and P. Calabrese, Entanglement asymmetry as a probe of symmetry breaking, Nature Commun. 14 (2023) 2036 [arXiv:2207.14693] [INSPIRE].
Acknowledgments
RB acknowledges support from the Croatian Science Foundation (HrZZ) project No. IP- 2019-4-3321. LC acknowledges support from ERC under Consolidator grant number 771536 (NEMO). The work of P. Panopoulos was supported by the Croatian Science Foundation Project “New Geometries for Gravity and Spacetime” (IP-2018-01-7615).
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Bonsignori, R., Capizzi, L. & Panopoulos, P. Boundary Symmetry Breaking in CFT and the string order parameter. J. High Energ. Phys. 2023, 27 (2023). https://doi.org/10.1007/JHEP05(2023)027
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DOI: https://doi.org/10.1007/JHEP05(2023)027