Abstract
We investigate how entanglement entropy behaves in a non-conformal scalar field system with a quantum phase transition, by the replica method. We study the σ-model in 3+1 dimensions which is O(N) symmetric as the mass squared parameter μ2 is positive, and undergoes spontaneous symmetry breaking while μ2 becomes negative. The area law leading divergence of the entanglement entropy is preserved in both of the symmetric and the broken phases. The spontaneous symmetry breaking changes the subleading divergence from log to log squared, due to the cubic interaction on the cone. At the leading order of the coupling constant expansion, the entanglement entropy reaches a cusped maximum at the quantum phase transition point μ2 = 0, and decreases while μ2 is tuned away from 0 into either phase.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865 [quant-ph/0702225] [INSPIRE].
V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74 (2002) 197 [quant-ph/0102094] [INSPIRE].
T.J. Osborne and M.A. Nielsen, Entanglement in a simple quantum phase transition, Phys. Rev. A 66 (2002) 032110 [quant-ph/0202162] [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].
C.-Y. Huang and F.-L. Lin, Multipartite entanglement measures and quantum criticality from matrix and tensor product states, Phys. Rev. A 81 (2010) 032304 [arXiv:0911.4670] [INSPIRE].
L. Borsten, M.J. Duff and P. Levay, The black-hole/qubit correspondence: an up-to-date review, Class. Quant. Grav. 29 (2012) 224008 [arXiv:1206.3166] [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].
J. Bardeen, L.N. Cooper and J.R. Schrieffer, Microscopic theory of superconductivity, Phys. Rev. 106 (1957) 162 [INSPIRE].
J. Bardeen, L.N. Cooper and J.R. Schrieffer, Theory of superconductivity, Phys. Rev. 108 (1957) 1175 [INSPIRE].
R.B. Laughlin, Anomalous quantum Hall effect: An Incompressible quantum fluid with fractional lycharged excitations, Phys. Rev. Lett. 50 (1983) 1395 [INSPIRE].
S.L. Sondhi, S.M. Girvin, J.P. Carini and D. Shahar, Continuous quantum phase transitions, Rev. Mod. Phys. 69 (1997) 315 [cond-mat/9609279] [INSPIRE].
J.-W. Chen, M. Huang, Y.-H. Li, E. Nakano and D.-L. Yang, Phase Transitions and the Perfectness of Fluids, Phys. Lett. B 670 (2008) 18 [arXiv:0709.3434] [INSPIRE].
J.-W. Chen, C.-T. Hsieh and H.-H. Lin, Minimum Shear Viscosity over Entropy Density at Phase Transition?: A Counterexample, Phys. Lett. B 701 (2011) 327 [arXiv:1010.3119] [INSPIRE].
L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80 (2008) 517 [quant-ph/0703044] [INSPIRE].
M.P. Hertzberg and F. Wilczek, Some Calculable Contributions to Entanglement Entropy, Phys. Rev. Lett. 106 (2011) 050404 [arXiv:1007.0993] [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].
J. Cardy and C.P. Herzog, Universal Thermal Corrections to Single Interval Entanglement Entropy for Two Dimensional Conformal Field Theories, Phys. Rev. Lett. 112 (2014) 171603 [arXiv:1403.0578] [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].
M.A. Metlitski, C.A. Fuertes and S. Sachdev, Entanglement Entropy in the O(N) Model, Phys. Rev. B 80 (2009) 115122 [arXiv:0904.4477] [INSPIRE].
M.P. Hertzberg, Entanglement Entropy in Scalar Field Theory, J. Phys. A 46 (2013) 015402 [arXiv:1209.4646] [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
A. Dobado, F.J. Llanes-Estrada and J.M. Torres-Rincon, Minimum of η/s and the phase transition of the linear sigma model in the large-N limit, Phys. Rev. D 80 (2009) 114015 [arXiv:0907.5483] [INSPIRE].
A. Dobado and J.M. Torres-Rincon, Bulk viscosity and the phase transition of the linear sigma model, Phys. Rev. D 86 (2012) 074021 [arXiv:1206.1261] [INSPIRE].
X.G. Wen, Topological Order in Rigid States, Int. J. Mod. Phys. B 4 (1990) 239 [INSPIRE].
H.-C. Jiang, Z. Wang and L. Balents, Identifying topological order by entanglement entropy, Nature Phys. 8 (2012) 902.
W. Li, A. Weichselbaum and J.v. Delft, Identifying Symmetry-protected Topological Order by Entanglement Entropy, Phys. Rev. B 88 (2013) 245121.
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
J.-W. Chen and J.-Y. Pang, On the Renormalization of Entanglement Entropy, arXiv:1709.03205 [INSPIRE].
O.A. Castro-Alvaredo and B. Doyon, Entanglement entropy of highly degenerate states and fractal dimensions, Phys. Rev. Lett. 108 (2012) 120401 [arXiv:1103.3247] [INSPIRE].
M.A. Metlitski and T. Grover, Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry, arXiv:1112.5166 [INSPIRE].
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: 1411.2916
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
Chen, JW., Dai, SH. & Pang, JY. Entanglement Entropy and Quantum Phase Transition in the O(N) σ-model. J. High Energ. Phys. 2021, 201 (2021). https://doi.org/10.1007/JHEP07(2021)201
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2021)201