Abstract
We compute universal finite corrections to entanglement entropy for generalised quantum Lifshitz models in arbitrary odd spacetime dimensions. These are generalised free field theories with Lifshitz scaling symmetry, where the dynamical critical exponent z equals the number of spatial dimensions d, and which generalise the 2+1-dimensional quantum Lifshitz model to higher dimensions. We analyse two cases: one where the spatial manifold is a d-dimensional sphere and the entanglement entropy is evaluated for a hemisphere, and another where a d-dimensional flat torus is divided into two cylinders. In both examples the finite universal terms in the entanglement entropy are scale invariant and depend on the compactification radius of the scalar field.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory: a non-technical introduction, Int. J. Quant. Inf.4 (2006) 429 [quant-ph/0505193] [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].
J. Eisert, Colloquium: area laws for the entanglement entropy, Rev. Mod. Phys.82 (2010) 277.
N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rept.646 (2016) 1.
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].
M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Lect. Notes Phys.931 (2017) 1 [arXiv:1609.01287].
T. Nishioka, Entanglement entropy: holography and renormalization group, Rev. Mod. Phys.90 (2018) 035007 [arXiv:1801.10352] [INSPIRE].
S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett.B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].
D.V. Fursaev, A. Patrushev and S.N. Solodukhin, Distributional geometry of squashed cones, Phys. Rev. D88 (2013) 044054 [arXiv:1306.4000] [INSPIRE].
V. Keranen, W. Sybesma, P. Szepietowski and L. Thorlacius, Correlation functions in theories with Lifshitz scaling, JHEP05 (2017) 033 [arXiv:1611.09371] [INSPIRE].
E. Ardonne, P. Fendley and E. Fradkin, Topological order and conformal quantum critical points, Annals Phys.310 (2004) 493 [cond-mat/0311466] [INSPIRE].
C. Brust and K. Hinterbichler, Free □ kscalar conformal field theory, JHEP02 (2017) 066 [arXiv:1607.07439] [INSPIRE].
M. Beccaria and A.A. Tseytlin, Partition function of free conformal fields in 3-plet representation, JHEP05 (2017) 053 [arXiv:1703.04460] [INSPIRE].
Y. Nakayama, Hidden global conformal symmetry without Virasoro extension in theory of elasticity, Annals Phys.372 (2016) 392 [arXiv:1604.00810] [INSPIRE].
T. Griffin, K.T. Grosvenor, P. Hořava and Z. Yan, Scalar field theories with polynomial shift symmetries, Commun. Math. Phys.340 (2015) 985 [arXiv:1412.1046] [INSPIRE].
T. Griffin, K.T. Grosvenor, P. Hořava and Z. Yan, Cascading multicriticality in nonrelativistic spontaneous symmetry breaking, Phys. Rev. Lett.115 (2015) 241601 [arXiv:1507.06992] [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP10 (2001) 034 [hep-th/0106113] [INSPIRE].
D. Anninos, T. Hartman and A. Strominger, Higher spin realization of the dS/CFT correspondence, Class. Quant. Grav.34 (2017) 015009 [arXiv:1108.5735] [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 and E. Fradkin, Universal behavior of entanglement in 2D quantum critical dimer models, J. Stat. Mech.1009 (2010) P09004 [arXiv:1006.1361] [INSPIRE].
J.-M. Stéphan, Shannon and entanglement entropies of one- and two-dimensional critical wave functions, Phys. Rev.B 80 (2009) 184421.
M. Oshikawa, Boundary conformal field theory and entanglement entropy in two-dimensional quantum Lifshitz critical point, arXiv:1007.3739 [INSPIRE].
M.P. Zaletel, J.H. Bardarson and J.E. Moore, Logarithmic terms in entanglement entropies of 2D quantum critical points and Shannon entropies of spin chains, Phys. Rev. Lett.107 (2011) 020402 [arXiv:1103.5452] [INSPIRE].
T. Zhou, X. Chen, T. Faulkner and E. Fradkin, Entanglement entropy and mutual information of circular entangling surfaces in the 2 + 1-dimensional quantum Lifshitz model, J. Stat. Mech.1609 (2016) 093101 [arXiv:1607.01771] [INSPIRE].
J.-M. Stephan, G. Misguich and V. Pasquier, Renyi entanglement entropies in quantum dimer models: from criticality to topological order, J. Stat. Mech.1202 (2012) P02003 [arXiv:1108.1699] [INSPIRE].
J.L. Cardy and I. Peschel, Finite size dependence of the free energy in two-dimensional critical systems, Nucl. Phys. B300 (1988) 377 [INSPIRE].
J.M. Stéphan, S. Furukawa, G. Misguich and V. Pasquier, Shannon and entanglement entropies of one- and two-dimensional critical wave functions, Phys. Rev. B80 (2009) 184421.
J.M. Stéphan, H. Ju, P. Fendley and R.G. Melko, Entanglement in gapless resonating-valence-bond states, New J. Phys.15 (2013) 015004.
X. Chen, G.Y. Cho, T. Faulkner and E. Fradkin, Scaling of entanglement in 2+1-dimensional scale-invariant field theories, J. Stat. Mech.1502 (2015) P02010 [arXiv:1412.3546] [INSPIRE].
X. Chen, W. Witczak-Krempa, T. Faulkner and E. Fradkin, Two-cylinder entanglement entropy under a twist, J. Stat. Mech.1704 (2017) 043104.
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V. Keränen and L. Thorlacius, Holographic geometries for condensed matter applications, in the proceedings of the 13thMarcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG13), July 1-7, Stockholm, Sweden (2015), arXiv:1307.2882 [INSPIRE].
S.A. Gentle and S. Vandoren, Lifshitz entanglement entropy from holographic cMERA, JHEP07 (2018) 013 [arXiv:1711.11509] [INSPIRE].
T. He, J.M. Magan and S. Vandoren, Entanglement entropy in Lifshitz theories, SciPost Phys.3 (2017) 034 [arXiv:1705.01147] [INSPIRE].
M.R. Mohammadi Mozaffar and A. Mollabashi, Entanglement in Lifshitz-type quantum field theories, JHEP07 (2017) 120 [arXiv:1705.00483] [INSPIRE].
J.S. Dowker, Determinants and conformal anomalies of GJMS operators on spheres, J. Phys.A 44 (2011) 115402 [arXiv:1010.0566] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer, Germany (1997).
P.H. Ginsparg, Applied conformal field theory, in the proceedings of the Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena, June 28-August 5, Les Houches, France (1988), hep-th/9108028 [INSPIRE].
L.J.M. C.R. Graham, R. Jenne and G.A.J. Sparling, Conformally invariant powers of the laplacian. I: existence, J. London Math. Soc.S2-46 (1992) 557.
T.P. Branson, P.B. Gilkey and D.V. Vassilevich, The asymptotics of the Laplacian on a manifold with boundary. 2, Boll. Union. Mat. Ital.11B (1997) 39 [hep-th/9504029] [INSPIRE].
A.R. Gover and K. Hirachi, Conformally invariant powers of the Laplacian: a complete non-existence theorem, J. Amer. Math. Soc.17 (2004) 389.
A. Gover, Laplacian operators and q-curvature on conformally einstein manifolds, math/0506037.
C.R. Graham, Conformal powers of the laplacian via stereographic projection, SIGMA3 (2007) 121 [arXiv:0711.4798].
C. Fefferman and C. R. Graham, Juhl’s formulae for gjms operators and q-curvatures, arXiv:1203.0360.
S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-riemannian manifolds (summary), SIGMA4 (2008) 036 [arXiv:0803.4331].
A. Juhl, On conformally covariant powers of the laplacian, arXiv:0905.3992.
A. Juhl, Explicit formulas for GJMS-operators and Q-curvatures, arXiv:1108.0273 [INSPIRE].
H. Baum and A. Juhl, Conformal differential geometry, Birkhäuser, Basel Switzerland (2010).
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys.A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
P. Chang and J. Dowker, Vacuum energy on orbifold factors of spheres, Nucl. Phys.B 395 (1993) 407.
J.S. Dowker, Effective action in spherical domains, Commun. Math. Phys.162 (1994) 633 [hep-th/9306154] [INSPIRE].
J.S. Dowker, Numerical evaluation of spherical GJMS determinants for even dimensions, arXiv:1310.0759 [INSPIRE].
J.S. Dowker, Functional determinants on spheres and sectors, J. Math. Phys.35 (1994) 4989 [Erratum ibid.36 (1995) 988] [hep-th/9312080] [INSPIRE].
J.S. Dowker, The boundary F-theorem for free fields, arXiv:1407.5909 [INSPIRE].
D. Gaiotto, Boundary F-maximization, arXiv:1403.8052 [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
D.E. Diaz, Polyakov formulas for GJMS operators from AdS/CFT, JHEP07 (2008) 103 [arXiv:0803.0571] [INSPIRE].
F. Bugini and D.E. Díaz, Holographic Weyl anomaly for GJMS operators: one Laplacian to rule them all, JHEP02 (2019) 188 [arXiv:1811.10380] [INSPIRE].
W. Witczak-Krempa, L.E. Hayward Sierens and R.G. Melko, Cornering gapless quantum states via their torus entanglement, Phys. Rev. Lett.118 (2017) 077202 [arXiv:1603.02684] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys.A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
P. Bueno and W. Witczak-Krempa, Holographic torus entanglement and its renormalization group flow, Phys. Rev.D 95 (2017) 066007 [arXiv:1611.01846] [INSPIRE].
J.S. Dowker, A technical note on the calculation of GJMS (Rac and Di) operator determinants, arXiv:1807.11872 [INSPIRE].
A.O. Barvinsky et al., Heat kernel methods for lifshitz theories, JHEP06 (2017) 063.
P. Bueno and R.C. Myers, Universal entanglement for higher dimensional cones, JHEP12 (2015) 168 [arXiv:1508.00587] [INSPIRE].
P. Bueno, R.C. Myers and W. Witczak-Krempa, Universality of corner entanglement in conformal field theories, Phys. Rev. Lett.115 (2015) 021602 [arXiv:1505.04804] [INSPIRE].
P. Bueno, R.C. Myers and W. Witczak-Krempa, Universal corner entanglement from twist operators, JHEP09 (2015) 091 [arXiv:1507.06997] [INSPIRE].
P. Bueno and R.C. Myers, Corner contributions to holographic entanglement entropy, JHEP08 (2015) 068 [arXiv:1505.07842] [INSPIRE].
P. Bueno and W. Witczak-Krempa, Bounds on corner entanglement in quantum critical states, Phys. Rev. B93 (2016) 045131 [arXiv:1511.04077] [INSPIRE].
I.R. Klebanov, T. Nishioka, S.S. Pufu and B.R. Safdi, On shape dependence and RG flow of entanglement entropy, JHEP07 (2012) 001 [arXiv:1204.4160] [INSPIRE].
H. Elvang and M. Hadjiantonis, Exact results for corner contributions to the entanglement entropy and Rényi entropies of free bosons and fermions in 3d, Phys. Lett.B 749 (2015) 383 [arXiv:1506.06729] [INSPIRE].
R.-X. Miao, A holographic proof of the universality of corner entanglement for CFTs, JHEP10 (2015) 038 [arXiv:1507.06283] [INSPIRE].
D.-W. Pang, Corner contributions to holographic entanglement entropy in non-conformal backgrounds, JHEP09 (2015) 133 [arXiv:1506.07979] [INSPIRE].
M. Alishahiha, A.F. Astaneh, P. Fonda and F. Omidi, Entanglement Entropy for Singular Surfaces in Hyperscaling violating Theories, JHEP09 (2015) 172 [arXiv:1507.05897] [INSPIRE].
T. Zhou, Entanglement entropy of local operators in quantum Lifshitz theory, J. Stat. Mech.1609 (2016) 093106.
M.R. Mohammadi Mozaffar and A. Mollabashi, Entanglement evolution in Lifshitz-type scalar theories, JHEP01 (2019) 137 [arXiv:1811.11470] [INSPIRE].
E. Plamadeala and E. Fradkin, Scrambling in the quantum Lifshitz model, J. Stat. Mech.1806 (2018) 063102.
H. Srivastava and J. Choi, Zeta and q-zeta functions and associated series and integrals, Elsevier, The Netherlands (2012).
E. Barnes, The theory of the moltiple Gamma function, Trans. Camb. Philos. Soc.19 (1904) 374.
V.S. Adamchik, Multiple gamma function and its application to computation of series, submitted to Ramanujan J. (2003), math/0308074.
E. Elizalde, Multidimensional extension of the generalized Chowla-Selberg formula, Commun. Math. Phys.198 (1998) 83 [hep-th/9707257] [INSPIRE].
E. Elizalde, Ten physical applications of spectral zeta functions, Lecture Notes in Physics volume 855, Springer, Germany (2012).
K. Kirsten, P. Loya and J. Park, Zeta functions of Dirac and Laplace-type operators over finite cylinders, Annals Phys.321 (2006) 1814 [INSPIRE].
E. Elizalde and M. Tierz, Multiplicative anomaly and zeta factorization, J. Math. Phys.45 (2004) 1168 [hep-th/0402186] [INSPIRE].
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1906.08252
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Angel-Ramelli, J., Puletti, V.G.M. & Thorlacius, L. Entanglement entropy in generalised quantum Lifshitz models. J. High Energ. Phys. 2019, 72 (2019). https://doi.org/10.1007/JHEP08(2019)072
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2019)072