Abstract
We study entanglement entropy in free Lifshitz scalar field theories holographically by employing the metrics proposed by Nozaki, Ryu and Takayanagi in [1] obtained from a continuous multi-scale entanglement renormalisation ansatz (cMERA). In these geometries we compute the minimal surface areas governing the entanglement entropy as functions of the dynamical exponent z and we exhibit a transition from an area law to a volume law analytically in the limit of large z. We move on to explore the effects of a massive deformation, obtaining results for any z in arbitrary dimension. We then trigger a renormalisation group flow between a Lifshitz theory and a conformal theory and observe a monotonic decrease in entanglement entropy along this flow. We focus on strip regions but also consider a disc in the undeformed theory.
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References
M. Nozaki, S. Ryu and T. Takayanagi, Holographic geometry of entanglement renormalization in quantum field theories, JHEP 10 (2012) 193 [arXiv:1208.3469] [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, 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].
E. Fradkin, Scaling of entanglement entropy at 2D quantum Lifshitz fixed points and topological fluids, J. Phys. A 42 (2009) 504011 [arXiv:0906.1569] [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].
D.E. Parker, R. Vasseur and J.E. Moore, Entanglement entropy in excited states of the quantum Lifshitz model, J. Phys. A 50 (2017) 254003 [arXiv:1702.07433] [INSPIRE].
X. Chen, W. Witczak-Krempa, T. Faulkner and E. Fradkin, Two-cylinder entanglement entropy under a twist, J. Stat. Mech. 1704 (2017) 043104 [arXiv:1611.01847] [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. B 80 (2009) 184421.
M. Oshikawa, Boundary conformal field theory and entanglement entropy in two-dimensional quantum Lifshitz critical point, arXiv:1007.3739 [INSPIRE].
X. Chen, E. Fradkin and W. Witczak-Krempa, Quantum spin chains with multiple dynamics, Phys. Rev. B 96 (2017) 180402 [arXiv:1706.02304] [INSPIRE].
X. Chen, E. Fradkin and W. Witczak-Krempa, Gapless quantum spin chains: multiple dynamics and conformal wavefunctions, J. Phys. A 50 (2017) 464002 [arXiv:1707.02317] [INSPIRE].
M.R. Mohammadi Mozaffar and A. Mollabashi, Entanglement in Lifshitz-type quantum field theories, JHEP 07 (2017) 120 [arXiv:1705.00483] [INSPIRE].
T. He, J.M. Magan and S. Vandoren, Entanglement entropy in Lifshitz theories, SciPost Phys. 3 (2017) 034 [arXiv:1705.01147] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].
S.A. Gentle and C. Keeler, On the reconstruction of Lifshitz spacetimes, JHEP 03 (2016) 195 [arXiv:1512.04538] [INSPIRE].
J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger invariance from Lifshitz isometries in holography and field theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].
J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement renormalization for quantum fields in real space, Phys. Rev. Lett. 110 (2013) 100402 [arXiv:1102.5524] [INSPIRE].
G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [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].
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].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
M.R. Mohammadi Mozaffar and A. Mollabashi, Logarithmic negativity in Lifshitz harmonic models, J. Stat. Mech. 1805 (2018) 053113 [arXiv:1712.03731] [INSPIRE].
H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
I. Arav, S. Chapman and Y. Oz, Lifshitz scale anomalies, JHEP 02 (2015) 078 [arXiv:1410.5831] [INSPIRE].
S. Pal and B. Grinstein, Weyl Consistency Conditions in Non-Relativistic Quantum Field Theory, JHEP 12 (2016) 012 [arXiv:1605.02748] [INSPIRE].
I. Arav, Y. Oz and A. Raviv-Moshe, Lifshitz anomalies, Ward identities and split dimensional regularization, JHEP 03 (2017) 088 [arXiv:1612.03500] [INSPIRE].
T. Griffin, P. Hořava and C.M. Melby-Thompson, Conformal Lifshitz gravity from holography, JHEP 05 (2012) 010 [arXiv:1112.5660] [INSPIRE].
M. Baggio, J. de Boer and K. Holsheimer, Anomalous breaking of anisotropic scaling symmetry in the quantum Lifshitz model, JHEP 07 (2012) 099 [arXiv:1112.6416] [INSPIRE].
I. Adam, I.V. Melnikov and S. Theisen, A non-relativistic Weyl anomaly, JHEP 09 (2009) 130 [arXiv:0907.2156] [INSPIRE].
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Gentle, S.A., Vandoren, S. Lifshitz entanglement entropy from holographic cMERA. J. High Energ. Phys. 2018, 13 (2018). https://doi.org/10.1007/JHEP07(2018)013
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DOI: https://doi.org/10.1007/JHEP07(2018)013