Abstract
We use the radial null energy condition to construct a monotonic a-function for a certain type of non-relativistic holographic RG flows. We test our a-function in three different geometries that feature a Boomerang RG flow, characterized by a domain wall between two AdS spaces with the same AdS radius, but with different (and sometimes direction-dependent) speeds of light. We find that the a-function monotonically decreases and goes to a constant in the asymptotic regimes of the geometry. Using the holographic dictionary in this asymptotic AdS spaces, we find that the a-function not only reads the fixed point central charge but also the speed of light, suggesting what the correct RG charge might be for non-relativistic RG flows.
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A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
Z. Komargodski, The Constraints of Conformal Symmetry on RG Flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Markov Property of the Conformal Field Theory Vacuum and the a Theorem, Phys. Rev. Lett. 118 (2017) 261602 [arXiv:1704.01870] [INSPIRE].
B. Swingle, Entanglement does not generally decrease under renormalization, J. Stat. Mech. 1410 (2014) P10041 [arXiv:1307.8117] [INSPIRE].
K. Jensen and A. O’Bannon, Constraint on Defect and Boundary Renormalization Group Flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].
Y. Wang, Defect a-theorem and a-maximization, JHEP 02 (2022) 061 [arXiv:2101.12648] [INSPIRE].
K. Jensen, A. O’Bannon, B. Robinson and R. Rodgers, From the Weyl Anomaly to Entropy of Two-Dimensional Boundaries and Defects, Phys. Rev. Lett. 122 (2019) 241602 [arXiv:1812.08745] [INSPIRE].
N. Kobayashi, T. Nishioka, Y. Sato and K. Watanabe, Towards a C-theorem in defect CFT, JHEP 01 (2019) 039 [arXiv:1810.06995] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, Entropic g Theorem in General Spacetime Dimensions, Phys. Rev. Lett. 130 (2023) 111603 [arXiv:2212.10575] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, Irreversibility, QNEC, and defects, JHEP 07 (2023) 004 [arXiv:2303.16935] [INSPIRE].
D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
C. Hoyos and P. Koroteev, On the Null Energy Condition and Causality in Lifshitz Holography, Phys. Rev. D 82 (2010) 084002 [Erratum ibid. 82 (2010) 109905] [arXiv:1007.1428] [INSPIRE].
J.T. Liu and Z. Zhao, Holographic Lifshitz flows and the null energy condition, arXiv:1206.1047 [INSPIRE].
A. González Lezcano et al., c-functions in flows across dimensions, JHEP 10 (2022) 083 [arXiv:2207.09360] [INSPIRE].
J.K. Ghosh, E. Kiritsis, F. Nitti and L.T. Witkowski, Holographic RG flows on curved manifolds and the F-theorem, JHEP 02 (2019) 055 [arXiv:1810.12318] [INSPIRE].
I. Arav, J.P. Gauntlett, M. Roberts and C. Rosen, Spatially modulated and supersymmetric deformations of ABJM theory, JHEP 04 (2019) 099 [arXiv:1812.11159] [INSPIRE].
C. Hoyos, N. Jokela, J.M. Penín and A.V. Ramallo, Holographic spontaneous anisotropy, JHEP 04 (2020) 062 [arXiv:2001.08218] [INSPIRE].
N. Jokela, J. Kastikainen, E. Kiritsis and F. Nitti, Flavored ABJM theory on the sphere and holographic F-functions, JHEP 03 (2022) 091 [arXiv:2112.08715] [INSPIRE].
E. Caceres and S. Shashi, Anisotropic flows into black holes, JHEP 01 (2023) 007 [arXiv:2209.06818] [INSPIRE].
E. Caceres, A. Kundu, A.K. Patra and S. Shashi, Trans-IR flows to black hole singularities, Phys. Rev. D 106 (2022) 046005 [arXiv:2201.06579] [INSPIRE].
E. Caceres, S. Shashi and H.-Y. Sun, Imprints of phase transitions on Kasner singularities, arXiv:2305.11177 [INSPIRE].
P. Chesler, A. Lucas and S. Sachdev, Conformal field theories in a periodic potential: results from holography and field theory, Phys. Rev. D 89 (2014) 026005 [arXiv:1308.0329] [INSPIRE].
A. Donos, J.P. Gauntlett and C. Pantelidou, Conformal field theories in d = 4 with a helical twist, Phys. Rev. D 91 (2015) 066003 [arXiv:1412.3446] [INSPIRE].
A. Donos, J.P. Gauntlett and O. Sosa-Rodriguez, Anisotropic plasmas from axion and dilaton deformations, JHEP 11 (2016) 002 [arXiv:1608.02970] [INSPIRE].
A. Donos, J.P. Gauntlett, C. Rosen and O. Sosa-Rodriguez, Boomerang RG flows in M-theory with intermediate scaling, JHEP 07 (2017) 128 [arXiv:1705.03000] [INSPIRE].
A. Donos, J.P. Gauntlett, C. Rosen and O. Sosa-Rodriguez, Boomerang RG flows with intermediate conformal invariance, JHEP 04 (2018) 017 [arXiv:1712.08017] [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].
K. Landsteiner and Y. Liu, The holographic Weyl semi-metal, Phys. Lett. B 753 (2016) 453 [arXiv:1505.04772] [INSPIRE].
K. Landsteiner, Y. Liu and Y.-W. Sun, Quantum phase transition between a topological and a trivial semimetal from holography, Phys. Rev. Lett. 116 (2016) 081602 [arXiv:1511.05505] [INSPIRE].
A.G. Grushin, Consequences of a condensed matter realization of Lorentz violating QED in Weyl semi-metals, Phys. Rev. D 86 (2012) 045001 [arXiv:1205.3722] [INSPIRE].
N. Grandi, V. Juričić, I. Salazar Landea and R. Soto-Garrido, Towards holographic flat bands, JHEP 05 (2021) 123 [arXiv:2103.01690] [INSPIRE].
N. Grandi, V. Juričić, I.S. Landea and R. Soto-Garrido, Engineering holographic flat fermionic bands, Phys. Rev. D 105 (2022) L081902 [arXiv:2112.12198] [INSPIRE].
S.S. Gubser and A. Nellore, Ground states of holographic superconductors, Phys. Rev. D 80 (2009) 105007 [arXiv:0908.1972] [INSPIRE].
J.P. Gauntlett and C. Rosen, Susy Q and spatially modulated deformations of ABJM theory, JHEP 10 (2018) 066 [arXiv:1808.02488] [INSPIRE].
S.S. Gubser, S.S. Pufu and F.D. Rocha, Quantum critical superconductors in string theory and M-theory, Phys. Lett. B 683 (2010) 201 [arXiv:0908.0011] [INSPIRE].
V. Balasubramanian and P. Kraus, Space-time and the holographic renormalization group, Phys. Rev. Lett. 83 (1999) 3605 [hep-th/9903190] [INSPIRE].
R.C. Myers and A. Singh, Comments on Holographic Entanglement Entropy and RG Flows, JHEP 04 (2012) 122 [arXiv:1202.2068] [INSPIRE].
N.P. Armitage, E.J. Mele and A. Vishwanath, Weyl and Dirac Semimetals in Three Dimensional Solids, Rev. Mod. Phys. 90 (2018) 015001 [arXiv:1705.01111] [INSPIRE].
K. Landsteiner, Y. Liu and Y.-W. Sun, Odd viscosity in the quantum critical region of a holographic Weyl semimetal, Phys. Rev. Lett. 117 (2016) 081604 [arXiv:1604.01346] [INSPIRE].
C. Copetti, J. Fernández-Pendás and K. Landsteiner, Axial Hall effect and universality of holographic Weyl semi-metals, JHEP 02 (2017) 138 [arXiv:1611.08125] [INSPIRE].
M. Baggioli, B. Padhi, P.W. Phillips and C. Setty, Conjecture on the Butterfly Velocity across a Quantum Phase Transition, JHEP 07 (2018) 049 [arXiv:1805.01470] [INSPIRE].
N. Grandi, V. Juričić, I. Salazar Landea and R. Soto-Garrido, Towards holographic flat bands, JHEP 05 (2021) 123 [arXiv:2103.01690] [INSPIRE].
D. Arean, M. Bertolini, C. Krishnan and T. Prochazka, Quantum Critical Superfluid Flows and Anisotropic Domain Walls, JHEP 09 (2011) 131 [arXiv:1106.1053] [INSPIRE].
E. D’Hoker and P. Kraus, Charged Magnetic Brane Solutions in AdS5 and the fate of the third law of thermodynamics, JHEP 03 (2010) 095 [arXiv:0911.4518] [INSPIRE].
Acknowledgments
The authors would like to thank Lorenzo Di Pietro, Carlos Pérez-Pardavila, Sanjit Shashi and Aaron Zimmerman for useful discussions. The work of EC is supported by the National Science Foundation under Grant Number PHY-2112725. The work of RCV is supported by the Robert N. Little Fellowship. K.L. is supported through the grants CEX2020-001007-S and PID2021-123017NB-100, PID2021-127726NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”. ISL is a CONICET and ICTP Associate (2023-2028) fellow. EC thanks the Instituto de Física Teórica (IFT) at UAM, Madrid, for hospitality during the initial stages of this work. ISL thanks to ICTP, IFT and Católica Chile U. for hospitality.
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Cáceres, E., Vásquez, R.C., Landsteiner, K. et al. Holographic a-functions and Boomerang RG flows. J. High Energ. Phys. 2024, 19 (2024). https://doi.org/10.1007/JHEP02(2024)019
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DOI: https://doi.org/10.1007/JHEP02(2024)019