Abstract
We study nonlinear energy diffusion in the SYK chain within the framework of Schwinger-Keldysh effective field theory. We analytically construct the corresponding effective action up to 40th order in the derivative expansion. According to this effective action, we calculate the first order loop correction of the energy density response function, whose pole is the dispersion relation of energy diffusion. As expected, the standard derivative expansion of the classical dispersion relation breaks down due to the long-time tails. However, we find that the nonlinear contributions are so that one can still derive the dispersion relation in the power series. In fact, due to the long-time tails, the classical dispersion relation is split into two series distinct from the derivative expansion, and we show they are convergent. The radius of convergence is proportional to the ratio of thermal conductivity to diffusion constant.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics, vol. 6, Pergamon Press, Oxford, U.K. (1987) [DOI].
L.D. Landau and E.M. Lifshitz, Statistical Physics Part 2, Course of Theoretical Physics, vol. 9, Butterworth-Heinemann, Oxford, U.K. (1980) [DOI].
M. Crossley, P. Glorioso and H. Liu, Effective field theory of dissipative fluids, JHEP 09 (2017) 095 [arXiv:1511.03646] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Adiabatic hydrodynamics: The eightfold way to dissipation, JHEP 05 (2015) 060 [arXiv:1502.00636] [INSPIRE].
K. Jensen, N. Pinzani-Fokeeva and A. Yarom, Dissipative hydrodynamics in superspace, JHEP 09 (2018) 127 [arXiv:1701.07436] [INSPIRE].
F.M. Haehl, R. Loganayagam and M. Rangamani, Effective Action for Relativistic Hydrodynamics: Fluctuations, Dissipation, and Entropy Inflow, JHEP 10 (2018) 194 [arXiv:1803.11155] [INSPIRE].
K. Jensen, R. Marjieh, N. Pinzani-Fokeeva and A. Yarom, A panoply of Schwinger-Keldysh transport, SciPost Phys. 5 (2018) 053 [arXiv:1804.04654] [INSPIRE].
S. Dubovsky, L. Hui, A. Nicolis and D.T. Son, Effective field theory for hydrodynamics: thermodynamics, and the derivative expansion, Phys. Rev. D 85 (2012) 085029 [arXiv:1107.0731] [INSPIRE].
S. Endlich, A. Nicolis, R.A. Porto and J. Wang, Dissipation in the effective field theory for hydrodynamics: First order effects, Phys. Rev. D 88 (2013) 105001 [arXiv:1211.6461] [INSPIRE].
S. Grozdanov and J. Polonyi, Viscosity and dissipative hydrodynamics from effective field theory, Phys. Rev. D 91 (2015) 105031 [arXiv:1305.3670] [INSPIRE].
P. Kovtun, G.D. Moore and P. Romatschke, Towards an effective action for relativistic dissipative hydrodynamics, JHEP 07 (2014) 123 [arXiv:1405.3967] [INSPIRE].
M. Harder, P. Kovtun and A. Ritz, On thermal fluctuations and the generating functional in relativistic hydrodynamics, JHEP 07 (2015) 025 [arXiv:1502.03076] [INSPIRE].
H. Liu and P. Glorioso, Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics, PoS TASI2017 (2018) 008 [arXiv:1805.09331] [INSPIRE].
P. Gao, P. Glorioso and H. Liu, Ghostbusters: Unitarity and Causality of Non-equilibrium Effective Field Theories, JHEP 03 (2020) 040 [arXiv:1803.10778] [INSPIRE].
X. Chen-Lin, L.V. Delacrétaz and S.A. Hartnoll, Theory of diffusive fluctuations, Phys. Rev. Lett. 122 (2019) 091602 [arXiv:1811.12540] [INSPIRE].
A. Jain and P. Kovtun, Late Time Correlations in Hydrodynamics: Beyond Constitutive Relations, Phys. Rev. Lett. 128 (2022) 071601 [arXiv:2009.01356] [INSPIRE].
N. Sogabe, N. Yamamoto and Y. Yin, Positive magnetoresistance induced by hydrodynamic fluctuations in chiral media, JHEP 09 (2021) 131 [arXiv:2105.10271] [INSPIRE].
M. Blake, H. Lee and H. Liu, A quantum hydrodynamical description for scrambling and many-body chaos, JHEP 10 (2018) 127 [arXiv:1801.00010] [INSPIRE].
M. Blake and H. Liu, On systems of maximal quantum chaos, JHEP 05 (2021) 229 [arXiv:2102.11294] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
M. Blake, R.A. Davison and D. Vegh, Horizon constraints on holographic Green’s functions, JHEP 01 (2020) 077 [arXiv:1904.12883] [INSPIRE].
S. Grozdanov, On the connection between hydrodynamics and quantum chaos in holographic theories with stringy corrections, JHEP 01 (2019) 048 [arXiv:1811.09641] [INSPIRE].
M. Natsuume and T. Okamura, Holographic chaos, pole-skipping, and regularity, PTEP 2020 (2020) 013B07 [arXiv:1905.12014] [INSPIRE].
M. Natsuume and T. Okamura, Nonuniqueness of Green’s functions at special points, JHEP 12 (2019) 139 [arXiv:1905.12015] [INSPIRE].
M. Natsuume and T. Okamura, Pole-skipping with finite-coupling corrections, Phys. Rev. D 100 (2019) 126012 [arXiv:1909.09168] [INSPIRE].
X. Wu, Higher curvature corrections to pole-skipping, JHEP 12 (2019) 140 [arXiv:1909.10223] [INSPIRE].
Y. Ahn, V. Jahnke, H.-S. Jeong and K.-Y. Kim, Scrambling in Hyperbolic Black Holes: shock waves and pole-skipping, JHEP 10 (2019) 257 [arXiv:1907.08030] [INSPIRE].
W. Li, S. Lin and J. Mei, Thermal diffusion and quantum chaos in neutral magnetized plasma, Phys. Rev. D 100 (2019) 046012 [arXiv:1905.07684] [INSPIRE].
N. Ceplak, K. Ramdial and D. Vegh, Fermionic pole-skipping in holography, JHEP 07 (2020) 203 [arXiv:1910.02975] [INSPIRE].
S. Das, B. Ezhuthachan and A. Kundu, Real time dynamics from low point correlators in 2d BCFT, JHEP 12 (2019) 141 [arXiv:1907.08763] [INSPIRE].
N. Abbasi and J. Tabatabaei, Quantum chaos, pole-skipping and hydrodynamics in a holographic system with chiral anomaly, JHEP 03 (2020) 050 [arXiv:1910.13696] [INSPIRE].
Y. Liu and A. Raju, Quantum Chaos in Topologically Massive Gravity, JHEP 12 (2020) 027 [arXiv:2005.08508] [INSPIRE].
Y. Ahn, V. Jahnke, H.-S. Jeong, K.-Y. Kim, K.-S. Lee and M. Nishida, Pole-skipping of scalar and vector fields in hyperbolic space: conformal blocks and holography, JHEP 09 (2020) 111 [arXiv:2006.00974] [INSPIRE].
Y. Ahn, V. Jahnke, H.-S. Jeong, K.-S. Lee, M. Nishida and K.-Y. Kim, Classifying pole-skipping points, JHEP 03 (2021) 175 [arXiv:2010.16166] [INSPIRE].
K.-Y. Kim, K.-S. Lee and M. Nishida, Holographic scalar and vector exchange in OTOCs and pole-skipping phenomena, JHEP 04 (2021) 092 [Erratum ibid. 04 (2021) 229] [arXiv:2011.13716] [INSPIRE].
K. Sil, Pole skipping and chaos in anisotropic plasma: a holographic study, JHEP 03 (2021) 232 [arXiv:2012.07710] [INSPIRE].
H. Yuan and X.-H. Ge, Pole-skipping and hydrodynamic analysis in Lifshitz, AdS2 and Rindler geometries, JHEP 06 (2021) 165 [arXiv:2012.15396] [INSPIRE].
N. Abbasi and M. Kaminski, Constraints on quasinormal modes and bounds for critical points from pole-skipping, JHEP 03 (2021) 265 [arXiv:2012.15820] [INSPIRE].
N. Ceplak and D. Vegh, Pole-skipping and Rarita-Schwinger fields, Phys. Rev. D 103 (2021) 106009 [arXiv:2101.01490] [INSPIRE].
H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, Bound of diffusion constants from pole-skipping points: spontaneous symmetry breaking and magnetic field, JHEP 07 (2021) 105 [arXiv:2104.13084] [INSPIRE].
H. Yuan and X.-H. Ge, Analogue of the pole-skipping phenomenon in acoustic black holes, Eur. Phys. J. C 82 (2022) 167 [arXiv:2110.08074] [INSPIRE].
M. Blake and R.A. Davison, Chaos and pole-skipping in rotating black holes, JHEP 01 (2022) 013 [arXiv:2111.11093] [INSPIRE].
K.-Y. Kim, K.-S. Lee and M. Nishida, Construction of bulk solutions for towers of pole-skipping points, arXiv:2112.11662 [INSPIRE].
F.M. Haehl and M. Rozali, Effective Field Theory for Chaotic CFTs, JHEP 10 (2018) 118 [arXiv:1808.02898] [INSPIRE].
D.M. Ramirez, Chaos and pole skipping in CFT2, JHEP 12 (2021) 006 [arXiv:2009.00500] [INSPIRE].
A. Jain, P. Kovtun, A. Ritz and A. Shukla, Hydrodynamic effective field theory and the analyticity of hydrostatic correlators, JHEP 02 (2021) 200 [arXiv:2011.03691] [INSPIRE].
J. Chao and T. Schaefer, Multiplicative noise and the diffusion of conserved densities, JHEP 01 (2021) 071 [arXiv:2008.01269] [INSPIRE].
N. Sogabe and Y. Yin, Off-equilibrium non-Gaussian fluctuations near the QCD critical point: an effective field theory perspective, JHEP 03 (2022) 124 [arXiv:2111.14667] [INSPIRE].
M. Baggioli and M. Landry, Effective Field Theory for Quasicrystals and Phasons Dynamics, SciPost Phys. 9 (2020) 062 [arXiv:2008.05339] [INSPIRE].
L.V. Delacrétaz, B. Goutéraux and V. Ziogas, Damping of Pseudo-Goldstone Fields, Phys. Rev. Lett. 128 (2022) 141601 [arXiv:2111.13459] [INSPIRE].
M.P. Heller, R.A. Janik and P. Witaszczyk, Hydrodynamic Gradient Expansion in Gauge Theory Plasmas, Phys. Rev. Lett. 110 (2013) 211602 [arXiv:1302.0697] [INSPIRE].
B. Withers, Short-lived modes from hydrodynamic dispersion relations, JHEP 06 (2018) 059 [arXiv:1803.08058] [INSPIRE].
S. Grozdanov, P.K. Kovtun, A.O. Starinets and P. Tadić, Convergence of the Gradient Expansion in Hydrodynamics, Phys. Rev. Lett. 122 (2019) 251601 [arXiv:1904.01018] [INSPIRE].
S. Grozdanov, P.K. Kovtun, A.O. Starinets and P. Tadić, The complex life of hydrodynamic modes, JHEP 11 (2019) 097 [arXiv:1904.12862] [INSPIRE].
P. Kovtun and L.G. Yaffe, Hydrodynamic fluctuations, long time tails, and supersymmetry, Phys. Rev. D 68 (2003) 025007 [hep-th/0303010] [INSPIRE].
P. Kovtun, G.D. Moore and P. Romatschke, The stickiness of sound: An absolute lower limit on viscosity and the breakdown of second order relativistic hydrodynamics, Phys. Rev. D 84 (2011) 025006 [arXiv:1104.1586] [INSPIRE].
P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, J. Phys. A 45 (2012) 473001 [arXiv:1205.5040] [INSPIRE].
A. Shukla, Hydrodynamic fluctuations and long-time tails in a fluid on an anisotropic background, Nucl. Phys. B 968 (2021) 115442 [arXiv:2101.10000] [INSPIRE].
L.V. Delacretaz, Heavy Operators and Hydrodynamic Tails, SciPost Phys. 9 (2020) 034 [arXiv:2006.01139] [INSPIRE].
Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125 [arXiv:1609.07832] [INSPIRE].
C.P. Herzog, P. Kovtun, S. Sachdev and D.T. Son, Quantum critical transport, duality, and M-theory, Phys. Rev. D 75 (2007) 085020 [hep-th/0701036] [INSPIRE].
C. Choi, M. Mezei and G. Sárosi, Pole skipping away from maximal chaos, JHEP 02 (2021) 207 [arXiv:2010.08558] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
K. Jensen, Chaos in AdS2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
E. Wang and U.W. Heinz, A Generalized fluctuation dissipation theorem for nonlinear response functions, Phys. Rev. D 66 (2002) 025008 [hep-th/9809016] [INSPIRE].
L.V. Delacretaz and P. Glorioso, Breakdown of Diffusion on Chiral Edges, Phys. Rev. Lett. 124 (2020) 236802 [arXiv:2002.08365] [INSPIRE].
P. Glorioso, J. Guo, J.F. Rodriguez-Nieva and A. Lucas, Breakdown of hydrodynamics below four dimensions in a fracton fluid, arXiv:2105.13365 [INSPIRE].
N. Abbasi and S. Tahery, Complexified quasinormal modes and the pole-skipping in a holographic system at finite chemical potential, JHEP 10 (2020) 076 [arXiv:2007.10024] [INSPIRE].
A. Jansen and C. Pantelidou, Quasinormal modes in charged fluids at complex momentum, JHEP 10 (2020) 121 [arXiv:2007.14418] [INSPIRE].
M. Baggioli, How small hydrodynamics can go, Phys. Rev. D 103 (2021) 086001 [arXiv:2010.05916] [INSPIRE].
D. Arean, R.A. Davison, B. Goutéraux and K. Suzuki, Hydrodynamic Diffusion and Its Breakdown near AdS2 Quantum Critical Points, Phys. Rev. X 11 (2021) 031024 [arXiv:2011.12301] [INSPIRE].
M.P. Heller, A. Serantes, M. Spaliński, V. Svensson and B. Withers, Hydrodynamic gradient expansion in linear response theory, Phys. Rev. D 104 (2021) 066002 [arXiv:2007.05524] [INSPIRE].
M. Asadi, H. Soltanpanahi and F. Taghinavaz, Critical behaviour of hydrodynamic series, JHEP 05 (2021) 287 [arXiv:2102.03584] [INSPIRE].
M. Baggioli, U. Gran and M. Tornsö, Collective modes of polarizable holographic media in magnetic fields, JHEP 06 (2021) 014 [arXiv:2102.09969] [INSPIRE].
N. Wu, M. Baggioli and W.-J. Li, On the universality of AdS2 diffusion bounds and the breakdown of linearized hydrodynamics, JHEP 05 (2021) 014 [arXiv:2102.05810] [INSPIRE].
S. Grozdanov, A.O. Starinets and P. Tadić, Hydrodynamic dispersion relations at finite coupling, JHEP 06 (2021) 180 [arXiv:2104.11035] [INSPIRE].
M.P. Heller, A. Serantes, M. Spaliński, V. Svensson and B. Withers, Hydrodynamic Gradient Expansion Diverges beyond Bjorken Flow, Phys. Rev. Lett. 128 (2022) 122302 [arXiv:2110.07621] [INSPIRE].
H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, The breakdown of magneto-hydrodynamics near AdS2 fixed point and energy diffusion bound, JHEP 02 (2022) 006 [arXiv:2105.03882] [INSPIRE].
K.-B. Huh, H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, Upper bound of the charge diffusion constant in holography, arXiv:2111.07515 [INSPIRE].
Y. Liu and X.-M. Wu, Breakdown of hydrodynamics from holographic pole collision, JHEP 01 (2022) 155 [arXiv:2111.07770] [INSPIRE].
C. Cartwright, M.G. Amano, M. Kaminski, J. Noronha and E. Speranza, Convergence of hydrodynamics in rapidly spinning strongly coupled plasma, arXiv:2112.10781 [INSPIRE].
I. Amado, C. Hoyos-Badajoz, K. Landsteiner and S. Montero, Hydrodynamics and beyond in the strongly coupled N = 4 plasma, JHEP 07 (2008) 133 [arXiv:0805.2570] [INSPIRE].
Y. Akamatsu, A. Mazeliauskas and D. Teaney, A kinetic regime of hydrodynamic fluctuations and long time tails for a Bjorken expansion, Phys. Rev. C 95 (2017) 014909 [arXiv:1606.07742] [INSPIRE].
S. Caron-Huot and O. Saremi, Hydrodynamic Long-Time tails From Anti de Sitter Space, JHEP 11 (2010) 013 [arXiv:0909.4525] [INSPIRE].
P. Glorioso, M. Crossley and H. Liu, A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems, arXiv:1812.08785 [INSPIRE].
G. Cheng and B. Swingle, Scrambling with conservation laws, JHEP 11 (2021) 174 [arXiv:2103.07624] [INSPIRE].
J. de Boer, M.P. Heller and N. Pinzani-Fokeeva, Holographic Schwinger-Keldysh effective field theories, JHEP 05 (2019) 188 [arXiv:1812.06093] [INSPIRE].
K. Skenderis and B.C. van Rees, Real-time gauge/gravity duality: Prescription, Renormalization and Examples, JHEP 05 (2009) 085 [arXiv:0812.2909] [INSPIRE].
J.K. Ghosh, R. Loganayagam, S.G. Prabhu, M. Rangamani, A. Sivakumar and V. Vishal, Effective field theory of stochastic diffusion from gravity, JHEP 05 (2021) 130 [arXiv:2012.03999] [INSPIRE].
Y. Bu, M. Fujita and S. Lin, Ginzburg-Landau effective action for a fluctuating holographic superconductor, JHEP 09 (2021) 168 [arXiv:2106.00556] [INSPIRE].
Y. Bu, T. Demircik and M. Lublinsky, All order effective action for charge diffusion from Schwinger-Keldysh holography, JHEP 05 (2021) 187 [arXiv:2012.08362] [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: 2112.12751
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
Abbasi, N. Long-time tails in the SYK chain from the effective field theory with a large number of derivatives. J. High Energ. Phys. 2022, 181 (2022). https://doi.org/10.1007/JHEP04(2022)181
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2022)181