Abstract
We compute the Krylov Complexity of a light operator \( \mathcal{O} \)L in an eigenstate of a 2d CFT at large central charge c. The eigenstate corresponds to a primary operator \( \mathcal{O} \)H under the state-operator correspondence. We observe that the behaviour of K-complexity is different (either bounded or exponential) depending on whether the scaling dimension of \( \mathcal{O} \)H is below or above the critical dimension hH = c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS3 geometry. Based on this feature, we hypothesize that the notions of operator growth and K-complexity for primary operators in 2d CFTs are closely related to the underlying entanglement structure of the state in which they are computed, thereby demonstrating explicitly their state-dependent nature. To provide further evidence for our hypothesis, we perform an analogous computation of K-complexity in a model of free massless scalar field theory in 2d, and in the integrable 2d Ising CFT, where there is no such transition in the spectrum of states.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Srednicki, Chaos and Quantum Thermalization, cond-mat/9403051 [https://doi.org/10.1103/PhysRevE.50.888] [INSPIRE].
C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rept. Prog. Phys. 79 (2016) 056001 [arXiv:1503.07538] [INSPIRE].
L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65 (2016) 239 [arXiv:1509.06411] [INSPIRE].
D.E. Parker et al., A Universal Operator Growth Hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [INSPIRE].
A. Avdoshkin, A. Dymarsky and M. Smolkin, Krylov complexity in quantum field theory, and beyond, arXiv:2212.14429 [INSPIRE].
H.A. Camargo, V. Jahnke, K.-Y. Kim and M. Nishida, Krylov complexity in free and interacting scalar field theories with bounded power spectrum, JHEP 05 (2023) 226 [arXiv:2212.14702] [INSPIRE].
A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104 (2021) L081702 [arXiv:2104.09514] [INSPIRE].
A. Dymarsky and A. Gorsky, Quantum chaos as delocalization in Krylov space, Phys. Rev. B 102 (2020) 085137 [arXiv:1912.12227] [INSPIRE].
A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2 (2020) 043234 [arXiv:1911.09672] [INSPIRE].
J.L.F. Barbón, E. Rabinovici, R. Shir and R. Sinha, On The Evolution Of Operator Complexity Beyond Scrambling, JHEP 10 (2019) 264 [arXiv:1907.05393] [INSPIRE].
S. Khetrapal, Chaos and operator growth in 2d CFT, JHEP 03 (2023) 176 [arXiv:2210.15860] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Krylov localization and suppression of complexity, JHEP 03 (2022) 211 [arXiv:2112.12128] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Krylov complexity from integrability to chaos, JHEP 07 (2022) 151 [arXiv:2207.07701] [INSPIRE].
V. Balasubramanian, P. Caputa, J.M. Magan and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106 (2022) 046007 [arXiv:2202.06957] [INSPIRE].
P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B 106 (2022) 195125 [arXiv:2205.05688] [INSPIRE].
B. Bhattacharjee, X. Cao, P. Nandy and T. Pathak, Operator growth in open quantum systems: lessons from the dissipative SYK, JHEP 03 (2023) 054 [arXiv:2212.06180] [INSPIRE].
P. Caputa et al., Spread complexity and topological transitions in the Kitaev chain, JHEP 01 (2023) 120 [arXiv:2208.06311] [INSPIRE].
M. Afrasiar et al., Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model, arXiv:2208.10520 [INSPIRE].
M. Alishahiha, On quantum complexity, Phys. Lett. B 842 (2023) 137979 [arXiv:2209.14689] [INSPIRE].
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
S. Chapman et al., Complexity and entanglement for thermofield double states, SciPost Phys. 6 (2019) 034 [arXiv:1810.05151] [INSPIRE].
E. Caceres et al., Complexity of Mixed States in QFT and Holography, JHEP 03 (2020) 012 [arXiv:1909.10557] [INSPIRE].
N. Chagnet, S. Chapman, J. de Boer and C. Zukowski, Complexity for Conformal Field Theories in General Dimensions, Phys. Rev. Lett. 128 (2022) 051601 [arXiv:2103.06920] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
L. Susskind, Three Lectures on Complexity and Black Holes, Springer, 2020, ISBN 978-3-030-45108-0 [https://doi.org/10.1007/978-3-030-45109-7] [arXiv:1810.11563] [INSPIRE].
L. Susskind, Why do Things Fall?, arXiv:1802.01198 [INSPIRE].
A. Chattopadhyay, A. Mitra and H.J.R. van Zyl, Spread complexity as classical dilaton solutions, Phys. Rev. D 108 (2023) 025013 [arXiv:2302.10489] [INSPIRE].
S. Chapman and G. Policastro, Quantum computational complexity from quantum information to black holes and back, Eur. Phys. J. C 82 (2022) 128 [arXiv:2110.14672] [INSPIRE].
P. Caputa and S. Datta, Operator growth in 2d CFT, JHEP 12 (2021) 188 [Erratum ibid. 09 (2022) 113] [arXiv:2110.10519] [INSPIRE].
A. Belin et al., Does Complexity Equal Anything?, Phys. Rev. Lett. 128 (2022) 081602 [arXiv:2111.02429] [INSPIRE].
A. Kar, L. Lamprou, M. Rozali and J. Sully, Random matrix theory for complexity growth and black hole interiors, JHEP 01 (2022) 016 [arXiv:2106.02046] [INSPIRE].
VS Viswanath and Gerhard Müllerller, The Recursion Method: Application to Many Body Dynamics, Springer (2008).
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP 11 (2015) 200 [arXiv:1501.05315] [INSPIRE].
L. Susskind, Complexity and Newton’s Laws, Front. in Phys. 8 (2020) 262 [arXiv:1904.12819] [INSPIRE].
T. Anous and J. Sonner, Phases of scrambling in eigenstates, SciPost Phys. 7 (2019) 003 [arXiv:1903.03143] [INSPIRE].
M.P. Mattis, Correlations in Two-dimensional Critical Theories, Nucl. Phys. B 285 (1987) 671 [INSPIRE].
M. Serbyn, D.A. Abanin and Z. Papić, Quantum many-body scars and weak breaking of ergodicity, Nature Phys. 17 (2021) 675 [arXiv:2011.09486] [INSPIRE].
S. Moudgalya, B.A. Bernevig and N. Regnault, Quantum many-body scars and Hilbert space fragmentation: a review of exact results, Rept. Prog. Phys. 85 (2022) 086501 [arXiv:2109.00548] [INSPIRE].
D. Banerjee and A. Sen, Quantum Scars from Zero Modes in an Abelian Lattice Gauge Theory on Ladders, Phys. Rev. Lett. 126 (2021) 220601 [arXiv:2012.08540] [INSPIRE].
S. Biswas, D. Banerjee and A. Sen, Scars from protected zero modes and beyond in U(1) quantum link and quantum dimer models, SciPost Phys. 12 (2022) 148 [arXiv:2202.03451] [INSPIRE].
B. Bhattacharjee, S. Sur and P. Nandy, Probing quantum scars and weak ergodicity breaking through quantum complexity, Phys. Rev. B 106 (2022) 205150 [arXiv:2208.05503] [INSPIRE].
M. Alishahiha and S. Banerjee, A universal approach to Krylov State and Operator complexities, SciPost Phys. 15 (2023) 080 [arXiv:2212.10583] [INSPIRE].
D. Das, R. Ghosh and K. Sengupta, Conformal Floquet dynamics with a continuous drive protocol, JHEP 05 (2021) 172 [arXiv:2101.04140] [INSPIRE].
X. Wen, Y. Gu, A. Vishwanath and R. Fan, Periodically, Quasi-periodically, and Randomly Driven Conformal Field Theories (II): Furstenberg’s Theorem and Exceptions to Heating Phases, SciPost Phys. 13 (2022) 082 [arXiv:2109.10923] [INSPIRE].
R. Fan, Y. Gu, A. Vishwanath and X. Wen, Floquet conformal field theories with generally deformed Hamiltonians, SciPost Phys. 10 (2021) 049 [arXiv:2011.09491] [INSPIRE].
X. Wen, R. Fan, A. Vishwanath and Y. Gu, Periodically, quasiperiodically, and randomly driven conformal field theories, Phys. Rev. Res. 3 (2021) 023044 [arXiv:2006.10072] [INSPIRE].
S. Das et al., Brane Detectors of a Dynamical Phase Transition in a Driven CFT, arXiv:2212.04201 [INSPIRE].
S. Das et al., Out-of-Time-Order correlators in driven conformal field theories, JHEP 08 (2022) 221 [arXiv:2202.12815] [INSPIRE].
P. Caputa and D. Ge, Entanglement and geometry from subalgebras of the Virasoro algebra, JHEP 06 (2023) 159 [arXiv:2211.03630] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
A. Banerjee, A. Kundu and R.R. Poojary, Rotating black holes in AdS spacetime, extremality, and chaos, Phys. Rev. D 102 (2020) 106013 [arXiv:1912.12996] [INSPIRE].
A. Banerjee, A. Kundu and R. Poojary, Maximal Chaos from Strings, Branes and Schwarzian Action, JHEP 06 (2019) 076 [arXiv:1811.04977] [INSPIRE].
A. Banerjee, A. Kundu and R.R. Poojary, Strings, branes, Schwarzian action and maximal chaos, Phys. Lett. B 838 (2023) 137632 [arXiv:1809.02090] [INSPIRE].
V. Malvimat and R.R. Poojary, Fast scrambling due to rotating shockwaves in BTZ, Phys. Rev. D 105 (2022) 126019 [arXiv:2112.14089] [INSPIRE].
V. Malvimat and R.R. Poojary, Fast scrambling of mutual information in Kerr-AdS4 spacetime, Phys. Rev. D 107 (2023) 026019 [arXiv:2207.13022] [INSPIRE].
V. Malvimat and R.R. Poojary, Fast scrambling of mutual information in Kerr-AdS5, JHEP 03 (2023) 099 [arXiv:2210.02950] [INSPIRE].
P. Gao, D.L. Jafferis and A.C. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].
A. Kundu, Wormholes and holography: an introduction, Eur. Phys. J. C 82 (2022) 447 [arXiv:2110.14958] [INSPIRE].
N. Lashkari, A. Dymarsky and H. Liu, Eigenstate Thermalization Hypothesis in Conformal Field Theory, J. Stat. Mech. 1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].
P. Banerjee, S. Datta and R. Sinha, Higher-point conformal blocks and entanglement entropy in heavy states, JHEP 05 (2016) 127 [arXiv:1601.06794] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, A bulk manifestation of Krylov complexity, arXiv:2305.04355 [INSPIRE].
W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nucl. Phys. B 984 (2022) 115948 [arXiv:2205.12815] [INSPIRE].
Acknowledgments
We thank Suman Das, Sabyasachi Maulik for initial collaboration on related topics; Debashis Banerjee, Diptarka Das, Shouvik Datta, Surbhi Khetrapal for various discussions and conversations related to this article; Pawel Caputa, Sumit Das, Anatoly Dymarsky, Julian Sonner for numerous comments and feedbacks on the manuscript. RS would like to thank SINP, Kolkata for hospitality during the initial stages of this work. AK is partially supported by CEFIPRA 6304–3, DAE-BRNS 58/14/12/2021-BRNS and CRG/2021/004539 of Govt. of India. RS is supported by the Royal Society-Newton International Fellowship, NIF/R1/221054-Royal Society. The work of VM was supported by the NRF grant funded by the Korea government (MSIT) (No. 2022R1A2C1003182) and by the Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea (RS-2023-00261799).
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: 2303.03426
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
Kundu, A., Malvimat, V. & Sinha, R. State dependence of Krylov complexity in 2d CFTs. J. High Energ. Phys. 2023, 11 (2023). https://doi.org/10.1007/JHEP09(2023)011
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2023)011