Abstract
We continue the analysis of the Krylov complexity in the IP matrix model. In a previous paper, [1], for a fundamental operator, it was shown that at zero temperature, the Krylov complexity oscillates and does not grow, but in the infinite temperature limit, the Krylov complexity grows exponentially in time as \( \sim \exp \left(\mathcal{O}\left(\sqrt{t}\right)\right) \). We study how the Krylov complexity changes from a zero-temperature oscillation to an infinite-temperature exponential growth. At low temperatures, the spectral density is approximated as collections of infinite Wigner semicircles. We showed that this infinite collection of branch cuts yields linear growth to the Lanczos coefficients and gives exponential growth of the Krylov complexity. Thus the IP model for any nonzero temperature shows exponential growth for the Krylov complexity even though the Green function decays by a power law in time. We also study the Lanczos coefficients and the Krylov complexity in the IOP matrix model taking into account the 1/N2 corrections. There, the Lanczos coefficients are constants and the Krylov complexity does not grow exponentially as expected.
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References
N. Iizuka and M. Nishida, Krylov complexity in the IP matrix model, JHEP 11 (2023) 065 [arXiv:2306.04805] [INSPIRE].
D.E. Parker et al., A Universal Operator Growth Hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [INSPIRE].
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].
A. Kitaev, A simple model of quantum holography, talks at KITP, 7 April 2015 and 27 May 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev, http://online.kitp.ucsb.edu/online/entangled15/kitaev2.
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].
A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2 (2020) 043234 [arXiv:1911.09672] [INSPIRE].
X. Cao, A statistical mechanism for operator growth, J. Phys. A 54 (2021) 144001 [arXiv:2012.06544] [INSPIRE].
S.-K. Jian, B. Swingle and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT gravity, JHEP 03 (2021) 014 [arXiv:2008.12274] [INSPIRE].
A. Dymarsky and A. Gorsky, Quantum chaos as delocalization in Krylov space, Phys. Rev. B 102 (2020) 085137 [arXiv:1912.12227] [INSPIRE].
D.J. Yates, A.G. Abanov and A. Mitra, Lifetime of Almost Strong Edge-Mode Operators in One-Dimensional, Interacting, Symmetry Protected Topological Phases, Phys. Rev. Lett. 124 (2020) 206803 [arXiv:2002.00098] [INSPIRE].
D.J. Yates, A.G. Abanov and A. Mitra, Dynamics of almost strong edge modes in spin chains away from integrability, Phys. Rev. B 102 (2020) 195419 [arXiv:2009.00057] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Operator complexity: a journey to the edge of Krylov space, JHEP 06 (2021) 062 [arXiv:2009.01862] [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].
D.J. Yates, A.G. Abanov and A. Mitra, Long-lived period-doubled edge modes of interacting and disorder-free Floquet spin chains, Commun. Phys. 5 (2022) 43 [arXiv:2105.13766] [INSPIRE].
D.J. Yates and A. Mitra, Strong and almost strong modes of Floquet spin chains in Krylov subspaces, Phys. Rev. B 104 (2021) 195121 [arXiv:2105.13246] [INSPIRE].
A. Dymarsky and M. Smolkin, Krylov complexity in conformal field theory, Phys. Rev. D 104 (2021) L081702 [arXiv:2104.09514] [INSPIRE].
J.D. Noh, Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field, Phys. Rev. E 104 (2021) 034112 [arXiv:2107.08287].
F.B. Trigueros and C.-J. Lin, Krylov complexity of many-body localization: Operator localization in Krylov basis, SciPost Phys. 13 (2022) 037 [arXiv:2112.04722] [INSPIRE].
C. Liu, H. Tang and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res. 5 (2023) 033085 [arXiv:2207.13603] [INSPIRE].
Z.-Y. Fan, Universal relation for operator complexity, Phys. Rev. A 105 (2022) 062210 [arXiv:2202.07220] [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].
P. Caputa, J.M. Magan and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res. 4 (2022) 013041 [arXiv:2109.03824] [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].
R. Heveling, J. Wang and J. Gemmer, Numerically probing the universal operator growth hypothesis, Phys. Rev. E 106 (2022) 014152 [arXiv:2203.00533] [INSPIRE].
K. Adhikari, S. Choudhury and A. Roy, Krylov Complexity in Quantum Field Theory, Nucl. Phys. B 993 (2023) 116263 [arXiv:2204.02250] [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].
B. Bhattacharjee, X. Cao, P. Nandy and T. Pathak, Krylov complexity in saddle-dominated scrambling, JHEP 05 (2022) 174 [arXiv:2203.03534] [INSPIRE].
B.-N. Du and M.-X. Huang, Krylov complexity in Calabi-Yau quantum mechanics, Int. J. Mod. Phys. A 38 (2023) 2350126 [arXiv:2212.02926] [INSPIRE].
A. Banerjee, A. Bhattacharyya, P. Drashni and S. Pawar, From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators, Phys. Rev. D 106 (2022) 126022 [arXiv:2205.15338] [INSPIRE].
W. Mück and Y. Yang, Krylov complexity and orthogonal polynomials, Nucl. Phys. B 984 (2022) 115948 [arXiv:2205.12815] [INSPIRE].
N. Hörnedal, N. Carabba, A.S. Matsoukas-Roubeas and A. del Campo, Ultimate Speed Limits to the Growth of Operator Complexity, Commun. Phys. 5 (2022) 207 [arXiv:2202.05006] [INSPIRE].
S. Guo, Operator growth in SU(2) Yang-Mills theory, arXiv:2208.13362 [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].
M. Alishahiha and S. Banerjee, A universal approach to Krylov State and Operator complexities, SciPost Phys. 15 (2023) 080 [arXiv:2212.10583] [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. Kundu, V. Malvimat and R. Sinha, State dependence of Krylov complexity in 2d CFTs, JHEP 09 (2023) 011 [arXiv:2303.03426] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, A bulk manifestation of Krylov complexity, JHEP 08 (2023) 213 [arXiv:2305.04355] [INSPIRE].
R. Zhang and H. Zhai, Universal Hypothesis of Autocorrelation Function from Krylov Complexity, arXiv:2305.02356 [INSPIRE].
A.A. Nizami and A.W. Shrestha, Krylov construction and complexity for driven quantum systems, arXiv:2305.00256 [INSPIRE].
K. Hashimoto, K. Murata, N. Tanahashi and R. Watanabe, Krylov complexity and chaos in quantum mechanics, JHEP 11 (2023) 040 [arXiv:2305.16669] [INSPIRE].
S. Nandy, B. Mukherjee, A. Bhattacharyya and A. Banerjee, Quantum state complexity meets many-body scars, arXiv:2305.13322 [INSPIRE].
P. Caputa and S. Liu, Quantum complexity and topological phases of matter, Phys. Rev. B 106 (2022) 195125 [arXiv:2205.05688] [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 [https://doi.org/10.1088/1742-5468/ad0032] [INSPIRE].
V. Balasubramanian, J.M. Magan and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107 (2023) 126001 [arXiv:2208.08452] [INSPIRE].
J. Erdmenger, S.-K. Jian and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, JHEP 08 (2023) 176 [arXiv:2303.12151] [INSPIRE].
A. Bhattacharya, P. Nandy, P.P. Nath and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, JHEP 12 (2022) 081 [arXiv:2207.05347] [INSPIRE].
S. He, P.H.C. Lau, Z.-Y. Xian and L. Zhao, Quantum chaos, scrambling and operator growth in \( T\overline{T} \) deformed SYK models, JHEP 12 (2022) 070 [arXiv:2209.14936] [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].
A. Bhattacharya, P. Nandy, P.P. Nath and H. Sahu, On Krylov complexity in open systems: an approach via bi-Lanczos algorithm, arXiv:2303.04175 [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].
K. Pal, K. Pal, A. Gill and T. Sarkar, Time evolution of spread complexity and statistics of work done in quantum quenches, Phys. Rev. B 108 (2023) 104311 [arXiv:2304.09636] [INSPIRE].
D. Patramanis and W. Sybesma, Krylov complexity in a natural basis for the Schrödinger algebra, arXiv:2306.03133 [INSPIRE].
A. Bhattacharyya, D. Ghosh and P. Nandi, Operator growth and Krylov Complexity in Bose-Hubbard Model, arXiv:2306.05542 [INSPIRE].
H.A. Camargo et al., Spectral and Krylov Complexity in Billiard Systems, arXiv:2306.11632 [INSPIRE].
P. Caputa, J.M. Magan, D. Patramanis and E. Tonni, Krylov complexity of modular Hamiltonian evolution, arXiv:2306.14732 [INSPIRE].
Z.-Y. Fan, Generalised Krylov complexity, arXiv:2306.16118 [INSPIRE].
M.J. Vasli et al., Krylov Complexity in Lifshitz-type Scalar Field Theories, arXiv:2307.08307 [INSPIRE].
A. Bhattacharyya et al., Krylov complexity and spectral form factor for noisy random matrix models, JHEP 23 (2020) 157 [arXiv:2307.15495] [INSPIRE].
M. Gautam et al., Spread complexity evolution in quenched interacting quantum systems, arXiv:2308.00636 [INSPIRE].
P. Suchsland, R. Moessner and P.W. Claeys, Krylov complexity and Trotter transitions in unitary circuit dynamics, arXiv:2308.03851 [INSPIRE].
N. Iizuka and J. Polchinski, A Matrix Model for Black Hole Thermalization, JHEP 10 (2008) 028 [arXiv:0801.3657] [INSPIRE].
N. Iizuka, T. Okuda and J. Polchinski, Matrix Models for the Black Hole Information Paradox, JHEP 02 (2010) 073 [arXiv:0808.0530] [INSPIRE].
B. Michel, J. Polchinski, V. Rosenhaus and S.J. Suh, Four-point function in the IOP matrix model, JHEP 05 (2016) 048 [arXiv:1602.06422] [INSPIRE].
T. Anegawa, N. Iizuka and M. Nishida, work in progress.
V.S. Viswanath and G. Müller, The Recursion Method: Application to Many-Body Dynamics. Springer Berlin, Heidelberg, Germany (1994).
D.S. Lubinsky, H.N. Mhaskar and E.B. Saff, A proof of Freud’s conjecture for exponential weights, Constructive Approximation 4 (1988) 65.
T.A. Elsayed, B. Hess and B.V. Fine, Signatures of chaos in time series generated by many-spin systems at high temperatures, Phys. Rev. E 90 (2014) 022910.
Acknowledgments
The work of NI was supported in part by JSPS KAKENHI Grant Number 18K03619 and also by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A “Extreme Universe” No. 21H05184. M.N. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2023-00245035).
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Iizuka, N., Nishida, M. Krylov complexity in the IP matrix model. Part II. J. High Energ. Phys. 2023, 96 (2023). https://doi.org/10.1007/JHEP11(2023)096
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DOI: https://doi.org/10.1007/JHEP11(2023)096