Abstract
We study Krylov complexity of a one-dimensional Bosonic system, the celebrated Bose-Hubbard Model. The Bose-Hubbard Hamiltonian consists of interacting bosons on a lattice, describing ultra-cold atoms. Apart from showing superfluid-Mott insulator phase transition, the model also exhibits both chaotic and integrable (mixed) dynamics depending on the value of the interaction parameter. We focus on the three-site Bose Hubbard Model (with different particle numbers), which is known to be highly mixed. We use the Lanczos algorithm to find the Lanczos coefficients and the Krylov basis. The orthonormal Krylov basis captures the operator growth for a system with a given Hamiltonian. However, the Lanczos algorithm needs to be modified for our case due to the instabilities instilled by the piling up of computational errors. Next, we compute the Krylov complexity and its early and late-time behaviour. Our results capture the chaotic and integrable nature of the system. Our paper takes the first step to use the Lanczos algorithm non-perturbatively for a discrete quartic bosonic Hamiltonian without depending on the auto-correlation method.
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C. von Keyserlingk, T. Rakovszky, F. Pollmann and S. Sondhi, Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8 (2018) 021013 [arXiv:1705.08910] [INSPIRE].
A. Nahum, S. Vijay and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8 (2018) 021014 [arXiv:1705.08975] [INSPIRE].
V. Khemani, A. Vishwanath and D.A. Huse, Operator spreading and the emergence of dissipation in unitary dynamics with conservation laws, Phys. Rev. X 8 (2018) 031057 [arXiv:1710.09835] [INSPIRE].
T. Rakovszky, F. Pollmann and C.W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, Phys. Rev. X 8 (2018) 031058 [arXiv:1710.09827] [INSPIRE].
S. Gopalakrishnan, D.A. Huse, V. Khemani and R. Vasseur, Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems, Phys. Rev. B 98 (2018) 220303 [arXiv:1809.02126] [INSPIRE].
A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. JETP 28 (1969) 1200.
S. Xu and B. Swingle, Locality, quantum fluctuations, and scrambling, Phys. Rev. X 9 (2019) 031048 [arXiv:1805.05376] [INSPIRE].
Y. Gu, A. Kitaev and P. Zhang, A two-way approach to out-of-time-order correlators, JHEP 03 (2022) 133 [arXiv:2111.12007] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, https://online.kitp.ucsb.edu/online/entangled15/kitaev/, University of California, Santa Barbara, CA, U.S.A., 7 April 2015.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, https://online.kitp.ucsb.edu/online/entangled15/kitaev2/, University of California, Santa Barbara, CA, U.S.A., 27 May 2015.
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].
B.V. Fine, T.A. Elsayed, C.M. Kropf and A.S. de Wijn, Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2, Phys. Rev. E 89 (2014) 012923.
S. Xu and B. Swingle, Accessing scrambling using matrix product operators, Nature Phys. 16 (2019) 199 [arXiv:1802.00801] [INSPIRE].
D.E. Parker et al., A universal operator growth hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [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].
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].
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, 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.
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].
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].
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].
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].
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].
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].
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, Phys. Rev. E 108 (2023) 054222 [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].
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].
V. Balasubramanian et al., Complexity growth in integrable and chaotic models, JHEP 07 (2021) 011 [arXiv:2101.02209] [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].
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, 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. Jaksch et al., Cold bosonic atoms in optical lattices, Phys. Rev. Lett. 81 (1998) 3108 [cond-mat/9805329] [INSPIRE].
S. Sachdev, Quantum phase transitions, Cambridge University Press, Cambridge, U.K. (1999).
K. Sengupta and N. Dupuis, Mott-insulator-to-superfluid transition in the Bose-Hubbard model: a strong-coupling approach, Phys. Rev. A 71 (2005) 033629.
S. Mossmann and C. Jung, Semiclassical approach to Bose-Einstein condensates in a triple well potential, Phys. Rev. A 74 (2006) 033601.
T.F. Viscondi and K. Furuya, Dynamics of a Bose-Einstein condensate in a symmetric triple-well trap, J. Phys. A 44 (2011) 175301.
M. Rautenberg and M. Gärttner, Classical and quantum chaos in a three-mode bosonic system, Phys. Rev. A 101 (2020) 053604 [arXiv:1907.04094] [INSPIRE].
M. Feingold and A. Peres, Regular and chaotic motion of coupled rotators, Physica D 9 (1983) 433.
M. Feingold, N. Moiseyev and A. Peres, Ergodicity and mixing in quantum theory. II, Phys. Rev. A 30 (1984) 509.
D. Wintgen and H. Friedrich, Classical and quantum-mechanical transition between regularity and irregularity in a Hamiltonian system, Phys. Rev. A 35 (1987) 1464.
G. Nakerst and M. Haque, Chaos in the three-site Bose-Hubbard model: classical versus quantum, Phys. Rev. E 107 (2023) 024210 [arXiv:2203.09953] [INSPIRE].
V. Balasubramanian, J.M. Magan and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107 (2023) 126001 [arXiv:2208.08452] [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].
M. Pinkwart, Thermalization and integrability in the one-dimensional Bose-Hubbard model, bachelor thesis, University of Cologne, Cologne, Germany (2014).
J.M. Zhang and R.X. Dong, Exact diagonalization: the Bose-Hubbard model as an example, Eur. J. Phys. 31 (2010) 591.
C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. B 45 (1950) 255 [INSPIRE].
V.S. Viswanath and G. Müller, The recursion method: application to many body dynamics, Springer, Berlin, Heidelberg, Germany (1994) [https://doi.org/10.1007/978-3-540-48651-0].
B.N. Parlett, The symmetric eigenvalue problem, Society for Industrial and Applied Mathematics, U.S.A. (1998).
H.D. Simon, The Lanczos algorithm with partial reorthogonalization, Math. Comput. 42 (1984) 115.
B.N. Parlett and D.S. Scott, The Lanczos algorithm with selective orthogonalization, Math. Comput. 33 (1979) 217.
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].
H.D. Simon, The Lanczos algorithm with partial reorthogonalization, Math. Comput. 42 (1984) 115.
A. Bhattacharyya et al., The multi-faceted inverted harmonic oscillator: chaos and complexity, SciPost Phys. Core 4 (2021) 002 [arXiv:2007.01232] [INSPIRE].
J. Martin, Cosmic inflation, quantum information and the pioneering role of John S. Bell in cosmology, Universe 5 (2019) 92 [arXiv:1904.00083] [INSPIRE].
M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum computation as geometry, Science 311 (2006) 1133 [quant-ph/0603161] [INSPIRE].
A. Bhattacharyya, D. Ghosh and P. Nandi, work in progress.
U. Sood and M. Kruczenski, Circuit complexity near critical points, J. Phys. A 55 (2022) 185301 [arXiv:2106.12648] [INSPIRE].
U. Sood and M. Kruczenski, Non-analyticity in holographic complexity near critical points, J. Phys. A 56 (2023) 045301 [arXiv:2211.00212] [INSPIRE].
W.-H. Huang, Complexity of Bose-Hubbard model: quantum phase transition, arXiv:2112.13066 [INSPIRE].
A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res. 2 (2020) 043234 [arXiv:1911.09672] [INSPIRE].
H. Zhao, J. Vovrosh, F. Mintert and J. Knolle, Quantum many-body scars in optical lattices, Phys. Rev. Lett. 124 (2020) 160604 [arXiv:2002.01746] [INSPIRE].
Q. Hummel, K. Richter and P. Schlagheck, Genuine many-body quantum scars along unstable modes in Bose-Hubbard systems, arXiv:2212.12046.
G.-X. Su et al., Observation of many-body scarring in a Bose-Hubbard quantum simulator, Phys. Rev. Res. 5 (2023) 023010 [arXiv:2201.00821] [INSPIRE].
C.A. Lamas et al., Statistics of holes and nature of superfluid phases in quantum dimer models, arXiv:1210.1270.
T. Sowiński, O. Dutta, P. Hauke, L. Tagliacozzo and M. Lewenstein, Dipolar molecules in optical lattices, Phys. Rev. Lett. 108 (2012) 115301 [arXiv:1109.4782].
B. Yang et al., Observation of gauge invariance in a 71-site Bose-Hubbard quantum simulator, Nature 587 (2020) 392 [arXiv:2003.08945] [INSPIRE].
Acknowledgments
D.G would like to thank the Physics department at the Indian Insititute of Technology Gandhinagar for their hospitality and Tanmoy Sengupta, Sumit Shaw, Ajit C. Balaram, Sashikanta Mohapatra, and Sayan Mukherjee for their useful suggestions and discussions. A.B like to thank the FISPAC Research Group, Department of Physics, University of Murcia, especially, Jose J. Fernández-Melgarejo for hospitality and useful discussion based on the seminar given on this paper. A.B and P.N is supported by Relevant Research Project grant (202011BRE03RP06633-BRNS) by the Board Of Research In Nuclear Sciences (BRNS), Department of Atomic Energy (DAE), India. A.B is supported by Mathematical Research Impact Centric Support Grant (MTR/2021/000490) by the Department of Science and Technology Science and Engineering Research Board (India). A.B and P.N thank the speakers and participants of the workshop “Quantum Information in QFT and AdS/CFT-III” organized at IIT Hyderabad between 16-18th September, 2022 and funded by SERB through a Seminar Symposia (SSY) grant (SSY/2022/000446) and “Quantum Information Theory in Quantum Field Theory and Cosmology” between 4-9th June, 2023 hosted by Banff International Research Centre at Canada for useful discussions. A.B. also acknowledge the associateship program of the Indian Academy of Science, Bengaluru.
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Bhattacharyya, A., Ghosh, D. & Nandi, P. Operator growth and Krylov complexity in Bose-Hubbard model. J. High Energ. Phys. 2023, 112 (2023). https://doi.org/10.1007/JHEP12(2023)112
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DOI: https://doi.org/10.1007/JHEP12(2023)112