Microcanonical windows on quantum operators

Silvia Pappalardi; Laura Foini; Jorge Kurchan

doi:10.22331/q-2024-01-11-1227

Microcanonical windows on quantum operators

Silvia Pappalardi^1,2, Laura Foini³, and Jorge Kurchan¹

¹Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
²Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany
³IPhT, CNRS, CEA, Université Paris Saclay, 91191 Gif-sur-Yvette, France

Published:	2024-01-11, volume 8, page 1227
Eprint:	arXiv:2304.10948v3
Doi:	https://doi.org/10.22331/q-2024-01-11-1227
Citation:	Quantum 8, 1227 (2024).

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Abstract

We discuss the construction of a microcanonical projection WOW of a quantum operator O induced by an energy window filter W, its spectrum, and the retrieval of canonical many-time correlations from it.

Featured image: Pictorial representation of the observable O and of the microcanonical projected WOW.

► BibTeX data

@article{Pappalardi2024microcanonical,
  doi = {10.22331/q-2024-01-11-1227},
  url = {https://doi.org/10.22331/q-2024-01-11-1227},
  title = {Microcanonical windows on quantum operators},
  author = {Pappalardi, Silvia and Foini, Laura and Kurchan, Jorge},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1227},
  month = jan,
  year = {2024}
}

► References

[1] J. M. Deutsch. Quantum statistical mechanics in a closed system. Physical Review A, 43 (4): 2046–2049, February 1991. URL https://doi.org/10.1103/physreva.43.2046.
https://doi.org/10.1103/physreva.43.2046

[2] Mark Srednicki. The approach to thermal equilibrium in quantized chaotic systems. Journal of Physics A: Mathematical and General, 32 (7): 1163–1175, January 1999. URL https://doi.org/10.1088/0305-4470/32/7/007.
https://doi.org/10.1088/0305-4470/32/7/007

[3] Luca D'Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Advances in Physics, 65 (3): 239–362, May 2016. URL https://doi.org/10.1080/00018732.2016.1198134.
https://doi.org/10.1080/00018732.2016.1198134

[4] Laura Foini and Jorge Kurchan. Eigenstate thermalization hypothesis and out of time order correlators. Physical Review E, 99 (4), April 2019. URL https://doi.org/10.1103/physreve.99.042139.
https://doi.org/10.1103/physreve.99.042139

[5] Yan V Fyodorov and Alexander D Mirlin. Scaling properties of localization in random band matrices: a $\sigma$-model approach. Physical review letters, 67 (18): 2405, 1991. URL https://doi.org/10.1103/PhysRevLett.124.120602.
https://doi.org/10.1103/PhysRevLett.124.120602

[6] M Kuś, M Lewenstein, and Fritz Haake. Density of eigenvalues of random band matrices. Physical Review A, 44 (5): 2800, 1991. URL https://doi.org/10.1103/PhysRevA.44.2800.
https://doi.org/10.1103/PhysRevA.44.2800

[7] Ya V Fyodorov, OA Chubykalo, FM Izrailev, and G Casati. Wigner random banded matrices with sparse structure: local spectral density of states. Physical review letters, 76 (10): 1603, 1996. URL https://doi.org/10.1103/PhysRevLett.76.1603.
https://doi.org/10.1103/PhysRevLett.76.1603

[8] Tomaz Prosen. Statistical properties of matrix elements in a hamilton system between integrability and chaos. Annals of Physics, 235 (1): 115–164, 1994. URL https://doi.org/10.1006/aphy.1994.1093.
https://doi.org/10.1006/aphy.1994.1093

[9] Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, and Beni Yoshida. Chaos, complexity, and random matrices. Journal of High Energy Physics, 2017 (11): 1–60, 2017. URL https://doi.org/10.1007/JHEP11(2017)048.
https://doi.org/10.1007/JHEP11(2017)048

[10] Anatoly Dymarsky and Hong Liu. New characteristic of quantum many-body chaotic systems. Phys. Rev. E, 99: 010102, Jan 2019. URL https://doi.org/10.1103/PhysRevE.99.010102.
https://doi.org/10.1103/PhysRevE.99.010102

[11] Anatoly Dymarsky. Mechanism of macroscopic equilibration of isolated quantum systems. Physical Review B, 99 (22): 224302, 2019. URL https://doi.org/10.1103/PhysRevB.99.224302.
https://doi.org/10.1103/PhysRevB.99.224302

[12] Anatoly Dymarsky. Bound on eigenstate thermalization from transport. Phys. Rev. Lett., 128: 190601, May 2022. URL https://doi.org/10.1103/PhysRevLett.128.190601.
https://doi.org/10.1103/PhysRevLett.128.190601

[13] Jonas Richter, Anatoly Dymarsky, Robin Steinigeweg, and Jochen Gemmer. Eigenstate thermalization hypothesis beyond standard indicators: Emergence of random-matrix behavior at small frequencies. Physical Review E, 102 (4), October 2020. URL https://doi.org/10.1103/physreve.102.042127.
https://doi.org/10.1103/physreve.102.042127

[14] Jiaozi Wang, Mats H Lamann, Jonas Richter, Robin Steinigeweg, Anatoly Dymarsky, and Jochen Gemmer. Eigenstate thermalization hypothesis and its deviations from random-matrix theory beyond the thermalization time. Physical Review Letters, 128 (18): 180601, 2022. URL https://doi.org/10.1103/PhysRevLett.128.180601.
https://doi.org/10.1103/PhysRevLett.128.180601

[15] Marlon Brenes, Silvia Pappalardi, Mark T. Mitchison, John Goold, and Alessandro Silva. Out-of-time-order correlations and the fine structure of eigenstate thermalization. Physical Review E, 104 (3), September 2021. URL https://doi.org/10.1103/physreve.104.034120.
https://doi.org/10.1103/physreve.104.034120

[16] Silvia Pappalardi and Jorge Kurchan. Quantum bounds on the generalized lyapunov exponents. Entropy, 25 (2): 246, 2023. URL https://doi.org/10.3390/e25020246.
https://doi.org/10.3390/e25020246

[17] Juan Maldacena, Stephen H. Shenker, and Douglas Stanford. A bound on chaos. Journal of High Energy Physics, 2016 (8), August 2016. URL https://doi.org/10.1007/jhep08(2016)106.
https://doi.org/10.1007/jhep08(2016)106

[18] Felix M Haehl, R Loganayagam, Prithvi Narayan, Amin A Nizami, and Mukund Rangamani. Thermal out-of-time-order correlators, kms relations, and spectral functions. Journal of High Energy Physics, 2017 (12): 1–55, 2017. URL https://doi.org/10.1007/jhep12(2017)154.
https://doi.org/10.1007/jhep12(2017)154

[19] Naoto Tsuji, Tomohiro Shitara, and Masahito Ueda. Bound on the exponential growth rate of out-of-time-ordered correlators. Physical Review E, 98 (1), July 2018. URL https://doi.org/10.1103/physreve.98.012216.
https://doi.org/10.1103/physreve.98.012216

[20] Silvia Pappalardi, Laura Foini, and Jorge Kurchan. Quantum bounds and fluctuation-dissipation relations. SciPost Physics, 12 (4), April 2022a. URL https://doi.org/10.21468/scipostphys.12.4.130.
https://doi.org/10.21468/scipostphys.12.4.130

[21] Silvia Pappalardi, Laura Foini, and Jorge Kurchan. Eigenstate thermalization hypothesis and free probability. Phys. Rev. Lett., 129: 170603, Oct 2022b. URL https://doi.org/10.1103/PhysRevLett.129.170603.
https://doi.org/10.1103/PhysRevLett.129.170603

[22] James A Mingo and Roland Speicher. Free probability and random matrices, volume 35. Springer, 2017. URL https://doi.org/10.1007/978-1-4939-6942-5.
https://doi.org/10.1007/978-1-4939-6942-5

[23] Tarek A. Elsayed, Benjamin Hess, and Boris V. Fine. Signatures of chaos in time series generated by many-spin systems at high temperatures. Phys. Rev. E, 90: 022910, Aug 2014. URL https://doi.org/10.1103/PhysRevE.90.022910.
https://doi.org/10.1103/PhysRevE.90.022910

[24] Daniel E Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, and Ehud Altman. A universal operator growth hypothesis. Physical Review X, 9 (4): 041017, 2019. URL.
https://doi.org/10.1103/PhysRevX.9.0410177

[25] Alexander Avdoshkin and Anatoly Dymarsky. Euclidean operator growth and quantum chaos. Physical Review Research, 2 (4): 043234, 2020. URL https://doi.org/10.1103/PhysRevResearch.2.043234.
https://doi.org/10.1103/PhysRevResearch.2.043234

[26] Chaitanya Murthy and Mark Srednicki. Bounds on chaos from the eigenstate thermalization hypothesis. Physical Review Letters, 123 (23), December 2019. URL https://doi.org/10.1103/physrevlett.123.230606.
https://doi.org/10.1103/physrevlett.123.230606

[27] Sirui Lu, Mari Carmen Bañuls, and J. Ignacio Cirac. Algorithms for quantum simulation at finite energies. PRX Quantum, 2: 020321, May 2021. URL https://doi.org/10.1103/PRXQuantum.2.020321.
https://doi.org/10.1103/PRXQuantum.2.020321

[28] Yilun Yang, J. Ignacio Cirac, and Mari Carmen Bañuls. Classical algorithms for many-body quantum systems at finite energies. Phys. Rev. B, 106: 024307, Jul 2022. URL https://doi.org/10.1103/PhysRevB.106.024307.
https://doi.org/10.1103/PhysRevB.106.024307

[29] Ehsan Khatami, Guido Pupillo, Mark Srednicki, and Marcos Rigol. Fluctuation-dissipation theorem in an isolated system of quantum dipolar bosons after a quench. Physical Review Letters, 111 (5), July 2013. URL https://doi.org/10.1103/physrevlett.111.050403.
https://doi.org/10.1103/physrevlett.111.050403

[30] MW Long, P Prelovšek, S El Shawish, J Karadamoglou, and X Zotos. Finite-temperature dynamical correlations using the microcanonical ensemble and the lanczos algorithm. Physical Review B, 68 (23): 235106, 2003. URL https://doi.org/10.1103/PhysRevB.68.235106.
https://doi.org/10.1103/PhysRevB.68.235106

[31] Xenophon Zotos. Microcanonical lanczos method. Philosophical Magazine, 86 (17-18): 2591–2601, 2006. URL https://doi.org/10.1080/14786430500227830.
https://doi.org/10.1080/14786430500227830

[32] Satoshi Okamoto, Gonzalo Alvarez, Elbio Dagotto, and Takami Tohyama. Accuracy of the microcanonical lanczos method to compute real-frequency dynamical spectral functions of quantum models at finite temperatures. Physical Review E, 97 (4): 043308, 2018. URL https://doi.org/10.1103/PhysRevE.97.043308.
https://doi.org/10.1103/PhysRevE.97.043308

[33] Marcos Rigol, Vanja Dunjko, and Maxim Olshanii. Thermalization and its mechanism for generic isolated quantum systems. Nature, 452 (7189): 854–858, April 2008. URL https://doi.org/10.1038/nature06838.
https://doi.org/10.1038/nature06838

[34] Peter Reimann. Typical fast thermalization processes in closed many-body systems. Nature communications, 7 (1): 1–10, 2016. URL https://doi.org/10.1038/ncomms10821.
https://doi.org/10.1038/ncomms10821

[35] Dieter Forster. Hydrodynamic fluctuations, broken symmetry, and correlation functions. CRC Press, 2018. URL https://doi.org/10.1201/9780429493683.
https://doi.org/10.1201/9780429493683

[36] Roland Speicher. Free probability theory and non-crossing partitions. Séminaire Lotharingien de Combinatoire [electronic only], 39: B39c–38, 1997. URL http://eudml.org/doc/119380.
http://eudml.org/doc/119380

[37] Kurusch Ebrahimi-Fard and Frédéric Patras. The combinatorics of green’s functions in planar field theories. Frontiers of Physics, 11: 1–23, 2016. URL https://doi.org/10.1007/s11467-016-0585-2.
https://doi.org/10.1007/s11467-016-0585-2

[38] Ludwig Hruza and Denis Bernard. Coherent fluctuations in noisy mesoscopic systems, the open quantum ssep, and free probability. Phys. Rev. X, 13: 011045, Mar 2023. URL https://doi.org/10.1103/PhysRevX.13.011045.
https://doi.org/10.1103/PhysRevX.13.011045

[39] Joël Bun, Jean-Philippe Bouchaud, and Marc Potters. Cleaning large correlation matrices: tools from random matrix theory. Physics Reports, 666: 1–109, 2017. URL https://doi.org/10.1016/j.physrep.2016.10.005.
https://doi.org/10.1016/j.physrep.2016.10.005

[40] Felix Fritzsch and Tomaž Prosen. Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions. Phys. Rev. E, 103: 062133, Jun 2021. URL https://doi.org/10.1103/PhysRevE.103.062133.
https://doi.org/10.1103/PhysRevE.103.062133

[41] Silvia Pappalardi, Felix Fritzsch, and Tomaž Prosen. General eigenstate thermalization via free cumulants in quantum lattice systems. arXiv preprint arXiv:2303.00713, 2023. URL https://doi.org/10.48550/arXiv.2303.00713.
https://doi.org/10.48550/arXiv.2303.00713
arXiv:2303.00713

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