{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T00:43:39Z","timestamp":1777941819810,"version":"3.51.4"},"reference-count":62,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Marie Sk\u0142odowska Curie Action GETQuantum","award":["101146632"],"award-info":[{"award-number":["101146632"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>The eigenstate thermalization hypothesis (ETH) is the leading interpretation in our current understanding of quantum thermalization. Recent results uncovered strong connections between quantum correlations in thermalizing systems and the structure of free probability theory, leading to the notion of full ETH. However, most studies have been performed for ergodic systems and it is still unclear whether or how full ETH manifests in ergodicity-breaking models. We fill this gap by studying standard probes of full ETH in ergodicity-breaking quantum circuits, presenting numerical and analytical results for interacting integrable systems. These probes can display distinct behavior and undergo a different scaling than the ones observed in ergodic systems. For the analytical results we consider an interacting integrable dual-unitary model and present the exact eigenstates, allowing us to analytically express common probes for full ETH. We discuss the underlying mechanisms responsible for these differences and show how the presence of solitons dictates the behavior of ETH-related quantities in the dual-unitary model. We show numerical evidence that this behavior is sufficiently generic away from dual-unitarity when restricted to the appropriate symmetry sectors.<\/jats:p>","DOI":"10.22331\/q-2025-12-15-1949","type":"journal-article","created":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T15:11:48Z","timestamp":1765811508000},"page":"1949","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Probes of Full Eigenstate Thermalization in Ergodicity-Breaking Quantum Circuits"],"prefix":"10.22331","volume":"9","author":[{"given":"Gabriel O.","family":"Alves","sequence":"first","affiliation":[{"name":"Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Felix","family":"Fritzsch","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pieter W.","family":"Claeys","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,12,15]]},"reference":[{"key":"0","unstructured":"H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1985)."},{"key":"1","doi-asserted-by":"publisher","unstructured":"H. Tasaki, Phys. Rev. Lett. 80, 1373 (1998).","DOI":"10.1103\/PhysRevLett.80.1373"},{"key":"2","doi-asserted-by":"publisher","unstructured":"A. Nazir and G. Schaller, Thermodynamics in the Quantum Regime, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Vol. 195 (Springer International Publishing, Cham, 2018) 1805.08307.","DOI":"10.1007\/978-3-319-99046-0"},{"key":"3","doi-asserted-by":"publisher","unstructured":"J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).","DOI":"10.1103\/physreva.43.2046"},{"key":"4","doi-asserted-by":"publisher","unstructured":"M. Srednicki, Phys. Rev. E 50, 888 (1994).","DOI":"10.1103\/PhysRevE.50.888"},{"key":"5","doi-asserted-by":"publisher","unstructured":"L. D&apos;Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016).","DOI":"10.1080\/00018732.2016.1198134"},{"key":"6","doi-asserted-by":"publisher","unstructured":"J. M. Deutsch, Rep. Prog. Phys. 81, 082001 (2018).","DOI":"10.1088\/1361-6633\/aac9f1"},{"key":"7","doi-asserted-by":"publisher","unstructured":"L. Foini and J. Kurchan, Phys. Rev. E 99, 042139 (2019).","DOI":"10.1103\/PhysRevE.99.042139"},{"key":"8","doi-asserted-by":"publisher","unstructured":"S. Pappalardi, L. Foini, and J. Kurchan, Phys. Rev. Lett. 129, 170603 (2022).","DOI":"10.1103\/PhysRevLett.129.170603"},{"key":"9","doi-asserted-by":"publisher","unstructured":"D. Voiculescu, in Operator Algebras and Their Connections with Topology and Ergodic Theory, edited by H. Araki, C. C. Moore, \u015e.-V. Stratila, and D.-V. Voiculescu (Springer, Berlin, Heidelberg, 1985) pp. 556\u2013588.","DOI":"10.1007\/BFb0074909"},{"key":"10","doi-asserted-by":"publisher","unstructured":"D. Voiculescu, J. Funct. Anal. 66, 323 (1986).","DOI":"10.1016\/0022-1236(86)90062-5"},{"key":"11","doi-asserted-by":"crossref","unstructured":"D. Voiculescu, J. Operat. Theor. 18, 223 (1987).","DOI":"10.1016\/0020-1383(87)90001-5"},{"key":"12","doi-asserted-by":"publisher","unstructured":"J. Novak and M. LaCroix, Three lectures on free probability (2012), arXiv:1205.2097.","DOI":"10.48550\/arXiv.1205.2097"},{"key":"13","doi-asserted-by":"publisher","unstructured":"J. A. Mingo and R. Speicher, Free Probability and Random Matrices, Fields Institute Monographs, Vol. 35 (Springer New York, New York, NY, 2017).","DOI":"10.1007\/978-1-4939-6942-5"},{"key":"14","doi-asserted-by":"publisher","unstructured":"S. Pappalardi, F. Fritzsch, and T. Prosen, Phys. Rev. Lett. 134, 140404 (2025).","DOI":"10.1103\/PhysRevLett.134.140404"},{"key":"15","doi-asserted-by":"publisher","unstructured":"S. Pappalardi, L. Foini, and J. Kurchan, Quantum 8, 1227 (2024).","DOI":"10.22331\/q-2024-01-11-1227"},{"key":"16","doi-asserted-by":"publisher","unstructured":"S. Jindal and P. Hosur, J. High Energ. Phys. 2024 (66), 66.","DOI":"10.1007\/JHEP09(2024)066"},{"key":"17","doi-asserted-by":"publisher","unstructured":"F. Fritzsch, T. Prosen, and S. Pappalardi, Phys. Rev. B 111, 054303 (2025).","DOI":"10.1103\/PhysRevB.111.054303"},{"key":"18","doi-asserted-by":"publisher","unstructured":"E. Vallini and S. Pappalardi, Long-time Freeness in the Kicked Top (2024), arXiv:2411.12050.","DOI":"10.48550\/arXiv.2411.12050"},{"key":"19","doi-asserted-by":"publisher","unstructured":"H. J. Chen and J. Kudler-Flam, Phys. Rev. B 111, 014311 (2025).","DOI":"10.1103\/PhysRevB.111.014311"},{"key":"20","doi-asserted-by":"publisher","unstructured":"H. A. Camargo, Y. Fu, V. Jahnke, K. Pal, and K.-Y. Kim, Quantum Signatures of Chaos from Free Probability (2025), arxiv:2503.20338.","DOI":"10.48550\/arXiv.2503.20338"},{"key":"21","doi-asserted-by":"publisher","unstructured":"M. Fava, J. Kurchan, and S. Pappalardi, Phys. Rev. X 15, 011031 (2025).","DOI":"10.1103\/PhysRevX.15.011031"},{"key":"22","doi-asserted-by":"publisher","unstructured":"S. Gopalakrishnan and A. Lamacraft, Phys. Rev. B 100, 064309 (2019).","DOI":"10.1103\/PhysRevB.100.064309"},{"key":"23","doi-asserted-by":"publisher","unstructured":"B. Bertini, P. Kos, and T. Prosen, Phys. Rev. Lett. 123, 210601 (2019).","DOI":"10.1103\/PhysRevLett.123.210601"},{"key":"24","doi-asserted-by":"publisher","unstructured":"B. Bertini, P. Kos, and T. Prosen, SciPost Phys. 8, 068 (2020).","DOI":"10.21468\/SciPostPhys.8.4.068"},{"key":"25","doi-asserted-by":"publisher","unstructured":"T. Gombor and B. Pozsgay, SciPost Phys. 12, 102 (2022).","DOI":"10.21468\/SciPostPhys.12.3.102"},{"key":"26","doi-asserted-by":"publisher","unstructured":"T. Holden-Dye, L. Masanes, and A. Pal, Quantum 9, 1615 (2025).","DOI":"10.22331\/q-2025-01-30-1615"},{"key":"27","doi-asserted-by":"publisher","unstructured":"G. M. Sommers, D. A. Huse, and M. J. Gullans, PRX Quantum 4, 030313 (2023).","DOI":"10.1103\/PRXQuantum.4.030313"},{"key":"28","doi-asserted-by":"publisher","unstructured":"P. W. Claeys and A. Lamacraft, J. Phys. A: Math. Theor. 57, 405301 (2024).","DOI":"10.1088\/1751-8121\/ad776a"},{"key":"29","doi-asserted-by":"publisher","unstructured":"A. Foligno, P. Calabrese, and B. Bertini, PRX Quantum 6, 010324 (2025).","DOI":"10.1103\/PRXQuantum.6.010324"},{"key":"30","doi-asserted-by":"publisher","unstructured":"N. Dowling, P. Kos, and K. Modi, Phys. Rev. Lett. 131, 180403 (2023).","DOI":"10.1103\/PhysRevLett.131.180403"},{"key":"31","doi-asserted-by":"publisher","unstructured":"M. A. Rampp and P. W. Claeys, Quantum 8, 1434 (2024).","DOI":"10.22331\/q-2024-08-08-1434"},{"key":"32","doi-asserted-by":"publisher","unstructured":"G. Giudice, G. Giudici, M. Sonner, J. Thoenniss, A. Lerose, D. A. Abanin, and L. Piroli, Phys. Rev. Lett. 128, 220401 (2022).","DOI":"10.1103\/PhysRevLett.128.220401"},{"key":"33","doi-asserted-by":"publisher","unstructured":"J. A. Monta\u00f1\u00e0 L\u00f3pez and P. Kos, J. Phys. A: Math. Theor. 57, 475301 (2024).","DOI":"10.1088\/1751-8121\/ad85b0"},{"key":"34","doi-asserted-by":"publisher","unstructured":"A. Chan, A. De Luca, and J. T. Chalker, Phys. Rev. Lett. 122, 220601 (2019).","DOI":"10.1103\/PhysRevLett.122.220601"},{"key":"35","doi-asserted-by":"publisher","unstructured":"D. Hahn, D. J. Luitz, and J. T. Chalker, Phys. Rev. X 14, 031029 (2024).","DOI":"10.1103\/PhysRevX.14.031029"},{"key":"36","doi-asserted-by":"publisher","unstructured":"R. Speicher, Math. Ann. 298, 611 (1994).","DOI":"10.1007\/BF01459754"},{"key":"37","doi-asserted-by":"publisher","unstructured":"M. Vanicat, L. Zadnik, and T. Prosen, Phys. Rev. Lett. 121, 030606 (2018).","DOI":"10.1103\/PhysRevLett.121.030606"},{"key":"38","doi-asserted-by":"publisher","unstructured":"M. Ljubotina, L. Zadnik, and T. Prosen, Phys. Rev. Lett. 122, 150605 (2019).","DOI":"10.1103\/PhysRevLett.122.150605"},{"key":"39","doi-asserted-by":"publisher","unstructured":"A. J. Friedman, A. Chan, A. De Luca, and J. T. Chalker, Phys. Rev. Lett. 123, 210603 (2019).","DOI":"10.1103\/PhysRevLett.123.210603"},{"key":"40","doi-asserted-by":"publisher","unstructured":"\u017d. Krajnik, E. Ilievski, and T. Prosen, SciPost Phys. 9, 038 (2020).","DOI":"10.21468\/SciPostPhys.9.3.038"},{"key":"41","doi-asserted-by":"publisher","unstructured":"P. W. Claeys, J. Herzog-Arbeitman, and A. Lamacraft, SciPost Phys. 12, 007 (2022).","DOI":"10.21468\/SciPostPhys.12.1.007"},{"key":"42","doi-asserted-by":"publisher","unstructured":"R. Suzuki, K. Mitarai, and K. Fujii, Quantum 6, 631 (2022).","DOI":"10.22331\/q-2022-01-24-631"},{"key":"43","doi-asserted-by":"publisher","unstructured":"T. Gombor and B. Pozsgay, SciPost Phys. 16, 114 (2024).","DOI":"10.21468\/SciPostPhys.16.4.114"},{"key":"44","doi-asserted-by":"publisher","unstructured":"E. Ilievski, E. Quinn, J. De Nardis, and M. Brockmann, J. Stat. Mech. 2016, 063101 (2016).","DOI":"10.1088\/1742-5468\/2016\/06\/063101"},{"key":"45","doi-asserted-by":"publisher","unstructured":"S. Gopalakrishnan and B. Zakirov, Quantum Sci. Technol. 3, 044004 (2018).","DOI":"10.1088\/2058-9565\/aad759"},{"key":"46","doi-asserted-by":"publisher","unstructured":"A. Maillard, L. Foini, A. L. Castellanos, F. Krzakala, M. M\u00e9zard, and L. Zdeborov\u00e1, J. Stat. Mech. , 113301 (2019).","DOI":"10.1088\/1742-5468\/ab4bbb"},{"key":"47","doi-asserted-by":"publisher","unstructured":"T. LeBlond, K. Mallayya, L. Vidmar, and M. Rigol, Phys. Rev. E 100, 062134 (2019).","DOI":"10.1103\/PhysRevE.100.062134"},{"key":"48","doi-asserted-by":"publisher","unstructured":"T. LeBlond and M. Rigol, Phys. Rev. E 102, 062113 (2020).","DOI":"10.1103\/PhysRevE.102.062113"},{"key":"49","doi-asserted-by":"publisher","unstructured":"F. H. L. Essler and A. J. J. M. de Klerk, Phys. Rev. X 14, 031048 (2024).","DOI":"10.1103\/PhysRevX.14.031048"},{"key":"50","doi-asserted-by":"publisher","unstructured":"M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Phys. Rev. X 10, 041017 (2020).","DOI":"10.1103\/PhysRevX.10.041017"},{"key":"51","doi-asserted-by":"publisher","unstructured":"M. Brenes, J. Goold, and M. Rigol, Phys. Rev. B 102, 075127 (2020).","DOI":"10.1103\/PhysRevB.102.075127"},{"key":"52","doi-asserted-by":"publisher","unstructured":"P. W. Claeys and A. Lamacraft, Phys. Rev. Lett. 126, 100603 (2021).","DOI":"10.1103\/PhysRevLett.126.100603"},{"key":"53","doi-asserted-by":"publisher","unstructured":"M. A. Rampp, R. Moessner, and P. W. Claeys, Phys. Rev. Lett. 130, 130402 (2023).","DOI":"10.1103\/PhysRevLett.130.130402"},{"key":"54","doi-asserted-by":"publisher","unstructured":"G. O. Alves, F. Fritzsch, and P. W. Claeys, Alves-gabriel\/dual_unitary_eth: First release, Zenodo (2025).","DOI":"10.5281\/ZENODO.15576372"},{"key":"55","doi-asserted-by":"publisher","unstructured":"J. B. Carrell, in Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear Algebra, edited by J. B. Carrell (Springer, New York, NY, 2017) pp. 337\u2013382.","DOI":"10.1007\/978-0-387-79428-0_11"},{"key":"56","doi-asserted-by":"publisher","unstructured":"B. Collins and P. \u015aniady, Commun. Math. Phys. 264, 773 (2006).","DOI":"10.1007\/s00220-006-1554-3"},{"key":"57","doi-asserted-by":"publisher","unstructured":"H. Fredricksen and J. Maiorana, Universitext. 23, 207 (1978).","DOI":"10.1016\/0012-365X(78)90002-X"},{"key":"58","doi-asserted-by":"publisher","unstructured":"H. Fredricksen and I. J. Kessler, Universitext. 61, 181 (1986).","DOI":"10.1016\/0012-365X(86)90089-0"},{"key":"59","doi-asserted-by":"publisher","unstructured":"J. Sawada, SIAM J. Comput. 31, 259 (2001).","DOI":"10.1137\/S0097539700377037"},{"key":"60","doi-asserted-by":"publisher","unstructured":"F. Ruskey, C. Savage, and T. Min Yih Wang, J. Algorithm. 13, 414 (1992).","DOI":"10.1016\/0196-6774(92)90047-G"},{"key":"61","doi-asserted-by":"publisher","unstructured":"R. Speicher, Lecture Notes on \"Free Probability Theory\" (2019), arxiv:1908.08125.","DOI":"10.48550\/arXiv.1908.08125"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-12-15-1949\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T15:12:15Z","timestamp":1765811535000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-12-15-1949\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,15]]},"references-count":62,"URL":"https:\/\/doi.org\/10.22331\/q-2025-12-15-1949","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,15]]},"article-number":"1949"}}