Entanglement of Disjoint Intervals in Dual-Unitary Circuits: Exact Results

Alessandro Foligno; Bruno Bertini

doi:10.22331/q-2025-03-26-1678

Entanglement of Disjoint Intervals in Dual-Unitary Circuits: Exact Results

Alessandro Foligno^1,2 and Bruno Bertini³

¹School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK
²Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK
³School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

Published:	2025-03-26, volume 9, page 1678
Editor:	Álvaro Alhambra
Eprint:	arXiv:2408.16750v3
Doi:	https://doi.org/10.22331/q-2025-03-26-1678
Citation:	Quantum 9, 1678 (2025).

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Abstract

The growth of the entanglement between two disjoint intervals and its complement after a quantum quench is regarded as a dynamical chaos indicator. Namely, it is expected to show qualitatively different behaviours depending on whether the underlying microscopic dynamics is chaotic or integrable. So far, however, this could only be verified in the context of conformal field theories. Here we present an exact confirmation of this expectation in a class of interacting microscopic Floquet systems on the lattice, i.e., dual-unitary circuits. These systems can either have $zero$ or a $\textit{super extensive}$ number of conserved charges: the latter case is achieved via fine-tuning. We show that, for $almost$ all dual unitary circuits on qubits and for a large family of dual-unitary circuits on qudits the asymptotic entanglement dynamics agrees with what is expected for chaotic systems. On the other hand, if we require the systems to have conserved charges, we find that the entanglement displays the qualitatively different behaviour expected for integrable systems. Interestingly, despite having many conserved charges, charge-conserving dual-unitary circuits are in general not Yang-Baxter integrable.

Featured image: Entanglement between a disjoint interval and the rest of the system after a quantum quench.

Popular summary

Despite its immense complexity in terms of microscopic components, matter at equilibrium is governed by a simple set laws: the glorious laws of thermodynamics. Developing the connection between these and the laws governing the microscopic dynamics has been one of the crowing achievements of 19th century’s physics. In the same way, a major quest keeping busy the theoretical physicists of our times is to deduce the macroscopic laws describing matter — especially quantum matter — when instead is away from an equilibrium state.

In this context, great attention is devoted to the phenomena that are “universal”, i.e., show very little dependence on the underlying microscopic dynamics, as they should be amenable to a simple “macroscopic” description. A prominent example of those is the way in which quantum correlations — entanglement — spread through a many body system. For example, considering a quantum many-body system prepared in a non-equilibrium state with no entanglement, one observes that during the evolution the entanglement between a connected subsystem and the rest grows linearly until it saturates signalling that the subsystem has reached an equilibrium state. In fact, the final value of the entanglement is related to the thermodynamic entropy of the stationary state of the subsystem.

In spite of this universality, however, the mechanisms governing the dynamics of entanglement are expected to depend on the nature of the microscopic dynamics and to be described by different phenomenological theories depending on whether the system is integrable or chaotic. This has led to the identification of a simple setting where a qualitative change should emerge: the evolution of the entanglement between a disjoint subsystem and the rest. Up to now, however, no exact result in clean, microscopic systems in the presence of interactions existed to prove or disprove these expectations. In this paper we fill this gap and present a rigorous proof of the occurrence of these different behaviours for dual-unitary circuits.

► BibTeX data

@article{Foligno2025entanglementof,
  doi = {10.22331/q-2025-03-26-1678},
  url = {https://doi.org/10.22331/q-2025-03-26-1678},
  title = {Entanglement of {D}isjoint {I}ntervals in {D}ual-{U}nitary {C}ircuits: {E}xact {R}esults},
  author = {Foligno, Alessandro and Bertini, Bruno},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1678},
  month = mar,
  year = {2025}
}

► References

[1] Pasquale Calabrese and John Cardy. ``Evolution of entanglement entropy in one-dimensional systems''. J. Stat. Mech.: Theory Exp. 2005, P04010 (2005).
https://doi.org/10.1088/1742-5468/2005/04/p04010

[2] Maurizio Fagotti and Pasquale Calabrese. ``Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field''. Phys. Rev. A 78, 010306 (2008).
https://doi.org/10.1103/PhysRevA.78.010306

[3] Vincenzo Alba and Pasquale Calabrese. ``Entanglement dynamics after quantum quenches in generic integrable systems''. SciPost Phys. 4, 17 (2018).
https://doi.org/10.21468/SciPostPhys.4.3.017

[4] Vincenzo Alba and Pasquale Calabrese. ``Entanglement and thermodynamics after a quantum quench in integrable systems''. Proc. Natl. Acad. Sci. U.S.A. 114, 7947–7951 (2017).
https://doi.org/10.1073/pnas.1703516114

[5] Gianluca Lagnese, Pasquale Calabrese, and Lorenzo Piroli. ``Entanglement dynamics of thermofield double states in integrable models''. J. Phys. A: Math. Theor. 55, 214003 (2022).
https://doi.org/10.1088/1751-8121/ac646b

[6] Hong Liu and S. Josephine Suh. ``Entanglement tsunami: Universal scaling in holographic thermalization''. Phys. Rev. Lett. 112, 011601 (2014).
https://doi.org/10.1103/PhysRevLett.112.011601

[7] Curtis T Asplund, Alice Bernamonti, Federico Galli, and Thomas Hartman. ``Entanglement scrambling in 2d conformal field theory''. J. High Energy Phys. 2015, 110 (2015).
https://doi.org/10.1007/JHEP09(2015)110

[8] Andreas M Läuchli and Corinna Kollath. ``Spreading of correlations and entanglement after a quench in the one-dimensional bose–hubbard model''. J. Stat. Mech.: Theory Exp. 2008, P05018 (2008).
https://doi.org/10.1088/1742-5468/2008/05/p05018

[9] Hyungwon Kim and David A. Huse. ``Ballistic spreading of entanglement in a diffusive nonintegrable system''. Phys. Rev. Lett. 111, 127205 (2013).
https://doi.org/10.1103/PhysRevLett.111.127205

[10] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. ``Quantum entanglement growth under random unitary dynamics''. Phys. Rev. X 7, 031016 (2017).
https://doi.org/10.1103/PhysRevX.7.031016

[11] Adam Nahum, Sagar Vijay, and Jeongwan Haah. ``Operator spreading in random unitary circuits''. Phys. Rev. X 8, 021014 (2018).
https://doi.org/10.1103/PhysRevX.8.021014

[12] C. W. von Keyserlingk, Tibor Rakovszky, Frank Pollmann, and S. L. Sondhi. ``Operator Hydrodynamics, OTOCs, and Entanglement Growth in Systems without Conservation Laws''. Phys. Rev. X 8, 021013 (2018).
https://doi.org/10.1103/PhysRevX.8.021013

[13] Rajarshi Pal and Arul Lakshminarayan. ``Entangling power of time-evolution operators in integrable and nonintegrable many-body systems''. Phys. Rev. B 98, 174304 (2018).
https://doi.org/10.1103/PhysRevB.98.174304

[14] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Entanglement spreading in a minimal model of maximal many-body quantum chaos''. Phys. Rev. X 9, 021033 (2019).
https://doi.org/10.1103/PhysRevX.9.021033

[15] Lorenzo Piroli, Bruno Bertini, J. Ignacio Cirac, and Tomaž Prosen. ``Exact dynamics in dual-unitary quantum circuits''. Phys. Rev. B 101, 094304 (2020).
https://doi.org/10.1103/PhysRevB.101.094304

[16] Sarang Gopalakrishnan and Austen Lamacraft. ``Unitary circuits of finite depth and infinite width from quantum channels''. Phys. Rev. B 100, 064309 (2019).
https://doi.org/10.1103/PhysRevB.100.064309

[17] Bruno Bertini, Katja Klobas, Vincenzo Alba, Gianluca Lagnese, and Pasquale Calabrese. ``Growth of rényi entropies in interacting integrable models and the breakdown of the quasiparticle picture''. Phys. Rev. X 12, 031016 (2022).
https://doi.org/10.1103/PhysRevX.12.031016

[18] Cheryne Jonay, David A. Huse, and Adam Nahum. ``Coarse-grained dynamics of operator and state entanglement''. (2018) arXiv:1803.00089.
arXiv:1803.00089

[19] Tianci Zhou and Adam Nahum. ``Entanglement membrane in chaotic many-body systems''. Phys. Rev. X 10, 031066 (2020).
https://doi.org/10.1103/PhysRevX.10.031066

[20] Vincenzo Alba and Pasquale Calabrese. ``Quantum information scrambling after a quantum quench''. Phys. Rev. B 100, 115150 (2019).
https://doi.org/10.1103/PhysRevB.100.115150

[21] Pasquale Calabrese, John Cardy, and Erik Tonni. ``Entanglement entropy of two disjoint intervals in conformal field theory''. J. Stat. Mech.: Theory Exp. 2009, P11001 (2009).
https://doi.org/10.1088/1742-5468/2009/11/P11001

[22] Maurizio Fagotti and Pasquale Calabrese. ``Entanglement entropy of two disjoint blocks in xy chains''. J. Stat. Mech.: Theory Exp. 2010, P04016 (2010).
https://doi.org/10.1088/1742-5468/2010/04/P04016

[23] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Exact correlation functions for dual-unitary lattice models in $1+1$ dimensions''. Phys. Rev. Lett. 123, 210601 (2019).
https://doi.org/10.1103/PhysRevLett.123.210601

[24] Tomaž Prosen. ``Many-body quantum chaos and dual-unitarity round-a-face''. Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 093101 (2021). arXiv:https://doi.org/10.1063/5.0056970.
https://doi.org/10.1063/5.0056970
arXiv:https://doi.org/10.1063/5.0056970

[25] Pavel Kos, Bruno Bertini, and Tomaž Prosen. ``Correlations in perturbed dual-unitary circuits: Efficient path-integral formula''. Phys. Rev. X 11, 011022 (2021).
https://doi.org/10.1103/PhysRevX.11.011022

[26] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Exact spectral statistics in strongly localized circuits''. Phys. Rev. B 105, 165142 (2022).
https://doi.org/10.1103/PhysRevB.105.165142

[27] Pavel Kos and Georgios Styliaris. ``Circuits of space and time quantum channels''. Quantum 7, 1020 (2023).
https://doi.org/10.22331/q-2023-05-24-1020

[28] Pieter W Claeys and Austen Lamacraft. ``Ergodic and non-ergodic dual-unitary quantum circuits with arbitrary local Hilbert space dimension''. Phys. Rev. Lett. 126, 100603 (2021).
https://doi.org/10.1103/PhysRevLett.126.100603

[29] Bruno Bertini, Pavel Kos, and Tomaž Prosen. ``Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits''. SciPost Phys. 8, 67 (2020).
https://doi.org/10.21468/SciPostPhys.8.4.067

[30] Bruno Bertini, Pavel Kos, and Tomaz Prosen. ``Operator entanglement in local quantum circuits ii: Solitons in chains of qubits''. SciPost Physics 8, 68 (2020).
https://doi.org/10.21468/scipostphys.8.4.068

[31] Tom Holden-Dye, Lluis Masanes, and Arijeet Pal. ``Fundamental charges for dual-unitary circuits'' (2023). arXiv:2312.14148.
https://doi.org/10.22331/q-2025-01-30-1615
arXiv:2312.14148

[32] Alessandro Foligno, Pasquale Calabrese, and Bruno Bertini. ``Nonequilibrium dynamics of charged dual-unitary circuits''. PRX Quantum 6, 010324 (2025).
https://doi.org/10.1103/PRXQuantum.6.010324

[33] S. Aravinda, Suhail Ahmad Rather, and Arul Lakshminarayan. ``From dual-unitary to quantum bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy''. Phys. Rev. Res. 3, 043034 (2021).
https://doi.org/10.1103/PhysRevResearch.3.043034

[34] Suhail Ahmad Rather, S. Aravinda, and Arul Lakshminarayan. ``Creating ensembles of dual unitary and maximally entangling quantum evolutions''. Phys. Rev. Lett. 125, 070501 (2020).
https://doi.org/10.1103/PhysRevLett.125.070501

[35] Tomaž Prosen. ``Many-body quantum chaos and dual-unitarity round-a-face''. Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 093101 (2021).
https://doi.org/10.1063/5.0056970

[36] Márton Borsi and Balázs Pozsgay. ``Construction and the ergodicity properties of dual unitary quantum circuits''. Phys. Rev. B 106, 014302 (2022).
https://doi.org/10.1103/PhysRevB.106.014302

[37] David B. A. Epstein. ``Almost all subgroups of a lie group are free''. Journal of Algebra 19, 261–262 (1971). url: https://api.semanticscholar.org/CorpusID:121800635.
https://api.semanticscholar.org/CorpusID:121800635

[38] W. Magnus, A. Karrass, and D. Solitar. ``Combinatorial group theory: Presentations of groups in terms of generators and relations''. Dover books on mathematics. Dover Publications. (2004). url: https://books.google.co.uk/books?id=1LW4s1RDRHQC.
https://books.google.co.uk/books?id=1LW4s1RDRHQC

[39] F.R. Gantmacher. ``The theory of matrices, vol ii''. Chelsea Publishing Company.

[40] M.A. Nielsen and I.L. Chuang. ``Quantum computation and quantum information: 10th anniversary edition''. Cambridge University Press. (2010). url: https://books.google.co.uk/books?id=-s4DEy7o-a0C.
https://books.google.co.uk/books?id=-s4DEy7o-a0C

[41] Pieter W Claeys and Austen Lamacraft. ``Operator dynamics and entanglement in space-time dual hadamard lattices''. Journal of Physics A: Mathematical and Theoretical 57, 405301 (2024).
https://doi.org/10.1088/1751-8121/ad776a

[42] Tibor Rakovszky, Frank Pollmann, and C. W. von Keyserlingk. ``Sub-ballistic growth of rényi entropies due to diffusion''. Phys. Rev. Lett. 122, 250602 (2019).
https://doi.org/10.1103/PhysRevLett.122.250602

[43] Yichen Huang. ``Dynamics of Rényi entanglement entropy in diffusive qudit systems''. IOP SciNotes 1, 035205 (2020).
https://doi.org/10.1088/2633-1357/abd1e2

Cited by

[1] Bruno Bertini, Pieter W. Claeys, and Tomaž Prosen, "Exactly solvable quantum many-body dynamics from space-time duality", Reviews of Modern Physics 98 2, 025001 (2026).

[2] Shachar Fraenkel and Colin Rylands, "Entanglement in dual unitary quantum circuits with impurities", Physical Review B 112 6, 064305 (2025).

[3] Bruno Bertini, "Non-equilibrium quantum many-body physics with quantum circuits", SciPost Physics Lecture Notes 124 (2026).

[4] Konstantinos Chalas, Pasquale Calabrese, and Colin Rylands, "Quench dynamics of entanglement from crosscap states", SciPost Physics 19 5, 132 (2025).

[5] Vincenzo Alba, "More on the operator space entanglement (OSE): Rényi OSE, revivals, and integrability breaking", Journal of Physics A Mathematical General 58 17, 175003 (2025).

[6] Ali Mollabashi and Mohammad-Javad Vasli, "Scrambling Without Chaos in Random Free-Fermionic Systems", arXiv:2510.21217, (2025).

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