Symmetry resolved out-of-time-order correlators of Heisenberg spin chains using projected matrix product operators

Martina Gisti; David J. Luitz; Maxime Debertolis

doi:10.22331/q-2025-10-01-1871

Symmetry resolved out-of-time-order correlators of Heisenberg spin chains using projected matrix product operators

Martina Gisti, David J. Luitz, and Maxime Debertolis

Institute of Physics, University of Bonn, Nußallee 12, 53115 Bonn, Germany

Published:	2025-10-01, volume 9, page 1871
Editor:	Thomas Elliott
Eprint:	arXiv:2503.20327v2
Doi:	https://doi.org/10.22331/q-2025-10-01-1871
Citation:	Quantum 9, 1871 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We extend the concept of operator charge in the context of an abelian $U(1)$ symmetry and apply this framework to symmetry-preserving matrix product operators (MPOs), enabling the description of operators projected onto specific sectors of the corresponding symmetry. Leveraging this representation, we study the effect of interactions on the scrambling of information in an integrable Heisenberg spin chain, by controlling the number of particles. Our focus lies on out-of-time order correlators (OTOCs) which we project on sectors with a fixed number of particles. This allows us to link the non-interacting system to the fully-interacting one by allowing more and more particle to interact with each other, keeping the interaction parameter fixed. While at short times, the OTOCs are almost not affected by interactions, the spreading of the information front becomes gradually faster and the OTOC saturate at larger values as the number of particle increases. We also study the behavior of finite-size systems by considering the OTOCs at times beyond the point where the front hits the boundary of the system. We find that in every sector with more than one particle, the OTOCs behave as if the local operator was rotated by a random unitary matrix, indicating that the presence of boundaries contributes to the maximal scrambling of local operators.

Featured image: Out-of-time-order correlators projected in different symmetry sectors, from the non-interacting limit (one particle sector) to the full interacting system (half-filled sector).

Popular summary

We introduce a representation of operators in quantum systems that respect a U(1) symmetry, such as those that conserve particle number. By extending the definition of the operator charge, we build a framework that lets us describe naturally operators acting only within specific particle-number sectors as a projected matrix product operator.

With this representation, we explore how interactions affect the scrambling of information (how local information spreads across a system) in an integrable Heisenberg spin chain. To do this, we look at out-of-time order correlators (OTOCs), which serve as a diagnostic of information spreading. By projecting OTOCs onto different particle-number sectors, we can smoothly connect the non-interacting limit (single-particle dynamics) to the fully interacting case, simply by increasing the number of particles while keeping the interaction strength fixed.

Our findings show that at early times, interactions hardly influence the OTOCs. At later times, as more particles are allowed to interact, information spreads faster and the OTOCs settle at higher values, signaling stronger scrambling. We also study finite systems at long times, when the spreading information front collides with system boundaries. Interestingly, in every case with more than one particle, the late-time behavior of OTOCs resembles what would happen if the local operator were randomly scrambled, suggesting that boundaries themselves play a crucial role in driving maximal information scrambling.

► BibTeX data

@article{Gisti2025symmetryresolvedout,
  doi = {10.22331/q-2025-10-01-1871},
  url = {https://doi.org/10.22331/q-2025-10-01-1871},
  title = {Symmetry resolved out-of-time-order correlators of {H}eisenberg spin chains using projected matrix product operators},
  author = {Gisti, Martina and Luitz, David J. and Debertolis, Maxime},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1871},
  month = oct,
  year = {2025}
}

► References

[1] Elliott H. Lieb and Derek W. Robinson. ``The finite group velocity of quantum spin systems''. Communications In Mathematical Physics 28, 251–257 (1972).
https://doi.org/10.1007/BF01645779

[2] M. B. Hastings. ``An area law for one-dimensional quantum systems''. Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007).
https://doi.org/10.1088/1742-5468/2007/08/P08024

[3] J. Eisert, M. Cramer, and M. B. Plenio. ``Colloquium: Area laws for the entanglement entropy''. Rev. Mod. Phys. 82, 277–306 (2010).
https://doi.org/10.1103/RevModPhys.82.277

[4] S. Bravyi, M. B. Hastings, and F. Verstraete. ``Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order''. Phys. Rev. Lett. 97, 050401 (2006).
https://doi.org/10.1103/PhysRevLett.97.050401

[5] David J. Luitz and Yevgeny Bar Lev. ``Information propagation in isolated quantum systems''. Phys. Rev. B 96, 020406 (2017).
https://doi.org/10.1103/PhysRevB.96.020406

[6] Michael Kastner. ``N-scaling of timescales in long-range n-body quantum systems''. Journal of Statistical Mechanics: Theory and Experiment 2017, 014003 (2017).
https://doi.org/10.1088/1742-5468/aa5119

[7] Anatoly I. Larkin and Yu. N. Ovchinnikov. ``Quasiclassical method in the theory of superconductivity''. Journal of Experimental and Theoretical Physics (1969). url: https://api.semanticscholar.org/CorpusID:117608877.
https://api.semanticscholar.org/CorpusID:117608877

[8] Juan Maldacena, Stephen H. Shenker, and Douglas Stanford. ``A bound on chaos''. Journal of High Energy Physics 2016, 106 (2016).
https://doi.org/10.1007/JHEP08(2016)106

[9] Ignacio García-Mata, Marcos Saraceno, Rodolfo A. Jalabert, Augusto J. Roncaglia, and Diego A. Wisniacki. ``Chaos signatures in the short and long time behavior of the out-of-time ordered correlator''. Phys. Rev. Lett. 121, 210601 (2018).
https://doi.org/10.1103/PhysRevLett.121.210601

[10] Jorge Chávez-Carlos, B. López-del Carpio, Miguel A. Bastarrachea-Magnani, Pavel Stránský, Sergio Lerma-Hernández, Lea F. Santos, and Jorge G. Hirsch. ``Quantum and Classical Lyapunov Exponents in Atom-Field Interaction Systems''. Phys. Rev. Lett. 122, 024101 (2019).
https://doi.org/10.1103/PhysRevLett.122.024101

[11] Arul Lakshminarayan. ``Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices''. Phys. Rev. E 99, 012201 (2019).
https://doi.org/10.1103/PhysRevE.99.012201

[12] Josef Rammensee, Juan Diego Urbina, and Klaus Richter. ``Many-body quantum interference and the saturation of out-of-time-order correlators''. Phys. Rev. Lett. 121, 124101 (2018).
https://doi.org/10.1103/PhysRevLett.121.124101

[13] Efim B. Rozenbaum, Sriram Ganeshan, and Victor Galitski. ``Lyapunov exponent and out-of-time-ordered correlator's growth rate in a chaotic system''. Phys. Rev. Lett. 118, 086801 (2017).
https://doi.org/10.1103/PhysRevLett.118.086801

[14] Ignacio García-Mata, Rodolfo A. Jalabert, and Diego A. Wisniacki. ``Out-of-time-order correlators and quantum chaos''. Scholarpedia 18, 55237 (2023).
https://doi.org/10.4249/scholarpedia.55237

[15] Tianrui Xu, Thomas Scaffidi, and Xiangyu Cao. ``Does scrambling equal chaos?''. Phys. Rev. Lett. 124, 140602 (2020).
https://doi.org/10.1103/PhysRevLett.124.140602

[16] Neil Dowling, Pavel Kos, and Kavan Modi. ``Scrambling is necessary but not sufficient for chaos''. Phys. Rev. Lett. 131, 180403 (2023).
https://doi.org/10.1103/PhysRevLett.131.180403

[17] 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

[18] C. W. Von Keyserlingk, Tibor Rakovszky, Frank Pollmann, and S. L. Sondhi. ``Operator hydrodynamics, $\text{OTOCs}$, and entanglement growth in systems without conservation laws''. Phys. Rev. X 8, 021013 (2018).
https://doi.org/10.1103/PhysRevX.8.021013

[19] Tibor Rakovszky, Frank Pollmann, and C. W. Von Keyserlingk. ``Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation''. Phys. Rev. X 8, 031058 (2018).
https://doi.org/10.1103/PhysRevX.8.031058

[20] Vedika Khemani, Ashvin Vishwanath, and David A. Huse. ``Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws''. Phys. Rev. X 8, 031057 (2018).
https://doi.org/10.1103/PhysRevX.8.031057

[21] Amos Chan, Andrea De Luca, and J. T. Chalker. ``Solution of a minimal model for many-body quantum chaos''. Phys. Rev. X 8, 041019 (2018).
https://doi.org/10.1103/PhysRevX.8.041019

[22] Yichen Huang, Fernando G. S. L. Brandão, and Yong-Liang Zhang. ``Finite-size scaling of out-of-time-ordered correlators at late times''. Phys. Rev. Lett. 123, 010601 (2019).
https://doi.org/10.1103/PhysRevLett.123.010601

[23] Vinitha Balachandran, Giuliano Benenti, Giulio Casati, and Dario Poletti. ``From the eigenstate thermalization hypothesis to algebraic relaxation of otocs in systems with conserved quantities''. Phys. Rev. B 104, 104306 (2021).
https://doi.org/10.1103/PhysRevB.104.104306

[24] Luca Capizzi, Jiaozi Wang, Xiansong Xu, Leonardo Mazza, and Dario Poletti. ``Hydrodynamics and the eigenstate thermalization hypothesis''. Phys. Rev. X 15, 011059 (2025).
https://doi.org/10.1103/PhysRevX.15.011059

[25] Yu Chen. ``Universal logarithmic scrambling in many body localization'' (2016). arXiv:1608.02765.
arXiv:1608.02765

[26] Brian Swingle, Gregory Bentsen, Monika Schleier-Smith, and Patrick Hayden. ``Measuring the scrambling of quantum information''. Phys. Rev. A 94, 040302 (2016).
https://doi.org/10.1103/PhysRevA.94.040302

[27] Yichen Huang, Yong-Liang Zhang, and Xie Chen. ``Out-of-time-ordered correlators in many-body localized systems''. Annalen der Physik 529, 1600318 (2017).
https://doi.org/10.1002/andp.201600318

[28] Ruihua Fan, Pengfei Zhang, Huitao Shen, and Hui Zhai. ``Out-of-time-order correlation for many-body localization''. Science Bulletin 62, 707–711 (2017).
https://doi.org/10.1016/j.scib.2017.04.011

[29] Kevin Slagle, Zhen Bi, Yi-Zhuang You, and Cenke Xu. ``Out-of-time-order correlation in marginal many-body localized systems''. Phys. Rev. B 95, 165136 (2017).
https://doi.org/10.1103/PhysRevB.95.165136

[30] Xiao Chen, Tianci Zhou, David A. Huse, and Eduardo Fradkin. ``Out-of-time-order correlations in many-body localized and thermal phases''. Annalen der Physik 529, 1600332 (2017).
https://doi.org/10.1002/andp.201600332

[31] Luis Colmenarez and David J. Luitz. ``Lieb-Robinson bounds and out-of-time order correlators in a long-range spin chain''. Phys. Rev. Res. 2, 043047 (2020).
https://doi.org/10.1103/PhysRevResearch.2.043047

[32] A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap. ``Scrambling and thermalization in a diffusive quantum many-body system''. New Journal of Physics 19, 063001 (2017).
https://doi.org/10.1088/1367-2630/aa719b

[33] Kévin Hémery, Frank Pollmann, and David J. Luitz. ``Matrix product states approaches to operator spreading in ergodic quantum systems''. Phys. Rev. B 100, 104303 (2019).
https://doi.org/10.1103/PhysRevB.100.104303

[34] Eiki Iyoda and Takahiro Sagawa. ``Scrambling of quantum information in quantum many-body systems''. Phys. Rev. A 97, 042330 (2018).
https://doi.org/10.1103/PhysRevA.97.042330

[35] Emiliano M. Fortes, Ignacio García-Mata, Rodolfo A. Jalabert, and Diego A. Wisniacki. ``Gauging classical and quantum integrability through out-of-time-ordered correlators''. Phys. Rev. E 100, 042201 (2019).
https://doi.org/10.1103/PhysRevE.100.042201

[36] Sarang Gopalakrishnan, David A. Huse, Vedika Khemani, and Romain Vasseur. ``Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems''. Phys. Rev. B 98, 220303 (2018).
https://doi.org/10.1103/PhysRevB.98.220303

[37] Cheng-Ju Lin and Olexei I. Motrunich. ``Out-of-time-ordered correlators in a quantum ising chain''. Phys. Rev. B 97, 144304 (2018).
https://doi.org/10.1103/PhysRevB.97.144304

[38] Cheng-Ju Lin and Olexei I. Motrunich. ``Out-of-time-ordered correlators in short-range and long-range hard-core boson models and in the luttinger-liquid model''. Phys. Rev. B 98, 134305 (2018).
https://doi.org/10.1103/PhysRevB.98.134305

[39] Jonathon Riddell and Erik S. Sørensen. ``Out of time ordered correlators and entanglement growth in the random field ${XX}$ spin chain''. Phys. Rev. B 99, 054205 (2019).
https://doi.org/10.1103/PhysRevB.99.054205

[40] Jonathon Riddell and Erik S. Sørensen. ``Out-of-time-order correlations in the quasiperiodic Aubry-André model''. Phys. Rev. B 101, 024202 (2020).
https://doi.org/10.1103/PhysRevB.101.024202

[41] Jiahui Bao and Cheng-Yong Zhang. ``Out-of-time-order correlators in one-dimensional ${XY}$ model''. Communications in Theoretical Physics 72, 085103 (2020).
https://doi.org/10.1088/1572-9494/ab8a28

[42] Shenglong Xu and Brian Swingle. ``Accessing scrambling using matrix product operators''. Nature Physics 16, 199–204 (2020).
https://doi.org/10.1038/s41567-019-0712-4

[43] Sandra Byju, Kinjalk Lochan, and S. Shankaranarayanan. ``Quenched Kitaev chain: Analogous model of gravitational collapse''. Phys. Rev. D 107, 105020 (2023).
https://doi.org/10.1103/PhysRevD.107.105020

[44] Jonathon Riddell, Wyatt Kirkby, D. H. J. O’Dell, and Erik S. Sørensen. ``Scaling at the $\text{OTOC}$ wavefront: free versus chaotic models''. Phys. Rev. B 108, L121108 (2023).
https://doi.org/10.1103/PhysRevB.108.L121108

[45] Jonathon Riddell and Erik S. Sørensen. ``Out-of-time ordered correlators and entanglement growth in the random-field ${XX}$ spin chain''. Phys. Rev. B 99, 054205 (2019).
https://doi.org/10.1103/PhysRevB.99.054205

[46] Javier Lopez-Piqueres, Brayden Ware, Sarang Gopalakrishnan, and Romain Vasseur. ``Operator front broadening in chaotic and integrable quantum chains''. Phys. Rev. B 104, 104307 (2021).
https://doi.org/10.1103/PhysRevB.104.104307

[47] Guifré Vidal. ``Efficient classical simulation of slightly entangled quantum computations''. Phys. Rev. Lett. 91, 147902 (2003).
https://doi.org/10.1103/PhysRevLett.91.147902

[48] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96–192 (2011).
https://doi.org/10.1016/j.aop.2010.09.012

[49] Sukhwinder Singh, Huan-Qiang Zhou, and Guifre Vidal. ``Simulation of one-dimensional quantum systems with a global $SU(2)$ symmetry''. New Journal of Physics 12, 033029 (2010).
https://doi.org/10.1088/1367-2630/12/3/033029

[50] Sebastian Paeckel, Thomas Köhler, and Salvatore R. Manmana. ``Automated construction of $U(1)$-invariant matrix-product operators from graph representations''. SciPost Physics 3, 035 (2017).
https://doi.org/10.21468/SciPostPhys.3.5.035

[51] Daniel E. Parker, Xiangyu Cao, and Michael P. Zaletel. ``Local matrix product operators: Canonical form, compression, and control theory''. Phys. Rev. B 102, 035147 (2020).
https://doi.org/10.1103/PhysRevB.102.035147

[52] Sukhwinder Singh, Robert N. C. Pfeifer, and Guifre Vidal. ``Tensor network states and algorithms in the presence of a global $U(1)$ symmetry''. Phys. Rev. B 83, 115125 (2011).
https://doi.org/10.1103/PhysRevB.83.115125

[53] Chu Guo and Dario Poletti. ``Matrix product states with adaptive global symmetries''. Phys. Rev. B 100, 134304 (2019).
https://doi.org/10.1103/PhysRevB.100.134304

[54] Johannes Hauschild and Frank Pollmann. ``Efficient numerical simulations with tensor networks: Tensor network python (TeNPy)''. SciPost Phys. Lect. NotesPage 5 (2018).
https://doi.org/10.21468/SciPostPhysLectNotes.5

[55] Matthew Fishman, Steven R. White, and E. Miles Stoudenmire. ``The ITensor Software Library for Tensor Network Calculations''. SciPost Phys. CodebasesPage 4 (2022).
https://doi.org/10.21468/SciPostPhysCodeb.4

[56] Gregory M. Crosswhite and Dave Bacon. ``Finite automata for caching in matrix product algorithms''. Phys. Rev. A 78, 012356 (2008).
https://doi.org/10.1103/PhysRevA.78.012356

[57] David Perez-Garcia, Frank Verstraete, Michael M. Wolf, and J. Ignacio Cirac. ``Matrix product state representations''. Quant. Inf. Comput. 7 (2007). arXiv:quant-ph/0608197.
https://doi.org/10.26421/QIC7.5-6-1
arXiv:quant-ph/0608197

[58] F. Verstraete, V. Murg, and J.I. Cirac. ``Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems''. Advances in Physics 57, 143–224 (2008).
https://doi.org/10.1080/14789940801912366

[59] Aniket Rath, Vittorio Vitale, Sara Murciano, Matteo Votto, Jérôme Dubail, Richard Kueng, Cyril Branciard, Pasquale Calabrese, and Benoı̂t Vermersch. ``Entanglement barrier and its symmetry resolution: Theory and experimental observation''. PRX Quantum 4, 010318 (2023).
https://doi.org/10.1103/PRXQuantum.4.010318

[60] Sara Murciano, Jérôme Dubail, and Pasquale Calabrese. ``More on symmetry resolved operator entanglement''. Journal of Physics A: Mathematical and Theoretical 57, 145002 (2024).
https://doi.org/10.1088/1751-8121/ad30d1

[61] T. Giamarchi and Oxford University Press. ``Quantum physics in one dimension''. International Series of Monographs on Physics. Clarendon Press. (2004).
https://doi.org/10.1093/acprof:oso/9780198525004.001.0001

[62] L. D. Faddeev. ``How algebraic bethe ansatz works for integrable model'' (1996). arXiv:hep-th/9605187.
arXiv:hep-th/9605187

[63] Adam Smith, Johannes Knolle, Roderich Moessner, and Dmitry L. Kovrizhin. ``Logarithmic spreading of out-of-time-ordered correlators without many-body localization''. Phys. Rev. Lett. 123, 086602 (2019).
https://doi.org/10.1103/PhysRevLett.123.086602

[64] Balázs Dóra and Roderich Moessner. ``Out-of-time-ordered density correlators in luttinger liquids''. Phys. Rev. Lett. 119, 026802 (2017).
https://doi.org/10.1103/PhysRevLett.119.026802

[65] A. Hallam, J. G. Morley, and A. G. Green. ``The lyapunov spectra of quantum thermalisation''. Nature Communications 10, 2708 (2019).
https://doi.org/10.1038/s41467-019-10336-4

[66] Sebastian Leontica and Andrew G. Green. ``Entanglement growth from squeezing on the mps manifold''. Nature Communications 16, 1832 (2025).
https://doi.org/10.1038/s41467-025-56959-8

[67] Ranjan Modak, Vincenzo Alba, and Pasquale Calabrese. ``Entanglement revivals as a probe of scrambling in finite quantum systems''. Journal of Statistical Mechanics: Theory and Experiment 2020, 083110 (2020).
https://doi.org/10.1088/1742-5468/aba9d9

[68] Vincenzo Alba. ``More on the operator space entanglement (ose): Rényi ose, revivals, and integrability breaking''. Journal of Physics A: Mathematical and Theoretical (2025).
https://doi.org/10.1088/1751-8121/adc9e6

[69] V. Hunyadi, Z. Rácz, and L. Sasvári. ``Dynamic scaling of fronts in the quantum $XX$ chain''. Phys. Rev. E 69, 066103 (2004).
https://doi.org/10.1103/PhysRevE.69.066103

[70] T. Platini and D. Karevski. ``Scaling and front dynamics in Ising quantum chains''. The European Physical Journal B - Condensed Matter and Complex Systems 48, 225–231 (2005).
https://doi.org/10.1140/epjb/e2005-00402-2

[71] Maurizio Fagotti. ``Higher-order generalized hydrodynamics in one dimension: The noninteracting test''. Phys. Rev. B 96, 220302 (2017).
https://doi.org/10.1103/PhysRevB.96.220302

[72] Mario Collura, Andrea De Luca, and Jacopo Viti. ``Analytic solution of the domain-wall nonequilibrium stationary state''. Phys. Rev. B 97, 081111 (2018).
https://doi.org/10.1103/PhysRevB.97.081111

[73] Vedika Khemani, David A. Huse, and Adam Nahum. ``Velocity-dependent Lyapunov exponents in many-body quantum, semiclassical, and classical chaos''. Phys. Rev. B 98, 144304 (2018).
https://doi.org/10.1103/PhysRevB.98.144304

[74] Yingfei Gu, Xiao-Liang Qi, and Douglas Stanford. ``Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models''. Journal of High Energy Physics 2017, 125 (2017).
https://doi.org/10.1007/JHEP05(2017)125

[75] Marko Ljubotina, Marko Žnidarič, and Tomaž Prosen. ``Spin diffusion from an inhomogeneous quench in an integrable system''. Nature Communications 8, 16117 (2017).
https://doi.org/10.1038/ncomms16117

[76] Marko Ljubotina, Marko Znidaric, and Tomaz Prosen. ``Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet''. Phys. Rev. Lett. 122, 210602 (2019).
https://doi.org/10.1103/PhysRevLett.122.210602

[77] Benoı̂t Collins and Ion Nechita. ``Random quantum channels I: Graphical calculus and the Bell state phenomenon''. Communications in Mathematical Physics 297, 345–370 (2010).
https://doi.org/10.1007/s00220-010-1012-0

[78] Milton Abramowitz and Irene A. Stegun. ``Handbook of mathematical functions with formulas, graphs, and mathematical tables''. Journal of Applied Mechanics. (1965).
https://doi.org/10.1115/1.3625776

Cited by

[1] Maxime Debertolis, "Natural super-orbitals representation of many-body operators", arXiv:2507.10690, (2025).

The above citations are from SAO/NASA ADS (last updated successfully 2026-08-09 13:51:01). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2026-08-09 13:50:59).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.