Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2411.07298 · Physical review. B./Physical review. B

Two-stage relaxation of operators through domain wall and magnon dynamics

Cheryne Jonay, Yue Li, Tianci Zhou

Published Jun 9, 2025• 3 citations
arXivHTMLPDF

Abstract

The out-of-time-ordered correlator (OTOC) is a popular probe for quantum information spreading and thermalization. In systems with local interactions, the OTOC defines a characteristic butterfly light cone that separates a regime unperturbed by chaos from one where time-evolved operators and the OTOC approach equilibrium. This relaxation has been shown to proceed in two stages. The first stage exhibits an extensive timescale and a decay rate known as the “phantom eigenvalue”, which is slower than the gap of the transfer matrix. In this paper, we investigate the two-stage relaxation of the OTOC towards its equilibrium value in various local quantum circuits. We apply a systematic framework based on an emergent statistical model, where the dynamics of two single-particle modes—a domain wall and a magnon—govern the decay rates. We show that a configuration with coexisting domain wall and magnon modes generates the phantom rate in the first stage, while competition between these two modes determines the second stage. We also examine this relaxation within the operator cluster picture. The magnon modes translate into a bound state of clusters, and the domain wall into a random operator, giving consistent rates. Finally, we extend our findings from random-in-time circuits to a broad class of Floquet models.

DOI
10.1103/physrevb.111.224304

Related papers

Two-step phantom relaxation of out-of-time-ordered correlations in random circuits

Jaš Bensa, Marko Žnidarič

We study out-of-time-ordered correlation (OTOC) functions in various random quantum circuits and show that the average dynamics is governed by a Markovian propagator. This is then used to study relaxation of OTOC to its long-time average value in circuits with random single-qubit unitaries, finding that relaxation in general proceeds in two steps: In the first phase that lasts upto an extensively long time the relaxation rate is given by a phantom eigenvalue of a nonsymmetric propagator, whereas in the second phase the rate is determined by the true 2nd largest propagator eigenvalue. We also obtain exact OTOC dynamics on the lightcone and an expression for the average OTOC in finite random circuits with random two-qubit gates.

arXiv:2112.07281•Physical Review Research•Mar 25, 2022•24 citations

Arbitrary relaxation rate under non-Hermitian matrix iterations

Jaš Bensa

We study the exponential relaxation of observables propagated with a non-Hermitian transfer matrix, an example being out-of-time-ordered correlations (OTOC) in brick-wall (BW) random quantum circuits. Until a time that scales as the system size, the exponential decay of observables is not usually determined by the second largest eigenvalue of the transfer matrix, as one can naively expect, but it is, in general, slower—this slower decay rate was dubbed “phantom eigenvalue.” Generally, this slower decay is given by the largest value in the pseudospectrum of the transfer matrix; however, we show that the decay rate can be an arbitrary value between the second largest eigenvalue and the largest value in the pseudospectrum. This arbitrary decay can be observed, for example, in the propagation of OTOC in periodic boundary conditions BW circuits. To explore this phenomenon, we study a matrix iteration made from a simple tridiagonal Toeplitz matrix. This setting can be used to propagate OTOC in random circuits with open boundary conditions and to describe a one-dimensional biased random walk with dissipation at the edges. Published by the American Physical Society 2024

arXiv:2312.14617•Physical Review Research•Jun 28, 2024•2 citations

Two-step relaxation in local many-body Floquet systems

Marko Žnidarič

Abstract We want to understand how relaxation process from an initial non-generic state proceeds towards a long-time typical state reached under unitary quantum evolution. One would expect that after some initial correlation time relaxation will be a simple exponential decay with constant decay rate. We show that this is not necessarily the case. Studying various Floquet systems with fixed two-qubit gates, and focusing on purity and out-of-time-ordered correlation functions, we find that in many situations relaxation proceeds in two phases of exponential decay having different relaxation rates. Namely, in the thermodynamic limit the relaxation rate exhibits a change at a critical time proportional to system’s size. The initial thermodynamically relevant rate can be slower or faster than the asymptotic one, demonstrating that the recently discovered phantom relaxation, in which the decay is slower than predicted by a nonzero transfer matrix gap, is not limited to only random circuits.

arXiv:2301.06395•Journal of Physics A Mathematical and Theoretical•Sep 21, 2023•9 citations

A Physical Theory of Two-stage Thermalization

Cheryne Jonay, Tianci Zhou

One indication of thermalization time is subsystem entanglement reaching thermal values. Recent studies on local quantum circuits reveal two exponential stages with decay rates $r_1$ and $r_2$ of the purity before and after thermalization. We provide an entanglement membrane theory interpretation, with $r_1$ corresponding to the domain wall free energy. Circuit geometry can lead to $r_1 < r_2$, producing a ``phantom eigenvalue". Competition between the domain wall and magnon leads to $r_2 < r_1$ when the magnon prevails. However, when the domain wall wins, this mechanism provides a practical approach for measuring entanglement growth through local correlation functions.

arXiv:2310.04491•Oct 6, 2023•1 citation

Physical theory of two-stage thermalization

Cheryne Jonay, Tianci Zhou

Thermalization time marks the scale at which the subsystem entanglement reaches its thermal value. It was recently shown to exhibit two distinct stages in a generic class of quantum circuits with decay rates ${r}_{1}$ and ${r}_{2}$ before and after thermalization. The two rates are observed in the Floquet dynamics of the state purity, entanglement spectra, and (local) correlation functions, both in clean and disordered systems. Entanglement membrane theory interprets ${r}_{1}$ as domain-wall energy and ${r}_{2}$ as either a geometric effect (phantom eigenvalue) or a competition with an emergent magnon mode. In the case when the domain wall dominates, this offers a practical measure for entanglement growth via local correlations.

Physical review. B./Physical review. B•Jul 22, 2024•9 citations

Probing chaos and thermalization through out-of-time-ordered correlators in random field spin chains

C Jisha, Shivam Mishra, Ravi Prakash

Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-$1/2$ chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law $(1/t)$ relaxation for integrable systems and a higher-degree power-law decay $(1/t^α; α\ge 1)$ followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.

quant-pharXiv:2606.18982•Jun 18, 2026•0 citations
The out-of-time-ordered correlator (OTOC) is a popular probe for quantum information spreading and thermalization. In systems with local interactions, the OTOC defines a characteristic butterfly light cone that separates a regime unperturbed by chaos from one where time-evolved operators and the OTOC approach equilibrium. This relaxation has been shown to proceed in two stages. The first stage exhibits an extensive timescale and a decay rate known as the “phantom eigenvalue”, which is slower than the gap of the transfer matrix. In this paper, we investigate the two-stage relaxation of the OTOC towards its equilibrium value in various local quantum circuits. We apply a systematic framework based on an emergent statistical model, where the dynamics of two single-particle modes—a domain wall and a magnon—govern the decay rates. We show that a configuration with coexisting domain wall and magnon modes generates the phantom rate in the first stage, while competition between these two modes determines the second stage. We also examine this relaxation within the operator cluster picture. The magnon modes translate into a bound state of clusters, and the domain wall into a random operator, giving consistent rates. Finally, we extend our findings from random-in-time circuits to a broad class of Floquet models.
We study out-of-time-ordered correlation (OTOC) functions in various random quantum circuits and show that the average dynamics is governed by a Markovian propagator. This is then used to study relaxation of OTOC to its long-time average value in circuits with random single-qubit unitaries, finding that relaxation in general proceeds in two steps: In the first phase that lasts upto an extensively long time the relaxation rate is given by a phantom eigenvalue of a nonsymmetric propagator, whereas in the second phase the rate is determined by the true 2nd largest propagator eigenvalue. We also obtain exact OTOC dynamics on the lightcone and an expression for the average OTOC in finite random circuits with random two-qubit gates.
We study the exponential relaxation of observables propagated with a non-Hermitian transfer matrix, an example being out-of-time-ordered correlations (OTOC) in brick-wall (BW) random quantum circuits. Until a time that scales as the system size, the exponential decay of observables is not usually determined by the second largest eigenvalue of the transfer matrix, as one can naively expect, but it is, in general, slower—this slower decay rate was dubbed “phantom eigenvalue.” Generally, this slower decay is given by the largest value in the pseudospectrum of the transfer matrix; however, we show that the decay rate can be an arbitrary value between the second largest eigenvalue and the largest value in the pseudospectrum. This arbitrary decay can be observed, for example, in the propagation of OTOC in periodic boundary conditions BW circuits. To explore this phenomenon, we study a matrix iteration made from a simple tridiagonal Toeplitz matrix. This setting can be used to propagate OTOC in random circuits with open boundary conditions and to describe a one-dimensional biased random walk with dissipation at the edges. Published by the American Physical Society 2024
Abstract We want to understand how relaxation process from an initial non-generic state proceeds towards a long-time typical state reached under unitary quantum evolution. One would expect that after some initial correlation time relaxation will be a simple exponential decay with constant decay rate. We show that this is not necessarily the case. Studying various Floquet systems with fixed two-qubit gates, and focusing on purity and out-of-time-ordered correlation functions, we find that in many situations relaxation proceeds in two phases of exponential decay having different relaxation rates. Namely, in the thermodynamic limit the relaxation rate exhibits a change at a critical time proportional to system’s size. The initial thermodynamically relevant rate can be slower or faster than the asymptotic one, demonstrating that the recently discovered phantom relaxation, in which the decay is slower than predicted by a nonzero transfer matrix gap, is not limited to only random circuits.
One indication of thermalization time is subsystem entanglement reaching thermal values. Recent studies on local quantum circuits reveal two exponential stages with decay rates r1r_1r1​ and r2r_2r2​ of the purity before and after thermalization. We provide an entanglement membrane theory interpretation, with r1r_1r1​ corresponding to the domain wall free energy. Circuit geometry can lead to r1<r2r_1 < r_2r1​<r2​, producing a ``phantom eigenvalue". Competition between the domain wall and magnon leads to r2<r1r_2 < r_1r2​<r1​ when the magnon prevails. However, when the domain wall wins, this mechanism provides a practical approach for measuring entanglement growth through local correlation functions.
Thermalization time marks the scale at which the subsystem entanglement reaches its thermal value. It was recently shown to exhibit two distinct stages in a generic class of quantum circuits with decay rates r1{r}_{1}r1​ and r2{r}_{2}r2​ before and after thermalization. The two rates are observed in the Floquet dynamics of the state purity, entanglement spectra, and (local) correlation functions, both in clean and disordered systems. Entanglement membrane theory interprets r1{r}_{1}r1​ as domain-wall energy and r2{r}_{2}r2​ as either a geometric effect (phantom eigenvalue) or a competition with an emergent magnon mode. In the case when the domain wall dominates, this offers a practical measure for entanglement growth via local correlations.
Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-1/21/21/2 chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law (1/t)(1/t)(1/t) relaxation for integrable systems and a higher-degree power-law decay (1/tα;α≥1)(1/t^α; α\ge 1)(1/tα;α≥1) followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.