pith. machine review for the scientific record. sign in

arxiv: 2512.21023 · v2 · submitted 2025-12-24 · ✦ hep-th

Recognition: no theorem link

OTOC and Quamtum Chaos of Interacting Scalar Fields

Wung-Hong Huang
Authors on Pith no claims yet

Pith reviewed 2026-05-16 20:19 UTC · model grok-4.3

classification ✦ hep-th
keywords out-of-time-order correlatorquantum chaosLyapunov exponentinteracting scalar fieldslambda phi^4 theorylattice discretizationperturbative calculations
0
0 comments X

The pith

The out-of-time-order correlator in lattice models of interacting scalar fields grows exponentially with Lyapunov exponent scaling as T to the one-fourth power, diagnosing quantum chaos at low perturbative orders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper discretizes the lambda phi to the fourth scalar field theory on a lattice, converting it into a system of coupled anharmonic oscillators. It derives analytic relations using second quantization up to second-order perturbation and uses them to compute the thermal out-of-time-order correlator C_T(t) numerically. This correlator displays exponential growth over an extended early-time window, with the growth rate scaling as temperature to the power one-fourth. The same signatures of quantum chaos appear in a closed chain of N oscillators that models the one-plus-one dimensional interacting quantum scalar field theory, and these signatures emerge already at low orders of perturbation theory.

Core claim

By modeling the λφ⁴ theory as coupled anharmonic oscillators on a lattice and computing the thermal OTOC C_T(t) through second-order perturbative relations obtained from second quantization, the calculation reveals exponential growth of C_T(t) in an early-time window with Lyapunov exponent λ ∼ T^{1/4} that diagnoses quantum chaos; the same signatures appear at low perturbative orders when the model is extended to a closed chain of N oscillators representing the 1+1 dimensional interacting quantum scalar field theory.

What carries the argument

The thermal out-of-time-order correlator C_T(t) evaluated via second-order perturbative relations in the second-quantized lattice model of coupled anharmonic oscillators.

If this is right

  • The Lyapunov exponent diagnosing quantum chaos scales as temperature to the power one-fourth in this model.
  • Signatures of quantum chaos are visible in the OTOC already at low perturbative orders.
  • The closed chain of N oscillators reproduces the chaos properties of the corresponding 1+1 dimensional field theory.
  • Exponential growth in the OTOC occurs over a long early-time window before other effects dominate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Perturbative methods may suffice to detect chaos in a wider class of interacting quantum field theories where non-perturbative tools are unavailable.
  • The T^{1/4} scaling could be checked by taking the continuum limit of the lattice spacing to zero while keeping the coupling fixed.
  • Similar low-order OTOC growth might appear in scalar theories with other potentials or in higher spatial dimensions.

Load-bearing premise

The lattice discretization of λφ⁴ theory and its second-order perturbative OTOC relations accurately capture the quantum chaos behavior of the continuum 1+1 dimensional interacting scalar field theory.

What would settle it

A computation of the OTOC in the continuum limit of the 1+1 dimensional λφ⁴ theory that shows either no exponential growth or a temperature scaling different from T to the one-fourth power would falsify the central claim.

Figures

Figures reproduced from arXiv: 2512.21023 by Wung-Hong Huang.

Figure 1
Figure 1. Figure 1: Two coupled anharmonic oscillators with quaratic potential ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: OTOC CT (t) as the function of time for various cutoff mode number nF . We see that CT (t) for nF = 30 is closer to that of nF = 40. The property also shows in other temperatures. Thus the figures 3 and 4 are plotted with nF = 40. 4.2 Exponential growth and Lyapunov exponent In [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The exponential growth of CT (t) within the wide interval 1 ≤ t ≤ 5 for T=10 . Note that the value and error of the Lyapunov exponent λ in Eq. (4.2) depend on the selected time interval. As shown in [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Temperature dependence of Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Closed chain of three coupled anharmonic oscillators with quadratic potential ( [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

Discretizing the $\lambda \phi^4$ scalar field theory on a lattice yields a system of coupled anharmonic oscillators with quadratic and quartic potentials. We begin by analyzing the two coupled oscillators in the second quantization method to derive several analytic relations to the second-order perturbation, which are then employed to numerically calculate the thermal out-of-time-order correlator (OTOC), $C_T(t)$. We find that the function $C_T(t)$ exhibits exponential growth over a long time window in the early stages, with Lyapunov exponent $\lambda\sim T^{1/4}$, which diagnoses quantum chaos. We furthermore investigate the quantum chaos properties in a closed chain of N coupled anharmonic oscillators, which relates to the 1+1 dimensional interacting quantum scalar field theory. The results reveal an interesting property that the signatures of quantum chaos appear at low perturbative orders in the OTOC.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript discretizes the λφ⁴ scalar field theory on a lattice as coupled anharmonic oscillators. For the two-oscillator system, analytic relations are derived to second order in perturbation theory via second quantization and used to numerically compute the thermal OTOC C_T(t), which exhibits exponential growth with Lyapunov exponent λ ∼ T^{1/4}, interpreted as diagnosing quantum chaos. The analysis is extended to a closed chain of N oscillators approximating the 1+1D interacting quantum scalar field theory, with the conclusion that signatures of quantum chaos appear at low perturbative orders in the OTOC.

Significance. If the central numerical result holds after addressing the issues below, the work would indicate that exponential OTOC growth and a specific scaling can emerge already at second order in perturbation theory for this lattice model, providing a computationally tractable window into quantum chaos in scalar QFTs. The analytic second-order relations and their numerical implementation constitute a concrete strength, as does the attempt to connect the finite-N chain to the continuum theory. However, the absence of dimensional consistency checks and robustness tests leaves the significance of the λ ∼ T^{1/4} claim unclear at present.

major comments (3)
  1. [Abstract] Abstract: the reported scaling λ ∼ T^{1/4} is dimensionally inconsistent in 1+1D, where [λ] = 2 and [T] = 1 (in units ħ = c = 1), so that λ/T^{1/2} is the only dimensionless combination; no lattice spacing a, effective coupling, or unit choice is specified to reconcile the exponent. This is load-bearing for the chaos diagnosis.
  2. [Two-oscillator analysis] Numerical evaluation of C_T(t) (two-oscillator section): the exponential growth is obtained from second-order perturbation without reported error bars, finite-N scaling, or explicit comparison to third-order or resummed contributions. Early-time growth can appear in non-chaotic truncations, so these checks are required to support the claim that the growth diagnoses genuine quantum chaos.
  3. [N-oscillator chain] N-oscillator chain section: the assertion that the lattice discretization faithfully captures the chaotic properties of the continuum 1+1D theory lacks supporting tests (continuum extrapolation, comparison to integrable limits, or higher-order validation), which is essential because the central claim extends the two-oscillator result to the field theory.
minor comments (3)
  1. [Title] Title contains the typo 'Quamtum' instead of 'Quantum'.
  2. [Abstract] Abstract: the range of temperatures T and coupling values λ used in the numerics should be stated explicitly, together with any cutoff or lattice-spacing dependence.
  3. [Introduction / Methods] Notation: the definition of the thermal OTOC C_T(t) and the precise operator ordering should be stated at first appearance; standard references on OTOCs in QFT (e.g., Maldacena-Shenker-Stanford) are missing.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We have addressed each major point below, making revisions where feasible to improve clarity, add checks, and strengthen the connection to the continuum limit. Our responses are provided point by point.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the reported scaling λ ∼ T^{1/4} is dimensionally inconsistent in 1+1D, where [λ] = 2 and [T] = 1 (in units ħ = c = 1), so that λ/T^{1/2} is the only dimensionless combination; no lattice spacing a, effective coupling, or unit choice is specified to reconcile the exponent. This is load-bearing for the chaos diagnosis.

    Authors: We thank the referee for identifying this dimensional issue. Our lattice discretization is performed in units where the lattice spacing a = 1, which sets the scale and renders the combination dimensionally consistent within the discrete theory. To clarify this, we will revise the abstract and relevant sections to explicitly state the choice of units, introduce the lattice spacing a, and express the scaling as λ ∼ (T/a)^{1/4} (with appropriate factors to match [λ] = 1 in energy units). We will also add a short paragraph on dimensional analysis in the lattice model, including the effective coupling strength. This revision directly addresses the concern and supports the chaos interpretation. revision: yes

  2. Referee: [Two-oscillator analysis] Numerical evaluation of C_T(t) (two-oscillator section): the exponential growth is obtained from second-order perturbation without reported error bars, finite-N scaling, or explicit comparison to third-order or resummed contributions. Early-time growth can appear in non-chaotic truncations, so these checks are required to support the claim that the growth diagnoses genuine quantum chaos.

    Authors: We agree that these robustness checks are valuable. In the revised manuscript we will include error bars on the numerical evaluation of C_T(t) obtained from the quadrature method used for the thermal trace. Although the two-oscillator system is fixed at N = 2, we will add a comparison of the OTOC growth to the free (λ = 0) case, where no exponential growth appears, and compute the leading third-order perturbative correction for a representative temperature to show that the Lyapunov exponent remains stable. We will also discuss why the long-time window of exponential growth is unlikely to be a truncation artifact. These additions will be incorporated, though a full resummation lies beyond the present scope. revision: partial

  3. Referee: [N-oscillator chain] N-oscillator chain section: the assertion that the lattice discretization faithfully captures the chaotic properties of the continuum 1+1D theory lacks supporting tests (continuum extrapolation, comparison to integrable limits, or higher-order validation), which is essential because the central claim extends the two-oscillator result to the field theory.

    Authors: We acknowledge the importance of these validation tests. In the revision we will add explicit comparisons to the integrable free-field limit (λ = 0), where the OTOC saturates without exponential growth, and present results for several increasing values of N to demonstrate that the λ ∼ T^{1/4} scaling persists. While a complete continuum extrapolation (N → ∞ at fixed physical volume) would require resources beyond those available for this study, the observed stability with N provides supporting evidence for the extension to the field theory. We will include this discussion and the corresponding figures in the updated manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the paper's derivation chain

full rationale

The paper begins with explicit lattice discretization of λφ⁴ theory to obtain coupled anharmonic oscillators, applies standard second-quantization perturbation theory to derive analytic relations to second order for the two-oscillator system, and then numerically evaluates the thermal OTOC C_T(t) from those relations to observe exponential growth with λ∼T^{1/4}. This scaling and the signatures at low orders emerge from the numerical computation on the derived expressions rather than being imposed by construction, fitted parameters, or redefinition of inputs. The extension to the N-oscillator chain follows identically without self-definitional loops, load-bearing self-citations, or ansatz smuggling. All steps remain independent of the final claim of diagnosing quantum chaos, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The work rests on standard lattice QFT discretization and perturbative quantum mechanics; no new free parameters are introduced or fitted to produce the scaling result, which is observed numerically.

free parameters (2)
  • interaction strength λ
    Parameter of the λφ⁴ theory; not fitted to data in the reported calculations.
  • temperature T
    Enters the thermal OTOC; the scaling with T is observed rather than imposed.
axioms (2)
  • domain assumption Discretization of λφ⁴ scalar field theory on a lattice produces a system of coupled anharmonic oscillators
    Standard lattice regularization invoked in the opening sentence.
  • domain assumption Second-order perturbation theory in the second-quantization formalism is sufficient to derive analytic relations for the two-oscillator OTOC
    Used to obtain relations employed in the subsequent numerical step.

pith-pipeline@v0.9.0 · 5440 in / 1518 out tokens · 51530 ms · 2026-05-16T20:19:29.748514+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 19 internal anchors

  1. [1]

    Quasiclassical method in the theory of superconductivity,

    A. I. Larkin and Y. N. Ovchinnikov, “Quasiclassical method in the theory of superconductivity,” JETP 28, 6 (1969) 1200

    work page 1969

  2. [2]

    A simple model of quantum holography,

    A. Kitaev, “A simple model of quantum holography,” in KITP Strings Seminar and Entanglement 2015 Program (2015)

    work page 2015

  3. [3]

    Hidden correlations in the Hawking radiation and thermal noise,

    A. Kitaev, “Hidden correlations in the Hawking radiation and thermal noise,” in Proceedings of the KITP (2015)

    work page 2015

  4. [4]

    Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet

    S. Sachdev and J. Ye, “Gapless spin fluid ground state in a random, quantum Heisenberg magnet,” Phys. Rev. Lett. 70, 3339 (1993) [cond-mat/9212030]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  5. [5]

    A bound on chaos

    J. Maldacena, S. H. Shenker, and D. Stanford, “A bound on chaos,” JHEP 08 (2016) 106 [arXiv:1503.01409]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  6. [6]

    Comments on the Sachdev-Ye-Kitaev model

    J. Maldacena and D. Stanford, “Remarks on the Sachdev-Ye-Kitaev model,” Phys. Rev. D 94, no. 10, 106002 (2016) [arXiv:1604.07818 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  7. [7]

    The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual

    A. Kitaev and S. J. Suh, “The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual,” JHEP 05 (2018)183 [ arXiv:1711.08467 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  8. [8]

    Black holes and the butterfly effect

    S. H. Shenker and D. Stanford, “Black holes and the butterfly effect,” JHEP 03 (2014) 067 [arXiv:1306.0622]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  9. [9]

    Stringy effects in scrambling

    S. H. Shenker and D. Stanford, “Stringy effects in scrambling,” JHEP. 05 (2015) 132 [arXiv:1412.6087]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  10. [10]

    Two-dimensional conformal field theory and the butterfly effect

    D. A. Roberts and D. Stanford, “Two-dimensional conformal field theory and the butterfly effect,” PRL. 115 (2015) 131603 [arXiv:1412.5123 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  11. [11]

    Multiple Shocks

    S.H. Shenker and D. Stanford, “Multiple Shocks,” JHEP 12 (2014) 046 [arXiv:1312.3296 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  12. [12]

    Localized shocks

    D. A. Roberts, D. Stanford and L. Susskind, “Localized shocks,” JHEP 1503 (2015) 051 [arXiv:1409.8180]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  13. [13]

    A Quantum Correction To Chaos

    A. L. Fitzpatrick and J. Kaplan, “A Quantum Correction To Chaos,” JHEP 05 (2016) 070 [arXiv:1601.06164]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  14. [14]

    On CFT and Quantum Chaos

    G. J. Turiaci and H. L. Verlinde, “On CFT and Quantum Chaos,” JHEP 1612 (2016) 110 [arXiv:1603.03020]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  15. [15]

    Chaos in AdS$_2$ holography

    Kristan Jensen, “Chaos in AdS2 holography,” Phys. Rev. Lett. 117 (2016) 111601 [arXiv: 1605.06098]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  16. [16]

    Out-of-time-order correlators in quantum mechanics

    K. Hashimoto, K. Murata and R. Yoshii, “Out-of-time-order correlators in quantum mechanics,” JHEP 1710, 138 (2017) [arXiv:1703.09435 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  17. [17]

    Out-of-time- order correlator in coupled harmonic oscillators,

    T. Akutagawa, K. Hashimoto, T. Sasaki, and R. Watanabe, “Out-of-time- order correlator in coupled harmonic oscillators,” JHEP 08 (2020) 013 [arXiv:2004.04381 [hep-th]]

    work page arXiv 2020

  18. [18]

    Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator,

    K. Hashimoto, K-B Huh, K-Y Kim, and R. Watanabe, “Exponential growth of out-of-time-order correlator without chaos: inverted harmonic oscillator,” JHEP 11 (2020) 068 [ arXiv:2007.04746]

    work page arXiv 2020

  19. [19]

    Universal level statistics of the out-of-time-ordered operator

    E. B. Rozenbaum, S. Ganeshan and V. Galitski, “Universal level statistics of the out-of-time-ordered operator,” Phys. Rev. B100, no. 3, 035112 (2019) [arXiv:1801.10591 [cond-mat.dis-nn]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  20. [20]

    Quantum and Classical Lyapunov Exponents in Atom-Field Interaction Systems

    J. Ch´ avez-Carlos, B. L´ opez-Del-Carpio, M. A. Bastarrachea-Magnani, P. Str´ ansk´ y, S. Lerma- Hern´ andez, L. F. Santos and J. G. Hirsch, “Quantum and Classical Lyapunov Exponents in Atom- Field Interaction Systems,” Phys. Rev. Lett.122, no. 2, 024101 (2019) [arXiv:1807.10292 [cond- mat.stat-mech]]. 16

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  21. [21]

    Scrambling in strongly chaotic weakly coupled bipartite systems: Universality beyond the Ehrenfest timescale,

    R. Prakash and A. Lakshminarayan, “Scrambling in strongly chaotic weakly coupled bipartite systems: Universality beyond the Ehrenfest timescale,” Phys. Rev. B101(2020) no.12, 121108 [arXiv:1904.06482 [quant-ph]]

    work page arXiv 2020

  22. [22]

    Out-of-time-order correlators in bipartite nonintegrable sys- tems,

    R. Prakash and A. Lakshminarayan, “Out-of-time-order correlators in bipartite nonintegrable sys- tems,” Acta Phys. Polon. A136(2019), 803-810 [arXiv:1911.02829 [quant-ph]]

    work page arXiv 2019

  23. [23]

    Recent developments in the holographic description of quantum chaos

    V. Jahnke, “Recent developments in the holographic description of quantum chaos,” Adv. High Energy Phys. 2019, 9632708 (2019) [arXiv:1811.06949[[hep-th]]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  24. [24]

    Generalised out-of-time-order correlator in supersymmetric quantum mechanics,

    R. N. Das, S. Dutta, and A. Maji, “Generalised out-of-time-order correlator in supersymmetric quantum mechanics,” JHEP 08 (2020) 013 [arXiv:2010.07089 [ quant-ph]]

    work page arXiv 2020

  25. [25]

    Quantum mechanical out-of-time-ordered-correlators for the anharmonic (quartic) oscillator,

    P. Romatschke, “Quantum mechanical out-of-time-ordered-correlators for the anharmonic (quartic) oscillator,” JHEP, 2101 (2021) 030 [arXiv:2008.06056 [hep-th]]

    work page arXiv 2021

  26. [26]

    Out-of-Time-Order Correlation at a Quantum Phase Transition

    H. Shen, P. Zhang, R. Fan, and H. Zhai, “Out-of-time-order correlation at a quantum phase tran- sition,” Phys. Rev. B 96 (2017) 054503 [arXiv:1608.02438 [cond-mat.quant-gas]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  27. [27]

    Onset of many-body chaos in the $O(N)$ model

    D. Chowdhury and B. Swingle, “Onset of many-body chaos in the O(N) model,” Phys. Rev D 96 (2017) 065005 [arXiv:1703.02545 [cond-mat.str-el]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  28. [28]

    Out-of-time-order Operators and the Butterfly Effect

    J. S. Cotler, D. Ding, and G. R. Penington, “Out-of-time-order Operators and the Butterfly Effect,” Annals Phys. 396 (2018) 318 [arXiv:1704.02979 [quant-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  29. [29]

    Early-time exponential instabilities in nonchaotic quantum systems,

    E. B. Rozenbaum, L. A. Bunimovich, and V. Galitski, “Early-time exponential instabilities in nonchaotic quantum systems,” Phys. Rev. Lett. 125 (2020) 014101 [arXiv:1902.05466]

    work page arXiv 2020

  30. [30]

    The Multi- faceted Inverted Harmonic Oscillator: Chaos and Complexity,

    A. Bhattacharyya, W. Chemissany, and S. S. Haque, J. Murugan, and B. Yan, “The Multi- faceted Inverted Harmonic Oscillator: Chaos and Complexity,” SciPost Phys. Core 4 (2021) 002 [arXiv:2007.01232 [hep-th]]

    work page arXiv 2021

  31. [31]

    Extracting classical Lyapunov exponent from one-dimensional quantum mechanics,

    T. Morita, “Extracting classical Lyapunov exponent from one-dimensional quantum mechanics,” Phys.Rev.D 106 (2022) 106001 [arXiv:2105.09603 [hep-th]]

    work page arXiv 2022

  32. [32]

    Perturbative OTOC and Quantum Chaos in Harmonic Oscillators : Second Quantization Method,

    Wung-Hong Huang, “Perturbative OTOC and Quantum Chaos in Harmonic Oscillators : Second Quantization Method,” [arXiv : 2306.03644 [hep-th]]

    work page arXiv

  33. [33]

    Second-order Perturbative OTOC of Anharmonic Oscillators,

    Wung-Hong Huang, “Second-order Perturbative OTOC of Anharmonic Oscillators,” [arXiv:2311.04541]

    work page arXiv

  34. [34]

    Third-order Perturbative OTOC of Anharmonic Oscillators,

    Wung-Hong Huang, “Third-order Perturbative OTOC of Anharmonic Oscillators,” [arXiv:2407.17500]

    work page arXiv

  35. [35]

    Perturbative complexity of interacting theory,

    Wung-Hong Huang, “Perturbative complexity of interacting theory,” Phys. Rev. D 103, (2021) 065002 [arXiv:2008.05944 [hep-th]]

    work page arXiv 2021

  36. [36]

    Stanford,Many-body chaos at weak coupling,JHEP10(2016) 009, [1512.07687]

    D. Stanford, “Many-body chaos at weak coupling,” JHEP 10 (2016) 009 [arXiv:1512.07687 [hep-th]]

    work page arXiv 2016

  37. [37]

    Classical and quantum butterfly effect in nonlinear vector me- chanics,

    N. Kolganov and D. A. Trunin, “Classical and quantum butterfly effect in nonlinear vector me- chanics,” Phys. Rev. D 106 (2022) , 025003 [arXiv:2205.05663 [hep-th]]. 17

    work page arXiv 2022