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Phase Transitions in Quasi-Periodically Driven Quantum Critical Systems: Analytical Results

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Pith's one-line read In a quasiperiodically driven quantum critical system, the boundary between heating and non-heating phases is fixed by one analytic condition, and the Lyapunov exponent controlling entanglement growth is obtained in closed form.

desk verdict Genuinely new type-II quasiperiodic-driven CFT setup with a clean Avila-based analysis; the exact phase diagram claim needs a caveat where uniform hyperbolicity occurs. read the letter →

arxiv 2501.04795 v1 pith:PQ7TETGC submitted 2025-01-08 cond-mat.stat-mech cond-mat.str-elhep-thmath-phmath.MPquant-ph

Jiyuan Fang , Qi Zhou , Xueda Wen This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.str-elhep-thmath-phmath.MPquant-ph MSC 81T4082B2637H15
keywords quasiperiodicdrivingconformalfieldtheoryheatingphasetransitionLyapunovexponentcocycleentanglemententropySL(2R)deformationdrivencriticalsystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that a one-dimensional quantum critical system described by a conformal field theory can undergo a sharp transition between a heating phase, where entanglement entropy grows linearly in time, and a non-heating phase, where it oscillates, when the driving Hamiltonians themselves vary quasi-periodically (type-II driving). The operator evolution in each step is an SU(1,1) matrix, so the whole drive becomes a one-frequency quasiperiodic cocycle; using the global theory of such cocycles, the authors derive the Lyapunov exponent and the complete phase diagram analytically. For the concrete example studied, the critical line is $|a|/\sqrt{1-a^2}\cdot\sinh\alpha = 1$, and outside the uniformly hyperbolic regime (where the exponential growth is uniform in the phase) the Lyapunov exponent is $\max\{\log(|a|/\sqrt{1-a^2}\cdot\sinh\alpha),0\}$. They further prove that the previously proposed type-I protocol, in which the duration of one Hamiltonian varies quasi-periodically, is always in the heating phase and has no phase transition. The CFT predictions are checked against free-fermion lattice simulations of entanglement entropy.

What carries the argument

The central object is a one-frequency analytic quasiperiodic cocycle---a sequence of $\mathrm{SL}(2,\mathbb{R})$ matrices indexed by a phase that advances by the irrational frequency each step---built as $A(x)=M_0(x)M_1$, where $M_0,M_1\in\mathrm{SU}(1,1)$ are the fractional-linear transformations describing how local operators evolve during one driving step. The argument complexifies the phase, $x\to x+i\epsilon$; the global theory of such cocycles guarantees that $\lambda_L(\epsilon)$ is convex and piecewise linear with integer slope (the acceleration), and the large-$\epsilon$ asymptotic is dominated by a constant matrix whose Lyapunov exponent is $\log|\beta_1\sinh\alpha|$ (or $\log|\alpha_1|$ for type-I). Combining the asymptotic value with the integer-slope constraint yields the max-formula for $\lambda_L$, and the vanishing of its argument locates the critical line. The same cocycle determines the physical diagnostic through $S_A(n)-S_A(0)=\frac{c}{3}(\log|\alpha_n+\beta_n|+\log|\alpha'_n+\beta'_n|)$.

What would settle it

Numerically compute $\lambda_L(0)$ from the matrix product (2.19) on a fine grid straddling the line $|a|/\sqrt{1-a^2}\cdot\sinh\alpha=1$; if any point on the predicted non-heating side has a positive Lyapunov exponent, or the entanglement entropy grows linearly there instead of oscillating, the phase boundary as stated is not exact. The paper's Fig. 9 already provides a parameter set where the max-formula misses a uniformly hyperbolic heating case, so the grid test would locate the true boundary.

Watch

Extended reading notes

Core claim

The paper's central claim is that in type-II quasiperiodic driving the phase diagram is controlled by the cocycle $A(x)=M_0(x)M_1$ and, for cocycles that are not uniformly hyperbolic, its complexified Lyapunov exponent has the form $\lambda_L(\epsilon)=\max\{\log|\beta_1\sinh\alpha|+\epsilon,0\}$. This makes the phase transition occur exactly when $|\beta_1\sinh\alpha|=1$, and gives the heating rate $\lambda_L(0)=\max\{\log|\beta_1\sinh\alpha|,0\}$; for the representative choice $a_0=1,a_+=a,a_-=0$ the condition becomes $|a|/\sqrt{1-a^2}\cdot\sinh\alpha=1$. In the heating phase the entanglement entropy grows as $S_A(n)\simeq (2c/3)\lambda_L n$, it grows logarithmically at the critical point, and it oscillates in the non-heating phase; the authors verify this behavior in lattice free-fermion calculations. For type-I driving, they prove $\lambda_L(0)=\log|\alpha_1|>0$ always, so no phase transition can appear. The paper thus asserts that type-II quasiperiodicity produces genuine, analytically determined heating transitions, while the earlier type-I quasiperiodicity cannot.

Load-bearing premise

The analytic phase boundary and Lyapunov exponent assume the type-II cocycle is not uniformly hyperbolic, a condition that the paper checks numerically at many parameter values but does not prove globally; where it fails, the formula can predict no heating when the actual Lyapunov exponent is positive, and the paper shows one such case.

Editorial extensions

If this is right

  • In the type-II protocol, the entire heating/non-heating phase diagram follows from one analytic formula, so no exhaustive parameter scan is needed to know where linear entanglement growth starts.
  • The linear growth rate of entanglement entropy in the heating phase is quantitatively fixed by the Lyapunov exponent, and the paper's lattice simulations confirm that prediction.
  • The previously studied type-I quasiperiodic protocol provably cannot host a heating-to-non-heating transition, so any non-heating behavior seen there would have to come from a different mechanism.
  • Phase transitions appear in all nine elliptic/parabolic/hyperbolic combinations of the two driving Hamiltonians, indicating that the transition is a generic feature of type-II driving rather than a special-case effect.
  • The uniform-versus-non-uniform hyperbolicity distinction is visible through subleading fluctuations of the entanglement entropy, giving a separate observable signature beyond the leading growth rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because formula (4.9) is only derived for non-uniformly hyperbolic cocycles, an exact global phase diagram would require locating uniformly hyperbolic islands; the paper's own Fig. 9 suggests these islands can sit near the predicted line, so the true boundary may differ there.
  • The same cocycle-plus-acceleration technique may transfer to other finite-dimensional symmetry groups or multi-band drives whenever the one-step evolution is an analytic matrix function of the quasi-periodic phase, though the integer-acceleration theorem used here is special to SL(2,R)-type cocycles.
  • A natural next step would be to look for the quantized acceleration $\omega_\lambda$ as a directly measurable quantity in the lattice model---for example, as a winding number or a quantized energy absorption rate---since the paper raises this as an open question without establishing it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies quasi-periodically driven (1+1)-dimensional CFTs with sl(2,R) deformations, building on earlier work on periodically, quasi-periodically, and randomly driven CFTs. Two driving protocols are considered: type-I, where the timing of a fixed Hamiltonian is quasiperiodic, and type-II, where the Hamiltonian itself depends quasiperiodically on the step index. Using Avila's global theory of one-frequency analytic SL(2,R) cocycles, the authors derive exact-looking formulas for the Lyapunov exponent λ_L and the acceleration ω_λ, and from them the heating/non-heating phase diagram. For type-I driving they show that λ_L>0 always, proving the absence of a phase transition. For type-II driving they obtain λ_L = max{log|β_1 sinh α|, 0} and a phase boundary |β_1 sinh α| = 1, which for their explicit choice becomes |a/√(1−a²) sinh α| = 1 (Eq. 4.10). The analytical predictions are compared with numerical Lyapunov exponents and with CFT/lattice entanglement-entropy evolution, with good agreement away from special parameter regions.

Significance. The paper makes a useful methodological contribution by bringing Avila's global theory into the study of driven CFTs, where it gives parameter-free formulas for the Lyapunov exponents without any fitting. The type-I result is a clean, correct explanation of the previously observed absence of phase transitions, and the type-II setup is a new and interesting way to obtain heating/non-heating transitions in quasiperiodic driving. The numerical checks, including lattice simulations of entanglement entropy, support the analysis in the regimes where the authors' assumptions hold. The main limitation is that the type-II phase diagram is not fully exact as claimed, because the analytic expression for λ_L is derived under the assumption that the cocycle is not uniformly hyperbolic, and the paper itself identifies uniformly hyperbolic regions where the formula fails and the phase boundary is shifted.

major comments (2)
  1. [Sec. IV.B–IV.C, Eqs. (4.9)–(4.10)] The central claim that the type-II phase boundary is exactly |a/√(1−a²) sinh α| = 1 is not supported, because Eq. (4.9) is derived under the assumption that the cocycle is not uniformly hyperbolic, and the manuscript itself reports a counterexample. In Fig. 9 (bottom) the parameters (a, α) = (0.25, 2) give |a/√(1−a²) sinh α| ≈ 0.936 < 1, so Eq. (4.10) places the system in the non-heating phase; however, the complexified Lyapunov exponent shown there is uniformly hyperbolic with λ_L(0) > 0, i.e., the system heats. The authors acknowledge 'subtle deviations' near the phase transitions but do not characterize the uniform-hyperbolic regions analytically, so the phase diagram in Fig. 7 and the abstract's statement that the phase diagram 'can be analytically obtained' are overstated. To make the claim exact, the authors need to either prove that uniformly hyperbolic heating does not occur on the non-heating side of (4.10), or provide an analytic description of the actual transition including those regions.
  2. [Appendix C and Abstract] The same non-uniform-hyperbolicity caveat applies to all the general type-II results in Appendix C: Eqs. (C6), (C9), and (C11) are stated conditional on the cocycle being non-uniformly hyperbolic, and no proof is given that the phase boundaries in Fig. 13 are valid in uniformly hyperbolic patches. Since the abstract and the conclusions of Sec. V make the unqualified claim that the phase diagrams and Lyapunov exponents are analytically obtained, the paper's central claim goes beyond what the derivation supports. The authors should either extend the analysis to cover uniformly hyperbolic cocycles or explicitly restrict the claim to the non-uniformly hyperbolic case and state the extent to which the plotted phase boundaries are proven.
minor comments (3)
  1. [Eq. (2.19)] The definition of the Lyapunov exponent in Eq. (2.19) appears to be missing the logarithm: as written it is the growth rate of the norm itself, but the subsequent formulas use λ_L as the exponential rate, i.e., λ_L = lim (1/n) log ||M_1⋯M_n||. Please correct the displayed equation.
  2. [Sec. IV.C, text near Fig. 9] The cross-references in the paragraph discussing non-uniform versus uniform hyperbolicity are inconsistent: the text says 'as shown in Fig. 10' and 'as seen in Fig. 10 (top)' / 'Fig. 10 (bottom)', but the relevant panels are in Fig. 9, which displays the complexified Lyapunov exponents; Fig. 10 shows entanglement entropy evolutions. Please fix the figure references.
  3. [Sec. IV.C, footnote 7] The footnote says that near the phase transition there could be uniformly hyperbolic cases with very small Lyapunov exponents, but the example shown in Fig. 9 for (a, α) = (0.25, 2) is not a case of a small Lyapunov exponent; it is a uniformly hyperbolic heating case where Eq. (4.9) gives subcritical behavior. Clarify the intended meaning, or adjust the footnote to match the actual counterexample.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Lyapunov exponents and phase boundaries are parameter-free analytic functions derived from Avila's external theory, not fitted inputs.

full rationale

No significant circularity. The paper's central results, the type-I Lyapunov exponent λL = log|α1*| and the type-II formula λL(ε) = max{log|β1 sinh α| + ε, 0}, are derived by applying Avila's global theory for one-frequency quasiperiodic SL(2,R) cocycles to the explicit SU(1,1) transfer matrices of the driven CFT. These are parameter-free functions of the driving parameters, not quantities fitted to the numerical data; the lattice and CFT numerics are used as verification, not as inputs. The type-II phase boundary |β1 sinh α| = 1 follows algebraically from setting the argument of the max to zero, and the paper openly states the condition under which this formula applies: if the cocycle is uniformly hyperbolic, (4.9) no longer holds, and the paper itself displays such a case in Fig. 9. That caveat is a correctness limitation on the claimed exactness of the analytic phase diagram, not a circular dependence. Self-citations to the authors' earlier work [1,25] supply the CFT framework and previously observed numerical facts, but the present derivation does not reduce to those citations; the load-bearing classification and Lyapunov-exponent formulas come from Avila's independently established theory. No fitted parameter is renamed as a prediction, and no result is defined in terms of the quantity it purports to explain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data; the driving parameters (a, alpha, theta, T0, T1) are physical inputs. The main unproved input is Avila's theorem, plus the CFT-to-cocycle mapping. The most fragile assumption is non-uniform hyperbolicity, which is not established analytically in type-II and is contradicted by the paper's own examples.

assumptions (5)
  • standard math Avila's global theory: for one-frequency analytic SL(2,R) cocycles, the complexified Lyapunov exponent lambda_L(epsilon) is convex and piecewise linear with integer slopes, and uniform hyperbolicity is equivalent to positive lambda_L with zero acceleration.
    Invoked in Sec. I.C and used to derive Eqs. (3.7), (4.4), and (4.5). Not proved in the paper.
  • domain assumption Operator evolution under sl(2,R) deformed CFT Hamiltonians is a Mobius transformation generated by SU(1,1) matrices, and the entanglement entropy is given by log|alpha_n + beta_n| in Eq. (2.20).
    Background from earlier driven CFT literature [1,15,21], used throughout the paper.
  • domain assumption The quasiperiodically driven evolution is equivalent to multiplying SU(1,1) matrices along a rotation of phase x = pi*omega*n on S1, so Avila's theory applies.
    Setup in Sec. III.A and IV.A; requires identifying the driving phase with the circle coordinate.
  • ad hoc to paper The cocycles in the type-II phase diagram are not uniformly hyperbolic in the regions where Eqs. (4.9) and (4.10) are used.
    Eq. (4.5) requires non-uniform hyperbolicity; the paper does not prove this and reports uniformly hyperbolic counterexamples, e.g., (a, alpha) = (0.25, 2) in Fig. 9.
  • domain assumption A positive Lyapunov exponent implies linear growth of entanglement entropy, and lambda_L = 0 at the transition implies logarithmic growth.
    Used in Sec. II and IV.D, based on prior CFT results [1,15].

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Pith. "Pith review of Phase Transitions in Quasi-Periodically Driven Quantum Critical Systems: Analytical Results." pith.science (2026). https://pith.science/paper/PQ7TETGC

@misc{pith2026250104795,
  author       = {Pith},
  title        = {Pith review of: Phase Transitions in Quasi-Periodically Driven Quantum Critical Systems: Analytical Results},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQ7TETGC}},
  note         = {Machine review of arXiv:2501.04795}
}
read the original abstract

In this work, we study analytically the phase transitions in quasi-periodically driven one dimensional quantum critical systems that are described by conformal field theories (CFTs). The phase diagrams and phase transitions can be analytically obtained by using Avila's global theory in one-frequency quasiperiodic cocycles. Compared to the previous works where the quasiperiodicity was introduced in the driving time and no phase transitions were observed [1], here we propose a setup where the quasiperiodicity is introduced in the driving Hamiltonians. In our setup, one can observe the heating phases, non-heating phases, and the phase transitions. The phase diagram as well as the Lyapunov exponents that determine the entanglement entropy evolution can be analytically obtained. In addition, based on Avila's theory, we prove there is no phase transition in the previously proposed setup of quasi-periodically driven CFTs [1]. We verify our field theory results by studying the time evolution of entanglement entropy on lattice models.

Figures

Figures reproduced from arXiv: 2501.04795 by the authors.

Figure 1
Figure 1. FIG. 1. Two different ways of doing quasiperiodic drivings as [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Complexified Lyapunov exponents [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Lyapunov exponents [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figures from the paper (9 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Complexified Lyapunov exponents [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Phase diagram of a quasi-periodically driven CFT [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of Lyapunov exponents obtained from [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Different behaviors of complexified Lyapunov ex [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of entanglement entropy evolution [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Entanglement entropy evolution for subsystem [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Phase diagram of a quasi-periodically driven CFT [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Comparison of entanglement entropy evolution for [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]

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Forward citations

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Works this paper leans on

88 extracted references · 42 canonical work pages · cited by 2 Pith papers

  1. [1]

    Can one have a setupofquasi-periodicallydrivenCFTswherethere are both heating phases and non-heating phases with phase transitions?

    For the setups of quasi-periodically driven CFTs thatarealreadystudiedintheliterature[1,12], only the heating phases were observed. Can one have a setupofquasi-periodicallydrivenCFTswherethere are both heating phases and non-heating phases with phase transitions?

  2. [2]

    For a general setup of quasi-periodically driven CFT, e.g., the Aubry-Andre-like driving in Ref.[1], can one find an analytical way to determine the phase diagram? Furthermore, in the heating phase, can one obtain analytically the Lyapunov expo- nents which determine the entanglement entropy evolution? In this work, we will give affirmative answers to the...

  3. [3]

    We generalize the setup of quasi-periodically driven CFTs in [1, 12] to the case where the driving Hamiltonians themselves depend on parameters in a quasi-periodical way, which is illustrated in the type-II driving in Fig.1.2 In this new setup, we find the phase transition between heating and non- heating phases is a generic feature

  4. [4]

    Both types of quasiperiodic drivings in Fig.1 can be studiedbasedonAvila’sglobaltheoryforquasiperi- odic cocycles. By applying Avilia’s global theory, one can evaluate the Lyapunov exponents as well as the acceleration analytically, based on which one can obtain the phase diagrams of quasi-periodically driven CFTs. For the type-I driving in Fig.1, we can ...

  5. [5]

    The subcritical regime:λL = 0 and ωλ = 0

  6. [6]

    The critical regime:λL = 0 and ωλ∈ Z+

  7. [7]

    Here Z+ denotepositiveintegers

    The supercritical regime:λL > 0 and ωλ∈ Z+. Here Z+ denotepositiveintegers. Pictorially, thefeatures of the complexified Lyapunov exponentsλL(ϵ) for these threedifferentcasesareshowninFig.2. Onecanfindthat althoughλL = 0 in both subcritical and critical regimes, it is stable in the subcritical regime and unstable in the critical regime under complexificat...

  8. [8]

    λ< 1: λL(E) = 0, andωλ(E) = 0

Show all 88 references
  1. [9]

    λ = 1: λL(E) = 0, andωλ(E) = 1

  2. [10]

    order param- eter

    λ> 1: λL(E) = log|λ|, andωλ(E) = 1. These three cases correspond to the subcritical, crit- ical, and supercritical regimes in Fig.2, and physically they correspond to the metallic phase, phase transition, and the insulating phase, respectively. The rest of this work is organiz...

  3. [11]

    More explicitly, the heating phase is determined by λL(ϵ = 0) > 0, i.e.,|β1· sinh(α)| > 1, and the phase transitions to the non-heating phases are determined by |β1· sinh(α)| = 1

    may be zero, indicating the possible phase transi- tions. More explicitly, the heating phase is determined by λL(ϵ = 0) > 0, i.e.,|β1· sinh(α)| > 1, and the phase transitions to the non-heating phases are determined by |β1· sinh(α)| = 1. (4.6) In the next subsection, we will g...

  4. [12]

    Here n denotes then-th time driving cycle

    Quasiperiodic Hamiltonians that are elliptic The driving Hamiltonians H0 are elliptic, which are deformed quasi-periodically in time as a0 = 1, a + =α cos(πω·n), a − =α sin(πω·n), (C1) where α∈ (0, 1), ω is an irrational number, andn∈ Z. Here n denotes then-th time driving cyc...

  5. [13]

    non-uniform

    Quasiperiodic Hamiltonians that are parabolic NowweconsidertheparabolicHamiltonians H0, which are deformed quasi-periodically in time as a0 = 1, a + = cos(πω·n), a − = sin(πω·n), (C12) 17 wheren denotes then-th driving cycle. The correspond- ing SU(1, 1) matrix in then-th driv...

  6. [15]

    Sachdev, Quantum Phase Transitions, 2nd ed

    S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cam- bridge University Press, 2011)

    2011

  7. [16]

    P. T. Dumitrescu, R. Vasseur, and A. C. Potter, Log- arithmically slow relaxation in quasiperiodically driven random spin chains, Phys. Rev. Lett.120, 070602 (2018)

    2018

  8. [17]

    Giergiel, A

    K. Giergiel, A. Kuroś, and K. Sacha, Discrete time qua- sicrystals, Phys. Rev. B99, 220303 (2019)

    2019

  9. [18]

    H. Zhao, F. Mintert, and J. Knolle, Floquet time spi- rals and stable discrete-time quasicrystals in quasiperiod- ically driven quantum many-body systems, Phys. Rev. B 100, 134302 (2019), arXiv:1906.06989 [cond-mat.dis-nn]

    2019 arXiv

  10. [19]

    D. R. Grempel, R. E. Prange, and S. Fishman, Quantum dynamics of a nonintegrable system, Phys. Rev. A29, 1639 (1984)

    1984

  11. [20]

    F. L. Moore, J. C. Robinson, C. Bharucha, P. E. Williams, and M. G. Raizen, Observation of dynamical localization in atomic momentum transfer: A new test- ing ground for quantum chaos, Phys. Rev. Lett.73, 2974 (1994)

    1994

  12. [21]

    Martin, G

    I. Martin, G. Refael, and B. Halperin, Topological fre- quency conversion in strongly driven quantum systems, Phys. Rev. X7, 041008 (2017)

    2017

  13. [22]

    Peng and G

    Y. Peng and G. Refael, Topological energy conversion through the bulk or the boundary of driven systems, Phys. Rev. B97, 134303 (2018)

    2018

  14. [23]

    D. V. Else, W. W. Ho, and P. T. Dumitrescu, Long- lived interacting phases of matter protected by multiple time-translation symmetries in quasiperiodically driven systems, Phys. Rev. X10, 021032 (2020)

    2020

  15. [24]

    A. J. Friedman, B. Ware, R. Vasseur, and A. C. Potter, Topological edge modes without symmetry in quasiperi- odically driven spin chains, Phys. Rev. B105, 115117 (2022), arXiv:2009.03314 [cond-mat.str-el]

    2022 arXiv

  16. [25]

    Lapierre, K

    B. Lapierre, K. Choo, A. Tiwari, C. Tauber, T. Neu- pert, and R. Chitra, The fine structure of heating in a quasiperiodically driven critical quantum system, arXiv e-prints , arXiv:2006.10054 (2020), arXiv:2006.10054 [cond-mat.str-el]

    2020 arXiv

  17. [26]

    Schmid, Y

    H. Schmid, Y. Peng, G. Refael, and F. von Oppen, Self- similar phase diagram of the Fibonacci-driven quantum Ising model, arXiv e-prints , arXiv:2410.18219 (2024), arXiv:2410.18219 [cond-mat.dis-nn]

    2024

  18. [27]

    Avila, Global theory of one-frequency Schrödinger op- erators, Acta Mathematica215, 1 (2015)

    A. Avila, Global theory of one-frequency Schrödinger op- erators, Acta Mathematica215, 1 (2015)

    2015

  19. [28]

    Wen and J.-Q

    X. Wen and J.-Q. Wu, Floquet conformal field theory, (2018), arXiv:1805.00031 [cond-mat.str-el]

    2018 arXiv

  20. [29]

    Wen and J.-Q

    X. Wen and J.-Q. Wu, Quantum dynamics in sine-square deformed conformal field theory: Quench from uniform to nonuniform conformal field theory, Physical Review B 97, 184309 (2018), arXiv:1802.07765 [cond-mat.str-el]

    2018 arXiv

  21. [30]

    R. Fan, Y. Gu, A. Vishwanath, and X. Wen, Emergent Spatial Structure and Entanglement Localization in Flo- quet Conformal Field Theory, Phys. Rev. X10, 031036 (2020), arXiv:1908.05289 [cond-mat.str-el]

    2020 arXiv

  22. [31]

    Lapierre, K

    B. Lapierre, K. Choo, C. Tauber, A. Tiwari, T. Neupert, and R. Chitra, Emergent black hole dynamics in criti- cal Floquet systems, Phys. Rev. Res.2, 023085 (2020), arXiv:1909.08618 [cond-mat.str-el]

    2020 arXiv

  23. [32]

    Lapierre and P

    B. Lapierre and P. Moosavi, Geometric approach to in- homogeneous Floquet systems, Physical Review B103, 224303 (2021), arXiv:2010.11268 [cond-mat.stat-mech]

    2021 arXiv

  24. [33]

    R. Fan, Y. Gu, A. Vishwanath, and X. Wen, Floquet conformal field theories with generally deformed Hamil- tonians, SciPost Phys.10, 049 (2021), arXiv:2011.09491 [hep-th]

    2021 arXiv

  25. [34]

    Han and X

    B. Han and X. Wen, Classification of S L2 deformed Flo- quet conformal field theories, Physical Review B 102, 205125 (2020), arXiv:2008.01123 [cond-mat.stat-mech]

    2020 arXiv

  26. [35]

    D. S. Ageev, A. A. Bagrov, and A. A. Iliasov, Determinis- tic chaos and fractal entropy scaling in Floquet conformal field theories, Physical Review B103, L100302 (2021), arXiv:2006.11198 [cond-mat.stat-mech]

    2021 arXiv

  27. [36]

    Andersen, F

    M. Andersen, F. Norfjand, and N. T. Zinner, Real-time correlation function of floquet conformal fields, Phys. Rev. D103, 056005 (2021)

    2021

  28. [37]

    D. Das, R. Ghosh, and K. Sengupta, Conformal Floquet dynamics with a continuous drive protocol, Journal of High Energy Physics2021, 172 (2021), arXiv:2101.04140 [hep-th]

    2021 arXiv

  29. [39]

    Sengupta, Out-of-Time-Order correlators in driven conformal field theories, Journal of High Energy Physics 2022, 221 (2022), arXiv:2202.12815 [hep-th]

    S.Das, B.Ezhuthachan, A.Kundu, S.Porey, B.Roy,and K. Sengupta, Out-of-Time-Order correlators in driven conformal field theories, Journal of High Energy Physics 2022, 221 (2022), arXiv:2202.12815 [hep-th]

    2022 arXiv

  30. [40]

    Bermond, M

    B. Bermond, M. Chernodub, A. G. Grushin, and D. Carpentier, Anomalous Luttinger equivalence be- tween temperature and curved spacetime: From black hole’s atmosphere to thermal quenches, arXiv e-prints , arXiv:2206.08784 (2022), arXiv:2206.08784 [cond- mat.stat-mech]

    2022 arXiv

  31. [41]

    K. Choo, B. Lapierre, C. Kuhlenkamp, A. Tiwari, T. Ne- upert, and R. Chitra, Thermal and dissipative effects on the heating transition in a driven critical system, arXiv e-prints , arXiv:2205.02869 (2022), arXiv:2205.02869 [cond-mat.str-el]

    2022 arXiv

  32. [42]

    X. Wen, R. Fan, and A. Vishwanath, Floquet’s Refrig- erator: Conformal Cooling in Driven Quantum Criti- cal Systems, arXiv e-prints , arXiv:2211.00040 (2022), arXiv:2211.00040 [cond-mat.str-el]

    2022 arXiv

  33. [44]

    Nozaki, K

    M. Nozaki, K. Tamaoka, and M. Tian Tan, In- homogeneous quenches as state preparation in two- dimensional conformal field theories, arXiv e-prints , arXiv:2310.19376 (2023), arXiv:2310.19376 [hep-th]

    2023 arXiv

  34. [47]

    K. Goto, M. Nozaki, S. Ryu, K. Tamaoka, and M. T. Tan, Scrambling and recovery of quantum information in inhomogeneous quenches in two-dimensional confor- mal field theories, Physical Review Research6, 023001 (2024), arXiv:2302.08009 [hep-th]

    2024 arXiv

  35. [49]

    K. Goto, T. Guo, T. Nosaka, M. Nozaki, S. Ryu, and K.Tamaoka,Spatialdeformationofmany-bodyquantum chaotic systems and quantum information scrambling, Phys. Rev. B 109, 054301 (2024), arXiv:2305.01019 [quant-ph]

    2024 arXiv

  36. [50]

    X. Wen, Y. Gu, A. Vishwanath, and R. Fan, Period- ically, Quasi-periodically, and Randomly Driven Con- formal Field Theories (II): Furstenberg’s Theorem and Exceptions to Heating Phases, SciPost Physics13, 082 (2022), arXiv:2109.10923 [cond-mat.stat-mech]

    2022 arXiv

  37. [51]

    Furstenberg, Noncommuting random products, Trans- actions of the American Mathematical Society108, 377 (1963)

    H. Furstenberg, Noncommuting random products, Trans- actions of the American Mathematical Society108, 377 (1963)

    1963

  38. [52]

    K. Goto, M. Nozaki, K. Tamaoka, M. Tian Tan, and S. Ryu, Non-Equilibrating a Black Hole with Inhomogeneous Quantum Quench, arXiv e-prints , arXiv:2112.14388 (2021), arXiv:2112.14388 [hep-th]

    2021 arXiv

  39. [53]

    Caputa and D

    P. Caputa and D. Ge, Entanglement and geometry from subalgebras of the Virasoro algebra, Journal of High En- ergy Physics 2023, 159 (2023), arXiv:2211.03630 [hep- th]

    2023 arXiv

  40. [54]

    de Boer, V

    J. de Boer, V. Godet, J. Kastikainen, and E. Keski- Vakkuri,QuantuminformationgeometryofdrivenCFTs, Journal of High Energy Physics 2023, 87 (2023), arXiv:2306.00099 [hep-th]

    2023 arXiv

  41. [55]

    Kudler-Flam, M

    J. Kudler-Flam, M. Nozaki, T. Numasawa, S. Ryu, and M. Tian Tan, Bridging two quantum quench problems – local joining quantum quench and Möbius quench – and their holographic dual descriptions, arXiv e-prints , arXiv:2309.04665 (2023), arXiv:2309.04665 [hep-th]

    2023 arXiv

  42. [56]

    Bernamonti, F

    A. Bernamonti, F. Galli, and D. Ge, Boundary- induced transitions in Möbius quenches of holo- graphic BCFT, arXiv e-prints , arXiv:2402.16555 (2024), arXiv:2402.16555 [hep-th]

    2024 arXiv

  43. [57]

    Miyata, M

    A. Miyata, M. Nozaki, K. Tamaoka, and M. Watan- abe, Hawking-Page and entanglement phase transition in 2d CFT on curved backgrounds, arXiv e-prints , arXiv:2406.06121 (2024), arXiv:2406.06121 [hep-th]

    2024 arXiv

  44. [58]

    Jiang and M

    H. Jiang and M. Mezei, New horizons for inhomo- geneous quenches and Floquet CFT, arXiv e-prints , arXiv:2404.07884 (2024), arXiv:2404.07884 [hep-th]

    2024 arXiv

  45. [59]

    Sengupta, Brane detectors of a dynamical phase transition in a driven CFT, SciPost Physics 15, 202 (2023), arXiv:2212.04201 [hep-th]

    S.Das, B.Ezhuthachan, A.Kundu, S.Porey, R.Baishali, and K. Sengupta, Brane detectors of a dynamical phase transition in a driven CFT, SciPost Physics 15, 202 (2023), arXiv:2212.04201 [hep-th]

    2023 arXiv

  46. [60]

    S. Das, B. Ezhuthachan, S. Porey, and B. Roy, Notes on heating phase dynamics in Floquet CFTs and Modular quantization, arXiv e-prints , arXiv:2406.10899 (2024), arXiv:2406.10899 [hep-th]

    2024 arXiv

  47. [61]

    C. Bai, A. Miyata, and M. Nozaki, Entanglement dynam- ics in 2d HCFTs on the curved background: the case of q-Möbius Hamiltonian, arXiv e-prints , arXiv:2408.06594 (2024), arXiv:2408.06594 [hep-th]

    2024 arXiv

  48. [62]

    Das and A

    J. Das and A. Kundu, Flowery Horizons & Bulk Ob- servers: sl(q)(2, R)Drivein 2dHolographicCFT,arXive- prints , arXiv:2412.18536 (2024), arXiv:2412.18536 [hep- th]

    2024 arXiv

  49. [63]

    D. Das, S. R. Das, A. Kundu, and K. Sengupta, Exactly Solvable Floquet Dynamics for Conformal Field Theo- ries in Dimensions Greater than Two, arXiv e-prints , arXiv:2311.13468 (2023), arXiv:2311.13468 [hep-th]

    2023 arXiv

  50. [64]

    Malvimat, S

    V. Malvimat, S. Porey, and B. Roy, Krylov Complexity in 2d CFTs with SL(2, R) deformed Hamiltonians, arXiv e-prints , arXiv:2402.15835 (2024), arXiv:2402.15835 [hep-th]

    2024 arXiv

  51. [65]

    Lapierre, T

    B. Lapierre, T. Numasawa, T. Neupert, and S. Ryu, Flo- quet engineered inhomogeneous quantum chaos in crit- ical systems, arXiv e-prints , arXiv:2405.01642 (2024), arXiv:2405.01642 [cond-mat.str-el]

    2024

  52. [66]

    W. Mao, M. Nozaki, K. Tamaoka, and M. Tian Tan, Lo- cal operator quench induced by two-dimensional inhomo- geneous and homogeneous CFT Hamiltonians, arXiv e- prints , arXiv:2403.15851 (2024), arXiv:2403.15851 [hep- th]

    2024 arXiv

  53. [67]

    X. Wen, Exactly solvable non-unitary time evolution in quantum critical systems I: effect of complex space- time metrics, Journal of Statistical Mechanics: Theory and Experiment2024, 103103 (2024), arXiv:2406.17059 [cond-mat.stat-mech]

    2024 arXiv

  54. [68]

    X. Liu, A. McDonald, T. Numasawa, B. Lian, and S. Ryu, Quantum Quenches of Conformal Field Theory with Open Boundary, arXiv e-prints , arXiv:2309.04540 (2023), arXiv:2309.04540 [cond-mat.stat-mech]

    2023 arXiv

  55. [69]

    Moosavi, M

    P. Moosavi, M. Christandl, G. M. Graf, and S. Sotiri- adis, Perfect Wave Transfer in Continuous Quan- tum Systems, arXiv e-prints , arXiv:2408.00723 (2024), arXiv:2408.00723 [quant-ph]

    2024

  56. [70]

    Banerjee, S

    T. Banerjee, S. Das, and K. Sengupta, Entan- glement asymmetry in periodically driven quantum systems, arXiv e-prints , arXiv:2412.03654 (2024), arXiv:2412.03654 [quant-ph]

    2024 arXiv

  57. [71]

    Caputa, J

    P. Caputa, J. M. Magan, and D. Patramanis, Geome- try of Krylov complexity, Physical Review Research4, 013041 (2022), arXiv:2109.03824 [hep-th]

    2022 arXiv

  58. [72]

    Allegra, J

    N. Allegra, J. Dubail, J.-M. Stéphan, and J. Viti, Inho- mogeneous field theory inside the arctic circle, Journal of Statistical Mechanics: Theory and Experiment5, 053108 (2016), arXiv:1512.02872 [cond-mat.stat-mech]

    2016 arXiv

  59. [73]

    Dubail, J.-M

    J. Dubail, J.-M. Stéphan, J. Viti, and P. Calabrese, Con- formal field theory for inhomogeneous one-dimensional quantum systems: the example of non-interacting Fermi gases, SciPost Physics 2, 002 (2017), arXiv:1606.04401 [cond-mat.str-el]

    2017 arXiv

  60. [74]

    J. Dubail, J.-M.Stéphan, andP.Calabrese,Emergence of curved light-cones in a class of inhomogeneous Luttinger 20 liquids, SciPost Physics3, 019 (2017), arXiv:1705.00679 [cond-mat.str-el]

    2017 arXiv

  61. [75]

    K.Gawędzki, E.Langmann,andP.Moosavi,Finite-Time Universality in Nonequilibrium CFT, Journal of Statis- tical Physics 172, 353 (2018), arXiv:1712.00141 [cond- mat.stat-mech]

    2018 arXiv

  62. [76]

    Langmann and P

    E. Langmann and P. Moosavi, Diffusive Heat Waves in Random Conformal Field Theory, Phys. Rev. Lett.122, 020201 (2019), arXiv:1807.10239 [cond-mat.stat-mech]

    2019 arXiv

  63. [77]

    X. Wen, R. Fan, A. Vishwanath, and Y. Gu, Periodi- cally, quasi-periodically, and randomly driven conformal field theories: Part i, (2020), arXiv:2006.10072 [cond- mat.stat-mech]

    2020 arXiv

  64. [78]

    A.Avila, J.You,andQ.Zhou,Sharpphasetransitionsfor the almost Mathieu operator, Duke Mathematical Jour- nal 166, 2697 (2017)

    2017

  65. [79]

    M. R. Herman, Une méthode pour minorer les exposants de lyapounov et quelques exemples montrant le caractere local dun théoreme d’arnold et de moser sur le tore de di- mension 2., Commentarii mathematici Helvetici58, 453 (1983)

    1983

  66. [80]

    Yoccoz, Some questions and remarks about sl (2, r) cocycles, Modern dynamical systems and applications , 447 (2004)

    J.-C. Yoccoz, Some questions and remarks about sl (2, r) cocycles, Modern dynamical systems and applications , 447 (2004)

    2004

  67. [81]

    Damanik, Schrödinger Operators with Dynami- cally Defined Potentials: A Survey, arXiv e-prints , arXiv:1410.2445 (2014), arXiv:1410.2445 [math.SP]

    D. Damanik, Schrödinger Operators with Dynami- cally Defined Potentials: A Survey, arXiv e-prints , arXiv:1410.2445 (2014), arXiv:1410.2445 [math.SP]

    2014 arXiv

  68. [82]

    C. M. Soukoulis and E. N. Economou, Localization in one-dimensional lattices in the presence of incommensu- rate potentials, Phys. Rev. Lett.48, 1043 (1982)

    1982

  69. [83]

    Simon, Almost periodic schrödinger operators: A re- view, Advances in Applied Mathematics3, 463 (1982)

    B. Simon, Almost periodic schrödinger operators: A re- view, Advances in Applied Mathematics3, 463 (1982)

    1982

  70. [84]

    Bourgain and S

    J. Bourgain and S. Jitomirskaya, Continuity of the Lya- punov Exponent for Quasiperiodic Operators with Ana- lytic Potential, Journal of Statistical Physics108, 1203 (2002), arXiv:math-ph/0110040 [math-ph]

    2002 arXiv

  71. [85]

    E.Witten,Coadjointorbitsofthevirasorogroup,Comm. Math. Phys.114, 1 (1988)

    1988

  72. [86]

    Ishibashi and T

    N. Ishibashi and T. Tada, Infinite circumference limit of conformal field theory, Journal of Physics A: Mathemat- ical and Theoretical48, 315402 (2015)

    2015

  73. [87]

    Ishibashi and T

    N. Ishibashi and T. Tada, Dipolar quantization and the infinite circumference limit of two-dimensional conformal field theories, International Journal of Modern Physics A 31, 1650170 (2016)

    2016

  74. [88]

    Tada, Time development of conformal field the- ories associated with l1 and l−1 operators, (2019), arXiv:1904.12414 [hep-th]

    T. Tada, Time development of conformal field the- ories associated with l1 and l−1 operators, (2019), arXiv:1904.12414 [hep-th]

    2019 arXiv

  75. [89]

    I. Peschel, LETTER TO THE EDITOR: Calculation of reduced density matrices from correlation functions, Journal of Physics A Mathematical General 36, L205 (2003), arXiv:cond-mat/0212631 [cond-mat]

    2003 arXiv

  76. [90]

    Y. Liu, Q. Zhou, and S. Chen, Localization transi- tion, spectrum structure, and winding numbers for one- dimensional non-hermitian quasicrystals, Phys. Rev. B 104, 024201 (2021)

    2021

  77. [91]

    X. Wang, Z. Wang, J. You, and Q. Zhou, Winding number, density of states and acceleration, International Mathematics Research Notices2024, 7972 (2024)

    2024

  78. [92]

    J. Fang, Q. Zhou, and X. Wen, In preparation

  79. [93]

    Viana, Lectures on Lyapunov exponents , Vol

    M. Viana, Lectures on Lyapunov exponents , Vol. 145 (Cambridge University Press, 2014)

    2014

  80. [94]

    Mañé, Quasi-anosov diffeomorphisms and hyperbolic manifolds, Transactions of the American Mathematical Society 229, 351 (1977)

    R. Mañé, Quasi-anosov diffeomorphisms and hyperbolic manifolds, Transactions of the American Mathematical Society 229, 351 (1977)

    1977

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