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Effects of non-integrability in a non-Hermitian time crystal

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Pith's one-line read Adding a non-integrable transverse interaction to a non-Hermitian Floquet Ising model shifts its phase boundaries and, above a critical strength $K_c\approx 0.085$, drives the steady state into a new x-ferromagnetic phase that mean-field…

desk verdict A solid extension of the non-Hermitian Floquet time-crystal program; the phase-boundary shift is well supported, but the new transition's location and the stability claim for K>Kc rest on a leading-order effective Hamiltonian that needs higher-order checks. read the letter →

arxiv 2412.04382 v2 pith:K2W5UULF submitted 2024-12-05 cond-mat.str-el cond-mat.stat-mechquant-ph

Weihua Xie , Michael Kolodrubetz , Vadim Oganesyan , Daniel P. Arovas This is my paper · ORCID
classification cond-mat.str-elcond-mat.stat-mechquant-ph
keywords non-HermitiantimecrystalFloquetsystemsintegrabilitybreakingXXZmodelsymmetry-breakingtransitionaverageHamiltoniantheoryquasi-long-rangeordertransverse-fieldIsing
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 asks whether the non-Hermitian Floquet time crystal survives the addition of an integrability-breaking interaction, and what new physics that interaction brings. Working with a non-Hermitian Floquet transverse-field Ising chain plus an $X$-$X$ coupling of strength $K$, the authors find two effects: the phase boundaries between paramagnet and time-crystalline phases shift with $K$, and above $K_c\approx 0.085$ a new symmetry-breaking transition appears at the special point $R=1$, visible as a jump in the steady-state transverse magnetization. Mean-field theory captures the boundary shift but completely misses the jump. Using a rotating-frame average-Hamiltonian expansion, the authors show that the full Floquet circuit reduces at leading order to a non-Hermitian XXZ model, and that the new transition is the easy-axis ferromagnetic transition of that model. They predict that the quasi-long-range time-crystalline (NFM1) phase survives finite $K$, even above $K_c$.

What carries the argument

The rotating-frame average-Hamiltonian (Floquet high-frequency) expansion around the fine-tuned point $\tanh\beta=e^{i\pi/3}$ with $J=K=0$. Because $U_0^6\propto 1$, one considers the super-cycle $U^6$, moves to the rotating frame, and keeps only the leading Baker--Campbell--Hausdorff term, obtaining the non-Hermitian XXZ Hamiltonian of Eq. (14). This effective Hamiltonian carries the argument: its easy-plane versus easy-axis phase structure explains the NFM1 time-crystal phase and the new x-ferromagnetic transition, and its steady-state magnetization reproduces the TEBD jump.

What would settle it

A decisive check is to compute the first neglected Baker--Campbell--Hausdorff correction to $H_{\text{eff}}$ at $K=0.1$, $J\approx 0.1$, and $\Delta\gamma=10^{-4}$ and compare its operator norm with the norms of $\Delta\gamma X_j$ and $\gamma K X_jX_{j+1}$: if the correction is not at least an order of magnitude smaller, the leading-order average Hamiltonian is not a faithful description and the predicted transition is unproven.

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Extended reading notes

Core claim

At the special point $R=1$, where the transverse field is purely imaginary with $\tanh\beta=e^{i\pi/3}$, the six-cycle Floquet operator $U_0^6$ is proportional to the identity, so $J$, $K$, and $\Delta\gamma=\gamma-\gamma_0$ can be treated as weak perturbations. Time-averaging the rotating-frame circuit at leading order yields an effective Hamiltonian $H_{\text{eff}}=i\sum_j[\Delta\gamma\,X_j+\gamma K\,X_jX_{j+1}+\tfrac12 J(Z_jZ_{j+1}+Y_jY_{j+1})]$, which is an anisotropic non-Hermitian XXZ chain with quantization axis $x$. In this description, the non-Hermitian time-crystal phase (NFM1) corresponds to the easy-plane Luttinger-liquid regime, while the integrability-breaking $X$-$X$ term drives an easy-axis ferromagnetic transition. TEBD data for the steady-state magnetization $\langle X\rangle_{\text{ss}}$ show a jump near $R=1$ for $K\gtrsim K_c\approx 0.085$, and the same jump appears in the steady state of $H_{\text{eff}}$. The paper concludes that this x-ferromagnetic phase is present in the full Floquet circuit, that mean-field theory misses it because it neglects the relevant fluctuations, and that the NFM1 time crystal remains stable for finite $K$, even above $K_c$.

Load-bearing premise

The whole analysis rests on the assumption that the leading-order rotating-frame average Hamiltonian of Appendix A (Eqs. A28--A29) faithfully describes the full Floquet circuit at the parameters used ($K=0.1$, $J\sim 0.1$, $|\Delta\gamma|\le 10^{-4}$ near $R=1$), with all higher-order correction terms in the Floquet expansion negligible; no error bound is given for that truncation.

Editorial extensions

If this is right

  • The PM--NFM1 phase boundaries move systematically with $K$, and a self-consistent mean-field decoupling that renormalizes the complex transverse field captures this shift at small and moderate $K$.
  • For $K>K_c\approx 0.085$, the steady state at $R=1$ develops a jump in $\langle X\rangle_{\text{ss}}$, a first-order-like symmetry-breaking transition into an x-ferromagnetic phase that is absent at $K=0$ and invisible to mean-field theory.
  • The NFM1 time-crystal phase, with quasi-long-range oscillatory temporal correlations, is predicted to survive finite $K$, including $K>K_c$, because it maps to the stable Luttinger-liquid regime of the effective XXZ chain.
  • Near $K_c$, the transition should show conventional XXZ criticality, with correlation functions possibly acquiring damping factors; weak symmetry-respecting perturbations, including higher-order Floquet terms, are expected not to destroy the phase structure.
  • The new x-ferromagnet is distinct from the $z$-ferromagnet present at $K=0$ and requires the integrability-breaking $X$-$X$ interaction to be realized.

Reading between the lines

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

  • If $H_{\text{eff}}$ is quantitatively reliable at $K\gtrsim K_c$, the same jump should appear in other steady-state observables, for example the $X$-$X$ structure factor or the overlap with a symmetry-broken product state; these provide a direct TEBD falsifiability check.
  • The super-cycle construction uses $\theta=\pi/3$ so that six Floquet steps return to the identity; for irrational $\theta$ no such finite super-cycle exists, and the paper leaves open whether the transition survives as a smooth boundary or fragments into a fractal one -- a numerical scan over nearby rational angles could settle this.
  • Because the mean-field decoupling misses the transition entirely, a symmetry-broken mean-field ansatz that allows spontaneous $x$-magnetization might capture the jump while keeping the free-fermion sector exact, isolating which fluctuations are essential.
  • The predicted survival of NFM1 above $K_c$ implies the temporal correlation exponent should remain power-law; measuring $C(N)$ at larger system sizes and bond dimensions for $K>K_c$ could confirm or refute that stability.
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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

3 major / 5 minor

Summary. The paper studies a non-Hermitian Floquet transverse-field Ising model with an added non-integrable interaction K∑XX. Using TEBD, mean-field theory, and average Hamiltonian theory, it claims two effects: a shift of the PM/NFM1 phase boundaries, and a new symmetry-breaking transition near R=1 that is absent in mean-field theory. The new transition is attributed to an x-ferromagnetic phase of a leading-order non-Hermitian XXZ effective Hamiltonian, with a critical point Kc≈0.085, and the NFM1 time-crystalline phase is predicted to survive for K>Kc.

Significance. If established, these results would show that integrability-breaking interactions can qualitatively change non-Hermitian Floquet phase diagrams and connect a new time-crystal-related transition to well-studied non-Hermitian XXZ physics. The paper is clearly written and combines complementary methods: TEBD data, a mean-field analysis that correctly captures the boundary shift, and an effective-Hamiltonian derivation with no fitted parameters. The main limitation is that the central new transition is inferred from a leading-order truncation of the Floquet expansion without controlling higher-order terms, and the TEBD evidence for the transition is not backed by finite-size scaling or error estimates.

major comments (3)
  1. [§III B and Appendix A, Eq. (14), Fig. 7] The identification Kc≈0.085 is made from the leading-order average Hamiltonian, but the validity of this truncation at the predicted transition is not controlled. At K=0.1 the expansion parameters are |J|≈0.096 and |γK|≈0.052, which are not small compared with the terms whose competition sets Kc, while the detuning used in Fig. 7 is |Re Δγ|=10^-4. Second-order BCH terms are of order JγK≈0.005, which is fifty times larger than this detuning. The paper needs an explicit estimate or computation of the leading higher-order corrections to Eq. (14), or a benchmark of Kc by exact numerics on the full circuit, before the transition can be attributed to the full Floquet dynamics.
  2. [§III B, Figs. 4 and 6] The new phase transition is inferred from a jump in the steady-state expectation value ⟨X⟩ at a single system size L=50, without finite-size scaling or error bars. A first-order transition in one dimension should sharpen with system size, and a jump at L=50 could in principle be a finite-size crossover. The authors should show ⟨X⟩ as a function of R for several system sizes (and confirm bond-dimension convergence at each size), or explicitly state the finite-size limitations of the evidence for the transition.
  3. [§III B, final paragraph and Fig. 8] The prediction that NFM1 survives for K>Kc is based on the stability of the Luttinger liquid against weak integrability-breaking perturbations. However, the leading-order Heff in Eq. (14) has a continuous U(1) symmetry about the x-axis and is integrable, while the exact Floquet circuit has neither property; the higher-order terms that break these properties are exactly the terms whose magnitude is not controlled. To support the survival claim, the paper should provide numerical evidence for power-law temporal correlations C(N) in the full circuit for K>Kc, or a controlled argument that the non-Hermitian steady state is insensitive to the U(1)-breaking perturbations.
minor comments (5)
  1. [§III A, Eq. (10)] There appears to be a factor-of-two inconsistency between the first line of Eq. (10), which has 2Kγ in the exponent, and the mean-field expression that follows, which has γK without the factor 2; please clarify the convention.
  2. [Fig. 4 caption] The caption says the time evolution of ⟨X⟩ is shown for a few values of R but does not state which R values are plotted; please include them in the caption or legend.
  3. [Fig. 6 caption] The caption 'Expectation value from Heff and H' is ambiguous; H should be identified as the full Floquet circuit or the TEBD result, and the curves should be clearly labeled.
  4. [Introduction and §III B] There are several typos, including 'considitions' in the introduction, 'differant' in §III, and 'demostrate' in §III B; these should be corrected.
  5. [§III B] The sentence 'This confirms that the phase structure of Heff is imprinted on that of the full interacting Floquet circuit' is too strong, because Fig. 7 shows only Heff; the comparison to the full circuit is made in Fig. 6 and is qualitative.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new symmetry-breaking transition and K_c are computed from an independently derived average Hamiltonian and checked against TEBD, not fitted or defined into existence.

full rationale

The central derivation chain is self-contained. Appendix A derives the effective Hamiltonian H_eff (Eqs. A28-A29, quoted as Eq. 14) from the full Floquet operator U_total by a rotating-frame transformation and a leading-order Baker-Campbell-Hausdorff average, with no parameter fitted to the TEBD data. The critical point K_c ≈ 0.085 is obtained from H_eff in Fig. 7, and Fig. 6 compares the H_eff steady state with TEBD as an independent check rather than as an input. The mean-field treatment also derives the phase-boundary shift through a self-consistency condition, not through fitting. The only self-citation is Ref. [18] (Basu et al., with overlapping authors), which supplies the base Floquet-Ising model, phase labels, and the NFM1 phase; this is legitimate background and is not the argument supporting the new interaction-induced x-ferromagnetic transition. The paper's own caveat that |Δγ|, |K|, and |J| are assumed much less than π, with no error bound on neglected higher-order BCH terms, is a correctness and approximation-validity concern, not a circularity: if higher-order terms matter, the prediction may fail, but that would be a broken approximation rather than a tautology. No equation in the paper reduces to its own output by construction.

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

No free parameters are fitted to data: K, J, and gamma are model parameters from the complex-temperature mapping, and K_c is computed from the effective Hamiltonian rather than adjusted to match TEBD. The paper rests on the prior phase diagram of Ref. [18], the validity of the leading-order Floquet expansion, the uniqueness of the non-Hermitian steady state, and the convergence of the TEBD sampling.

assumptions (5)
  • domain assumption The K=0 non-Hermitian Floquet Ising model and its phase diagram, including the NFM1 quasi-long-range time-crystalline phase, are as described in Ref. [18].
    Used throughout Sections II and III as the baseline; Ref. [18] shares authors with this paper, so this is self-cited background rather than independent verification.
  • domain assumption The leading-order average Hamiltonian is sufficient: higher-order Baker-Campbell-Hausdorff terms are negligible for small J, K, and Delta gamma.
    Appendix A, Eqs. (A28)-(A29), truncates the Floquet expansion at first order; no estimate of the neglected terms is provided.
  • domain assumption The non-Hermitian Floquet evolution has a unique steady state, so late-time properties are independent of the initial state and can be compared to equilibrium ground states.
    Section II, Eqs. (3)-(5), and Section III B where the steady state is compared to the ground state of a Hermitian XXZ chain.
  • domain assumption Random z-basis sampling approximates the infinite-temperature trace accurately enough for the correlation functions and steady-state expectation values.
    Section II B describes sampling and the slope-fit normalization (Fig. 2), but no sampling error or convergence criterion is given beyond one bond-dimension check in Appendix B.
  • domain assumption The non-Hermitian XXZ Luttinger liquid is stable to weak symmetry-respecting integrability-breaking perturbations, so the NFM1 phase survives for K > K_c.
    Section III B and Discussion; this extrapolates Ref. [37] and standard Luttinger liquid stability to the Floquet steady state without a proof.

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Cite this review

Pith. "Pith review of Effects of non-integrability in a non-Hermitian time crystal." pith.science (2026). https://pith.science/paper/K2W5UULF

@misc{pith2026241204382,
  author       = {Pith},
  title        = {Pith review of: Effects of non-integrability in a non-Hermitian time crystal},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K2W5UULF}},
  note         = {Machine review of arXiv:2412.04382}
}
read the original abstract

Time crystals are systems that spontaneously break time-translation symmetry, exhibiting repeating patterns in time. Recent work has shown that non-Hermitian Floquet systems can host a time crystalline phase with quasi-long-range order. In this work, we investigate the effect of introducing a non-integrable interaction term into this non-Hermitian time crystal model. Using a combination of numerical TEBD simulations, mean-field analysis, and perturbation theory, we find that the interaction term has two notable effects. First, it induces a shift in the phase diagram, moving the boundaries between different phases. Second, a sufficiently strong interaction induces an unexpected symmetry-breaking transition, which is not captured by the mean-field approach. Within average Hamiltonian theory, we trace this back to a ferromagnetic transition in the anisotropic non-Hermitian XXZ model. Our results demonstrate that the interplay between non-Hermitian dynamics and many-body interactions can lead to novel symmetry breaking.

Figures

Figures reproduced from arXiv: 2412.04382 by the authors.

Figure 1
Figure 1. FIG. 1. Phase diagram of different spatial/temporal order of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Example fit of log [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Upper panels are two-time correlation function, and [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Expectation value from mean-field treatment with [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Expectation value from [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Expectation value from [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Phase diagram after adding interaction term. [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Example of convergence with bond dimension in PM phas [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

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Forward citations

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Reference graph

Works this paper leans on

45 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Spatial power-law decay has been documented[18] but the precise variation of time (space) exponents α 1( α ′

    work page

  2. [2]

    Similarly, for the NFM2 phase, C(y) ∼ y−α 2 cos (a2y)

    has not been understood in detail. Similarly, for the NFM2 phase, C(y) ∼ y−α 2 cos (a2y). For the ferro- magnetic phase, C(N ) and C(y) converge to a non-zero values at large N or y. A. Interaction term The Hermitian transverse-field Ising model is inte- grable. This is most readily seen by performing a Jordan- Wigner transformation to free Majorana fermio...

    work page

  3. [3]

    Tuning away from this point, the PM phase is achieved upon either increasing or decreasing R by a sufficient amount

    The point R = 1, which lies in the middle of the NFM1 phase, is special because it corre- 4 sponds to a purely real (Hermitian) magnetic field term. Tuning away from this point, the PM phase is achieved upon either increasing or decreasing R by a sufficient amount. Beginning with the PM phase, the correlations clearly decay exponentially in both cases, but t...

    work page

  4. [4]

    As predicted from analyzing the two-time correlation functions, the phase boundary shifts upward to larger R values when K > 0 is added

    Extracting the late-time ⟨X⟩ss, we see that there are kinks at the two PM to NFM1 transitions for all values of K shown. As predicted from analyzing the two-time correlation functions, the phase boundary shifts upward to larger R values when K > 0 is added. However, we also see an unexpected new phenomenon, namely a jump in ⟨X⟩ss near R = 1. This jump is ...

    work page

  5. [5]

    Floquet Hamiltonian

    The shifts of the phase diagram are accurately captured by the this mean-field treatment at small and moderate K, meaning that this physics is just described by a self-consistent shift of the (complex) transverse field. Note that this works despite known is- sues with mean-field theory in one dimension because the remaining parts of the model are treated exa...

    work page

  6. [6]

    0001 as illustrated in fig

    0002 where Re (∆ γ) = − 0. 0001 as illustrated in fig. 7. The magnetization shows a clear Luttinger liquid to x- ferromagnet transition at Kc ≈ 0. 085. This confirms that the phase structure of Heff is imprinted on that of the full interacting Floquet circuit. Combine the results from above, the phase diagram of our model as illustrated in Fig. 8. Crucially,...

    work page

  7. [7]

    Manzano, A short introduction to the lindblad master equation, AIP Advances 10, 025106 (2020)

    D. Manzano, A short introduction to the lindblad master equation, AIP Advances 10, 025106 (2020)

    work page 2020

  8. [8]

    Naghiloo, M

    M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit, Nature Physics 15, 1232 (2019)

    work page 2019

Show all 45 references
  1. [9]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-hermitian systems, Rev. Mod. Phys. 93, 015005 (2021)

    2021

  2. [10]

    Yao and Z

    S. Yao and Z. Wang, Edge states and topo- logical invariants of non-hermitian systems, Phys. Rev. Lett. 121, 086803 (2018)

    2018

  3. [11]

    Cui and Y

    X.-D. Cui and Y. Zheng, Geometric phases in non-hermitian quantum mechanics, Phys. Rev. A 86, 064104 (2012)

    2012

  4. [12]

    Doppler, A

    J. Doppler, A. A. Mailybaev, J. B¨ ohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Dynamically encircling an exceptional point for asymmetric mode switching, Nature 537, 76 (2016)

    2016

  5. [13]

    H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topo- logical energy transfer in an optomechanical system with exceptional points, Nature 537, 80 (2016)

    2016

  6. [14]

    L. Guo, L. Du, C. Yin, Y. Zhang, and S. Chen, Dynam- ical evolutions in non-hermitian triple-well systems with a complex potential, Phys. Rev. A 97, 032109 (2018)

    2018

  7. [15]

    C. M. Bender, D. C. Brody, and H. F. Jones, Complex extension of quantum mechanics, Phys. Rev. Lett. 89, 270401 (2002)

    2002

  8. [16]

    M. V. Berry, Optical polarization evo- lution near a non-hermitian degeneracy, Journal of Optics 13, 115701 (2011)

    2011

  9. [17]

    Longhi, Bloch oscillations in complex crystals with PT symmetry, Phys

    S. Longhi, Bloch oscillations in complex crystals with PT symmetry, Phys. Rev. Lett. 103, 123601 (2009)

    2009

  10. [18]

    C. T. West, T. Kottos, and T. c. v. Prosen, PT -symmetric wave chaos, Phys. Rev. Lett. 104, 054102 (2010)

    2010

  11. [19]

    C. E. R¨ uter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of parity–time symmetry in optics, Nature Physics 6, 192 (2010)

    2010

  12. [20]

    J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, Observation of parity-time symmetry breaking transitions in a dissipative floquet system of ultracold atoms, Nature Communications 10, 855 (2019)

    2019

  13. [21]

    Miri and A

    M.-A. Miri and A. Al` u, Exceptional points in op- tics and photonics, Science 363, eaar7709 (2019) , https://www.science.org/doi/pdf/10.1126/science.aar7709

    2019 doi

  14. [22]

    T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. H¨ ofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, Observation of non- hermitian degeneracies in a chaotic exciton-polariton bi...

    2015

  15. [23]

    Zhang, X.-Q

    D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, Observation of the ex- ceptional point in cavity magnon-polaritons, Nature Communications 8, 1368 (2017)

    2017

  16. [24]

    S. Basu, D. P. Arovas, S. Gopalakrishnan, C. A. Hoo- ley, and V. Oganesyan, Fisher zeros and persistent temporal oscillations in nonunitary quantum circuits, Phys. Rev. Research 4, 013018 (2022) . 11

    2022

  17. [25]

    C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. i. theory of condensation, Phys. Rev. 87, 404 (1952)

    1952

  18. [26]

    T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. ii. lattice gas and ising model, Phys. Rev. 87, 410 (1952)

    1952

  19. [27]

    R. B. Griffiths, Nonanalytic behavior above the critical point in a random ising ferromagnet, Phys. Rev. Lett. 23, 17 (1969)

    1969

  20. [28]

    Wilczek, Quantum time crystals, Phys

    F. Wilczek, Quantum time crystals, Phys. Rev. Lett. 109, 160401 (2012)

    2012

  21. [29]

    Watanabe and M

    H. Watanabe and M. Oshikawa, Absence of quantum time crystals, Phys. Rev. Lett. 114, 251603 (2015)

    2015

  22. [30]

    C. W. von Keyserlingk and S. L. Sondhi, Phase structure of one-dimensional interact- ing floquet systems. ii. symmetry-broken phases, Phys. Rev. B 93, 245146 (2016)

    2016

  23. [31]

    D. V. Else, B. Bauer, and C. Nayak, Floquet time crys- tals, Phys. Rev. Lett. 117, 090402 (2016)

    2016

  24. [32]

    Rovny, R

    J. Rovny, R. L. Blum, and S. E. Barrett, Observation of discrete-time-crystal signatures in an ordered dipolar many-body system, Phys. Rev. Lett. 120, 180603 (2018)

    2018

  25. [33]

    Autti, V

    S. Autti, V. B. Eltsov, and G. E. Volovik, Observation of a time quasicrystal and its transition to a superfluid time crystal, Phys. Rev. Lett. 120, 215301 (2018)

    2018

  26. [34]

    S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. von Keyserlingk, N. Y. Yao, E. Demler, and M. D. Lukin, Ob- servation of discrete time-crystalline order in a disorder ed dipolar many-body system, Nature 543, 221 (2017)

    2017

  27. [35]

    X. Mi, M. Ippoliti, C. Quintana, et al., Time- crystalline eigenstate order on a quantum processor, Nature 601, 531 (2022)

    2022

  28. [36]

    Khemani, A

    V. Khemani, A. Lazarides, R. Moessner, and S. L. Sondhi, Phase structure of driven quantum systems, Phys. Rev. Lett. 116, 250401 (2016)

    2016

  29. [37]

    Yousefjani, A

    R. Yousefjani, A. Carollo, K. Sacha, S. Al-Kuwari, and A. Bayat, Non-hermitian discrete time crystals (2024), arXiv:2410.22713 [quant-ph]

    2024 arXiv

  30. [38]

    Sachdev, Quantum Phase Transitions (Cambridge University Press, 2011)

    S. Sachdev, Quantum Phase Transitions (Cambridge University Press, 2011)

    2011

  31. [39]

    Hauschild and F

    J. Hauschild and F. Pollmann, Efficient numerical sim- ulations with Tensor Networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes , 5 (2018)

    2018

  32. [40]

    While it may seem that ⟨X⟩ is conserved during time evo- lution, as it commutes with the XX -interacting Hamilto- nian, this is is not true for non-Hermitian time evolution because the different X-sectors can pick up different nor- malizations

  33. [41]

    Sabetta and G

    T. Sabetta and G. Misguich, Nonequilibrium steady states in the quantum xxz spin chain, Phys. Rev. B 88, 245114 (2013)

    2013

  34. [42]

    Giuliano, D

    D. Giuliano, D. Rossini, P. Sodano, and A. Trom- bettoni, Xxz spin- 1 2 representation of a finite- u bose-hubbard chain at half-integer filling, Phys. Rev. B 87, 035104 (2013)

    2013

  35. [43]

    Yamamoto, M

    K. Yamamoto, M. Nakagawa, M. Tezuka, M. Ueda, and N. Kawakami, Universal prop- erties of dissipative tomonaga-luttinger liquids: Case study of a non-hermitian xxz spin chain, Phys. Rev. B 105, 205125 (2022)

    2022

  36. [44]

    Artuso, P

    R. Artuso, P. Cvitanovi´ c, and B. G. Kenny, Phase transitions on strange irrational sets, Phys. Rev. A 39, 268 (1989)

    1989

  37. [45]

    W. Xie, M. Kolodrubetz, and V. Oganesyan, Effect of noise on quantum circuit realization of non-hermit ian time (2024), arXiv:2409.06113 [cond-mat.str-el]

    2024 arXiv

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Reviewed August 11, 2026 · model on record in the stance chip above.