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REVIEW 3 major objections 5 minor 69 references

Detecting Many-Body Scars from Fisher Zeros

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper conjectures that quantum many-body scars always leave a continuous line of Fisher zeros off the imaginary axis of the complex inverse-temperature plane, extending to the long-time limit.

desk verdict A genuinely new diagnostic proposal for QMBS with consistent numerics, but the long-time extension is extrapolated and the word 'confirmed' overshoots the evidence. read the letter →

arxiv 2501.09478 v3 pith:EJX27NQS submitted 2025-01-16 cond-mat.str-el cond-mat.quant-gascond-mat.stat-mechhep-latquant-ph

Yuchen Meng , Songtai Lv , Yang Liu , Zefan Tan , Erhai Zhao , Haiyuan Zou This is my paper · ORCID
classification cond-mat.str-elcond-mat.quant-gascond-mat.stat-mechhep-latquant-ph
keywords quantummany-bodyscarsFisherzerospartitionfunctionthermofielddoubleeigenstatethermalizationtensorrenormalizationgroupPXPmodelergodicitybreaking
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

Quantum many-body scars (QMBSs) — rare eigenstates that refuse to thermalize and produce long-lived oscillations — are usually found by hand-picking candidate states and monitoring their dynamics. This paper proposes a bulk diagnostic: analytically continue the partition function $Z = \mathrm{Tr}\, e^{-\beta H}$ to complex inverse temperature $\beta = \beta_r + i\beta_i$, where the imaginary part acts as time, and look at where $Z$ vanishes (its Fisher zeros). The central conjecture is that any scarred system has at least one continuous line of Fisher zeros off the $\beta_i$ axis, extending to large $\beta_i$ and separating regions of the plane with different long-time thermalization behavior. The conjecture is supported by tensor-network (HOTRG) computations and exact results on three models: the generalized $\bar{P}X\bar{P}$ (PXP-type) model, the Ising chain with transverse and longitudinal fields, and a cluster spin model related to lattice gauge theory. If correct, the geometry of Fisher zeros becomes a unified statistical-mechanics classification that tells QMBS apart from strong ergodicity breaking, dynamical quantum phase transitions, and thermal phase transitions without inspecting individual eigenstates.

What carries the argument

The central object is the Fisher zero: a point $\beta = \beta_r + i\beta_i$ in the complex inverse-temperature plane where the analytically continued partition function $Z(\beta) = \mathrm{Tr}\, e^{-\beta H}$ vanishes, located here as the intersection of the contours $\mathrm{Re}\,Z = 0$ and $\mathrm{Im}\,Z = 0$. Two constructions carry the argument. First, the thermofield double identification: purifying the thermal state with a copy of the system turns $Z(\beta_r, \beta_i)$ into the return amplitude of the doubled state evolved for time $\beta_i$, which ties the pattern of zeros to the long-time thermalization of the original system. Second, the $S_0 = |Z(\beta_r, t)/Z(0,0)|^2 = 1$ contour, which converges to a fixed-point line for the renormalization-group flow on the complex $\beta$ plane; Fisher zeros form a 'mountain range' that blocks the flow to this $S_0 = 1$ 'water basin', and an off-axis zero line blocking the flow is the operative signature of QMBS. The zeros are obtained numerically by Trotterizing the partition function into a two-dimensional tensor network and contracting it at complex $\beta$ with the higher-order tensor renormalization group (HOTRG), for system sizes up to $L = 128$.

What would settle it

In the transverse-field Ising chain at $h_l = 0$, where the zero-line equation is known exactly, one can check analytically whether any zero line extends to arbitrarily large $\beta_i$; if all exact lines terminate or close into loops at finite $\beta_i$, the conjecture's long-time feature fails in the one model where it can be checked in closed form. Numerically, the same check can be run for the $\bar{P}X\bar{P}$ model at $g = 0$ by contracting $Z$ at $\beta_i$ well beyond $2\pi$ and testing whether the off-axis line still connects through the computed region and whether the nearest-zero spacing keeps decaying with system size; a line that bends back to the $\beta_i$ axis, or a spacing that saturates, would falsify the conjecture as stated.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is a conjecture stated as a defining signature: a system with QMBS possesses at least one line of Fisher zeros that lies off the $\beta_i$ axis and extends to large $\beta_i$, the long-time limit. The reasoning starts from the thermofield double construction — purifying a thermal state by doubling the Hilbert space makes $Z(\beta_r, \beta_i)$ the return amplitude of a pure state evolving under the physical Hamiltonian, so $\beta_i$ is time and the weights $e^{-\beta_r E_n/2}$ control the overlap with scar states; because scarred components evade thermalization, the return amplitude cannot change smoothly across the plane, and zero lines mark the boundary between regions with qualitatively different long-time behavior. The numerical evidence is that in the $\bar{P}X\bar{P}$ model the Fisher zeros detach from the $\beta_i$ axis once the blockade parameter drops below about $g \approx 0.3$, exactly where exact-diagonalization dynamics shows the antiferromagnetic state oscillating at a scar-related frequency while the ferromagnetic state does not; in the Ising chain with both transverse and longitudinal fields the same off-axis zero line appears and coexists with broken quasiparticle-confinement loops; and a cluster spin model reproduces the pattern. The paper also contrasts the geometry with strong ergodicity breaking: SBETH zeros repeatedly cross the $\beta_i$ axis, so the two forms of ergodicity violation are distinguished by whether the zero line detaches from the axis.

Load-bearing premise

The long-time half of the conjecture is assumed rather than computed: the numerics reach imaginary temperatures only around $\beta_i \approx 2\pi$ and system sizes only up to $L = 128$, so the load-bearing premise is that the short off-axis zero segments seen there persist as continuous lines out to arbitrarily large $\beta_i$ in the thermodynamic limit.

Editorial extensions

If this is right

  • Scars become detectable from the partition function alone, at the cost of contracting a tensor network rather than enumerating eigenstates, so the diagnostic scales to systems where exact diagonalization is impossible.
  • The $\beta$-plane geometry yields a unified classification: QMBS zeros live off the imaginary axis and run to long times, SBETH zeros cross the axis, DQPT points sit on it, and thermal critical points pinch it — one statistical-mechanics picture for all ergodicity-breaking phenomena.
  • The $\bar{P}X\bar{P}$ study places the QMBS-to-SBETH crossover at $g \approx 0.3$, where the zero lines detach from the $\beta_i$ axis; this is a concrete, falsifiable boundary parameter extracted from the zero geometry.
  • In the Ising chain with both fields, the coexistence of an off-axis scar line with broken meson-confinement loops shows that scar physics and quasiparticle physics leave independent, separately readable signatures in the same zero pattern.
  • The $S_0 = 1$ fixed-point line extends the known result that Fisher zeros bound complex renormalization-group flows, giving a fully complex-$\beta$ RG interpretation of weak ergodicity breaking as flow-blocking.

Reading between the lines

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

  • The paper argues the conjecture only in the direction 'scars imply an off-axis zero line'; the converse — a persistent off-axis line implies scars — is the version that would make the diagnostic a practical screening tool, and the paper does not test it.
  • Because the identification of $Z$ with a return amplitude needs only the spectrum, the same zero-line diagnostic should transfer to non-unitary settings such as monitored circuits; the paper mentions this direction but does not propose a concrete detection protocol.
  • A quantitative step the paper leaves open: the density or scaling exponent of zeros along the scar line could encode the scar tower's level spacing, converting a yes/no flag into a measurement of scar structure.
  • The continuity argument relies on power-law decay of nearest-zero distances with $L$, which suggests an independent numerical test — check that the zero spacing along an off-axis line vanishes as $L \to \infty$ in any candidate model before trusting the line.
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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 proposes a new diagnostic for quantum many-body scars (QMBS) based on the pattern of Fisher zeros in the complex inverse-temperature plane. The central conjecture is that, for a system with QMBS, a continuous line of Fisher zeros lies off the imaginary β axis and extends to large βi (the long-time limit). This is motivated by interpreting the analytically continued partition function as the return amplitude of a thermofield double state. The authors test the conjecture numerically using the generalized PXP (bar-P X bar-P) model, the Ising chain with transverse and longitudinal fields, and a cluster spin model in the supplemental material. They also contrast the QMBS scenario with strong ergodicity breaking (SBETH), where Fisher zeros intersect the imaginary axis. The paper includes finite-size scaling of zero spacings and distances, exact results for special Ising limits, and a renormalization-group flow interpretation in terms of a fixed line S0=1. The conclusion is that Fisher zeros offer a route to detect scars without inspecting individual eigenstates.

Significance. If the conjecture is correct, the work provides a conceptually new and potentially practical diagnostic for QMBS, connecting them to the statistical-mechanics framework of partition-function zeros. This would be a valuable addition to the field, as current methods often require examining eigenstates or dynamics directly. The numerical evidence is fairly extensive for a finite βi window, and the analytic solution of the longitudinal-field Ising limit is a useful check. The paper makes falsifiable predictions about zero configurations that could guide future experiments. However, the load-bearing claim about the extension of the zero lines to large βi is not substantiated by the presented data, and the numerical method lacks convergence tests. These issues currently limit the certainty with which the conjecture can be accepted.

major comments (3)
  1. [Generalization and outlook; main conjecture (Introduction)] The conjecture is specifically about the existence of an off-axis Fisher zero line that 'extends to large βi (the long-time limit).' However, all reported zeros are computed only in a finite window βi ≲ 2π (Figs. 2, 3, and S1–S5), with system sizes up to L=128. The finite-size scaling in Fig. S1(g–i) concerns distances between neighboring zeros or the distance from the βi axis for zeros at small βi; it does not address the existence of zeros at parametrically large βi. Therefore, the statement in the Generalization section that 'We have confirmed our conjecture' is an overstatement. The large-βi part of the conjecture remains an extrapolation, and the paper should either provide direct numerical evidence at larger βi, or at least a rigorous or quantitative argument for why the zero lines can be continued to infinity, or explicitly caveat the conjecture as unverified in that limit.
  2. [P XP model: numerical methods paragraph] The HOTRG calculation fixes N=1024 Trotter steps and identifies Fisher zeros as intersections of Re Z=0 and Im Z=0, but no convergence or error analysis is provided. Since the zero locations are derived from the numerical partition function, truncation errors in the tensor network contraction directly shift the zero positions. The paper should include, for example, a comparison of zero locations for different bond dimensions or different Trotter step numbers, and estimate error bars for the zero coordinates. Without such checks, the power-law fits in Fig. S1 could be fitting numerical artifacts rather than genuine finite-size scaling.
  3. [Fig. 2 and text around it; 'diagnostic' claim] The classification of parameter regimes as QMBS, crossover, or SBETH is based on the same dynamical quantities (frequency spectra and oscillations of AFM and FM states, Fig. 2(b,d,f)) that the Fisher zeros are meant to diagnose. This creates a circularity: the zero patterns are shown to differ in regimes already known to be scarred or non-scarred from dynamics, but the paper does not demonstrate that an independent observer using only the zero configurations (without knowing the dynamic classification) would correctly identify the scarred regime. To support the claim that Fisher zeros provide a 'diagnostic,' the authors should either show a predictive use of the zero patterns (e.g., using them to guess the dynamics without prior classification) or at least discuss this limitation explicitly.
minor comments (5)
  1. [Supplemental Material, Eq. (S2)] The notation in Eq. (S2), S0(βi+1, L) = S0(βi, L/2), uses βi both for the imaginary part of β and as a subscript in an iteration index; this is confusing. Please rename the iterated variable (e.g., β_n) to avoid ambiguity.
  2. [Introduction, first paragraph] There are two typos: 'In his seminar work' should be 'seminal work', and 'motivated from' in the abstract should be 'motivated by'.
  3. [End Matter, first paragraph] The phrase 'the singularities of fi = ln |Zi|' is used, but the text also refers to 'the singularities of f' in the SBETH case; it would be helpful to clarify the distinction between singularities of f and those of fi, and to define the shaded regions in Fig. 4(a) more explicitly.
  4. [Ising chain section, Fig. 3(c)] The caption for Fig. 3(c) says 'Only the zeros describing the scars are marked by black dots,' but the main text later refers to 'the emergent line extending to large βi' without specifying how large. Please quantify the maximum βi shown and state whether this line is observed to terminate or is cut off by the plotted range.
  5. [Generalization and outlook] The phrase 'These three model studies offer strong evidence that the unifying link between Fisher zeros and thermalization breakdown is rather general' uses 'rather general' too vaguely; consider rephrasing to specify the scope of generality claimed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Fisher-zero patterns are computed directly from Z and compared with independent ED dynamics; the long-βi extension is an extrapolated conjecture, not an input.

full rationale

The central claim is a conjecture, not a derivation from the data it is tested against. The Fisher zeros are obtained by HOTRG contraction of the Trotterized tensor-network representation of Z = Tr e^{-βH}, and the scar/SBETH labels come from independent exact-diagonalization dynamics (AFM versus FM oscillations, frequency peaks) and from known model properties (PXP scars, meson confinement). No parameter is fitted to the zero locations and then recycled as a prediction of those same zeros. The off-axis zero lines are read off as intersections of Re Z = 0 and Im Z = 0; their claimed continuation to large βi is supported only within a finite-βi window, which is an evidential gap rather than a circular reduction. The End Matter RG-flow discussion is explicitly illustrative: S0 is defined from the same Z, so the statement that flows terminate on the S0 = 1 line is partly definitional, but it is not used as the proof of the QMBS conjecture; the main evidence is the direct zero computation. Self-citations ([36-38] for transverse-field Ising Fisher zeros and [58,59] for the RG-boundary relation) are background and not load-bearing: they do not assume the QMBS-zero link. Hence no step reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim introduces no fitted parameters; all plotted quantities are computed directly from the Hamiltonians. The auxiliary power-law fits in the supplemental (e.g., exponents 0.67, 0.93) are descriptive and do not feed into the conjecture. The main burden falls on the TFD purification assumption and on the unverified thermodynamic-limit extrapolation of the zero lines to large βi.

assumptions (4)
  • standard math Analytic continuation of the partition function to complex inverse temperature β is valid, and zeros of Z in the complex β-plane encode phase transition information (Fisher/Lee-Yang theory).
    Invoked in the Introduction to define Fisher zeros and in the interpretation of QMBS zero lines as marking phase boundaries.
  • domain assumption In the thermofield double construction, the auxiliary copy R acts as a bath, so scars in L continue to evade thermalization in the doubled system; hence Z(βr,t) as a return amplitude encodes scar dynamics.
    Section 'Introduction', third paragraph, where the TFD state and the bath argument are introduced; this is the physical motivation for why zeros should move off the imaginary axis.
  • domain assumption Finite-size Fisher zeros of these 1D models form continuous lines in the thermodynamic limit, so the finite-size HOTRG zero sets can be extrapolated to lines.
    Stated after Eq. (1) with citation to prior work; the continuum extrapolation is essential for the conjecture's line-of-zeros language.
  • ad hoc to paper The HOTRG contraction with N=1024 Trotter steps yields Z accurately enough to locate zeros; no convergence or error analysis is provided.
    The numerical method section fixes N=1024 but does not report bond dimension, Trotter time step, or truncation error; the zero locations are used without error bars.

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Cite this review

Pith. "Pith review of Detecting Many-Body Scars from Fisher Zeros." pith.science (2026). https://pith.science/paper/EJX27NQS

@misc{pith2026250109478,
  author       = {Pith},
  title        = {Pith review of: Detecting Many-Body Scars from Fisher Zeros},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EJX27NQS}},
  note         = {Machine review of arXiv:2501.09478}
}
abstract

The far-from-equilibrium dynamics of certain interacting quantum systems still defy precise understanding. One example is the so-called quantum many-body scars (QMBSs), where a set of energy eigenstates evade thermalization to give rise to long-lived oscillations. Despite the success of viewing scars from the perspectives of symmetry, commutant algebra, and quasiparticles, it remains a challenge to elucidate the mechanism underlying all QMBS and to distinguish them from other forms of ergodicity breaking. In this work, we introduce an alternative route to detect and diagnose QMBS based on Fisher zeros, i.e., the patterns of zeros of the analytically continued partition function $Z$ on the complex $\beta$ (inverse temperature) plane. For systems with scars, a continuous line of Fisher zeros will appear off the imaginary $\beta$ axis and extend upward, separating the $\beta$ plane into regions with distinctive thermalization behaviors. This conjecture is motivated from interpreting the complex $Z$ as the return amplitude of the thermofield double state, and it is validated by analyzing two models with QMBS, the $\bar{P}X\bar{P}$ model and the Ising chain in external fields. These models also illustrate the key difference between QMBS and strong ergodicity breaking including their distinctive renormalization group flows on the complex $\beta$ plane. This ``statistical mechanics" approach places QMBS within the same framework of thermal and dynamical phase transitions. It has the advantage of spotting scars without exhaustively examining each individual quantum state.

Figures

Figures reproduced from arXiv: 2501.09478 by the authors.

Figure 1
Figure 1. FIG. 1. (Schematic) A synopsis of possible locations of Fisher [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Fisher zeroes and real-time dynamics in the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Fisher zeros and real-time dynamics of the Ising chain [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Key differences between SBETH and QMBS in the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Three-dimensional contour plots of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Works this paper leans on

69 extracted references · 46 canonical work pages

  1. [1]

    J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991)

    1991

  2. [2]

    shadow ef- fect

    In the SBETH case (a, g = 0 .9), f becomes nonanalytic over a broad interval ofβi. In the QMBS case (b, g = 0), both f and f ′ evolve smoothly and f exhibits only an inflection point. Right: the RG flows (arrows) for g = 0.9 (c) and g = 0 (d). The black lines connecting to the origin are the fixed points defined by S0 = 1. The contour lines are for differ...

    work page

  3. [3]

    C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´ c, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018)

    2018

  4. [4]

    Srednicki, Chaos and quantum thermalization, Phys

    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994)

    1994

  5. [5]

    Schecter and T

    M. Schecter and T. Iadecola, Weak ergodicity breaking and quantum many-body scars in spin-1 XY magnets, Phys. Rev. Lett. 123, 147201 (2019)

    2019

  6. [6]

    Moudgalya, N

    S. Moudgalya, N. Regnault, and B. A. Bernevig, Entan- glement of exact excited states of Affleck-Kennedy-Lieb- Tasaki models: Exact results, many-body scars, and vio- 6 lation of the strong eigenstate thermalization hypothesis, Phys. Rev. B 98, 235156 (2018)

    work page 2018

  7. [7]

    P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Poll- mann, Ergodicity breaking arising from Hilbert space frag- mentation in dipole-conserving Hamiltonians, Phys. Rev. X 10, 011047 (2020)

    work page 2020

  8. [8]

    A. J. A. James, R. M. Konik, and N. J. Robinson, Non- thermal states arising from confinement in one and two dimensions, Phys. Rev. Lett. 122, 130603 (2019)

    work page 2019

Show all 69 references
  1. [9]

    Bluvstein, A

    D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Se- meghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, S. Choi, M. Serbyn, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Controlling quantum many- body dynamics in driven Rydberg atom arrays, Science 371, 1355 (2021)

    2021

  2. [10]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Probing many- body dynamics on a 51-atom quantum simulator, Nature (London) 551, 579 (2017)

    2017

  3. [11]

    G.-X. Su, H. Sun, A. Hudomal, J.-Y. Desaules, Z.-Y. Zhou, B. Yang, J. C. Halimeh, Z.-S. Yuan, Z. Papi´ c, and J.-W. Pan, Observation of many-body scarring in a Bose- Hubbard quantum simulator, Phys. Rev. Res. 5, 023010 (2023)

    2023

  4. [12]

    Quan- tum Simulations of Fundamental Physics

    The vertical dashed lines correspond to the first energy gap. The insets of each panel display the real-time oscillation curves of the AFM (upper) and FM states (lower), with time in units of 2 π. (g) The two distinct regions of thermalization behavior: QMBS (0 ≤ g ≲ 0.3) and ...

  5. [13]

    Zhang, H

    P. Zhang, H. Dong, Y. Gao, L. Zhao, J. Hao, J.-Y. De- saules, Q. Guo, J. Chen, J. Deng, B. Liu, W. Ren, Y. Yao, X. Zhang, S. Xu, K. Wang, F. Jin, X. Zhu, B. Zhang, H. Li, C. Song, Z. Wang, F. Liu, Z. Papi´ c, L. Ying, H. Wang, and Y.-C. Lai, Many-body Hilbert space scarring on...

    2022

  6. [14]

    Serbyn, D

    M. Serbyn, D. A. Abanin, and Z. Papi´ c, Quantum many- body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021)

    2021

  7. [15]

    Moudgalya, B

    S. Moudgalya, B. A. Bernevig, and N. Regnault, Quan- tum many-body scars and Hilbert space fragmentation: A review of exact results, Rep. Prog. Phys. 85, 086501 (2022)

    2022

  8. [16]

    Moudgalya and O

    S. Moudgalya and O. I. Motrunich, Exhaustive charac- terization of quantum many-body scars using commutant algebras, Phys. Rev. X 14, 041069 (2024)

    2024

  9. [17]

    Shiraishi and T

    N. Shiraishi and T. Mori, Systematic construction of counterexamples to the eigenstate thermalization hypoth- esis, Phys. Rev. Lett. 119, 030601 (2017)

    2017

  10. [18]

    Chandran, T

    A. Chandran, T. Iadecola, V. Khemani, and R. Moessner, Quantum many-body scars: A quasiparticle perspective, Annu. Rev. Condens. Matter Phys. 14, 443 (2023)

    2023

  11. [19]

    M. C. Ba˜ nuls, J. I. Cirac, and M. B. Hastings, Strong and weak thermalization of infinite nonintegrable quan- tum systems, Phys. Rev. Lett. 106, 050405 (2011)

    2011

  12. [20]

    Kormos, M

    M. Kormos, M. Collura, G. Tak´ acs, and P. Calabrese, Real-time confinement following a quantum quench to a non-integrable model, Nat. Phys. 13, 246 (2016)

    2016

  13. [21]

    Lin and O

    C.-J. Lin and O. I. Motrunich, Quasiparticle explanation of the weak-thermalization regime under quench in a non- integrable quantum spin chain, Phys. Rev. A 95, 023621 (2017)

    2017

  14. [22]

    Collura, M

    M. Collura, M. Kormos, and G. Tak´ acs, Dynamical man- ifestation of the Gibbs paradox after a quantum quench, Phys. Rev. A 98, 053610 (2018)

    2018

  15. [23]

    Wilczek, Quantum time crystals, Phys

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

    2012

  16. [24]

    D. V. Else, B. Bauer, and C. Nayak, Prethermal phases of matter protected by time-translation symmetry, Phys. Rev. X 7, 011026 (2017)

    2017

  17. [25]

    Medenjak, B

    M. Medenjak, B. Buˇ ca, and D. Jaksch, Isolated Heisen- berg magnet as a quantum time crystal, Phys. Rev. B102, 041117(R) (2020)

    2020

  18. [26]

    Kongkhambut, J

    P. Kongkhambut, J. Skulte, L. Mathey, J. G. Cosme, A. Hemmerich, and H. Keßler, Observation of a continuous time crystal, Science 377, 670 (2022)

    2022

  19. [27]

    Liu, J.-Y

    T. Liu, J.-Y. Ou, K. F. MacDonald, and N. I. Zheludev, Photonic metamaterial analogue of a continuous time crys- tal, Nat. Phys. 19, 986 (2023)

    2023

  20. [28]

    Greilich, N

    A. Greilich, N. E. Kopteva, A. N. Kamenskii, P. S. Sokolov, V. L. Korenev, and M. Bayer, Robust continu- ous time crystal in an electron–nuclear spin system, Nat. Phys. 20, 631 (2024)

    2024

  21. [29]

    X. Wu, Z. Wang, F. Yang, R. Gao, C. Liang, M. K. Tey, X. Li, T. Pohl, and L. You, Dissipative time crystal in a strongly interacting Rydberg gas, Nat. Phys. 20, 1389 (2024)

    2024

  22. [30]

    K. Bull, A. Hallam, Z. Papi´ c, and I. Martin, Tuning be- tween continuous time crystals and many-body scars in long-range XY Zspin chains, Phys. Rev. Lett.129, 140602 (2022)

    2022

  23. [31]

    Buˇ ca, Unified theory of local quantum many-body dynamics: Eigenoperator thermalization theorems, Phys

    B. Buˇ ca, Unified theory of local quantum many-body dynamics: Eigenoperator thermalization theorems, Phys. Rev. X 13, 031013 (2023)

    2023

  24. [32]

    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

  25. [33]

    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

  26. [34]

    Fisher and W

    M. Fisher and W. Brittin, Statistical Physics, Weak In- teractions, Field Theory, Lectures in Theoretical Physics (University of Colorado Press, Boulder, 1965), Vol. VIIC

    1965

  27. [35]

    Maldacena, S

    J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08(2016)106

    2016

  28. [36]

    Takahashi and H

    Y. Takahashi and H. Umezawa, Thermo field dynamics, Int. J. Mod. Phys. B 10, 1755 (1996)

    1996

  29. [37]

    del Campo, J

    A. del Campo, J. Molina-Vilaplana, and J. Sonner, Scrambling the spectral form factor: Unitarity constraints and exact results, Phys. Rev. D 95, 126008 (2017)

    2017

  30. [38]

    Y. Liu, S. Lv, Y. Yang, and H. Zou, Signatures of quan- tum criticality in the complex inverse temperature plane, Chin. Phys. Lett. 40, 050502 (2023)

    2023

  31. [39]

    Y. Liu, S. Lv, Y. Meng, Z. Tan, E. Zhao, and H. Zou, Exact fisher zeros and thermofield dynamics across a quan- tum critical point, Phys. Rev. Res. 6, 043139 (2024)

    2024

  32. [40]

    Y. Liu, E. Zhao, and H. Zou, From complexification to self-similarity: New aspects of quantum criticality, Chin. Phys. Lett. 41, 100501 (2024)

    2024

  33. [41]

    M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model, Phys. Rev. Lett. 110, 135704 (2013)

    2013

  34. [42]

    A. A. Zvyagin, Dynamical quantum phase transitions (re- view article), Low Temp. Phys. 42, 971 (2016)

    2016

  35. [43]

    Heyl, Dynamical quantum phase transitions: A re- view, Rep

    M. Heyl, Dynamical quantum phase transitions: A re- view, Rep. Prog. Phys. 81, 054001 (2018)

    2018

  36. [44]

    Van Damme, J.-Y

    M. Van Damme, J.-Y. Desaules, Z. Papi´ c, and J. C. Hal- imeh, Anatomy of dynamical quantum phase transitions, Phys. Rev. Res. 5, 033090 (2023). 7

    2023

  37. [45]

    Lesanovsky and H

    I. Lesanovsky and H. Katsura, Interacting fibonacci anyons in a Rydberg gas, Phys. Rev. A 86, 041601(R) (2012)

    2012

  38. [46]

    C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´ c, Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018)

    2018

  39. [47]

    W. W. Ho, S. Choi, H. Pichler, and M. D. Lukin, Periodic orbits, entanglement, and quantum many-body scars in constrained models: Matrix product state approach, Phys. Rev. Lett. 122, 040603 (2019)

    2019

  40. [48]

    Zhang, D

    S.-Y. Zhang, D. Yuan, T. Iadecola, S. Xu, and D.-L. Deng, Extracting quantum many-body scarred eigenstates with matrix product states, Phys. Rev. Lett. 131, 020402 (2023)

    2023

  41. [49]

    Ivanov and O

    A. Ivanov and O. Motrunich, Many exact area-law scar eigenstates in the nonintegrable PXP and related models, arXiv:2503.16327

    arXiv

  42. [50]

    Or´ us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann

    R. Or´ us, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (Amsterdam) 349, 117 (2014)

    2014

  43. [51]

    J. I. Cirac, D. P´ erez-Garc ´ ıa, N. Schuch, and F. Ver- straete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021)

    2021

  44. [52]

    Meurice, R

    Y. Meurice, R. Sakai, and J. Unmuth-Yockey, Tensor lat- tice field theory for renormalization and quantum comput- ing, Rev. Mod. Phys. 94, 025005 (2022)

    2022

  45. [53]

    M. Suzuki, Relationship between d-dimensional quantal spin systems and (d+1)-dimensional Ising systems: Equiv- alence, critical exponents and systematic approximants of the partition function and spin correlations, Prog. Theor. Phys. 56, 1454 (1976)

    1976

  46. [54]

    Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, Coarse-graining renormalization by higher-order singular value decomposition, Phys. Rev. B 86, 045139 (2012)

    2012

  47. [55]

    Denbleyker, Y

    A. Denbleyker, Y. Liu, Y. Meurice, M. P. Qin, T. Xiang, Z. Y. Xie, J. F. Yu, and H. Zou, Controlling sign problems in spin models using tensor renormalization, Phys. Rev. D 89, 016008 (2014)

    2014

  48. [56]

    The Supplemental Material also contains Refs.[36,37,55-59]

    See Supplemental Material at http://link.aps.org/supplemental/10.1103/glc5-hv2m for the details on (a) Fisher zeros of the ¯P X¯P model and the fixed-point line, (b) Fisher zeros of Zi in the ¯P X¯P model, (c) Fisher zeros of a lattice gauge field related model, (d) the Ising ...

    doi

  49. [57]

    Borla, R

    U. Borla, R. Verresen, F. Grusdt, and S. Moroz, Confined phases of one-dimensional spinless fermions coupled to Z2 gauge theory, Phys. Rev. Lett. 124, 120503 (2020)

    2020

  50. [58]

    Iadecola and M

    T. Iadecola and M. Schecter, Quantum many-body scar states with emergent kinetic constraints and finite- entanglement revivals, Phys. Rev. B 101, 024306 (2020)

    2020

  51. [59]

    J. C. Halimeh, L. Barbiero, P. Hauke, F. Grusdt, and A. Bohrdt, Robust quantum many-body scars in lattice gauge theories, Quantum 7, 1004 (2023)

    2023

  52. [60]

    Denbleyker, D

    A. Denbleyker, D. Du, Y. Liu, Y. Meurice, and H. Zou, Fisher’s zeros as the boundary of renormalization group flows in complex coupling spaces, Phys. Rev. Lett. 104, 251601 (2010)

    2010

  53. [61]

    Meurice and H

    Y. Meurice and H. Zou, Complex renormalization group flows for 2d nonlinear o(n) sigma models, Phys. Rev. D 83, 056009 (2011)

    2011

  54. [62]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rig- orous results on valence-bond ground states in antiferro- magnets, Phys. Rev. Lett. 59, 799 (1987)

    1987

  55. [63]

    R. Shen, Y. Guo, and S. Yang, Construction of non-Hermitian parent Hamiltonian from matrix product states, Phys. Rev. Lett. 130, 220401 (2023)

    2023

  56. [64]

    Peter Zoller, Closing remarks at the 2nd Confer- ence on ‘Quantum Simulations of Fundamental Physics’ (https://m.koushare.com/live/details/37543) and the International Workshop on ‘Quantum Systems with Novel Spatiotemporal Control’ (https://m.koushare.com/live/details/38269)

  57. [65]

    Francis, D

    A. Francis, D. Zhu, C. Huerta Alderete, S. Johri, X. Xiao, J. K. Freericks, C. Monroe, N. M. Linke, and A. F. Kem- per, Many-body thermodynamics on quantum comput- ers via partition function zeros, Sci. Adv. 7, 10.1126/sci- adv.abf2447 (2021)

    2021 doi

  58. [66]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Ran- dom quantum circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023)

    2023

  59. [67]

    Granet, C

    E. Granet, C. Zhang, and H. Dreyer, Volume-law to area-law entanglement transition in a nonunitary periodic Gaussian circuit, Phys. Rev. Lett. 130, 230401 (2023)

    2023

  60. [68]

    X. Peng, H. Zhou, B.-B. Wei, J. Cui, J. Du, and R.-B. Liu, Experimental observation of Lee-Yang zeros, Phys. Rev. Lett. 114, 010601 (2015)

    2015

  61. [69]

    𝛽" 𝐿=32 𝐿 𝑑#𝑑$ 0 0.5 1 1.5 20 0.5 1 1.5 2 •••••••••• •••••••••• 00.511.520 0.5 1 1.5 2 𝐿=128(c) (f) 𝑔=0.9 𝑔=0.9 00.010.020.030.040.050.060 0.02 0.04 0.06 0.08(i) 𝑔=0.9 𝛽!(𝜋) 𝛽

    Y. Meng, S. Lv, Y. Liu, Z. Tan, E. Zhao, and H. Zou, Figure data for detecting many-body scars from Fisher zeros (https://doi.org/10.5281/zenodo.16616679) (2025) . Supplementary information for ‘Detecting Many-Body Scars from Fisher Zeros’ FISHER ZEROS OF THE¯P X¯P MODEL AND T...

    2025 doi

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Reviewed August 10, 2026 · model on record in the stance chip above.