Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

At the O(3) quantum critical point, Rényi defects split into distinct universality classes, and the extraordinary defect undergoes an ordering transition as the Rényi index is tuned.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 15:05 UTC pith:6VOX3OGK

load-bearing objection Solid evidence for multiple Rényi-defect universality classes at n=2,3; the finite-n transition on the extraordinary defect is a suggestive but unproven extra. the 3 major comments →

arxiv 2605.00104 v2 pith:6VOX3OGK submitted 2026-04-30 cond-mat.str-el cond-mat.stat-mechhep-thquant-ph

Criticality on R\'enyi defects at (2+1)d O(3) quantum critical points

Yanzhang Zhu , Zhe Wang , Meng Cheng , Zheng Yan This is my paper
classification cond-mat.str-el cond-mat.stat-mechhep-thquant-ph
keywords Rényi entanglement entropyRényi defectconical defectO(3) Wilson-Fisherdefect universality classquantum Monte Carlosurface criticalityBinder cumulant
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that the Rényi (or conical) defect that encodes entanglement entropy is not a single universal object: at the (2+1)d O(3) Wilson-Fisher quantum critical point, microscopically different entanglement cuts flow to three distinct defect universality classes, named ordinary, special, and extraordinary. By measuring equal-time spin correlations and Binder cumulants on the n-sheeted replica manifold, the authors extract different defect scaling dimensions for each class and find evidence that the extraordinary defect transitions from disordered to ferromagnetically ordered as the Rényi index n grows past roughly 3–4. If correct, this explains why supposedly universal Rényi-entropy coefficients depend on lattice details: different cuts are really different defect fixed points. It also makes the Rényi index itself a tunable parameter that can drive a phase transition on an entanglement defect.

Core claim

For a fixed Rényi index n, the Rényi defect created by an entanglement cut at the (2+1)d O(3) quantum critical point is not unique. Choosing ordinary, special, or extraordinary entanglement bipartitions yields three distinct Rényi-defect universality classes, characterized by different scaling dimensions Δ̂_φ for the O(3) order parameter on the defect (n=2: 0.358(5), 0.278(2), 0.317(2); n=3: 0.324(6), 0.154(1), 0.178(1)). Ordinary and special defects remain disordered for the accessible Rényi indices, while the extraordinary defect shows a Binder-cumulant crossing near n_c≈3–4, with disordered behavior at small n and clear ferromagnetic order on the defect at larger n, as seen in finite-size

What carries the argument

The central object is the Rényi defect: the codimension-two conical singularity introduced when a subsystem is traced out, represented in simulations by an n-sheeted replica manifold. Observables localized on the defect are measured through the Rényi expectation value ⟨O⟩_n = Tr(O ρ_A^n)/Tr ρ_A^n, with long-distance defect physics captured by the defect scaling dimension Δ̂_φ defined through C_s(L) ∼ a L^{-2Δ̂_φ}(1+b/L). The classification borrows the ordinary/special/extraordinary language of surface criticality, and the Binder cumulant of the defect magnetization serves as the diagnostic for ordering on the defect as n is varied.

Load-bearing premise

The load-bearing premise is that the apparent order on the extraordinary Rényi defect at n≥4 is genuine spontaneous symmetry breaking on the defect line, not a finite-size artifact of a very small defect scaling dimension—a possibility the paper itself explicitly flags.

What would settle it

Compute the extraordinary-defect spin correlation and Binder cumulant at n=4 for system sizes much larger than L=80, say L=128–256, and check whether the Binder crossing remains near n_c≈3–4 and whether the order parameter extrapolates to a nonzero value. If the crossing shifts with system size or the order parameter vanishes, the finite-n ordering transition is a finite-size effect; if true spontaneous breaking of a continuous symmetry on a line defect is confirmed, it would confront the general argument against such breaking in relativistic CFTs.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The known sensitivity of Rényi-entropy scaling to microscopic cut geometry can be understood as different entanglement cuts flowing to different defect universality classes in the infrared.
  • Measured defect scaling dimensions can serve as fingerprints for identifying which defect fixed point a given lattice entanglement cut realizes.
  • For the extraordinary cut, Rényi entanglement observables should inherit a nonanalyticity near n_c≈3–4, so the shape dependence of Rényi entropy changes qualitatively with Rényi index.
  • Ordinary and special defects remain disordered through the studied range (n=2, 3, and up to n=10 for the special cut), while the extraordinary defect orders at large n.
  • The classification gives a unified vocabulary for boundary criticality and entanglement cuts, suggesting that the same three classes should control Rényi-entropy subleading corrections in other O(N) critical models.

Where Pith is reading between the lines

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

  • Editorial: If the extraordinary-defect ordering transition is genuine, the Rényi index acts as a relevant control parameter in an effective defect renormalization-group flow, implying a new kind of entanglement-driven phase transition that is invisible in the bulk correlation functions.
  • Editorial: The same numerical protocol applied to O(2) or Ising critical points would test whether the three-defect-class pattern is generic for Wilson-Fisher fixed points; the predicted exponents should differ if the classification is universal.
  • Editorial: A decisive missing check is a scaling collapse of the extraordinary-defect Binder cumulant across the n_c≈3–4 crossing using L^{1/ν}; the current data show a crossing but do not yet establish a divergent correlation-length exponent.
  • Editorial: The close-to-critical extrapolations at n=4 give small but nonzero order parameters, so the transition could still be a very weak first-order or near-BKT transition rather than a conventional continuous one; longer correlation functions along the defect would discriminate.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies Rényi defects at the (2+1)d O(3) quantum critical point by measuring local spin correlations and Binder cumulants on the n-sheeted replica manifold with quantum Monte Carlo. Three microscopic entanglement bipartitions—called ordinary, special, and extraordinary—are argued to realize distinct Rényi-defect universality classes. Table I reports scaling dimensions Δφ̂ for n=2 and n=3 that differ across cuts, and the paper further claims a finite-n transition on the extraordinary defect near n_c≈3–4, with a ferromagnetically ordered phase for larger n. Cross-checks in bilayer and staggered dimer models are presented in the Supplemental Material, along with fit-stability tables.

Significance. If the distinct-exponent result holds, it is significant: it would show that microscopic details of an entanglement cut can flow to different defect universality classes, thereby explaining the observed cut-dependence of Rényi entropy subleading terms. The paper has real strengths: it uses large-scale QMC on replica manifolds, provides cross-model checks in two additional lattice realizations, and reports Lmin-stability of the n=2 fits. I do not see a circularity problem: the boundary-criticality names are an analogy, and the extracted exponents come from raw correlation data. The weaker, and load-bearing, part is the claimed extraordinary-defect phase transition, which the paper itself concedes may be a finite-size artifact.

major comments (3)
  1. [Extraordinary Rényi defects (Figs. 3, 4) and Discussions] The finite-n transition on the extraordinary defect is not established by the presented evidence. The Binder crossing in Fig. 4(c) uses only L=16,24,32, with no error bars shown, and there is no scaling collapse, no crossing at larger L, and no critical-exponent extraction. The order-parameter evidence is polynomial extrapolation to small intercepts: Cs(∞)=0.02244(10), Mz^2(∞)=0.02506(8) at n=4, and 0.04165(11)/0.04177(10) at n=10. These small intercepts are exactly what a disorder operator with very small but nonzero Δφ̂ would produce on finite systems. The authors themselves concede in the Discussions that the apparent order "could be a finite-size manifestation ... with a very small Δφ̂", and cite [66] arguing against continuous-symmetry breaking on line defects. Since the abstract and conclusion present this transition as a central result, this is a load-bearing issue. Either direct
  2. [Table I, Eq. (3), and SM Table S1] The "distinct universality classes" claim rests on fits to Eq. (3), which assumes a single L^{-1} correction. SM Table S1 shows a systematic drift for the ordinary exponent from 0.358(5) at Lmin=16 to 0.40(3) at Lmin=40, and for the extraordinary exponent from 0.317(2) to 0.34(2). At the largest Lmin the ordinary–extraordinary difference is only about 1.7σ, much weaker than at Lmin=16. For n=3, no stability analysis is reported. Since the central claim is that the three exponents are distinct at fixed n, please provide Lmin-stability for n=3 and test whether the distinctions survive the inclusion of a second correction term, such as c L^{-2} or a fixed subleading exponent. The current evidence is sufficient to distinguish ordinary from special at n=2, but the full three-way distinction needs this robustness check.
  3. [Discussions] The comparison with analytic large-N results relies on unpublished numbers attributed to [65] ("Some representative values from the large-N solution are Δφ̂≈0.34 for n=2 and 0.23 for n=3 [10,65]"). These values cannot be checked by the reader. If this comparison is intended to support the numerics, the details should be made available in a preprint or in the paper itself; otherwise the comparison should be presented more qualitatively, or the reliance on [65] removed. This is not fatal to the main claims, but it is an unverifiable input.
minor comments (4)
  1. [Fig. 4] The Binder cumulant panels show no visible error bars. If the statistical errors are smaller than the symbol size, state this explicitly; otherwise include them, especially for the claimed crossing in Fig. 4(c).
  2. [Abstract and Conclusion] The phrase "ferromagnetically ordered phase" and the unqualified statement of a finite-n transition are stronger than the evidence and than the caveat in the Discussions. The wording should match the level of support: if the transition claim is retained, it should be stated as suggestive rather than established.
  3. [SM Table S1] The text says the reduced chi-squared values are "close to unity", but the extraordinary-defect fit gives χ²_red=1.606. This is not close to unity in the usual sense and may indicate that the single-correction form in Eq. (3) is less adequate for that cut. Please comment.
  4. [General] The paper should clarify that the quoted C_s(L) is the equal-time correlation on the Rényi manifold, and not a standard ground-state correlation; this distinction is important for readers outside the replica-QMC subfield.

Circularity Check

0 steps flagged

No load-bearing circularity; exponents are direct fits, and the extraordinary-transition caveat is a correctness concern, not a circular step.

full rationale

The derivation chain is self-contained. The Rényi defect scaling dimensions in Table I are obtained by direct finite-size fits of the raw correlation Cs(L) to Eq. (3); the same quantity is not used as both input and output. The ordinary/special/extraordinary classification is an analogy to boundary criticality, with the cuts defined geometrically in Fig. 1, not a theorem imported from the paper's own prior work. The claimed finite-n transition on the extraordinary defect is an extrapolation from Cs(L), Mz^2, and Binder crossings; the paper explicitly concedes that the apparent order 'could be a finite-size manifestation of the defect order parameter with a very small Δφ' and cites [66] against line-defect symmetry breaking, so the transition is presented as evidence with an acknowledged alternative, not as a consequence forced by construction. The only mild self-citation issue is reference [65] (Cheng, Myerson-Jain, Shankar 2026, unpublished), used for large-N comparison values; it is not load-bearing because the main claims do not rest on those numbers and the paper's own simulations are the evidence. Thus score 2: one minor non-load-bearing self-citation, no circular reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 1 invented entities

The central multi-class claim rests mainly on the fit form (3) and the assumed universality of the three boundary classes. The transition claim rests on the interpretation of the Binder crossing and the validity of the ordered phase. The only truly invented entity is the extraordinary-defect ordered phase, which currently lacks independent confirmation.

free parameters (2)
  • Correction-to-scaling exponent in Eq. (3) = fixed at 1 (b L^{-1} term)
    The correlation fits assume the leading finite-size correction is L^{-1}; if the true exponent differs, the extracted Δφ shifts. SM Table S1 shows the ordinary exponent drifting from 0.358(5) to 0.40(3) as Lmin increases, a drift comparable to inter-class differences.
  • Lower size cutoff Lmin in fits = 16
    Reported Δφ values use Lmin=16 with Lmax=80; the SM shows a mild systematic drift with Lmin, largest for the ordinary cut. This hand-chosen cutoff affects the central exponent claim.
axioms (5)
  • domain assumption The dimerized Heisenberg model at J'/J=1.9096 realizes the 3D O(3) Wilson-Fisher QCP.
    Used for the bulk model; taken from prior literature [47], not re-verified in this paper.
  • domain assumption The Rényi-index expectation ⟨O⟩_n = Tr(Oρ_A^n)/Tr ρ_A^n can be computed directly in replica QMC and equals the defect expectation value in the CFT.
    Foundation of the entire measurement scheme; supported by refs [48–52], several of which are by the same group.
  • standard math The conical defect boundary condition (Eq. 4) and bulk-to-defect OPE (Eq. 5) describe the IR limit, with bulk Δφ≈0.52.
    Standard CFT input, cited in the text.
  • domain assumption The large-N reference values Δφ≈0.34 (n=2) and 0.23 (n=3) from [10,65] are correct.
    Used to position the results as "beyond large-N"; [65] is unpublished and self-authored, so this benchmark is not independently verifiable.
  • domain assumption The restriction of [66] against continuous symmetry breaking on line defects is relevant to the observed extraordinary-defect order.
    The authors use this to frame the ordered phase as surprising, and offer two mutually exclusive escape routes without deciding between them.
invented entities (1)
  • Extraordinary Rényi-defect ferromagnetic phase no independent evidence
    purpose: Describes the apparent spontaneously symmetry-broken state on the tilted cut at n≥4 and in the large-n limit.
    Identified only in this numerical study; no independent falsifiable handle yet. The paper itself notes it may be a finite-size manifestation of a small Δφ, and that continuous symmetry breaking on line defects is disfavored by [66].

pith-pipeline@v1.3.0-alltime-deepseek · 13274 in / 14892 out tokens · 148722 ms · 2026-08-02T15:05:58.965166+00:00 · methodology

0 comments
read the original abstract

At a quantum critical point, the universal scaling behavior of R\'enyi entanglement entropy is controlled by the universality class of the codimension-two R\'enyi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along R\'enyi defect lines in (2+1)d quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed R\'enyi index $n$, there exist multiple R\'enyi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the R\'enyi index. Our results highlight the key role of defect universality classes in determining the universal scaling of R\'enyi entropy, and provide a framework for understanding the previously observed dependence of R\'enyi entropy scaling on microscopic lattice details.

Figures

Figures reproduced from arXiv: 2605.00104 by Meng Cheng, Yanzhang Zhu, Zheng Yan, Zhe Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the columnar dimerized [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Finite-size scaling of observables for ordinary and spe [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite-size scaling of observables for the extraordinary [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Review history (2 revisions) →

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Determination of thermodynamics from entanglement entropy in the finite-density O(N) model

    hep-th 2026-07 accept novelty 7.0

    The derivative of entanglement entropy with respect to subregion volume equals the thermal entropy density in the large-subregion limit, verified via lattice simulations of the finite-density O(4) model using dual wor...

Reference graph

Works this paper leans on

66 extracted references · 6 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.2004, P06002 (2004)

    2004

  2. [2]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entan- glement in many-body systems, Rev. Mod. Phys.80, 517 (2008)

    2008

  3. [3]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010)

    2010

  4. [4]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A42, 504005 (2009)

    2009

  5. [5]

    Nishioka, Entanglement entropy: holography and renormalization group, Rev

    T. Nishioka, Entanglement entropy: holography and renormalization group, Rev. Mod. Phys.90, 035007 (2018)

    2018

  6. [6]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B424, 443 (1994)

    1994

  7. [7]

    V. E. Korepin, Universality of entropy scaling in one di- mensional gapless models, Phys. Rev. Lett.92, 096402 (2004)

    2004

  8. [8]

    Laflorencie, Quantum entanglement in condensed matter systems, Phys

    N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rep.646, 1 (2016)

    2016

  9. [9]

    A. B. Kallin, M. B. Hastings, R. G. Melko, and R. R. Singh, Anomalies in the entanglement properties of the square-lattice heisenberg model, Phys. Rev. B84, 165134 (2011)

    2011

  10. [10]

    M. A. Metlitski, C. A. Fuertes, and S. Sachdev, Entan- glement Entropy in the O(N) model, Phys. Rev. B80, 115122 (2009)

    2009

  11. [11]

    M. A. Metlitski and T. Grover, Entanglement entropy of systems with spontaneously broken continuous symme- try, (2011), arXiv:1112.5166 [cond-mat.str-el]

    Pith/arXiv arXiv 2011

  12. [12]

    Li and F

    H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states, Phys. Rev. Lett.101, 010504 (2008)

    2008

  13. [13]

    Kitaev and J

    A. Kitaev and J. Preskill, Topological entanglement en- tropy, Phys. Rev. Lett.96, 110404 (2006)

    2006

  14. [14]

    Casini and M

    H. Casini and M. Huerta, Universal terms for the entan- glement entropy in 2+1 dimensions, Nucl. Phys. B764, 183 (2007)

    2007

  15. [15]

    Fradkin and J

    E. Fradkin and J. E. Moore, Entanglement entropy of 2d conformal quantum critical points: Hearing the shape of a quantum drum, Phys. Rev. Lett.97, 050404 (2006)

    2006

  16. [16]

    Z. Deng, L. Liu, W. Guo, and H.-Q. Lin, Diagnosing quantumphasetransitionorderanddeconfinedcriticality via entanglement entropy, Phys. Rev. Lett.133, 100402 (2024)

    2024

  17. [17]

    Y.-C. Lin, F. Iglói, and H. Rieger, Entanglement entropy at infinite-randomness fixed points in higher dimensions, Phys. Rev. Lett.99, 147202 (2007)

    2007

  18. [18]

    A. B. Kallin, E. Stoudenmire, P. Fendley, R. R. Singh, and R. G. Melko, Corner contribution to the entangle- ment entropy of an o (3) quantum critical point in 2+ 1 dimensions, J. Stat. Mech.2014, P06009 (2014)

    2014

  19. [19]

    Helmes and S

    J. Helmes and S. Wessel, Entanglement entropy scaling in the bilayer heisenberg spin system, Phys. Rev. B89, 245120 (2014)

    2014

  20. [20]

    Park, Logarithmic corrections to the entanglement en- tropy, Phys

    C. Park, Logarithmic corrections to the entanglement en- tropy, Phys. Rev. D92, 126013 (2015)

    2015

  21. [21]

    Whitsitt, W

    S. Whitsitt, W. Witczak-Krempa, and S. Sachdev, En- tanglement entropy of large-nwilson-fisher conformal field theory, Phys. Rev. B95, 045148 (2017)

    2017

  22. [22]

    P.Bueno, R.C.Myers,andW.Witczak-Krempa,Univer- sality of corner entanglement in conformal field theories, Phys. Rev. Lett.115, 021602 (2015)

    2015

  23. [23]

    Helmes, L

    J. Helmes, L. E. Hayward Sierens, A. Chandran, W. Witczak-Krempa, and R. G. Melko, Universal corner entanglement of dirac fermions and gapless bosons from the continuum to the lattice, Phys. Rev. B94, 125142 (2016)

    2016

  24. [24]

    Zhao, Y.-C

    J. Zhao, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, Scaling of entanglement entropy at deconfined quantum criticality, Phys. Rev. Lett.128, 010601 (2022)

    2022

  25. [25]

    M. Song, J. Zhao, Z. Y. Meng, C. Xu, and M. Cheng, Extracting subleading corrections in entanglement en- tropy at quantum phase transitions, SciPost Phys.17, 010 (2024)

    2024

  26. [26]

    M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Measuring renyi entanglement entropy in quan- tum monte carlo simulations, Phys. Rev. Lett.104, 157201 (2010)

    2010

  27. [27]

    D’Emidio, Entanglement entropy from nonequilibrium work, Phys

    J. D’Emidio, Entanglement entropy from nonequilibrium work, Phys. Rev. Lett.124, 110602 (2020)

    2020

  28. [28]

    Grover, Entanglement of interacting fermions in quan- tum monte carlo calculations, Phys

    T. Grover, Entanglement of interacting fermions in quan- tum monte carlo calculations, Phys. Rev. Lett.111, 130402 (2013)

    2013

  29. [29]

    Humeniuk and T

    S. Humeniuk and T. Roscilde, Quantum monte carlo cal- culationofentanglementrényientropiesforgenericquan- 6 tum systems, Phys. Rev. B86, 235116 (2012)

    2012

  30. [30]

    A. B. Kallin, K. Hyatt, R. R. P. Singh, and R. G. Melko, Entanglement at a two-dimensional quantum crit- ical point: A numerical linked-cluster expansion study, Phys. Rev. Lett.110, 135702 (2013)

    2013

  31. [31]

    Zhao, B.-B

    J. Zhao, B.-B. Chen, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, Measuring rényi entanglement entropy withhighefficiencyandprecisioninquantummontecarlo simulations, npj Quantum Mater.7, 69 (2022)

    2022

  32. [32]

    Da Liao, M

    Y. Da Liao, M. Song, J. Zhao, and Z. Y. Meng, Ex- tracting universal corner entanglement entropy during the quantum monte carlo simulation, Phys. Rev. B110, 235111 (2024)

    2024

  33. [33]

    Z. Wang, Z. Deng, Z. Liu, Z. Wang, Y.-M. Ding, L. Zhang, W. Guo, and Z. Yan, Universal behavior in entanglement entropy reveals quantum criticality and underlying symmetry breaking, Chin. Phys. Lett.42, 110712 (2025)

    2025

  34. [34]

    Z. Wang, Z. Wang, Y.-M. Ding, B.-B. Mao, and Z. Yan, Bipartite reweight-annealing algorithm of quan- tum monte carlo to extract large-scale data of entangle- ment entropy and its derivative, Nat. Commun.16, 5880 (2025)

    2025

  35. [35]

    Z. Wang, C. Guo, B.-B. Mao, and Z. Yan, Universal and non-universal contributions of entanglement under differ- entbipartitions, (2026),arXiv:2601.12365[cond-mat.str- el]

    arXiv 2026

  36. [36]

    D’Emidio, R

    J. D’Emidio, R. Orús, N. Laflorencie, and F. de Juan, Universal features of entanglement entropy in the hon- eycomb hubbard model, Phys. Rev. Lett.132, 076502 (2024)

    2024

  37. [37]

    D’Emidio and A

    J. D’Emidio and A. W. Sandvik, Entanglement entropy and deconfined criticality: Emergent so(5) symmetry and proper lattice bipartition, Phys. Rev. Lett.133, 166702 (2024)

    2024

  38. [38]

    Y. Zhu, Z. Liu, Z. Wang, Y.-C. Wang, and Z. Yan, Bi- partite entanglement and surface criticality: The extra contribution of the nonordinary edge in entanglement, Phys. Rev. Lett.136, 046501 (2026)

    2026

  39. [39]

    Liu, R.-Z

    Z. Liu, R.-Z. Huang, Z. Yan, and D.-X. Yao, Demon- strating the wormhole mechanism of the entanglement spectrum via a perturbed boundary, Phys. Rev. B109, 094416 (2024)

    2024

  40. [40]

    Bianchi, M

    L. Bianchi, M. Meineri, R. C. Myers, and M. Smolkin, Rényi entropy and conformal defects, JHEP07, 076, arXiv:1511.06713 [hep-th]

    Pith/arXiv arXiv

  41. [41]

    Andrei, A

    N. Andrei, A. Bissi, M. Buican, J. Cardy, P. Dorey, N. Drukker, J. Erdmenger, D. Friedan, D. Fursaev, A. Konechny,et al., Boundary and defect cft: open prob- lems and applications, J. Phys. A: Math. Theor.53, 453002 (2020)

    2020

  42. [42]

    G.Cuomo, Z.Komargodski,andA.Raviv-Moshe,Renor- malization group flows on line defects, Phys. Rev. Lett. 128, 021603 (2022)

    2022

  43. [43]

    Z. Wang, Z. Wang, Y.-M. Ding, Z. Liu, Z. Yan, and L. Zhang, Boundary renormalization group flow of en- tanglement entropy at a (2 + 1)-dimensional quantum critical point, Phys. Rev. B113, L161104 (2026)

    2026

  44. [44]

    I. Affleck, Conformal field theory approach to quan- tum impurity problems, inField Theories for Low- Dimensional Condensed Matter Systems: Spin Systems and Strongly Correlated Electrons(Springer, 2000) pp. 117–141

    2000

  45. [45]

    Cuomo, Z

    G. Cuomo, Z. Komargodski, and M. Mezei, Localized magnetic field in the o (n) model, J. High Energy Phys. 2022(2), 134

    2022

  46. [46]

    Y.-H. Wu, Y. Zhang, H.-H. Tu, and M. Cheng, Impurity screening by defects in(1 + 1)dquantum critical systems, Phys. Rev. Lett.136, 036502 (2026)

    2026

  47. [47]

    C. Ding, L. Zhang, and W. Guo, Engineering surface crit- ical behavior of (2 + 1)-dimensional o(3) quantum critical points, Phys. Rev. Lett.120, 235701 (2018)

    2018

  48. [48]

    Yan and Z

    Z. Yan and Z. Y. Meng, Unlocking the general relation- ship between energy and entanglement spectra via the wormhole effect, Nat. Commun.14, 2360 (2023)

    2023

  49. [49]

    Yan, Relevant long-range interaction of the entan- glement hamiltonian emerges from a short-range gapped system, Phys

    C.Li, R.-Z.Huang, Y.-M.Ding, Z.Y.Meng, Y.-C.Wang, and Z. Yan, Relevant long-range interaction of the entan- glement hamiltonian emerges from a short-range gapped system, Phys. Rev. B109, 195169 (2024)

    2024

  50. [50]

    M. Song, J. Zhao, Z. Yan, and Z. Y. Meng, Different tem- perature dependence for the edge and bulk of the entan- glement hamiltonian, Phys. Rev. B108, 075114 (2023)

    2023

  51. [51]

    S. Wu, X. Ran, B. Yin, Q.-F. Li, B.-B. Mao, Y.-C. Wang, and Z. Yan, Classical model emerges in quantum en- tanglement: Quantum monte carlo study for an ising- heisenberg bilayer, Phys. Rev. B107, 155121 (2023)

    2023

  52. [52]

    Z. Liu, Z. Wang, D.-X. Yao, and Z. Yan, Worldline de- confinement and emergent long-range interaction in the entanglement hamiltonian and in the entanglement spec- trum, Phys. Rev. B113, 144425 (2026)

    2026

  53. [53]

    Cardy,Scaling and renormalization in statistical physics, Cambridge lecture notes in physics, Vol

    J. Cardy,Scaling and renormalization in statistical physics, Cambridge lecture notes in physics, Vol. 5 (Cam- bridge University Press, Cambridge, 1996) p. 238 pages

    1996

  54. [54]

    Binder and P

    K. Binder and P. C. Hohenberg, Surface effects on mag- netic phase transitions, Phys. Rev. B9, 2194 (1974)

    1974

  55. [55]

    Y. Deng, H. W. J. Blöte, and M. P. Nightingale, Surface and bulk transitions in three-dimensionalO(n)models, Phys. Rev. E72, 016128 (2005)

    2005

  56. [56]

    Zhang and F

    L. Zhang and F. Wang, Unconventional surface critical behavior induced by a quantum phase transition from the two-dimensional affleck-kennedy-lieb-tasaki phase to a néel-ordered phase, Phys. Rev. Lett.118, 087201 (2017)

    2017

  57. [57]

    Weber, F

    L. Weber, F. Parisen Toldin, and S. Wessel, Nonordinary edge criticality of two-dimensional quantum critical mag- nets, Phys. Rev. B98, 140403 (2018)

    2018

  58. [58]

    C. Ding, W. Zhu, W. Guo, and L. Zhang, Special tran- sition and extraordinary phase on the surface of a two- dimensional quantum Heisenberg antiferromagnet, Sci- Post Phys.15, 012 (2023)

    2023

  59. [59]

    Wang, S.-Q

    Z. Wang, S.-Q. Ning, Z. Liu, J. Rong, Y.-C. Wang, Z. Yan, and W. Guo, Surface phase transitions in a (1 + 1)-dimensionalsu(2)1 conformal field theory bound- ary coupled to a(2 + 1)-dimensional‡2 bulk, Phys. Rev. B110, 115122 (2024)

    2024

  60. [60]

    Z. Wang, L. Lu, S.-Q. Ning, Z. Liu, Y.-C. Wang, Z. Yan, and W. Guo, Detecting underlying symmetry-protected topological phases via strange correlators and edge engi- neering, Phys. Rev. B113, 054405 (2026). [61]Salso contains the density component of the O(3) cur- rent. But it is more irrelevant (∆J = 2) than the order parameter (∆ϕ ≈0.52). These two oper...

    2026

  61. [62]

    Billó, M

    M. Billó, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri, and R. Pellegrini, Line defects in the 3d Ising model, JHEP07, 055, arXiv:1304.4110 [hep-th]

    Pith/arXiv arXiv

  62. [63]

    Billò, V

    M. Billò, V. Gonçalves, E. Lauria, and M. Meineri, Defects in conformal field theory, JHEP04, 091, arXiv:1601.02883 [hep-th]. 7

    Pith/arXiv arXiv

  63. [64]

    Here for simplicity, we suppress theθdependence con- trolled by the transverse spin, as well as other less rele- vant terms in the expansion

  64. [65]

    Cheng, N

    M. Cheng, N. Myerson-Jain, and G. Shankar (2026), un- published

    2026

  65. [66]

    Cuomo and S

    G. Cuomo and S. Zhang, Spontaneous symmetry break- ing on surface defects, JHEP03, 022, arXiv:2306.00085 [hep-th]

    Pith/arXiv arXiv

  66. [67]

    J. Wang, Z. Liu, B.-B. Mao, X. Tian, Z. Xiong, Z. Wang, and Z. Yan, Spontaneous continuous-symmetry break- ing and tower of states in a comb chain, (2025), arXiv:2506.23258 [cond-mat.str-el]. S1 Criticality on Rényi defect at (2+1)dO(3) quantum critical points -Supplemental Material- Yanzhang Zhu,1,2,3 Zhe Wang,1,2 Meng Cheng,4 and Zheng Yan1,2 1Departmen...

    Pith/arXiv arXiv 2025