Pith. sign in

REVIEW 3 major objections 5 minor 53 references

Continuous phase transition between bosonic integer quantum Hall liquid and trivial insulator: evidences for deconfined quantum criticality

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A bosonic integer quantum Hall state on a honeycomb lattice is shown to pass continuously into a trivial insulator under a periodic potential, with the critical point described by emergent QED3 with two Dirac flavors.

desk verdict A credible DMRG case for a continuous BIQH-to-Mott transition with c≈1, but the finite-width Ly=8 cylinder carries the whole quantitative load and the QED3 identification is underdetermined. read the letter →

arxiv 1908.02490 v2 pith:LJ4DYTLU submitted 2019-08-07 cond-mat.str-el

Tian-Sheng Zeng , D. N. Sheng , W. Zhu This is my paper · ORCID
classification cond-mat.str-el
keywords bosonicintegerquantumHalleffectdeconfinedcriticalityQED3density-matrixrenormalizationgrouptopologicalKmatrixsymmetryprotectedphasehoneycomblatticecontinuoustransition
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

The paper sets out to establish that a bosonic integer quantum Hall (BIQH) state on a correlated honeycomb lattice turns into a topologically trivial Mott insulator through a continuous phase transition as an imbalanced periodic chemical potential is increased, and that the critical point is described by an emergent $\mathrm{QED}_3$ with two flavors of Dirac fermions. Using infinite density-matrix renormalization group (DMRG) calculations, the authors identify the BIQH phase through its topological $K$ matrix and quantized drag Hall conductance, then observe smooth evolution of energy and density observables across the transition while the correlation length peaks at a single critical point. At that point, the entanglement entropy scales with the correlation length with central charge $c\simeq 1$, which matches the prediction for two Dirac flavors coupled to a dynamical gauge field. This would be a rare two-dimensional example of deconfined quantum criticality between a symmetry-protected topological phase and a trivial phase.

What carries the argument

The argument rests on two pieces of machinery. The first is the topological $K$ matrix of the two-component bosonic system, a symmetric integer matrix whose inverse gives the Chern number matrix $C = K^{-1}$; through quantized charge pumping under flux insertion in DMRG, the BIQH phase is pinned to $C = \begin{pmatrix}0&1\\1&0\end{pmatrix}$, meaning the off-diagonal drag Hall conductance is unity. The second is the entanglement-scaling diagnostic at the critical point, built on the Calabrese--Cardy relation $S = (c/6)\log\xi$ between entanglement entropy and the infinite-DMRG correlation length $\xi$. The identification of the critical theory as $\mathrm{QED}_3$ with $N_f=2$ uses the fermionic parton construction of bosons: the two Dirac flavors are coupled to a noncompact $U(1)$ gauge field, and the gauge coupling removes the total charge mode, reducing the central charge to $c = N_f - 1 = 1$.

What would settle it

A converged DMRG calculation on wider cylinders ($L_y=12$ or larger) would settle the claim: if the correlation-length peak as a function of $\mu$ splits into two peaks (an intermediate phase) or sharpens into a jump, the continuous $\mathrm{QED}_3$ critical point is a finite-width artifact rather than a true two-dimensional transition.

Watch

Extended reading notes

Core claim

The central discovery is that the BIQH-to-trivial-insulator transition driven by the periodic potential $\mu$ is continuous and preserves all symmetries: the first derivative of the ground-state energy and the density imbalance evolve smoothly, with no level crossing in exact diagonalization and no discontinuity in DMRG. The correlation length reaches a bond-dimension-dependent peak at a unique critical point, and at that point the single-particle correlation function $\langle b_0^\dagger b_r\rangle$ decays algebraically while the entanglement entropy satisfies $S = (c/6)\log\xi$ with $c\simeq 1$. The authors argue that the value $c=1$, rather than $c=2$ for two free Dirac flavors, is the signature that the critical theory is massless $\mathrm{QED}_3$ with $N_f=2$ fermions coupled to a noncompact gauge field, which gaps out the total charge mode; this is a characteristic signature of deconfined quantum criticality. In contrast, the transition driven by the nearest-neighbor interaction $V$ is first-order and breaks the $U(1)\times U(1)$ protecting symmetry down to a global $U(1)$, which the authors interpret as natural because symmetric-protected phases cannot undergo a continuous transition when the protecting symmetry is lost.

Load-bearing premise

The load-bearing assumption is that an infinitely long cylinder only eight sites wide, with periodic boundary conditions in the transverse direction, faithfully represents the two-dimensional thermodynamic limit of the transition.

Editorial extensions

If this is right

  • If the transition is truly continuous, this is a concrete two-dimensional lattice realization of deconfined quantum criticality between a symmetry-protected topological phase and a trivial insulator.
  • The measured central charge $c\simeq 1$ at the critical point supports the emergent $\mathrm{QED}_3$ with $N_f=2$ description, implying the deconfined gauge field is the organizing principle of the criticality.
  • The transition preserves the $U(1)\times U(1)$ symmetry, so no local order parameter changes; the phase transition is therefore Landau-forbidden in the sense of symmetry-breaking order.
  • The first-order $V$-driven transition that breaks the protecting symmetry indicates that the order of the transition depends on whether the symmetry protecting the SPT phase is preserved.
  • The algebraic decay of single-particle correlations at the critical point, $\langle b_0^\dagger b_r\rangle \sim r^{-\eta}$, provides a concrete signature that can be looked for in other candidate models.

Reading between the lines

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

  • A natural extension is to compute the correlation-length exponent $\nu$ and the anomalous dimension $\eta$ at the $\mu$-driven critical point on larger cylinders; if the transition is genuinely $\mathrm{QED}_3$, these exponents should match the predictions for $N_f=2$ fermionic QED3, not the mean-field values.
  • The same entanglement-scaling diagnostic—central charge $c=1$ for two Dirac flavors—could be transferred to other candidate deconfined critical points in bosonic or fermionic lattice models to distinguish emergent gauge-field criticality from free-fermion or conventional criticality.
  • If the continuous transition survives in the two-dimensional limit, the periodic-potential strength $\mu$ becomes a tunable knob for studying the crossover between confined and deconfined gauge dynamics, and possibly for engineering interfaces between a bosonic SPT phase and a trivial Mott insulator.
  • The result suggests that symmetry-preserving perturbations, unlike symmetry-breaking ones, may generically convert first-order SPT-to-trivial transitions into continuous ones in two dimensions, motivating similar searches in fermionic SPT systems with the same $U(1)\times U(1)$ symmetry.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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. This paper studies a hardcore-boson correlated-hopping model on the honeycomb lattice at half filling. Using exact diagonalization and infinite DMRG, the authors identify a bosonic integer quantum Hall (BIQH) phase through the Chern number matrix obtained from charge-pumping calculations, and then study transitions out of this phase. They claim that a staggered periodic chemical potential drives a continuous transition from the BIQH phase to a featureless trivial Mott insulator, with all local observables evolving smoothly, while a nearest-neighbor repulsion drives a first-order transition to a spin-polarized insulator. At the continuous critical point, the entanglement entropy versus correlation length gives a central charge c≃1, which they interpret as evidence for emergent QED3 with two flavors of Dirac fermions (c=Nf−1=1) and hence for deconfined quantum criticality between an SPT phase and a trivial insulator.

Significance. If the central claim is correct, the paper provides a rare two-dimensional microscopic example of deconfined quantum criticality involving a symmetry-protected topological phase, connecting to recent duality-based proposals and potentially stimulating further numerical and analytical work. The K-matrix identification via two independent charge-pumping protocols is clean, well documented, and does not rely on the critical-scaling analysis. The authors are also candid about the computational limitations of their DMRG calculations. The main weakness is that the quantitative evidence for the continuous transition and for the QED3 identification rests almost entirely on a single cylinder width (Ly=8) and on a small number of bond dimensions, with no width extrapolation; this limits the strength of the conclusions that can be drawn about the true two-dimensional limit.

major comments (3)
  1. [Sec. IV, Figs. 4(c),(d)] The claim of a continuous two-dimensional transition rests almost entirely on Ly=8 data. The text explicitly states that Ly=12 is "computationally difficult" and that the authors "focus our following discussions on a relatively small system width Ly=8 in our quasi-one-dimensional infinite cylinder," and Fig. 4(d) shows only two bond dimensions with a much weaker peak. No data for other widths and no extrapolation in 1/Ly or finite-entanglement scaling collapse are provided. The broad correlation-length peak in Fig. 4(c) with only M=1500, 1800, and 2000 does not by itself establish a diverging correlation length at a unique critical point. As a result, the observed continuous-looking behavior could be a quasi-one-dimensional finite-width effect, and the central claim that the BIQH-to-Mott transition is continuous in two dimensions is not secured. Please either add controlled Ly=6, 10, and 12 data with matched truncation errors or substantially soften the two-dimensional claim.
  2. [Sec. IV, Eq. (4), Fig. 5(b)] The identification of the critical theory as QED3 with two flavors of Dirac fermions is based on c≈1 from S=(c/6)logξ. This is not a unique diagnostic: any single gapless mode on the cylinder, for example the quasi-one-dimensional version of an ordinary bosonic critical point, would also give c=1, and the algebraic decay of ⟨b†b⟩ is the superfluid-type response generic to such states. The comparison with the predicted c=Nf−1=1 is therefore necessary but not sufficient. Additional discriminators are needed, such as extraction of critical exponents, the full low-energy spectrum, or the scaling of the topological/drag response across the critical point, to substantiate the emergent-gauge-field interpretation.
  3. [Sec. IV, Fig. 5] The S-versus-logξ scaling analysis is not described with enough detail to be reproducible or robust: the manuscript does not state the number of μ points used, the fitting window, or the uncertainty of the fitted slope, and "c≃1" is quoted without error bars. Because the correlation-length peak in Fig. 4(c) is broad, the choice of fitting window can materially shift the fitted central charge. Please provide the fit protocol, the number of points, and an error estimate for the extracted slope.
minor comments (5)
  1. [Sec. IV, Fig. 4] The value of the critical potential μc is not given anywhere; the black dashed line in Fig. 4 is not quantified. Please state μc and the criterion used to determine it, as this is needed for reproducibility and for any scaling analysis.
  2. [Sec. IV] The sentence describing ⟨b†ibj⟩ as "off-diagonal long range correlations" should read "quasi-long-range (algebraic) correlations," since the data show power-law rather than true long-range order.
  3. [Sec. IV, Fig. 5] The phrase "for different system sizes, the smooth behavior ... supports a continuous phase transition" overstates the evidence, since only Ly=8 and Ly=12 are shown and only one width is used for the critical scaling; consider rewording to "is consistent with."
  4. [Sec. II and Fig. 5] The text and abstract state that up to M=6500 states are kept, but all reported scaling results use M≤2500. Please clarify the maximum bond dimension actually used in each figure.
  5. [Eq. (1)] In the second hopping term of Eq. (1), the index k is not defined in the equation; it should be identified as the B-site index of the nearest-neighbor pair.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the central claims are supported by independent numerical observables and external theoretical benchmarks, with self-citations only in methodological roles.

full rationale

The paper's derivation chain is not circular. The BIQH phase is identified by topological probes that are independent of the conclusion: charge pumping under inserted flux is a raw DMRG observable, and the resulting pumped charges (Delta Q=2 for both-component flux and Delta Q~1 for one-component flux) are compared with, not defined by, the expected Chern-number matrix C=K^{-1}=((0,1),(1,0)). The continuous-transition evidence consists of smooth first-order energy derivatives, smooth density imbalance, a correlation-length peak that grows with bond dimension, and algebraic single-particle correlations; none of these quantities is constructed from the QED3 claim. The central charge is extracted by fitting S=(c/6)log(xi) to DMRG data (Fig. 5b), giving c~1, and then compared with the external theoretical prediction c=Nf-1=1 for emergent QED3 with two fermion flavors, as computed in Ref. [53]; the fit is not defined in terms of that prediction. Self-citations (Refs. [14,46,47]) appear only as background for the model and for the charge-pumping method, and they do not carry the central claim. The quasi-1D Ly=8 limitation and the fact that c=1 could also be consistent with other single-mode critical theories are robustness and interpretation concerns, not circularity. No equation is equivalent to its input by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard K-matrix theory, a 1+1D entanglement scaling relation applied to a 2D cylinder, and the parton-construction duality to QED3. The only fitted quantity is the effective central charge c extracted from the entropy slope; the chiral flux alpha = pi/2 is a fixed model choice. No new entities are introduced.

free parameters (2)
  • effective central charge c = about 1
    Slope of S versus log xi fitted at the critical point in Fig. 5(b), then compared with the theoretical prediction c = Nf - 1 = 1 for QED3 with two Dirac fermion flavors.
  • chiral flux alpha = pi/2
    Fixed value of the background chiral flux in Eq. 1; the phase diagram and critical behavior are established only at this value.
assumptions (4)
  • domain assumption The K-matrix classification and the relation C = K^{-1} apply to this two-component bosonic system, so measured charge pumping values uniquely identify the BIQH state.
    Used in Sec. III to translate the measured pumping (Delta Q = 2 for both components, Delta Q = 1 for one component) into the Chern number matrix of Eq. 2.
  • domain assumption The 1+1D CFT formula S = (c/6) log xi remains valid for extracting an effective central charge on a finite-width infinite cylinder at a 2D quantum critical point.
    Used in Sec. IV and Fig. 5(b) to extract c = 1 and to distinguish free Dirac fermions (c = 2) from QED3 (c = 1).
  • domain assumption The transfer-matrix correlation length computed in infinite DMRG is a faithful proxy for the physical correlation length, and its broad peak marks the quantum critical point.
    Used in Sec. IV to locate the phase boundary; convergence with bond dimension M is shown but no extrapolation to M to infinity is performed.
  • domain assumption The parton-construction duality mapping this boson model to QED3 with two Dirac fermion flavors is correct for this transition.
    Borrowed from Refs. [32,36,37]; the measured central charge is interpreted through this mapping.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Continuous phase transition between bosonic integer quantum Hall liquid and trivial insulator: evidences for deconfined quantum criticality." pith.science (2026). https://pith.science/paper/LJ4DYTLU

@misc{pith2026190802490,
  author       = {Pith},
  title        = {Pith review of: Continuous phase transition between bosonic integer quantum Hall liquid and trivial insulator: evidences for deconfined quantum criticality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LJ4DYTLU}},
  note         = {Machine review of arXiv:1908.02490}
}
abstract

The deconfined quantum critical point, a prototype Landau-forbidden transition, could exist in principle in the phase transitions involving symmetry protected topological phase, however, examples of such kinds of transition in physical systems are rare beyond one-dimensional systems. Here, using density-matrix renormalization group calculation, we unveil a bosonic integer quantum Hall phase in two-dimensional correlated honeycomb lattice, by full identification of its internal structure from the topological $\mathbf{K}$ matrix. Moreover we demonstrate that imbalanced periodic chemical potentials can destroy the bosonic integer quantum Hall state and drive it into a featureless trivial (Mott) insulator, where all physical observables evolve smoothly across the critical point. At the critical point the entanglement entropy reveals a characteristic scaling behavior, which is consistent with the critical field theory as an emergent QED$_3$ with two flavors of Dirac fermions.

Figures

Figures reproduced from arXiv: 1908.02490 by the authors.

Figure 1
Figure 1. FIG. 1. Possible phase transitions between a SPT phase and [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) The schematic plot of the correlated honeycomb [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) The charge transfer in the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Numerical DMRG results near the critical point [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 29 canonical work pages

  1. [1]

    Landau, Zh

    Lev D. Landau, Zh. Eksp. Teor. Fiz. 7: 19-32 (1937)

    work page 1937

  2. [2]

    Wen, Rev

    X.-G. Wen, Rev. Mod. Phys. 89, 041004 (2017)

    2017

  3. [3]

    Senthil, Annual Review of Condensed Matter Physics 6, 299 (2015)

    T. Senthil, Annual Review of Condensed Matter Physics 6, 299 (2015)

    2015

  4. [4]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005)

    2005

  5. [5]

    F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983). 6

    work page 1983

  6. [6]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 87, 155114 (2013)

    2013

  7. [7]

    Senthil and M

    T. Senthil and M. Levin, Phys. Rev. Lett. 110, 046801 (2013)

    work page 2013

  8. [8]

    Furukawa and M

    S. Furukawa and M. Ueda, Phys. Rev. Lett. 111, 090401 (2013)

    work page 2013

Show all 53 references
  1. [9]

    Regnault and T

    N. Regnault and T. Senthil, Phys. Rev. B 88, 161106(R) (2013)

    2013

  2. [10]

    Wu and J

    Y.-H. Wu and J. K. Jain, Phys. Rev. B 87, 245123 (2013)

    2013

  3. [11]

    S. D. Geraedts and O. I. Motrunich, Ann. of Phys. 334, 288 (2013)

    2013

  4. [12]

    Y.-C. He, S. Bhattacharjee, R. Moessner, and F. Poll- mann, Phys. Rev. Lett. 115, 116803 (2015)

    2015

  5. [13]

    Sterdyniak, N

    A. Sterdyniak, N. R. Cooper, and N. Regnault, Phys. Rev. Lett. 115, 116802 (2015)

    2015

  6. [14]

    T.-S. Zeng, W. Zhu, and D. N. Sheng, Phys. Rev. B 93, 195121 (2016)

    2016

  7. [15]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)

    2011

  8. [16]

    Lu and A

    Y.-M. Lu and A. Vishwanath, Phys. Rev. B 86, 125119 (2012)

    2012

  9. [17]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Science 338, 1604 (2012)

    2012

  10. [18]

    Vishwanath and T

    A. Vishwanath and T. Senthil, Phys. Rev. X 3, 011016 (2013)

    2013

  11. [19]

    Xu and T

    C. Xu and T. Senthil, Phys. Rev. B 87, 174412 (2013)

    2013

  12. [20]

    Schuch, D

    N. Schuch, D. Perez-Garcia, and I. Cirac, Phys. Rev. B 84, 165139 (2011)

    2011

  13. [21]

    Gu and X.-G

    Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009)

    2009

  14. [22]

    Pollmann, A

    F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010)

    2010

  15. [23]

    Z.-X. Liu, M. Liu, and X.-G. Wen, Phys. Rev. B 84, 075135 (2011)

    2011

  16. [24]

    den Nijs and K

    M. den Nijs and K. Rommelse, Phys. Rev. B 40, 4709 (1989)

    1989

  17. [25]

    X. Chen, F. Wang, Y.-M. Lu, and D.-H. Lee, Nucl. Phys. B 873, 248 (2013)

    2013

  18. [26]

    L. Tsui, Y. Huang, H.-C. Jiang, and D.-H. Lee, Nucl. Phys. B 919, 470 (2017)

    2017

  19. [27]

    Lahtinen and E

    V. Lahtinen and E. Ardonne, Phys. Rev. Lett. 115, 237203 (2015)

    2015

  20. [28]

    Verresen, R

    R. Verresen, R. Moessner, and Frank Pollmann, Phys. Rev. B 96, 165124 (2017)

    2017

  21. [29]

    Senthil, A

    T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004)

    2004

  22. [30]

    Senthil, L

    T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Phys. Rev. B 70, 144407 (2004)

    2004

  23. [31]

    Karch and D

    A. Karch and D. Tong, Phys. Rev. X 6, 031043 (2016)

    2016

  24. [32]

    C. Wang, A. Nahum, M. A. Metlitski, C. Xu, and T. Senthil, Phys. Rev. X 7, 031051 (2017)

    2017

  25. [33]

    Senthil, D

    T. Senthil, D. T. Son, C. Wang, and C. Xu, Phys. Rep. 827, 1 (2019)

    2019

  26. [34]

    Y. Q. Qin, Y.-Y. He, Y.-Z. You, Z.-Y. Lu, A. Sen, A. W. Sandvik, C. Xu, and Z. Y. Meng, Phys. Rev. X 7, 031052 (2017)

    2017

  27. [35]

    Geraedts and Olexei I

    S. Geraedts and Olexei I. Motrunich, Phys. Rev. B 96, 115137 (2017)

    2017

  28. [36]

    Grover and A

    T. Grover and A. Vishwanath, Phys. Rev. B 87, 045129 (2013)

    2013

  29. [37]

    Lu and D.-H

    Y.-M. Lu and D.-H. Lee, Phys. Rev. B 89, 195143 (2014)

    2014

  30. [38]

    S. C. Morampudi, C. von Keyserlingk, and F. Pollmann, Phys. Rev. B 90, 035117 (2014)

    2014

  31. [39]

    Fuji, Y.-C

    Y. Fuji, Y.-C. He, S. Bhattacharjee, and F. Pollmann, Phys. Rev. B 93, 195143 (2016)

    2016

  32. [40]

    S. R. White, Phys. Rev. Lett. 69, 2863 (1992)

    1992

  33. [41]

    I. P. McCulloch, arxiv: 0804.2509

    arXiv

  34. [42]

    Cincio and G

    L. Cincio and G. Vidal, Phys. Rev. Lett. 110, 067208 (2013)

    2013

  35. [43]

    D. N. Sheng, L. Balents, and Z. Wang, Phys. Rev. Lett. 91, 116802 (2003)

    2003

  36. [44]

    D. N. Sheng, Z.-Y. Weng, L. Sheng, and F. D. M. Hal- dane, Phys. Rev. Lett. 97, 036808 (2006)

    2006

  37. [45]

    S. S. Gong, W. Zhu, and D. N. Sheng, Sci. Rep. 4, 6317 (2014)

    2014

  38. [46]

    T.-S. Zeng, W. Zhu, and D. N. Sheng, Phys. Rev. B 95, 125134 (2017)

    2017

  39. [47]

    Zeng and D

    T.-S. Zeng and D. N. Sheng, Phys. Rev. B 97, 035151 (2018); T.-S. Zeng, D. N. Sheng, and W. Zhu, Phys. Rev. B 100, 075106 (2019)

    2018

  40. [48]

    Wen and Y.-S

    X.-G. Wen and Y.-S. Wu, Phys. Rev. Lett. 70, 1501 (1993)

    1993

  41. [49]

    W. Chen, M. P. A. Fisher, and Y.-S. Wu, Phys. Rev. B 48, 13749 (1993)

    1993

  42. [50]

    Kol and N

    A. Kol and N. Read, Phys. Rev. B 48, 8890 (1993)

    1993

  43. [51]

    J. Y. Lee, C. Wang, M. P. Zaletel, A. Vishwanath, and Y.-C. He, Phys. Rev. X 8, 031015 (2018)

    2018

  44. [52]

    Calabrese and J

    P. Calabrese and J. Cardy, J. Phys. A 42, 504005 (2009)

    2009

  45. [53]

    S. D. Geraedts, M. P. Zaletel, R. S. K. Mong, M. A. Metlitski, A. Vishwanath, and O. I. Motrunich, Science 352, 197 (2016)

    2016

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.