pith. machine review for the scientific record. sign in

arxiv: 2603.18278 · v2 · submitted 2026-03-18 · ❄️ cond-mat.str-el · hep-lat· hep-th

Recognition: no theorem link

Symmetric Mass Generation in a Bilayer Honeycomb Lattice with SU(2)timesSU(2)timesSU(2)/mathbb{Z}₂ Symmetry

Cheng-Hao He , Yi-Zhuang You , Xiao Yan Xu
Authors on Pith no claims yet

Pith reviewed 2026-05-15 08:18 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-lathep-th
keywords symmetric mass generationDirac fermionsquantum Monte Carlocritical exponentsanomalous dimensionnon-Abelian symmetryhoneycomb latticebilayer model
0
0 comments X

The pith

DQMC simulations yield the first unbiased estimate of the fermion anomalous dimension at the symmetric mass generation critical point as 0.071(1).

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

The paper investigates the non-perturbative critical theory of symmetric mass generation, where strong interactions gap Dirac fermions in 2+1 dimensions without symmetry breaking or topological order. Large-scale determinant quantum Monte Carlo simulations of a bilayer honeycomb lattice model with SU(2)×SU(2)×SU(2)/Z2 symmetry establish a direct continuous transition at critical coupling Jc ≈ 2.6. The work measures the correlation length exponent ν = 1.14(2) and obtains the fermion anomalous dimension η_ψ = 0.071(1), which deviates from large-N and variational Monte Carlo predictions. Comparison with a related model that develops an intermediate excitonic phase shows that pure non-Abelian symmetry stabilizes the direct transition. These results supply quantitative constraints on candidate critical theories for SMG.

Core claim

Determinant quantum Monte Carlo simulations of a bilayer honeycomb lattice with SU(2)×SU(2)×SU(2)/Z2 symmetry demonstrate a direct continuous symmetric mass generation transition at Jc ≈ 2.6, where single-particle and bosonic gaps open simultaneously with correlation length exponent ν = 1.14(2) and fermion anomalous dimension η_ψ = 0.071(1). An exhaustive check of all 19 symmetry-inequivalent fermion bilinear order parameters finds no symmetry breaking. The measured η_ψ differs substantially from the large-N value ≈0.595 and variational Monte Carlo value ≈0.62. A contrasting Spin(5)×U(1)/Z2 model develops an intermediate excitonic phase, indicating that pure non-Abelian symmetry is decisive,

What carries the argument

Large-scale determinant quantum Monte Carlo simulations on the bilayer honeycomb lattice model with SU(2)×SU(2)×SU(2)/Z2 symmetry, which track simultaneous gap opening and perform finite-size scaling to extract the fermion anomalous dimension without bias from symmetry-breaking operators.

Load-bearing premise

Finite-size scaling of the DQMC data accurately captures the thermodynamic-limit exponents and no relevant symmetry-breaking operators are generated by the lattice regularization.

What would settle it

A simulation on larger lattices or with different regularization that measures a fermion anomalous dimension near 0.6 or detects order in any of the 19 fermion bilinear channels at the transition would falsify the direct-transition claim.

Figures

Figures reproduced from arXiv: 2603.18278 by Cheng-Hao He, Xiao Yan Xu, Yi-Zhuang You.

Figure 2
Figure 2. Figure 2: FIG. 2. (a) Single-particle gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. (a) Bilayer honeycomb lattice. (b) Phase diagram of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Structure factors of ferromagnetic (FM), charge den [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Finite-size scaling analysis for ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

A central question beyond the Landau paradigm is the non-perturbative critical theory of the symmetric mass generation (SMG) transition, where strong interactions gap Dirac fermions in (2+1) dimensions without triggering spontaneous symmetry breaking or topological order. While previous studies have already provided evidence for direct SMG transitions in (2+1) dimensions, the fermion scaling dimension -- the key observable for distinguishing candidate critical theories -- has not been determined in a controlled unbiased way. In this Letter, using large-scale determinant quantum Monte Carlo (DQMC) simulations of a bilayer honeycomb lattice model with $\mathrm{SU}(2)\times\mathrm{SU}(2)\times\mathrm{SU}(2)/\mathbb{Z}_2$ symmetry, we establish a direct continuous transition by observing the simultaneous opening of single-particle and bosonic gaps at a critical coupling $J_c \approx 2.6$ with correlation length exponent $\nu = 1.14(2)$, while an exhaustive search over all 19 symmetry-inequivalent fermion bilinear order parameters confirms the absence of any symmetry breaking. We further obtain the first controlled unbiased estimate of the fermion anomalous dimension, $\eta_\psi = 0.071(1)$, which deviates significantly from the large-$N$ prediction ($\eta_\psi \approx 0.595$) and variational Monte Carlo estimates ($\eta_\psi \approx 0.62$), thereby placing direct quantitative constraints on SMG criticality. By contrasting with a related $\mathrm{Spin}(5)\times\mathrm{U}(1)/\mathbb{Z}_2$ model that develops an intermediate excitonic phase, we show that pure non-Abelian symmetry plays a decisive role in stabilizing the direct SMG transition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript uses large-scale determinant quantum Monte Carlo simulations of a bilayer honeycomb lattice model with SU(2)×SU(2)×SU(2)/Z₂ symmetry to study the symmetric mass generation (SMG) transition. It reports a direct continuous transition at J_c ≈ 2.6 with correlation-length exponent ν = 1.14(2), simultaneous opening of single-particle and bosonic gaps, absence of symmetry breaking after checking all 19 symmetry-inequivalent fermion bilinears, and the first controlled estimate of the fermion anomalous dimension η_ψ = 0.071(1). This value deviates from large-N (≈0.595) and variational Monte Carlo (≈0.62) predictions. The work contrasts the model with a related Spin(5)×U(1)/Z₂ realization that exhibits an intermediate excitonic phase, arguing that the non-Abelian symmetry stabilizes the direct SMG transition.

Significance. If the finite-size scaling results hold, the paper supplies the first unbiased numerical value for the fermion anomalous dimension at an SMG critical point in (2+1) dimensions. This quantitative benchmark can discriminate among candidate continuum theories for SMG and highlights the role of symmetry in avoiding intermediate phases. The exhaustive bilinear search and direct comparison to a gapped-phase model are additional strengths.

major comments (2)
  1. [Finite-size scaling analysis] Finite-size scaling and exponent extraction: The headline result η_ψ = 0.071(1) is extracted from the fermion Green's function at the estimated J_c ≈ 2.6 using ν = 1.14(2). The manuscript must demonstrate that the simulated lattice sizes reach the asymptotic scaling regime, that corrections-to-scaling (O(L^{-ω})) are either negligible or explicitly included in the fits, and that the uncertainty in J_c does not mix gapped or ordered regimes into the data. Without such controls the small η_ψ could be an artifact rather than the continuum SMG value.
  2. [Order-parameter analysis] Symmetry-breaking search: The claim of no symmetry breaking rests on an exhaustive check of 19 fermion bilinears. The manuscript should explicitly list these operators, confirm they exhaust all symmetry-allowed bilinears permitted by the SU(2)×SU(2)×SU(2)/Z₂ group, and address whether the lattice regularization could generate additional weakly relevant operators not captured by the 19-bilinear set.
minor comments (1)
  1. [Methods and symmetry section] The abstract states that 19 order parameters were checked but the main text should include a compact table or appendix listing their explicit forms and symmetry quantum numbers for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work's significance, and constructive comments. We address each major point below. Where the comments identify gaps in the presentation of our finite-size scaling and symmetry analysis, we have revised the manuscript to provide the requested details and controls.

read point-by-point responses

  1. Referee: Finite-size scaling analysis: The headline result η_ψ = 0.071(1) is extracted from the fermion Green's function at the estimated J_c ≈ 2.6 using ν = 1.14(2). The manuscript must demonstrate that the simulated lattice sizes reach the asymptotic scaling regime, that corrections-to-scaling (O(L^{-ω})) are either negligible or explicitly included in the fits, and that the uncertainty in J_c does not mix gapped or ordered regimes into the data. Without such controls the small η_ψ could be an artifact rather than the continuum SMG value.

    Authors: We agree that a more explicit demonstration of the asymptotic regime is needed. In the revised manuscript we have added (i) scaling collapses for the fermion Green's function and bosonic susceptibility using L = 6–24, (ii) explicit two-parameter fits that include the leading correction-to-scaling term with ω ≈ 0.8 extracted from the data collapse, and (iii) a consistency check in which J is varied within the quoted uncertainty of J_c while monitoring the extracted η_ψ; the value remains stable to within the reported error bar. These additions confirm that the simulated sizes are in the scaling regime and that the small η_ψ is not an artifact of finite-size or off-critical effects. We note that still larger lattices would further strengthen the result but are currently beyond our computational resources. revision: yes

  2. Referee: Symmetry-breaking search: The claim of no symmetry breaking rests on an exhaustive check of 19 fermion bilinears. The manuscript should explicitly list these operators, confirm they exhaust all symmetry-allowed bilinears permitted by the SU(2)×SU(2)×SU(2)/Z₂ group, and address whether the lattice regularization could generate additional weakly relevant operators not captured by the 19-bilinear set.

    Authors: We have added a new supplementary table that explicitly enumerates all 19 symmetry-inequivalent fermion bilinears, classified by the irreducible representations of SU(2)×SU(2)×SU(2)/Z₂. A short group-theoretic argument (now included in the main text) shows that these operators exhaust the complete set of bilinears allowed by the symmetry. Concerning lattice artifacts, the model is constructed so that the full non-Abelian symmetry is preserved at every lattice site; any additional operator generated by the regularization would have to be a singlet under the symmetry group and is therefore either forbidden or irrelevant at the critical point. We have expanded the discussion of this point in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: purely numerical extraction of SMG exponents from DQMC data

full rationale

The central results (J_c ≈ 2.6, ν = 1.14(2), η_ψ = 0.071(1)) are obtained by direct finite-size scaling fits to determinant quantum Monte Carlo observables on the bilayer honeycomb lattice. The critical point is located from the simultaneous closing of single-particle and bosonic gaps in the simulated data; the anomalous dimension is read off from the power-law decay of the fermion Green's function at that point. No equation defines a quantity in terms of itself, no fitted parameter is relabeled as an independent prediction, and no load-bearing step reduces to a self-citation chain. The derivation chain is therefore self-contained against the external benchmark of the raw Monte Carlo measurements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the DQMC algorithm for this sign-problem-free model and on the accuracy of finite-size scaling to extract universal exponents; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Determinant quantum Monte Carlo converges to the correct thermal ensemble for the chosen lattice model without a sign problem.
    Standard assumption for DQMC in fermionic systems with appropriate particle-hole symmetry.
  • domain assumption Finite-size scaling analysis on accessible lattice sizes yields the correct thermodynamic-limit critical exponents.
    Common assumption in lattice Monte Carlo studies of critical phenomena.

pith-pipeline@v0.9.0 · 5646 in / 1444 out tokens · 83880 ms · 2026-05-15T08:18:10.381879+00:00 · methodology

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. Continuous symmetry analysis and systematic identification of candidate order parameters for interacting fermion models

    cond-mat.str-el 2026-03 unverdicted novelty 6.0

    A systematic method maps interacting fermion Hamiltonians to Majorana space, classifies their continuous symmetries with semisimple Lie algebra theory, and enumerates order parameters by decomposing induced representa...

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    P. W. Anderson, Plasmons, gauge invariance, and mass, Phys. Rev.130, 439 (1963)

    work page 1963

  2. [2]

    Englert and R

    F. Englert and R. Brout, Broken symmetry and the mass of gauge vector mesons, Phys. Rev. Lett.13, 321 (1964)

    work page 1964

  3. [3]

    P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett.13, 508 (1964)

    work page 1964

  4. [4]

    G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Global conservation laws and massless particles, Phys. Rev. Lett. 13, 585 (1964)

    work page 1964

  5. [5]

    Wang and Y.-Z

    J. Wang and Y.-Z. You, Symmetric mass generation, Symmetry14, 1475 (2022)

    work page 2022

  6. [6]

    You, Y.-C

    Y.-Z. You, Y.-C. He, C. Xu, and A. Vishwanath, Sym- metric fermion mass generation as deconfined quantum criticality, Phys. Rev. X8, 011026 (2018)

    work page 2018

  7. [7]

    Tong, Comments on symmetric mass generation in 2d and 4d, Journal of High Energy Physics2022, 1 (2022)

    D. Tong, Comments on symmetric mass generation in 2d and 4d, Journal of High Energy Physics2022, 1 (2022)

    work page 2022

  8. [8]

    M. Zeng, Z. Zhu, J. Wang, and Y.-Z. You, Symmetric mass generation in the 1 + 1 dimensional chiral fermion 3-4-5-0 model, Phys. Rev. Lett.128, 185301 (2022)

    work page 2022

  9. [9]

    D.-C. Lu, M. Zeng, J. Wang, and Y.-Z. You, Fermi surface symmetric mass generation, Phys. Rev. B107, 195133 (2023)

    work page 2023

  10. [10]

    Xu and C

    Y. Xu and C. Xu, Green’s function zero and symmet- ric mass generation, arXiv preprint arXiv:2103.15865 (2021)

    work page arXiv 2021

  11. [11]

    You, Y.-C

    Y.-Z. You, Y.-C. He, A. Vishwanath, and C. Xu, From bosonic topological transition to symmetric fermion mass generation, Phys. Rev. B97, 125112 (2018)

    work page 2018

  12. [12]

    Eichten and J

    E. Eichten and J. Preskill, Chiral gauge theories on the lattice, Nucl. Phys. B268, 179 (1986)

    work page 1986

  13. [13]

    I.-H. Lee, J. Shigemitsu, and R. E. Shrock, Study of dif- ferent lattice formulations of a Yukawa model with a real scalar field, Nucl. Phys. B334, 265 (1990)

    work page 1990

  14. [14]

    I.-H. Lee, J. Shigemitsu, and R. E. Shrock, Lattice study of a Yukawa theory with a real scalar field, Nuclear Physics B330, 225 (1990)

    work page 1990

  15. [15]

    W. Bock, A. K. De, K. Jansen, J. Jers´ ak, T. Neuhaus, and J. Smit, Phase diagram of a lattice su(2)⊗su(2) scalar-fermion model with naive and wilson fermions, Nucl. Phys. B344, 207 (1990)

    work page 1990

  16. [16]

    Bock and A

    W. Bock and A. K. De, Unquenched investigation of fermion masses in a chiral fermion theory on the lattice, Phys. Lett. B245, 207 (1990)

    work page 1990

  17. [17]

    Hasenfratz, P

    A. Hasenfratz, P. Hasenfratz, K. Jansen, J. Kuti, and Y. Shen, The equivalence of the top quark condensate and the elementary Higgs field, Nucl. Phys. B365, 79 (1991)

    work page 1991

  18. [18]

    Banks and A

    T. Banks and A. Dabholkar, Decoupling a fermion whose mass comes from a Yukawa coupling: Nonperturbative considerations, Phys. Rev. D46, 4016 (1992)

    work page 1992

  19. [19]

    M. F. L. Golterman, D. N. Petcher, and E. Rivas, Ab- sence of chiral fermions in the Eichten-Preskill model, Nucl. Phys. B395, 596 (1993)

    work page 1993

  20. [20]

    You and C

    Y.-Z. You and C. Xu, Symmetry-protected topological states of interacting fermions and bosons, Phys. Rev. B 90, 245120 (2014)

    work page 2014

  21. [21]

    Fidkowski and A

    L. Fidkowski and A. Kitaev, Effects of interactions on the topological classification of free fermion systems, Phys. Rev. B81, 134509 (2010)

    work page 2010

  22. [22]

    Fidkowski and A

    L. Fidkowski and A. Kitaev, Topological phases of fermions in one dimension, Phys. Rev. B83, 075103 (2011)

    work page 2011

  23. [23]

    Wang and X.-G

    J. Wang and X.-G. Wen, Non-perturbative regulariza- tion of 1+ 1d anomaly-free chiral fermions and bosons: On the equivalence of anomaly matching conditions and boundary gapping rules, arXiv preprint arXiv:1307.7480 (2013)

    work page arXiv 2013

  24. [24]

    Slagle, Y.-Z

    K. Slagle, Y.-Z. You, and C. Xu, Exotic quantum phase transitions of strongly interacting topological insulators, Phys. Rev. B91, 115121 (2015)

    work page 2015

  25. [25]

    He, H.-Q

    Y.-Y. He, H.-Q. Wu, Y.-Z. You, C. Xu, Z. Y. Meng, and Z.-Y. Lu, Quantum critical point of dirac fermion mass generation without spontaneous symmetry break- ing, Phys. Rev. B94, 241111 (2016)

    work page 2016

  26. [26]

    Hou and Y.-Z

    W. Hou and Y.-Z. You, Variational monte carlo study 6 of symmetric mass generation in a bilayer honeycomb lattice model, Phys. Rev. B108, 125130 (2023)

    work page 2023

  27. [27]

    Z. H. Liu, Y. Da Liao, G. Pan, M. Song, J. Zhao, W. Jiang, C.-M. Jian, Y.-Z. You, F. F. Assaad, Z. Y. Meng, and C. Xu, Disorder operator and r´ enyi entangle- ment entropy of symmetric mass generation, Phys. Rev. Lett.132, 156503 (2024)

    work page 2024

  28. [28]

    Catterall, Fermion mass without symmetry breaking, Journal of High Energy Physics2016, 1 (2016)

    S. Catterall, Fermion mass without symmetry breaking, Journal of High Energy Physics2016, 1 (2016)

    work page 2016

  29. [29]

    Ayyar and S

    V. Ayyar and S. Chandrasekharan, Massive fermions without fermion bilinear condensates, Phys. Rev. D91, 065035 (2015)

    work page 2015

  30. [30]

    Ayyar and S

    V. Ayyar and S. Chandrasekharan, Origin of fermion masses without spontaneous symmetry breaking, Phys. Rev. D93, 081701 (2016)

    work page 2016

  31. [31]

    Maiti, D

    S. Maiti, D. Banerjee, S. Chandrasekharan, and M. K. Marinkovic, Phase diagram of a lattice fermion model with symmetric mass generation, arXiv preprint arXiv:2602.18360 (2026)

    work page arXiv 2026

  32. [32]

    S. S. Razamat and D. Tong, Gapped chiral fermions, Phys. Rev. X11, 011063 (2021)

    work page 2021

  33. [33]

    Blankenbecler, D

    R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Monte Carlo calculations of coupled boson-fermion sys- tems. I, Phys. Rev. D24, 2278 (1981)

    work page 1981

  34. [34]

    D. J. Scalapino and R. L. Sugar, Monte Carlo calculations of coupled boson-fermion systems. II, Phys. Rev. B24, 4295 (1981)

    work page 1981

  35. [35]

    Assaad and H

    F. Assaad and H. Evertz, World-line and determinan- tal quantum monte carlo methods for spins, phonons and electrons, inComputational Many-Particle Physics, edited by H. Fehske, R. Schneider, and A. Weiße (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) pp. 277–356

    work page 2008

  36. [36]

    Sugiyama and S

    G. Sugiyama and S. E. Koonin, Auxiliary field Monte- Carlo for quantum many-body ground states, Annals of Physics168, 1 (1986)

    work page 1986

  37. [37]

    Sorella, S

    S. Sorella, S. Baroni, R. Car, and M. Parrinello, A novel technique for the simulation of interacting fermion sys- tems, Europhysics Letters (EPL)8, 663 (1989)

    work page 1989

  38. [38]

    S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. E. Gubernatis, and R. T. Scalettar, Numerical study of the two-dimensional Hubbard model, Physical Review B40, 506 (1989)

    work page 1989

  39. [39]

    F.-H. Wang, F. Sun, C. He, and X. Y. Xu, Resolving Quantum Criticality in the Honeycomb Hubbard Model (2026), arXiv:2602.03656 [cond-mat]

    work page arXiv 2026

  40. [40]

    See Supplemental Material(SM) for more details

    work page

  41. [41]

    Continuous symmetry analysis and systematic identification of candidate order parameters for interacting fermion models

    C.-H. He, Y.-Z. You, and X. Y. Xu, Continuous symme- try analysis and systematic identification of candidate order parameters for interacting fermion models, arXiv preprint arXiv:2603.18285 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  42. [42]

    Harada, Bayesian inference in the scaling analysis of critical phenomena, Physical Review E84, 56704 (2011)

    K. Harada, Bayesian inference in the scaling analysis of critical phenomena, Physical Review E84, 56704 (2011)

    work page 2011

  43. [43]

    Harada, Kernel method for corrections to scaling, Physical Review E92, 012106 (2015)

    K. Harada, Kernel method for corrections to scaling, Physical Review E92, 012106 (2015)

    work page 2015

  44. [44]

    R. K. Kaul and S. Sachdev, Quantum criticality of u(1) gauge theories with fermionic and bosonic matter in two spatial dimensions, Phys. Rev. B77, 155105 (2008)

    work page 2008

  45. [45]

    Li, Y.-K

    Z.-X. Li, Y.-K. Yu, Z.-X. Li, and S. Yin, Symmetric mass generation transition and its nonequilibrium criti- cal dynamics in a bilayer honeycomb lattice model, arXiv preprint arXiv:2603.22736 (2026)

    work page arXiv 2026

  46. [46]

    Chang, S

    W.-X. Chang, S. Guo, Y.-Z. You, and Z.-X. Li, Fermi surface symmetric mass generation: A quantum monte- carlo study (2023), arXiv:2311.09970 [cond-mat]

    work page arXiv 2023

  47. [47]

    F. F. Assaad, Depleted kondo lattices: Quantum monte carlo and mean-field calculations, Physical Review B65, 115104 (2002)

    work page 2002

  48. [48]

    Li, Y.-F

    Z.-X. Li, Y.-F. Jiang, S.-K. Jian, and H. Yao, Fermion- induced quantum critical points, Nature Communica- tions8, 314 (2017)

    work page 2017

  49. [49]

    Da Liao, Z

    Y. Da Liao, Z. Y. Meng, and X. Y. Xu, Valence bond orders at charge neutrality in a possible two-orbital extended hubbard model for twisted bilayer graphene, Physical Review Letters123, 157601 (2019)

    work page 2019

  50. [50]

    Hawashin, M

    B. Hawashin, M. M. Scherer, and L. Janssen, Gross- neveu-xy quantum criticality in moir\’e dirac materials, Physical Review B111, 205129 (2025)

    work page 2025

  51. [51]

    N. Zerf, L. N. Mihaila, P. Marquard, I. F. Herbut, and M. M. Scherer, Four-loop critical exponents for the gross- neveu-yukawa models, Physical Review D96, 096010 (2017)

    work page 2017

  52. [52]

    J. A. Gracey, Critical exponentηatO(1/N 3) in the chiral xy model using the largeNconformal bootstrap, Physi- cal Review D103, 065018 (2021)

    work page 2021

  53. [53]

    Classen, I

    L. Classen, I. F. Herbut, and M. M. Scherer, Fluctuation- induced continuous transition and quantum criticality in dirac semimetals, Physical Review B96, 115132 (2017)

    work page 2017

  54. [54]

    Hasenbusch, Eliminating leading and subleading cor- rections to scaling in the three-dimensional xy universal- ity class, Physical Review B112, 184512 (2025)

    M. Hasenbusch, Eliminating leading and subleading cor- rections to scaling in the three-dimensional xy universal- ity class, Physical Review B112, 184512 (2025)

    work page 2025

  55. [55]

    c† A,1,↑c† A,2,↓ −c † A,1,↓c† A,2,↑ + c† B,1,↑c† B,2,↓ −c † B,1,↓c† B,2,↑ + (cA,1,↑cA,2,↓ −c A,1,↓cA,2,↑) + (cB,1,↑cB,2,↓ −c B,1,↓cB,2,↑) #

    S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, Carving out ope space and precise o(2) model critical exponents, Journal of High Energy Physics2020, 142 (2020). 7 Supplemental Material: Symmetric Mass Generation in a Bilayer Honeycomb Lattice withSU(2)×SU(2)×SU(2)/Z 2 Symmetry PROJECTION QUANTUM MONTE CARLO METHOD To i...

    work page 2020

  56. [56]

    (S46) with an applied magnetic flux Φ = 10 −4Φ0

    For the rangeJ= 2.3–2.8, the trial wave function is chosen as the ground state of Eq. (S46) with an applied magnetic flux Φ = 10 −4Φ0. We set the projection length to 2Θ = max{17, L+ 5}and the Trotter time step to ∆τ= 0.1

    work page

  57. [57]

    The ground state ofH ′′ 0 is adopted as the trial wave function

    For the rangeJ= 3.2–3.8, we construct a Hamiltonian that introduces disorder into the hopping matrix elements: H ′′ 0 =− X ⟨i,j⟩,α (t+δ r i,j)c † i,αcj,α + h.c.(S49) whereδdenotes the disorder strength, set toδ= 10 −4tin our calculations, andr i,j represents a random number uniformly distributed within the interval (0,1). The ground state ofH ′′ 0 is adop...

    work page