pith. sign in

arxiv: 2606.29242 · v1 · pith:W7VI3PC7new · submitted 2026-06-28 · ❄️ cond-mat.str-el

Winding-Sector Transitions and Thermodynamic Incommensurability in Helical Valence Bond Phase under Tilted Boundary Conditions

Yan Liu , Jie Lou , Yan Chen This is my paper

Pith reviewed 2026-06-30 02:49 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords helical valence bond phaseincommensurate phasetilted boundary conditionsdomain wall densityJ-Q3 modelvalence bond solidprojector quantum Monte Carlowinding sector transitions
0
0 comments X

The pith

The helical valence bond phase is a genuine two-dimensional incommensurate phase with long-range bond-bond order in the thermodynamic limit.

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

The paper studies the S=1/2 staircase J-Q3 model in its maximally anisotropic limit with projector quantum Monte Carlo. It applies a 45° tilted periodic boundary condition to remove boundary-induced finite-size effects on spatially modulated structures. Defining a domain wall density to track the modulation, the authors show that both this density and the characteristic wavevector vary continuously with the coupling ratio and exhibit no lock-in to commensurate values. Thermodynamic extrapolations confirm that winding-sector transitions are finite-size artifacts, while the helical valence bond phase maintains long-range bond-bond order without commensurability. The transition from the columnar valence bond solid occurs at g_c = 0.046(2).

Core claim

By implementing 45° tilted periodic boundary conditions in projector quantum Monte Carlo simulations of the maximally anisotropic staircase J-Q3 model, the authors demonstrate that both the domain wall density and the characteristic wavevector of the helical valence bond phase evolve continuously with the coupling ratio g, showing no commensurate lock-in behavior. Thermodynamic extrapolations confirm long-range bond-bond order, establishing the helical valence bond phase as a genuine two-dimensional incommensurate phase. Winding-sector transitions are identified as finite-size effects enforced by boundary commensurability, with the columnar to helical transition at g_c = 0.046(2).

What carries the argument

The 45° tilted periodic boundary condition, which removes intermediate phases and boundary commensurability constraints, together with the domain wall density that quantifies the spatial modulation of the helical valence bond phase.

If this is right

  • The helical valence bond phase exhibits continuous, non-locked evolution of its modulation wavevector in the thermodynamic limit.
  • Winding-sector transitions vanish as finite-size effects once boundary commensurability is removed.
  • Long-range bond-bond order persists throughout the incommensurate helical phase.
  • The phase boundary between columnar valence bond solid and helical valence bond phase sits at g_c = 0.046(2).

Where Pith is reading between the lines

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

  • The same tilted-boundary method could be tested on other candidate incommensurate valence-bond or spin-liquid phases to separate true incommensurability from boundary artifacts.
  • Continuous wavevector evolution implies the phase may support gapless sliding modes or floating character that standard periodic boundaries obscure.
  • Response functions or entanglement measures computed under these boundaries might reveal whether the incommensurate order couples to other degrees of freedom.

Load-bearing premise

The 45° tilted periodic boundary condition fully eliminates boundary-induced finite-size ambiguities and provides unbiased access to the thermodynamic limit without introducing new commensurability constraints or artifacts.

What would settle it

A clear lock-in of the wavevector to a rational multiple of 2π at sufficiently large system sizes under the same tilted boundaries, or the reappearance of winding-sector transitions independent of boundary choice, would falsify the thermodynamic incommensurability claim.

Figures

Figures reproduced from arXiv: 2606.29242 by Jie Lou, Yan Chen, Yan Liu.

Figure 1
Figure 1. Figure 1: (a) Schematic illustration of the Q terms. (b) Schematic of the L × L × 2 lattice geometry with 45◦ tilted periodic boundary conditions. Thick blue lines and dashed black lines represent the strong (J(1+h)) and weak (J(1−h)) bonds, respectively. arXiv:2606.29242v1 [cond-mat.str-el] 28 Jun 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic illustrations of VBS configurations cor [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Order parameters and corresponding Binder cumulants as a function of the coupling ratio [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Ground-state energy curves of the cVBS phase [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Domain wall density ρ as a function of coupling ratio g for L = 32 and 64. (b) Behavior of ρ versus 1/L at a fixed g = 0.27. The horizontal dashed line denotes the thermodynamic limit value (ρ = 0.08531(5)) extracted from the fit in (c). (c) The ground-state energy E0 as a function of ρ at g = 0.27. The solid curve is a quadratic fit and the vertical dashed line marks the energy minimum. sector. Within… view at source ↗
Figure 6
Figure 6. Figure 6: The extrapolated thermodynamic limit of (a) the [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

We investigate the ground states of the $S = 1/2$ staircase $J$-$Q_3$ model in the maximally anisotropic limit by employing projector quantum Monte Carlo simulations. To overcome boundary-induced finite-size ambiguities inherent in the study of spatially modulated structures, we implement a $45^{\circ}$ tilted periodic boundary condition that eliminates intermediate phases and provides direct access to winding-sector transitions of the system. By defining a domain wall density to quantify the spatial modulation of the helical valence bond phase, we perform thermodynamic extrapolations and demonstrate that both the domain wall density and the characteristic wavevector evolve continuously with the coupling ratio, exhibiting no commensurate lock-in behavior. Our results establish that the helical valence bond phase is a genuine two-dimensional incommensurate phase with long-range bond-bond order in the thermodynamic limit, clarifying that winding-sector transitions are finite-size effects enforced by boundary commensurability. Furthermore, we determine the phase transition point between columnar valence bond solid phase and helical valence bond phase to be $g_c = 0.046(2)$.

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 / 2 minor

Summary. The paper studies the S=1/2 staircase J-Q3 model in the maximally anisotropic limit via projector quantum Monte Carlo. It introduces 45° tilted periodic boundary conditions to remove boundary-induced finite-size ambiguities for modulated structures, defines a domain-wall density to characterize the helical valence bond phase, and reports thermodynamic extrapolations showing that both domain-wall density and characteristic wavevector evolve continuously with the coupling ratio g without commensurate lock-in. The central claims are that the helical valence bond phase is a genuine two-dimensional incommensurate phase possessing long-range bond-bond order in the thermodynamic limit, that winding-sector transitions are finite-size artifacts, and that the columnar-to-helical transition occurs at g_c = 0.046(2).

Significance. If the numerical evidence is robust, the work would provide concrete support for the existence of a stable incommensurate helical valence-bond phase in two dimensions and demonstrate a practical route (tilted PBC) for studying spatially modulated quantum phases without artificial commensurability constraints. The continuous evolution and absence of lock-in constitute falsifiable predictions that can be tested by other methods.

major comments (2)
  1. [Methods and Results] § Methods (tilted PBC implementation) and Results (domain-wall extrapolations): the claim that the 45° tilt fully removes boundary commensurability constraints and permits unbiased access to incommensurate wavevectors rests on the assumption that allowed modulations become dense in the thermodynamic limit. The manuscript must explicitly show the allowed helical wavevectors under the tilted supercell (e.g., via the discrete set of winding sectors) and demonstrate that the observed continuous g-dependence is not an artifact of residual discretization; otherwise the absence of lock-in cannot be distinguished from a boundary-induced selection effect.
  2. [Results] Results (thermodynamic extrapolations of domain-wall density and wavevector): the central claim of long-range bond-bond order and continuous incommensurability in the thermodynamic limit is supported only by extrapolations whose details (system sizes, fitting forms, χ^{2} values, and error propagation) are not reported in sufficient detail. Without these, it is impossible to assess whether the reported continuous evolution and g_c = 0.046(2) are robust or sensitive to the largest accessible sizes.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the range of system sizes used for each extrapolation and the functional form assumed for the finite-size scaling.
  2. [Methods] Notation for the domain-wall density should be defined once in the main text with a clear relation to the bond-bond correlation function.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below and will revise the manuscript to incorporate the requested clarifications and details.

read point-by-point responses

  1. Referee: [Methods and Results] § Methods (tilted PBC implementation) and Results (domain-wall extrapolations): the claim that the 45° tilt fully removes boundary commensurability constraints and permits unbiased access to incommensurate wavevectors rests on the assumption that allowed modulations become dense in the thermodynamic limit. The manuscript must explicitly show the allowed helical wavevectors under the tilted supercell (e.g., via the discrete set of winding sectors) and demonstrate that the observed continuous g-dependence is not an artifact of residual discretization; otherwise the absence of lock-in cannot be distinguished from a boundary-induced selection effect.

    Authors: We agree that an explicit demonstration of the allowed wavevectors is required to fully substantiate the claim. In the revised manuscript we will add a new subsection (or figure) in Methods that enumerates the discrete winding sectors permitted by the 45° tilted supercell and maps them onto the corresponding helical wavevectors. We will further include a finite-size scaling argument, supported by additional data, showing that the density of accessible sectors increases with linear size L, rendering the observed continuous g-dependence robust against residual discretization. This addition will directly address the concern that the absence of lock-in might be a boundary artifact. revision: yes

  2. Referee: [Results] Results (thermodynamic extrapolations of domain-wall density and wavevector): the central claim of long-range bond-bond order and continuous incommensurability in the thermodynamic limit is supported only by extrapolations whose details (system sizes, fitting forms, χ^{2} values, and error propagation) are not reported in sufficient detail. Without these, it is impossible to assess whether the reported continuous evolution and g_c = 0.046(2) are robust or sensitive to the largest accessible sizes.

    Authors: We acknowledge that the extrapolation details were not reported with sufficient transparency. In the revision we will expand the Results section and add a dedicated appendix that (i) lists all system sizes used, (ii) specifies the fitting forms (e.g., linear or quadratic in 1/L), (iii) reports the χ² values for each fit, and (iv) describes the error-propagation procedure. These additions will allow readers to evaluate the robustness of the continuous evolution and the quoted transition point g_c = 0.046(2). revision: yes

Circularity Check

0 steps flagged

No circularity; numerical extrapolations are independent of inputs

full rationale

The paper's derivation consists of projector QMC simulations under 45° tilted PBC, definition of domain wall density as a direct measure of modulation, and thermodynamic extrapolation of density and wavevector versus coupling ratio g. These quantities are computed from the sampled configurations and extrapolated to the limit; the continuous evolution and absence of lock-in are data-driven conclusions, not forced by redefinition or by fitting a parameter to a subset and relabeling it a prediction. No self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the load-bearing steps. The boundary-condition assumption is tested rather than presupposed, yielding a self-contained numerical result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work depends on the validity of projector QMC for ground-state sampling and the assumption that tilted boundaries remove rather than replace finite-size effects; g_c is obtained from data extrapolation.

free parameters (1)
  • g_c = 0.046(2)
    Critical coupling ratio obtained via thermodynamic extrapolation of simulation data.
axioms (1)
  • domain assumption Projector quantum Monte Carlo accurately samples the ground state of the S=1/2 staircase J-Q3 model without sign problem or ergodicity issues.
    Standard assumption invoked for all QMC results in the abstract.

pith-pipeline@v0.9.1-grok · 5722 in / 1255 out tokens · 42393 ms · 2026-06-30T02:49:02.539899+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

59 extracted references · 2 canonical work pages

  1. [1]

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

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

    2011

  2. [2]

    S. L. Sondhi, S. Girvin, J. Carini, and D. Shahar, Con- 7 tinuous quantum phase transitions, Reviews of modern physics69, 315 (1997)

    1997

  3. [3]

    P. W. Anderson, More is different: broken symmetry and the nature of the hierarchical structure of science., Sci- ence177, 393 (1972)

    1972

  4. [4]

    P. M. Chaikin, T. C. Lubensky, and T. A. Witten,Prin- ciples of condensed matter physics, Vol. 10 (Cambridge university press Cambridge, 1995)

    1995

  5. [5]

    L. D. Landau and E. M. Lifshitz,Statistical physics: vol- ume 5, Vol. 5 (Elsevier, 2013)

    2013

  6. [6]

    Pokrovsky and A

    V. Pokrovsky and A. Talapov, Ground state, spectrum, and phase diagram of two-dimensional incommensurate crystals, Physical Review Letters42, 65 (1979)

    1979

  7. [7]

    M. E. Fisher and W. Selke, Infinitely many commensu- rate phases in a simple ising model, Physical Review Let- ters44, 1502 (1980)

    1980

  8. [8]

    Villain and P

    J. Villain and P. Bak, Two-dimensional ising model with competing interactions: floating phase, walls and dislo- cations, Journal de Physique42, 657 (1981)

    1981

  9. [9]

    Bak, Commensurate phases, incommensurate phases and the devil’s staircase, Reports on Progress in Physics 45, 587 (1982)

    P. Bak, Commensurate phases, incommensurate phases and the devil’s staircase, Reports on Progress in Physics 45, 587 (1982)

    1982

  10. [10]

    Selke, The ANNNI model—theoretical analysis and experimental application, Physics Reports170, 213 (1988)

    W. Selke, The ANNNI model—theoretical analysis and experimental application, Physics Reports170, 213 (1988)

    1988

  11. [11]

    Overhauser, Observability of charge-density waves by neutron diffraction, Physical Review B3, 3173 (1971)

    A. Overhauser, Observability of charge-density waves by neutron diffraction, Physical Review B3, 3173 (1971)

    1971

  12. [12]

    P. Lee, T. Rice, and P. Anderson, Conductivity from charge or spin density waves, Solid State Communica- tions14, 703 (1974)

    1974

  13. [13]

    W. L. McMillan, Theory of discommensurations and the commensurate-incommensurate charge-density-wave phase transition, Physical Review B14, 1496 (1976)

    1976

  14. [14]

    Blinc, Magnetic resonance and relaxation in struc- turally incommensurate systems, Physics Reports79, 331 (1981)

    R. Blinc, Magnetic resonance and relaxation in struc- turally incommensurate systems, Physics Reports79, 331 (1981)

    1981

  15. [15]

    Woods, L

    C. Woods, L. Britnell, A. Eckmann, R. Ma, J. Lu, H. Guo, X. Lin, G. Yu, Y. Cao, R. V. Gorbachev,et al., Commensurate–incommensurate transition in graphene on hexagonal boron nitride, Nature physics10, 451 (2014)

    2014

  16. [16]

    O. A. Starykh, Unusual ordered phases of highly frus- trated magnets: a review, Reports on Progress in Physics 78, 052502 (2015)

    2015

  17. [17]

    Whitsitt, R

    S. Whitsitt, R. Samajdar, and S. Sachdev, Quantum field theory for the chiral clock transition in one spatial dimen- sion, Physical Review B98, 205118 (2018)

    2018

  18. [18]

    Chepiga, I

    N. Chepiga, I. Affleck, and F. Mila, Floating, critical, and dimerized phases in a frustrated spin- 3 2 chain, Physical Review B101, 174407 (2020)

    2020

  19. [19]

    Nyckees and F

    S. Nyckees and F. Mila, Commensurate-incommensurate transition in the chiral ashkin-teller model, Physical Re- view Research4, 013093 (2022)

    2022

  20. [20]

    Savary and L

    L. Savary and L. Balents, Quantum spin liquids: a re- view, Reports on Progress in Physics80, 016502 (2017)

    2017

  21. [21]

    Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Reviews of Modern Physics89, 025003 (2017)

    2017

  22. [22]

    A. W. Sandvik, Computational studies of quantum spin systems, inAIP Conf. Proc, Vol. 1297 (2010) p. 135

    2010

  23. [23]

    Zhang and P

    K. Zhang and P. Charbonneau, Monte carlo approach for studying microphases applied to the axial next-nearest- neighbor ising and the ising-coulomb models, Physical Review B83, 214303 (2011)

    2011

  24. [24]

    Jordan, R

    J. Jordan, R. Or´ us, G. Vidal, F. Verstraete, and J. I. Cirac, Classical simulation of infinite-size quantum lat- tice systems in two spatial dimensions, Physical Review Letters101, 250602 (2008)

    2008

  25. [25]

    A. W. Sandvik, Stochastic series expansion methods, arXiv preprint arXiv:1909.10591 (2019)

    work page arXiv 1909

  26. [26]

    B. Zhao, J. Takahashi, and A. W. Sandvik, Multicritical deconfined quantum criticality and lifshitz point of a he- lical valence-bond phase, Physical Review Letters125, 257204 (2020)

    2020

  27. [27]

    A. W. Sandvik, Evidence for deconfined quantum criti- cality in a two-dimensional heisenberg model with four- spin interactions, Physical Review Letters98, 227202 (2007)

    2007

  28. [28]

    J. Lou, A. W. Sandvik, and N. Kawashima, Anti- ferromagnetic to valence-bond-solid transitions in two- dimensional SU(N) heisenberg models with multispin in- teractions, Physical Review B80, 180414 (2009)

    2009

  29. [29]

    Sen and A

    A. Sen and A. W. Sandvik, Example of a first-order N´ eel to valence-bond-solid transition in two dimensions, Phys- ical Review B82, 174428 (2010)

    2010

  30. [30]

    A. W. Sandvik, Finite-size scaling and boundary effects in two-dimensional valence-bond solids, Physical Review B85, 134407 (2012)

    2012

  31. [31]

    Shu, S.-K

    Y.-R. Shu, S.-K. Jian, and S. Yin, Nonequilibrium dy- namics of deconfined quantum critical point in imaginary time, Physical Review Letters128, 020601 (2022)

    2022

  32. [32]

    Y.-C. Wang, N. Ma, M. Cheng, and Z. Y. Meng, Scaling of the disorder operator at deconfined quantum critical- ity, SciPost Physics13, 123 (2022)

    2022

  33. [33]

    Zhao, Y.-C

    J. Zhao, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, Scaling of entanglement entropy at deconfined quantum criticality, Physical Review Letters128, 010601 (2022)

    2022

  34. [34]

    Z. Deng, L. Liu, W. Guo, and H.-Q. Lin, Diagnosing quantum phase transition order and deconfined criticality via entanglement entropy, Physical Review Letters133, 100402 (2024)

    2024

  35. [35]

    D’Emidio and A

    J. D’Emidio and A. W. Sandvik, Entanglement entropy and deconfined criticality: emergent SO(5) symmetry and proper lattice bipartition, Physical Review Letters 133, 166702 (2024)

    2024

  36. [36]

    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, Physical Review Letters136, 046501 (2026)

    2026

  37. [37]

    Zhang, N

    F. Zhang, N. Desai, W. Guo, and R. K. Kaul, Phase diagrams ofs=1/2 bilayer models of SU(2) symmetric antiferromagnets, Physical Review B113, 115120 (2026)

    2026

  38. [38]

    Senthil, A

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

    2004

  39. [39]

    R. G. Melko and R. K. Kaul, Scaling in the fan of an unconventional quantum critical point, Physical Review Letters100, 017203 (2008)

    2008

  40. [40]

    K. Chen, Y. Huang, Y. Deng, A. Kuklov, N. Prokof’ev, and B. Svistunov, Deconfined criticality flow in the heisenberg model with ring-exchange interactions, Phys- ical Review Letters110, 185701 (2013)

    2013

  41. [41]

    Nahum, J

    A. Nahum, J. Chalker, P. Serna, M. Ortu˜ no, and A. So- moza, Deconfined quantum criticality, scaling violations, and classical loop models, Physical Review X5, 041048 (2015)

    2015

  42. [42]

    Nahum, P

    A. Nahum, P. Serna, J. Chalker, M. Ortu˜ no, and A. Somoza, Emergent SO(5) symmetry at the n´ eel to valence-bond-solid transition, Physical Review Letters 8 115(2015)

    2015

  43. [43]

    H. Shao, W. Guo, and A. W. Sandvik, Quantum critical- ity with two length scales, Science352, 213 (2016)

    2016

  44. [44]

    Y. Q. Qin, Y.-Y. He, Y.-Z. You, Z.-Y. Lu, A. Sen, A. W. Sandvik, C. Xu, and Z. Y. Meng, Duality between the deconfined quantum-critical point and the bosonic topo- logical transition, Physical Review X7, 031052 (2017)

    2017

  45. [45]

    Ma, G.-Y

    N. Ma, G.-Y. Sun, Y.-Z. You, C. Xu, A. Vishwanath, A. W. Sandvik, and Z. Y. Meng, Dynamical signature of fractionalization at a deconfined quantum critical point, Physical Review B98, 174421 (2018)

    2018

  46. [46]

    Takahashi, H

    J. Takahashi, H. Shao, B. Zhao, W. Guo, and A. W. Sandvik, SO(5) multicriticality in two-dimensional quan- tum magnets, arXiv:2405.06607 (2024)

    work page arXiv 2024

  47. [47]

    M. Song, J. Zhao, M. Cheng, C. Xu, M. Scherer, L. Janssen, and Z. Y. Meng, Evolution of entanglement entropy at SU(N) deconfined quantum critical points, Science Advances11, eadr0634 (2025)

    2025

  48. [48]

    A. W. Sandvik, Stochastic series expansion method with operator-loop update, Physical Review B59, R14157 (1999)

    1999

  49. [49]

    A. W. Sandvik, Ground state projection of quantum spin systems in the valence-bond basis, Physical Review Let- ters95, 207203 (2005)

    2005

  50. [50]

    A. W. Sandvik and H. G. Evertz, Loop updates for vari- ational and projector quantum monte carlo simulations in the valence-bond basis, Physical Review B82, 024407 (2010)

    2010

  51. [51]

    Beach and A

    K. Beach and A. W. Sandvik, Some formal results for the valence bond basis, Nuclear Physics B750, 142 (2006)

    2006

  52. [52]

    Schulz, Critical behavior of commensurate- incommensurate phase transitions in two dimensions, Physical Review B22, 5274 (1980)

    H. Schulz, Critical behavior of commensurate- incommensurate phase transitions in two dimensions, Physical Review B22, 5274 (1980)

    1980

  53. [53]

    Tang and A

    Y. Tang and A. W. Sandvik, Method to characterize spinons as emergent elementary particles, Physical Re- view Letters107, 157201 (2011)

    2011

  54. [54]

    A. W. Sandvik, V. N. Kotov, and O. P. Sushkov, Ther- modynamics of a gas of deconfined bosonic spinons in two dimensions, Physical Review Letters106, 207203 (2011)

    2011

  55. [55]

    Tang and A

    Y. Tang and A. W. Sandvik, Confinement and decon- finement of spinons in two dimensions, Physical Review Letters110, 217213 (2013)

    2013

  56. [56]

    H. Shao, W. Guo, and A. W. Sandvik, Emergent topolog- ical excitations in a two-dimensional quantum spin sys- tem, Physical Review B91, 094426 (2015)

    2015

  57. [57]

    H. Suwa, A. Sen, and A. W. Sandvik, Level spectroscopy in a two-dimensional quantum magnet: Linearly dispers- ing spinons at the deconfined quantum critical point, Physical Review B94, 144416 (2016)

    2016

  58. [58]

    L. Liu, H. Shao, Y.-C. Lin, W. Guo, and A. W. Sand- vik, Random-singlet phase in disordered two-dimensional quantum magnets, Physical Review X8, 041040 (2018)

    2018

  59. [59]

    Beccaria, M

    M. Beccaria, M. Campostrini, and A. Feo, Evidence for a floating phase of the transverse annni model at high frustration, Physical Review B76, 094410 (2007)

    2007