Pith. sign in

REVIEW 4 major objections 5 minor 40 references

Landscape of incompressible crystals of hard-core bosons on the square-kagome lattice

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper argues that the square-kagome lattice hosts a hierarchy of incompressible phases beyond the conventional six-site-unit-cell description, with a stable $\rho=3/4$ phase and additional phases at $\rho=13/24$, $17/24$, and $19/24$…

desk verdict A useful HMFT study that makes a solid case for the 3/4 plateau but leaves the new 24-site plateaus unproven; deserves review with a demand for converged numerics. read the letter →

arxiv 2608.05927 v1 pith:AQBJ4LCC submitted 2026-08-06 cond-mat.str-el

Pratyay Ghosh This is my paper · ORCID
classification cond-mat.str-el
keywords square-kagomelatticehard-corebosonscompactlocalizedstatesmagnetizationplateaushierarchicalclustermean-fieldtheoryXXZmodelfrustratedmagnetismincompressiblephases
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

This paper argues that the square-kagome lattice supports more incompressible phases than the conventional six-site-unit-cell description reveals. Using a hierarchical cluster mean-field approximation, the author reproduces the known localized-magnon crystal phases at densities $2/3$ and $5/6$ and finds additional incompressible states at $13/24$, $3/4$, $17/24$, and $19/24$. The $3/4$ phase persists in both 12-site and 24-site clusters and is identified as a robust half-magnetization plateau in the spin-$1/2$ XXZ model. The enlarged-cluster results are then applied to two square-kagome compounds, reproducing measured plateaus and predicting new ones, notably a $1/4$ plateau in KCu$_6$AlBiO$_4$(SO$_4$)$_5$Cl.

What carries the argument

The argument rests on hierarchical cluster mean-field theory (HMFT), in which the lattice is partitioned into identical clusters, intra-cluster correlations are treated exactly by numerical diagonalization, and inter-cluster couplings are decoupled through self-consistent mean fields. Three cluster geometries are used: a 12-site cluster and two 24-site clusters (Setup 1 and Setup 2) chosen to accommodate the compact localized states and the approximate 20-site localized configuration. The Matsubara-Matsuda mapping converts hard-core boson densities into spin magnetizations, so each incompressible density plateau becomes a magnetization plateau. The key object is the approximate 20-site localized state, a nearly destructive-interference eigenstate with finite leakage that is not an exact compact localized state but nonetheless drives the $17/24$, $3/4$, and $19/24$ phases.

What would settle it

A concrete test is to compute the same hard-core boson or XXZ model on the square-kagome lattice with an unbiased method on larger systems, such as a tensor-network or exact-diagonalization calculation on 36- or 48-site clusters: if the $13/24$, $17/24$, $19/24$, or $3/4$ plateaus disappear, the central claim fails. For the material prediction, a magnetization measurement of KCu$_6$AlBiO$_4$(SO$_4$)$_5$Cl up to roughly 200 T that shows no plateau at $m/m_s=1/4$ near 75 T would falsify the predicted additional plateau.

Watch

Extended reading notes

Core claim

The central claim is that the conventional six-site unit-cell description of the square-kagome lattice is incomplete: it captures only the compact-localized-state phases at $\rho=2/3$ and $5/6$, whereas a hierarchical cluster mean-field treatment with larger clusters stabilizes a richer set of incompressible crystals. The $\rho=3/4$ phase, corresponding to the $m/m_s=1/2$ magnetization plateau, appears already at the 12-site level and survives both 24-site cluster geometries, making it a robust property of the model rather than a finite-size artifact. The phases at $\rho=13/24$, $17/24$, and $19/24$ appear only when the variational cluster is enlarged, and they are associated with an approximately localized 20-site configuration rather than an exact compact localized state. Through the Matsubara-Matsuda mapping these phases correspond to magnetization plateaus at $m/m_s=1/12$, $5/12$, and $7/12$ in the spin-$1/2$ XXZ model.

Load-bearing premise

The load-bearing premise is that the product-state cluster mean-field ansatz on the 12- and 24-site clusters represents the true infinite-lattice ground state of these localized crystal phases, so the new plateaus are physical rather than artifacts of the chosen cluster size and geometry.

Editorial extensions

If this is right

  • The six-site unit-cell picture is incomplete; enlarged-cluster treatments are needed to enumerate the incompressible phases of the square-kagome lattice.
  • The $\rho=3/4$ phase corresponds to a stable $m/m_s=1/2$ magnetization plateau that should appear in the spin-$1/2$ square-kagome Heisenberg and XXZ models.
  • The ideal square-kagome model should also show plateaus at $m/m_s=1/12$, $5/12$, and $7/12$, whose stability depends on the localized configurations accessible at larger clusters.
  • KCu$_6$AlBiO$_4$(SO$_4$)$_5$Cl should show a magnetization plateau at $m/m_s=1/4$ around 75 T, between the existing low-field data and the observed $1/3$ plateau.
  • Strong diagonal exchange in Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$Cl$_3$ suppresses the low-field plateau and shifts the highest plateau to $m/m_s=5/6$, qualitatively different from the ideal lattice.

Reading between the lines

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

  • One can test the $3/4$ plateau's stability directly by running the same HMFT hierarchy on 36- and 48-site clusters; if it persists while the neighboring $13/24$ and $19/24$ phases shift or vanish, the hierarchy suggests a critical cluster-size scale for each phase.
  • The same beyond-unit-cell logic could be applied to other flat-band lattices with multiple compact-localized-state sizes, where enlarged clusters may reveal similar hierarchies of approximate localized crystals.
  • If the 20-site approximate localized state is the true microscopic origin, the plateau widths should scale with $V/t$ in a predictable way, so varying the XXZ anisotropy along the $t'$ axis provides a tunable test of the mechanism.
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

4 major / 5 minor

Summary. The manuscript studies hard-core bosons on the square-kagome lattice with nearest-neighbor hopping and repulsion, using a hierarchical cluster mean-field method (HMFT) on 12-site and two distinct 24-site clusters. It reports that the conventional six-site unit-cell description captures only the known 2/3 and 5/6 incompressible phases, while enlarged clusters reveal additional plateaus at 3/4, 13/24, 17/24, and 19/24, which map via the Matsubara-Matsuda transformation to magnetization plateaus of a spin-1/2 XXZ model. The paper also applies the method to two square-kagome compounds with first-principles exchange parameters, reproducing observed 1/3 and 2/3 plateaus in KCu6AlBiO4(SO4)5Cl and predicting a 1/4 plateau near 75 T, while Na6Cu7BiO4(PO4)4Cl3 is argued to show plateaus at 1/3 and 5/6.

Significance. If the central claim is correct, the paper would show that the square-kagome lattice supports a substantially richer hierarchy of incompressible crystals than previously recognized, and it would provide concrete, falsifiable magnetization-plateau predictions for two candidate materials. The strongest point in favor of the paper is the rho=3/4 phase: it appears in both 12-site and 24-site calculations and is independently supported by earlier exact-diagonalization signatures on 24-, 36-, and 48-site clusters, even though those signatures were not interpreted as a robust plateau. The material predictions are also genuinely falsifiable, and the phase diagrams involve no free parameters fitted to target plateaus. These strengths are, however, concentrated on the 3/4 phase; the claimed intrinsic character of the 13/24, 17/24, and 19/24 phases rests on much weaker evidence, as detailed in the major comments.

major comments (4)
  1. [Sec. V, Figs. 3(b) and 4(a)] The central claim that the 13/24, 17/24, and 19/24 phases are intrinsic to the infinite lattice is not established because the two 24-site clusters give different plateau sets: Setup 2 adds a 17/24 plateau that is absent in Setup 1 and reduces the width of the 19/24 plateau. Since these densities correspond to integer particle numbers on a 24-site cluster, they are exactly the values one would expect from finite-cluster quantization even if the thermodynamic limit has no such gap. The paper does not provide a 36- or 48-site calculation, no exact-diagonalization data at these densities, and no estimate of the charge gap as a function of cluster size. I request either larger-cluster HMFT results, an independent method (e.g., exact diagonalization or tensor-network simulation) for at least the 13/24 and 19/24 densities, or an explicit quantitative argument that the 24-site discretization error cannot account for the plateaus.
  2. [Sec. III B] Section III B states that only spatial structures compatible with the chosen cluster geometry can be represented. The clusters were deliberately chosen to accommodate the candidate localized states, including the approximate 20-site state with finite leakage shown in Fig. 1(c). Finding that state in the enlarged variational space is therefore partly a consequence of the variational ansatz, not an unbiased search. To support the intrinsic character of the new phases, the paper should demonstrate that these states remain lowest in energy when represented in a cluster-independent basis, for example by comparing against a larger cluster that does not have the 20-site construction built in, or by quantifying the leakage amplitude and showing that it decays with cluster size.
  3. [Sec. V, rho=3/4 paragraph] The rho=3/4 phase is the best-supported new phase, but its status as a robust incompressible phase would be strengthened by a direct measurement of the charge gap and its scaling with cluster size. The paper cites exact-diagonalization signatures on 24-, 36-, and 48-site clusters, but those earlier works attributed the feature to finite-size effects, and the present manuscript does not provide a convergence analysis of the gap or the local density pattern across cluster sizes. Without such quantitative support, the claim that rho=3/4 is a true thermodynamic plateau remains plausible but not fully demonstrated.
  4. [Sec. VI A, Fig. 5(a)] The predicted 1/4 magnetization plateau in KCu6AlBiO4(SO4)5Cl at roughly 75 T is computed with Setup 2 only. Given that Setup 1 and Setup 2 already disagree on the plateau structure of the ideal model at nearby densities (e.g., 17/24 and 19/24), the material prediction should be checked with at least one alternative cluster geometry, and preferably with a small perturbation of the exchange parameters, before it is presented as a concrete experimental target.
minor comments (5)
  1. [Sec. I] There is a typo in the introduction: 'Heisenebrg' should be 'Heisenberg'.
  2. [Sec. VI B and Fig. 5] The chemical formula of Na6Cu7BiO4(PO4)4Cl3 is written inconsistently: the text uses Na6Cu7BiO4(PO4)4Cl3, while the figure caption and one occurrence in the text use 'Na6Cu7BiO4(PO)4Cl3'. Please unify the notation.
  3. [Fig. 4(b)] The caption states that the 20-site localized state has 'only weak hopping processes extending beyond the cluster boundary', but no numerical measure of the leakage is given. Defining a leakage parameter and reporting its value would make the approximate-localization claim quantitative and testable.
  4. [Sec. V, Fig. 4(a)] The lower-density region of the Setup 2 phase diagram is not shown and is described only as 'qualitatively identical' to Setup 1. Since the discrepancy between Setups 1 and 2 is a central issue, the full phase diagram or a quantitative comparison of plateau boundaries should be provided.
  5. [Sec. III A] The convergence criterion is stated as a change of all mean fields below 10^-6, but there is no mention of the number of self-consistent iterations, the dependence on initial conditions, or the possible presence of multiple local minima. A brief discussion of these numerical aspects would improve reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the derivation is self-contained, with only a minor non-load-bearing self-citation; the finite-cluster limitations are a convergence concern, not a circular reduction.

full rationale

The paper's phase diagram is produced by an explicit variational method, not by fitting parameters to the target plateaus. The known phases at rho=2/3 and 5/6 are reproduced from the same Hamiltonian, and the rho=3/4 claim is cross-checked against external exact-diagonalization studies (Refs. [10,23]). Material predictions use exchange couplings from external first-principles/experimental papers (Refs. [31,40]) and are falsifiable, so they do not reduce to fitted inputs. The only self-citation in a load-bearing position is Ref. [29], used to motivate going beyond compact localized states; however, the paper independently constructs the 20-site approximate localized state, and the HMFT calculation is a nontrivial self-consistent solution, so the citation is not load-bearing. The choice of 24-site clusters is admittedly tailored to accommodate candidate localized structures (Sec. III B), and Setup 1 versus Setup 2 give different plateau sets; this is a genuine convergence/finite-size limitation that weakens the claim that rho=13/24, 17/24, and 19/24 are intrinsic, but it is a correctness risk rather than a circular reduction of the derivation to its inputs. No equation in the paper is equivalent by construction to a predicted plateau, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results rest on the HMFT truncation and on cluster geometries chosen by hand to accommodate candidate localized structures. The material predictions additionally assume the accuracy of external exchange parameters. No free parameters are fitted by the paper itself, but the method has uncontrolled approximation error, and the new high-density phases depend on the specific 24-site clusters used.

assumptions (5)
  • domain assumption The hard-core boson Hamiltonian with nearest-neighbor hopping t, repulsion V, and chemical potential mu is a faithful model for the square-kagome compounds discussed.
    Invoked in Eq. (1) and in Section VI, where real materials are modeled with additional exchange bonds and anisotropies taken from first-principles references.
  • domain assumption The product-state cluster mean-field ansatz, with inter-cluster hopping and density decoupled at the mean-field level, faithfully represents the ground state of the infinite lattice.
    Assumed in Section III A, Eqs. (5)-(8); the approximation is uncontrolled and its error is not estimated.
  • ad hoc to paper The chosen 12-site and 24-site cluster geometries define a variational space large enough to capture the true thermodynamic phases.
    Section III B states that clusters are selected to accommodate the candidate localized states, so the appearance of those states in the phase diagram is partly built into the calculation.
  • ad hoc to paper The approximate 20-site localized state with finite leakage remains stable in the thermodynamic limit and is not destroyed by inter-cluster correlations.
    Introduced in Section II and Fig. 1(c) and used in Section V to explain the 19/24, 3/4, and 17/24 plateaus; the paper provides no exact proof of stability.
  • domain assumption The first-principles exchange parameters taken from Refs. [31,40] are accurate enough for quantitative magnetization predictions.
    Used in Section VI for the two material-specific magnetization curves; any error in these parameters directly shifts the predicted plateau fields.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Landscape of incompressible crystals of hard-core bosons on the square-kagome lattice." pith.science (2026). https://pith.science/paper/AQBJ4LCC

@misc{pith2026260805927,
  author       = {Pith},
  title        = {Pith review of: Landscape of incompressible crystals of hard-core bosons on the square-kagome lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AQBJ4LCC}},
  note         = {Machine review of arXiv:2608.05927}
}
abstract

We investigate a hard-core boson model on the square-kagome lattice using hierarchical mean-field theory beyond the conventional unit-cell description. We show that the conventional description of the square-kagome lattice based on its smallest unit cell does not fully capture the hierarchy of incompressible states supported by the lattice, but only captures the two compact-localized-state-based phases at densities $5/6$ and $2/3$. Within our approach, we reproduce these previously established phases and uncover additional incompressible states enabled by enlarging the variational cluster. Among these, the $\rho=3/4$ phase is found to be particularly robust, which, through the Matsubara-Matsuda mapping, corresponds to a half-magnetization plateau in the spin-$1/2$ XXZ model. We further apply our approach to two experimentally relevant square-kagome compounds using exchange parameters obtained from first-principles calculations. The calculated magnetization processes are in good agreement with available experimental results and predict additional plateau structures in these materials.

Figures

Figures reproduced from arXiv: 2608.05927 by the authors.

Figure 1
Figure 1. FIG. 1. Compact localized states and approximate localized structures on the square-kagome lattice. (a) Square-kagome [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Cluster geometries used in the hierarchical mean-field calculations. (a) The 12-site cluster, which represents the smallest [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. We therefore consider two additional 24-site clus [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Ground-state phase diagrams obtained from cluster mean-field theory. (a) Phase diagram obtained using the 12-site [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Comparison of the phase diagram obtained using the alternative 24-site cluster (Setup 2) and the corresponding local [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Magnetic properties of square-kagome compounds. (a) Exchange network and calculated magnetization curve of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 31 canonical work pages

  1. [10]

    Nakano, Y

    H. Nakano, Y. Hasegawa, and T. Sakai, Magnetiza- tion Jump in the Magnetization Process of the Spin- 1/2 Heisenberg Antiferromagnet on a Distorted Square- Kagome Lattice, Journal of the Physical Society of Japan 84, 114703 (2015)

    work page 2015

  2. [23]

    Richter, O

    J. Richter, O. Derzhko, and J. Schnack, Thermodynamics of the spin-half square kagome lattice antiferromagnet, Physical Review B105, 144427 (2022)

    2022

  3. [1]

    Sutherland, Localization of electronic wave functions due to local topology, Physical Review B34, 5208 (1986)

    B. Sutherland, Localization of electronic wave functions due to local topology, Physical Review B34, 5208 (1986)

    work page 1986

  4. [2]

    D. L. Bergman, C. Wu, and L. Balents, Band touching from real-space topology in frustrated hopping models, Physical Review B78, 125104 (2008)

    2008

  5. [3]

    Schulenburg, A

    J. Schulenburg, A. Honecker, J. Schnack, J. Richter, and H.-J. Schmidt, Macroscopic Magnetization Jumps due to Independent Magnons in Frustrated Quantum Spin Lat- tices, Physical Review Letters88, 167207 (2002)

    work page 2002

  6. [4]

    Richter, J

    J. Richter, J. Schulenburg, A. Honecker, J. Schnack, and H.-J. Schmidt, Exact eigenstates and macroscopic magnetization jumps in strongly frustrated spin lattices, Journal of Physics: Condensed Matter16, S779 (2004)

    2004

  7. [5]

    Derzhko and J

    O. Derzhko and J. Richter, Universal low-temperature behavior of frustrated quantum antiferromagnets in the vicinity of the saturation field, The European Physical 8 Journal B52, 23 (2006)

    work page 2006

  8. [6]

    Derzhko, J

    O. Derzhko, J. Richter, A. Honecker, M. Maksymenko, and R. Moessner, Low-temperature properties of the Hubbard model on highly frustrated one-dimensional lat- tices, Physical Review B81, 014421 (2010)

    work page 2010

Show all 40 references
  1. [7]

    Y. Chen, J. Huang, K. Jiang, and J. Hu, Decoding flat bands from compact localized states, Science Bulletin68, 3165–3171 (2023)

    2023

  2. [8]

    N. Ara, A. Banerjee, R. Basu, and B. Krishnan, Flat bands and compact localised states: A Carrollian roadmap, SciPost Physics19, 046 (2025)

    2025

  3. [9]

    Richter, Localized-magnon states in strongly frus- trated quantum spin lattices, Low Temperature Physics 31, 695 (2005)

    J. Richter, Localized-magnon states in strongly frus- trated quantum spin lattices, Low Temperature Physics 31, 695 (2005)

    2005

  4. [11]

    Hasegawa, H

    Y. Hasegawa, H. Nakano, and T. Sakai, Metamagnetic jump in the spin- 1 2 antiferromagnetic Heisenberg model on the square kagome lattice, Physical Review B98, 014404 (2018)

    2018

  5. [12]

    C. K. Majumdar and D. K. Ghosh, On next-nearest- neighbor interaction in linear chain. ii, Journal of Math- ematical Physics10, 1399–1402 (1969)

    1969

  6. [13]

    Sriram Shastry and B

    B. Sriram Shastry and B. Sutherland, Exact ground state of a quantum mechanical antiferromagnet, Physica B+C 108, 1069–1070 (1981)

    1981

  7. [14]

    M. P. Gelfand, Linked-tetrahedra spin chain: Exact ground state and excitations, Phys. Rev. B43, 8644 (1991)

    1991

  8. [15]

    Capponi, Numerical study of magnetization plateaus in the spin- 1 2 heisenberg antiferromagnet on the checker- board lattice, Phys

    S. Capponi, Numerical study of magnetization plateaus in the spin- 1 2 heisenberg antiferromagnet on the checker- board lattice, Phys. Rev. B95, 014420 (2017)

    2017

  9. [16]

    Ghosh, T

    P. Ghosh, T. M¨ uller, and R. Thomale, Another exact ground state of a two-dimensional quantum antiferro- magnet, Phys. Rev. B105, L180412 (2022)

    2022

  10. [17]

    Ghosh, J

    P. Ghosh, J. Seufert, T. M¨ uller, F. Mila, and R. Thomale, Maple leaf antiferromagnet in a magnetic field, Phys. Rev. B108, L060406 (2023)

    2023

  11. [18]

    Schnack, H.-J

    J. Schnack, H.-J. Schmidt, A. Honecker, J. Schulenburg, and J. Richter, Exact eigenstates of highly frustrated spin lattices probed in high fields, Journal of Physics: Con- ference Series51, 43 (2006)

    2006

  12. [19]

    Siddharthan and A

    R. Siddharthan and A. Georges, Square kagome quantum antiferromagnet and the eight-vertex model, Physical Re- view B65, 014417 (2001)

    2001

  13. [20]

    Rousochatzakis, R

    I. Rousochatzakis, R. Moessner, and J. van den Brink, Frustrated magnetism and resonating valence bond physics in two-dimensional kagome-like magnets, Phys. Rev. B88, 195109 (2013)

    2013

  14. [21]

    Derzhko, J

    O. Derzhko, J. Richter, O. Krupnitska, and T. Krokhmal- skii, The square-kagome quantum Heisenberg antiferro- magnet at high magnetic fields: The localized-magnon paradigm and beyond, Low Temperature Physics40, 513 (2014)

    2014

  15. [22]

    Lugan, L

    T. Lugan, L. D. C. Jaubert, and A. Ralko, Topological nematic spin liquid on the square kagome lattice, Physi- cal Review Research1, 033147 (2019)

    2019

  16. [24]

    Richter and J

    J. Richter and J. Schnack, Magnetism of theS= 1/2J 1- J2 square-kagome lattice antiferromagnet, Physical Re- view B107, 245115 (2023)

    2023

  17. [25]

    Astrakhantsev, F

    N. Astrakhantsev, F. Ferrari, N. Niggemann, T. M¨ uller, A. Chauhan, A. Kshetrimayum, P. Ghosh, N. Regnault, R. Thomale, J. Reuther, T. Neupert, and Y. Iqbal, Pin- wheel valence bond crystal ground state of the spin- 1 2 Heisenberg antiferromagnet on the shuriken lattice, Phys...

    2021

  18. [26]

    S. S. Jahromi and Y. Iqbal, Quantum phase diagram of the spin- 1 2 Heisenberg antiferromagnet on the square- kagome lattice: A tensor network study, Physical Review B113, L140402 (2026)

    2026

  19. [27]

    Matter Phys.12, 507 (2009)

    Richter, Schulenburg, Tomczak, and Schmalfuß, The heisenberg antiferromagnet on the square-kagom´ e lattice, Condens. Matter Phys.12, 507 (2009)

    2009

  20. [28]

    Schmoll, A

    P. Schmoll, A. Kshetrimayum, J. Naumann, J. Eisert, and Y. Iqbal, Tensor network study of the spin- 1 2 Heisenberg antiferromagnet on the shuriken lattice, Phys- ical Review B107, 064406 (2023)

    2023

  21. [29]

    Ghosh and F

    P. Ghosh and F. Mila, Simplex crystal ground state and magnetization plateaus in the spin-1/2 heisenberg model on the ruby lattice (2025), arXiv:2512.14173 [cond- mat.str-el]

    2025

  22. [30]

    Matsubara and H

    T. Matsubara and H. Matsuda, A Lattice Model of Liquid Helium, I, Progress of Theoretical Physics16, 569 (1956)

    1956

  23. [31]

    Fujihala, K

    M. Fujihala, K. Morita, R. Mole, S. Mitsuda, T. To- hyama, S.-i. Yano, D. Yu, S. Sota, T. Kuwai, A. Koda, H. Okabe, H. Lee, S. Itoh, T. Hawai, T. Masuda, H. Sagayama, A. Matsuo, K. Kindo, S. Ohira-Kawamura, and K. Nakajima, Gapless spin liquid in a square- kagome lattice antif...

    2020

  24. [32]

    O. V. Yakubovich, L. V. Shvanskaya, G. V. Kiriukhina, A. S. Volkov, O. V. Dimitrova, and A. N. Vasiliev, Hy- drothermal synthesis and a composite crystal structure of Na6Cu7BiO4(PO)4[Cl,(OH)] 3 as a candidate for quan- tum spin liquid, Inorganic Chemistry60, 11450–11457 (2021)

    2021

  25. [33]

    B. Liu, Z. Zeng, A. Xu, Y. Sun, O. Yakubovich, L. Shvan- skaya, S. Li, and A. Vasiliev, Low-temperature specific- heat studies on two square-kagome antiferromagnets, Physical Review B105, 155153 (2022)

    2022

  26. [34]

    Ortiz and C

    G. Ortiz and C. D. Batista, Hierarchical mean-field the- ory in quantum statistical mechanics: A bosonic exam- ple, Phys. Rev. B67, 134301 (2003)

    2003

  27. [35]

    C. D. Batista and G. Ortiz, Algebraic approach to inter- acting quantum systems, Adv. Phys.53, 1 (2004)

    2004

  28. [36]

    Isaev, G

    L. Isaev, G. Ortiz, and J. Dukelsky, Hierarchical mean- field approach to theJ 1−J2 heisenberg model on a square lattice, Phys. Rev. B79, 024409 (2009)

    2009

  29. [37]

    Isaev and G

    L. Isaev and G. Ortiz, Frustrated magnets and quantum paramagnetic phases at finite temperature, Phys. Rev. B 86, 100402(R) (2012)

    2012

  30. [38]

    Huerga, J

    D. Huerga, J. Dukelsky, N. Laflorencie, and G. Or- tiz, Chiral phases of two-dimensional hard-core bosons with frustrated ring exchange, Phys. Rev. B89, 094401 (2014)

    2014

  31. [39]

    Huerga, S

    D. Huerga, S. Capponi, J. Dukelsky, and G. Ortiz, Stair- case of crystal phases of hard-core bosons on the kagome lattice, Physical Review B94, 165124 (2016)

    2016

  32. [40]

    Niggemann, N

    N. Niggemann, N. Astrakhantsev, A. Ralko, F. Ferrari, A. Maity, T. M¨ uller, J. Richter, R. Thomale, T. Neu- pert, J. Reuther, Y. Iqbal, and H. O. Jeschke, Quan- tum paramagnetism in the decorated square-kagome an- 9 tiferromagnet Na 6Cu7BiO4(PO4)4Cl3, Physical Review B108, L2...

    2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.