Pith. sign in

REVIEW 2 major objections 4 minor 53 references

Optimized Tensor-Network Renormalization for Quantum Dynamics: Resolving the Spectral Function of $\mathrm{K_2Co(SeO_3)_2}$

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read Folding excitation states into the renormalization step makes tensor-network spectra stable and reproduces inelastic neutron-scattering measurements for the triangular-lattice supersolid K2Co(SeO3)2.

desk verdict ET-CTMRG is a genuinely new fix for a known instability in iPEPS spectral functions, with strong benchmarks and a mostly convincing material application; the main gap is an overstatement of the K2Co(SeO3)2 agreement. read the letter →

arxiv 2608.02473 v1 pith:JHXYIVEZ submitted 2026-08-03 cond-mat.str-el quant-ph

Jiahang Hu , Runze Chi , B. Normand , Hai-Jun Liao , T. Xiang This is my paper
classification cond-mat.str-elquant-ph
keywords tensornetworksexcitationspectradynamicalstructurefactorcorner-transfermatrixrenormalizationgroupiPEPSspinsupersolidtriangularlatticeXXZmodelinelasticneutronscattering
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 argues that numerical instabilities in tensor-network computations of excitation spectra for two-dimensional quantum magnets come from one specific step: the renormalization tensors used to coarse-grain the environment are built only from the ground state, so they systematically discard the excited-state subspace. It introduces ET-CTMRG, in which the truncation projectors are optimized using both ground-state and single-excitation tensors folded into one matrix K. On benchmark Heisenberg antiferromagnets the truncation error drops by orders of magnitude and the spectrum converges as more states are kept; for the strongly anisotropic triangular-lattice material K2Co(SeO3)2 the method produces spectra that match inelastic neutron scattering, where the ground-state-only version fails. The paper pinpoints the effective Hamiltonian matrix, not the norm matrix, as the source of instability.

What carries the argument

The central object is the matrix K (and its partner K̃), built from the upper and lower halves of the corner-transfer-matrix network: it aggregates the ground-state half-block M0 with half-blocks containing one excitation tensor B, one B†, and both B and B† (M1, M2, M3). Replacing M0 by K in the CTMRG cost function—so that the truncation projectors P and Q come from the truncated SVD of K†K̃—carries excitation information into every renormalization step at essentially the same computational cost as the ground-state-only version.

What would settle it

A decisive check is to compute εphy, the physical Frobenius error using only the physical combinations of Mi and M̃j, for the triangular-lattice XXZ model of K2Co(SeO3)2 at increasing χ. If εphy plateaus at a large value while ε keeps decreasing, the optimal projectors are being driven by nonphysical cross terms and the method's accuracy guarantee for that regime fails.

Watch

Extended reading notes

Core claim

The central claim is that the numerical instability in iPEPS spectral functions is caused by CTMRG truncation projectors being constructed from ground-state tensors alone. The paper replaces the ground-state half-block matrix M0 with an aggregated matrix K built from all half-block configurations containing zero, one, or two excitation tensors, so the projectors P and Q solve the minimization problem with cost ||K†(I−PQ)K̃||. The optimal rank-χ projectors are the truncated SVD factors of K†K̃. This excitation-tailored renormalization makes the effective Hamiltonian matrix accurate enough that solving the generalized eigenvalue problem Heff v = E Neff v yields stable, converged spectra. Bench

Load-bearing premise

The method's accuracy rests on the premise that the rank-χ projectors obtained from the SVD of K†K̃ faithfully preserve the excited-state subspace that actually controls Heff—even though the optimized cost function contains nonphysical cross terms whose effect is only bounded by the physical truncation error, not guaranteed to match it.

Editorial extensions

If this is right

  • ET-CTMRG yields stable, convergent excitation spectra for models where ground-state-only CTMRG is unstable, enabling full-momentum studies with large retained-state counts.
  • For Heisenberg antiferromagnets, the truncation error is reduced by orders of magnitude at the same CTM bond dimension, and the spectrum converges systematically as more states are kept.
  • The instability in spectra is traced to the effective Hamiltonian matrix, not to ill-conditioning of the norm matrix; replacing only the norm matrix does not cure the instability.
  • For K2Co(SeO3)2 in the supersolid Y phase, the computed spectral function agrees quantitatively with inelastic neutron-scattering data, including the two low-energy magnon branches and the two-magnon continuum.
  • The method avoids unit-cell enlargement and second-order derivatives while keeping the computational structure close to that of ground-state CTMRG.

Reading between the lines

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

  • The same principle—building truncation projectors from excited-state configurations rather than only the reference state—could be tested in other tensor-network settings, such as fermionic or bosonic response-function calculations, to probe its generality beyond spin models.
  • The aggregated cost function includes nonphysical cross terms; a practical check would be to monitor the physical Frobenius error εphy separately for strongly anisotropic or frustrated models, where the numerical coincidence of ε and εphy seen in one benchmark may not persist.
  • Because excitation-tailored projectors must be recomputed at every CTMRG step, the method forgoes the cached-projector speed-up available in ground-state-only schemes; a low-rank update or recycling of the previous step's projectors could recover some speed while retaining excitation information.
  • The paper's suggested M-point dip energy near 0.06 meV, roughly half the measured intensity peak, implies that the model parameters (such as Jxy/Jz, determined thermodynamically) could be refined by fitting the full computed spectrum to the neutron data.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces ET-CTMRG, a modification of the CTMRG projector construction used in iPEPS-based calculations of dynamical spectral functions. Instead of building the truncation projectors P and Q solely from the ground-state half-block M0 as in GS-CTMRG, the method defines aggregated half-block matrices K and K̃ from all configurations containing zero or one B/B† excitation, and obtains the optimal rank-χ projectors from the SVD of K†K̃. The authors show analytically that this minimizes a combined Frobenius cost ε, demonstrate numerically that for the square-lattice Heisenberg antiferromagnet the truncation error decreases by many orders of magnitude with χ and the spectrum stabilizes with the number of retained states Nr, and apply the method to the triangular-lattice XXZ model of K2Co(SeO3)2, where GS-CTMRG is severely unstable. The computed spectra are compared with inelastic neutron scattering data at three fields in the supersolid Y phase; good overall agreement is reported, with a caveat in the Supplemental Material about the M-point mode energy.

Significance. If the claims hold, ET-CTMRG is a valuable methodological advance: it addresses a known instability of iPEPS spectral-function calculations, is applicable at arbitrary momenta without unit-cell enlargement, and retains the computational structure of GS-CTMRG. The paper is transparent: the core derivation is presented in the main text and SM, benchmarks include several paradigmatic models, and the material comparison uses Hamiltonian parameters taken from previous thermodynamic/INS work rather than fitted to the target spectra. The stable convergence with Nr and χ, and the order-of-magnitude error reduction, are concrete falsifiable strengths. The main limitations are the lack of a formal guarantee that the aggregated cost preserves the physical excited-state subspace (only an upper-bound relation is proven) and a small but explicit discrepancy in the application that moderates the "excellent quantitative agreement" headline.

major comments (2)
  1. [SM S4, Fig. S8] The statement in SM S2B that "εphy must be smaller than ε, and hence minimizing ε optimizes εphy simultaneously" is not logically correct. From ε² = εphy² + εnonphys², minimizing ε minimizes an upper bound on εphy, but it does not guarantee that εphy is minimized or even reduced; the nonphysical pairs (1,1) and (2,2) could in principle dominate the SVD and bias the projectors. Fig. S1 shows near-equality of ε and εphy for the SLHAF at one momentum and D=3, which is useful empirical evidence but not a general guarantee. For K2Co(SeO3)2, the model where GS-CTMRG fails most dramatically, no ε/εphy comparison is shown. Since the central claim is that ET-CTMRG "faithfully" renormalizes the excited-state manifold, the authors should either prove a stronger relation or explicitly state that the optimality is with respect to the aggregated cost, and provide ε/εphy comparisons for the models and
  2. [SM S4, Fig. S8] The manuscript's own extrapolation in Fig. S8 gives an M-point lower-mode energy of approximately 0.06 meV, which the text notes is "approximately half of the energy at which the maximum in the INS intensity appears in Fig. 5(d)"; the QMC value quoted is about 0.08 meV. This is a significant quantitative discrepancy on the lowest, most prominent branch at a symmetry point. The abstract's "excellent quantitative agreement" and the main text's "quantitative accuracy" are therefore overstated. While the overall spectral shape and field dependence agree well, the M-point branch energy is off by roughly a factor of two from the experimental peak position. The authors should qualify the agreement claim (e.g., "good qualitative and semi-quantitative agreement") and discuss whether this discrepancy is within the expected finite-D error or points to the parameter refinement suggested in the SM.
minor comments (4)
  1. [Main text, 'Benchmark and application'] The phrase "this leads to a strategy call" is unclear; presumably a strategic choice or decision is meant. Please rephrase.
  2. [SM S4, first paragraph] The text refers to "Eq. (10) of the main text" for the INS intensity, but in the main text the expression appears as Eq. (9). Please correct the cross-reference.
  3. [SM S2B, around Eq. (S9)] The sentence "the norm [Eq. (S10)] and energy [Eq. (S9)]" is imprecise: Eq. (S9) is the effective Hamiltonian matrix, and the excitation energy is obtained from the generalized eigenvalue problem of Eq. (S11). Please rephrase to avoid suggesting that Eq. (S9) itself is the energy.
  4. [Throughout] No data-availability or code-availability statement is provided. For a numerical-methods paper, a statement on whether the ET-CTMRG implementation is publicly available would improve reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: ET-CTMRG projectors are derived from excitation-tailored half-blocks, not fitted to target spectra; minor non-load-bearing self-citations only.

full rationale

The central derivation is self-contained. ET-CTMRG projectors P,Q are obtained by minimizing the Frobenius cost of Eq. (7) over K and K̃ built from ground-state and excitation half-block tensors, via Eckart–Young–Mirsky SVD (Eq. (8) and SM Eq. (S19)); they are not fit to the computed spectral function. The K2Co(SeO3)2 benchmark uses Hamiltonian parameters (Jz=3.1 meV, Jxy=0.217 meV, gc=7.9) taken from earlier experimental work [24], not fitted to the iPEPS spectra, and compares against independent INS data [23–25]. The paper explicitly benchmarks against external models (SLHAF, TLHAF, KHAF) and shows systematic convergence in χ and Nr, satisfying the 'self-contained against external benchmarks' criterion. Self-citations [10,11,21] are used for prior spectra, context, and a GS-CTMRG speed-up technique; [21] is explicitly stated not to apply to ET-CTMRG ('This approach cannot be applied to ET-CTMRG'), so it is not load-bearing. The only notable flagged passage is SM Eqs. (S20)–(S22), where the paper admits ε contains nonphysical components and asserts that since ε_phy ≤ ε, minimizing ε optimizes ε_phy simultaneously. That is an unproven bound-based argument and a potential correctness/convergence weakness, but it is not a circular reduction of the method's output to its input; minimizing an upper bound is not equivalent to defining the prediction in terms of the fit. No pattern of self-definition, fitted-input-called-prediction, self-citation load-bearing, imported uniqueness, ansatz-smuggling, or renaming is present.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities. The only inputs are numerical truncation parameters and standard model parameters taken from Ref. [24]. The principal burden is the unproven equivalence between minimizing the aggregated error ε (containing nonphysical cross terms) and minimizing the physical error ε_phy; the paper supports this numerically in Fig. S1. The single-mode ansatz and CTMRG environment approximation are inherited from the prior iPEPS framework.

free parameters (4)
  • iPEPS bond dimension D = D=4 for main K2Co(SeO3)2 spectra; D=3,4,5 for gap extrapolations
    Controls tensor-network accuracy; not fitted to target data, standard convergence parameter.
  • CTM bond dimension χ = χ=100 (K2Co), χ=40-200 (benchmarks)
    Controls environment truncation; chosen by hand; ET-CTMRG convergence demonstrated in χ.
  • Retained excitation states N_r = up to 2000
    Number of eigenstates kept in spectral representation; large values needed for continua; convergence in N_r is the central stability test.
  • Lorentzian broadening η = 0.02 meV
    Applied to iPEPS spectra for INS comparison; chosen by hand, not fitted.
assumptions (5)
  • standard math Eckart–Young–Mirsky theorem: the optimal rank-χ projector pair is obtained from the truncated SVD of K†K̃.
    Used to derive Eq. (8) from Eq. (7); standard result, no independent evidence needed.
  • domain assumption Single-mode/tangent-space excitation ansatz |Φ_k(B)⟩ spans the relevant low-energy excitations.
    Used in Eq. (1); standard in iPEPS spectral methods (Refs. [3,4,6,7]); restricts to single local perturbation.
  • domain assumption The triangular-lattice XXZ Hamiltonian with Jz=3.1 meV, Jxy=0.217 meV, gc=7.9 accurately describes K2Co(SeO3)2.
    Parameters adopted from Ref. [24] (thermodynamic/INS); if incomplete, the comparison to INS is not a test of ET-CTMRG.
  • domain assumption CTMRG with boundary bond dimension χ gives an accurate environment for the infinite iPEPS contraction.
    Standard numerical assumption; convergence in χ is demonstrated for benchmarks but not exhaustively for K2Co.
  • ad hoc to paper Minimizing the aggregated Frobenius error ε (including nonphysical cross terms) also optimizes the physical error ε_phy.
    The paper shows ε_phy ≤ ε and numerically that ε and ε_phy behave similarly (Fig. S1), but no proof that nonphysical terms do not bias projectors in all cases; this is load-bearing for the method.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimized Tensor-Network Renormalization for Quantum Dynamics: Resolving the Spectral Function of $\mathrm{K_2Co(SeO_3)_2}$." pith.science (2026). https://pith.science/paper/JHXYIVEZ

@misc{pith2026260802473,
  author       = {Pith},
  title        = {Pith review of: Optimized Tensor-Network Renormalization for Quantum Dynamics: Resolving the Spectral Function of $\mathrmK_2Co(SeO_3)_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JHXYIVEZ}},
  note         = {Machine review of arXiv:2608.02473}
}
abstract

Tensor-network methods have opened a powerful route for the study of dynamical spectral functions in two-dimensional quantum systems. However, existing approaches within the framework of infinite projected entangled-pair states construct the required renormalization tensors solely from the ground-state environment and can suffer from severe numerical instability. We identify the origin of this instability and introduce an excitation-tailored corner-transfer-matrix renormalization-group (ET-CTMRG) method to resolve it. By incorporating excitation tensors into the renormalization procedure, the method constructs a substantially more accurate effective Hamiltonian matrix and thereby yields reliable and well-converged excitation spectra. For Heisenberg antiferromagnets, it reduces truncation errors by orders of magnitude and for the particularly complex case of the supersolid phase in the triangular-lattice XXZ magnet $\mathrm{K_2Co(SeO_3)_2}$, it achieves excellent quantitative agreement with inelastic neutron-scattering measurements. ET-CTMRG therefore provides a robust framework for investigating the dynamical properties of strongly correlated quantum systems.

Figures

Figures reproduced from arXiv: 2608.02473 by the authors.

Figure 1
Figure 1. FIG. 1. Graphical representation of half-CTM tensors and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Frobenius error, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Comparison of the excitation spectra computed for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Spin excitation spectra of K [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

53 extracted references · 1 linked inside Pith

  1. [26]

    Ponsioen, J

    B. Ponsioen, J. Hasik, and P. Corboz, Improved sum- mations ofn-point correlation functions of projected entangled-pair states, Phys. Rev. B108, 195111 (2023)

    2023

  2. [1]

    ¨Ostlund and S

    S. ¨Ostlund and S. Rommer, Thermodynamic Limit of Density Matrix Renormalization, Phys. Rev. Lett.75, 3537 (1995)

    1995

  3. [2]

    Haegeman, T

    J. Haegeman, T. J. Osborne, and F. Verstraete, Post- matrix product state methods: To tangent space and beyond, Phys. Rev. B88, 075133 (2013)

    2013

  4. [3]

    Vanderstraeten, M

    L. Vanderstraeten, M. Mari¨ en, F. Verstraete, and J. Haegeman, Excitations and the tangent space of pro- jected entangled-pair states, Phys. Rev. B92, 201111 (2015)

    2015

  5. [4]

    Vanderstraeten, J

    L. Vanderstraeten, J. Haegeman, and F. Verstraete, Sim- ulating excitation spectra with projected entangled-pair 6 states, Phys. Rev. B99, 165121 (2019)

    2019

  6. [5]

    Liao, J.-G

    H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Differen- tiable Programming Tensor Networks, Phys. Rev. X9, 031041 (2019)

    2019

  7. [6]

    Ponsioen and P

    B. Ponsioen and P. Corboz, Excitations with projected entangled pair states using the corner transfer matrix method, Phys. Rev. B101, 195109 (2020)

    2020

  8. [7]

    Ponsioen, F

    B. Ponsioen, F. F. Assaad, and P. Corboz, Automatic differentiation applied to excitations with projected en- tangled pair states, SciPost Phys.12, 006 (2022)

    2022

Show all 53 references
  1. [8]

    W.-L. Tu, L. Vanderstraeten, N. Schuch, H.-Y. Lee, N. Kawashima, and J.-Y. Chen, Generating Function for Projected Entangled-Pair States, PRX Quantum5, 010335 (2024)

    2024

  2. [9]

    J. D. Arias Espinoza and P. Corboz, Spectral functions with infinite projected entangled-pair states, Phys. Rev. B110, 094314 (2024)

    2024

  3. [10]

    R. Chi, Y. Liu, Y. Wan, H.-J. Liao, and T. Xiang, Spin Excitation Spectra of Anisotropic Spin-1/2 Triangular Lattice Heisenberg Antiferromagnets, Phys. Rev. Lett. 129, 227201 (2022)

    2022

  4. [11]

    R. Chi, J. Hu, H.-J. Liao, and T. Xiang, Dynamical spec- tra of spin supersolid states in triangular antiferromag- nets, Phys. Rev. B110, L180404 (2024)

    2024

  5. [12]

    Zhang, R

    X.-Y. Zhang, R. Chi, Y. Liu, and L. Wang, Two- dimensional excitation information by a matrix prod- uct state method on infinite helices, Phys. Rev. B109, 075129 (2024)

    2024

  6. [13]

    Y. Xu, J. Hasik, B. Ponsioen, and A. H. Nevidomskyy, Simulating spin dynamics of supersolid states in a quan- tum Ising magnet, Phys. Rev. B111, L060402 (2025)

    2025

  7. [14]

    Sheng, J

    J. Sheng, J. Hu, L. Xu, L. Wang, X. Shi, R. Chi, D. Yu, A. Podlesnyak, P. Piyawongwatthana, N. Murai, S. Ohira-Kawamura, H. Yuan, L. Wang, J.-W. Mei, H.-J. Liao, T. Xiang, L. Wu, and Z. Wang, Possible Observa- tion of Quadrupole Waves in Spin Nematics, Phys. Rev. Lett.135, 156...

    2025

  8. [15]

    Nishino and K

    T. Nishino and K. Okunishi, Corner Transfer Matrix Renormalization Group Method, J. Phys. Soc. Jpn.65, 891 (1996)

    1996

  9. [16]

    Or´ us, Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems, Phys

    R. Or´ us, Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems, Phys. Rev. B85, 205117 (2012)

    2012

  10. [17]

    Corboz, T

    P. Corboz, T. M. Rice, and M. Troyer, Competing States in thet-JModel: Uniformd-Wave State versus Stripe State, Phys. Rev. Lett.113, 046402 (2014)

    2014

  11. [18]

    Y. Liu, S. Shao, S. He, Z. Y. Xie, J.-W. Mei, H.-G. Luo, and J. Zhao, Quantum dynamics in a spin- 1 2 square lat- ticeJ 1−J2−δaltermagnet, Phys. Rev. B111, 245117 (2025)

    2025

  12. [19]

    Tan, J.-Y

    Y. Tan, J.-Y. Chen, D. Poilblanc, and J.-W. Mei, Gauge symmetry of excited states in projected entangled-pair state simulations, Phys. Rev. Res.6, 023102 (2024)

    2024

  13. [20]

    K. Wang, S. Feng, P. Zhu, R. Chi, H.-J. Liao, N. Trivedi, and T. Xiang, Fractionalization signatures in the dynam- ics of quantum spin liquids, Phys. Rev. B111, L100402 (2025)

    2025

  14. [21]

    J. Hu, R. Chi, Y. Guo, B. Normand, H.-J. Liao, and T. Xiang, Dynamical Spectral Function of the Kagome Quantum Spin Liquid (2025), to appear in Phys. Rev. Lett., arXiv:2512.18831

    2025

  15. [22]

    Zhong, S

    R. Zhong, S. Guo, and R. J. Cava, Frustrated magnetism in the layered triangular lattice materials K 2Co(SeO3)2 and Rb2Co(SeO3)2, Phys. Rev. Mater.4, 084406 (2020)

    2020

  16. [23]

    M. Zhu, V. Romerio, N. Steiger, S. D. Nabi, N. Mu- rai, S. Ohira-Kawamura, K. Y. Povarov, Y. Skourski, R. Sibille, L. Keller, Z. Yan, S. Gvasaliya, and A. Zhe- ludev, Continuum Excitations in a Spin Supersolid on a Triangular Lattice, Phys. Rev. Lett.133, 186704 (2024)

    2024

  17. [24]

    M. Zhu, L. M. Chinellato, V. Romerio, N. Murai, S. Ohira-Kawamura, C. Balz, Z. Yan, S. Gvasaliya, Y. Kato, C. D. Batista, and A. Zheludev, Wannier states and spin supersolid physics in the triangular antifer- romagnet K 2Co(SeO3)2, npj Quantum Mater.10, 74 (2025)

    2025

  18. [25]

    T. Chen, A. Ghasemi, J. Zhang, L. Shi, Z. Tagay, Y. Chen, L. Chen, E. S. Choi, M. Jaime, M. Lee, Y. Hao, H. Cao, B. L. Winn, A. A. Podlesnyak, D. M. Pajerowski, R. Zhong, X. Xu, N. P. Armitage, R. Cava, and C. Bro- holm, Phase diagram and spectroscopic signatures of a supersol...

    2026

  19. [27]

    Vanderstraeten, J

    L. Vanderstraeten, J. Haegeman, and F. Verstraete, Tangent-space methods for uniform matrix product states, Phys. Rev. B99, 165121 (2019)

    2019

  20. [28]

    See the Supplemental Material, which contains Refs. [44– 51], for detailed discussions of (S1) the GS-CTMRG method, (S2) the ET-CTMRG method, (S3) GS- and ET-CTMRG benchmarks for some paradigm quantum magnetic models, and (S4) further analysis of the spec- tral function for K ...

  21. [29]

    Eckart and G

    C. Eckart and G. Young, The Approximation of One Matrix by Another of Lower Rank, Psychometrika1, 211–218 (1936)

    1936

  22. [30]

    Mirsky, Symmetric gauge functions and unitarily in- variant norms, Q

    L. Mirsky, Symmetric gauge functions and unitarily in- variant norms, Q. J. Math.11, 50 (1960)

    1960

  23. [31]

    Wessel and M

    S. Wessel and M. Troyer, Supersolid Hard-Core Bosons on the Triangular Lattice, Phys. Rev. Lett.95, 127205 (2005)

    2005

  24. [32]

    R. G. Melko, A. Paramekanti, A. A. Burkov, A. Vish- wanath, D. N. Sheng, and L. Balents, Supersolid Order from Disorder: Hard-Core Bosons on the Triangular Lat- tice, Phys. Rev. Lett.95, 127207 (2005)

    2005

  25. [33]

    Heidarian and K

    D. Heidarian and K. Damle, Persistent Supersolid Phase of Hard-Core Bosons on the Triangular Lattice, Phys. Rev. Lett.95, 127206 (2005)

    2005

  26. [34]

    Boninsegni and N

    M. Boninsegni and N. V. Prokof’ev, Supersolid phase of hardcore bosons on a triangular lattice, Phys. Rev. Lett. 95, 237204 (2005)

    2005

  27. [35]

    Heidarian and A

    D. Heidarian and A. Paramekanti, Supersolidity in the Triangular Lattice Spin-1/2 XXZ Model: A Variational Perspective, Phys. Rev. Lett.104, 015301 (2010)

    2010

  28. [36]

    Sellmann, X.-F

    D. Sellmann, X.-F. Zhang, and S. Eggert, Phase diagram of the antiferromagnetic XXZ model on the triangular lattice, Phys. Rev. B91, 081104(R) (2015)

    2015

  29. [37]

    G. H. Wannier, Antiferromagnetism. The Triangular Ising Net, Phys. Rev.79, 357 (1950)

    1950

  30. [38]

    R. M. F. Houtappel, Order-disorder in hexagonal lattices, Physica16, 425 (1950)

    1950

  31. [39]

    Flores-Calder´ on, R

    R. Flores-Calder´ on, R. Moessner, and F. Pollmann, Un- conventional spin dynamics and supersolid excitations in the triangular-lattice XXZ model, Phys. Rev. B112, 184423 (2025)

    2025

  32. [40]

    Mauri, S

    A. Mauri, S. Roose, and F. Mila, Non-linear spin wave 7 theory in the strong easy-axis limit of the triangular XXZ model, Phys. Rev. B113, 174433 (2026)

    2026

  33. [41]

    A. Bose, K. Hart, R. Sutcliffe, and A. Paramekanti, Mod- ified large-Napproach to gapless spin liquids, magnetic orders, and dynamics: Application to triangular lattice antiferromagnets, Phys. Rev. B111, 214410 (2025)

    2025

  34. [42]

    Lin and T

    X. Lin and T. Shi, Pseudo-Goldstone Modes at Finite Temperature (2025), arXiv:2505.07229

    2025 arXiv

  35. [43]

    G. L. Squires,Introduction to the Theory of Thermal Neutron Scattering, 3rd ed. (Cambridge University Press, 2012)

    2012

  36. [44]

    Dalla Piazza, M

    B. Dalla Piazza, M. Mourigal, N. B. Christensen, G. J. Nilsen, P. Tregenna-Piggott, T. G. Perring, M. Enderle, D. F. McMorrow, D. A. Ivanov, and H. M. Rønnow, Frac- tional excitations in the square-lattice quantum antifer- romagnet, Nat. Phys.11, 62 (2014)

    2014

  37. [45]

    K. W. Plumb, A. T. Savici, G. E. Granroth, F. C. Chou, and Y. Kim, High-energy continuum of magnetic exci- tations in the two-dimensional quantum antiferromagnet Sr2CuO2Cl2, Phys. Rev. B89, 180410(R) (2014)

    2014

  38. [46]

    Powalski, G

    M. Powalski, G. S. Uhrig, and K. P. Schmidt, Roton Min- imum as a Fingerprint of Magnon-Higgs Scattering in Or- dered Quantum Antiferromagnets, Phys. Rev. Lett.115, 207202 (2015)

    2015

  39. [47]

    H. Shao, Y. Q. Qin, S. Capponi, S. Chesi, Z. Y. Meng, and A. W. Sandvik, Nearly Deconfined Spinon Excita- tions in the Square-Lattice Spin-1/2 Heisenberg Antifer- romagnet, Phys. Rev. X7, 041072 (2017)

    2017

  40. [48]

    M. B. Stone, I. A. Zaliznyak, T. Hong, C. L. Broholm, and D. H. Reich, Quasiparticle breakdown in a quantum spin liquid, Nature440, 187 (2005)

    2005

  41. [49]

    Verresen, R

    R. Verresen, R. Moessner, and F. Pollmann, Avoided quasiparticle decay from strong quantum interactions, Nat. Phys.15, 750 (2019)

    2019

  42. [50]

    J. A. Hern´ andez, A. A. Eberharter, M. Schuler, J. Lass, D. G. Mazzone, R. Sibille, S. Raymond, K. W. Kr¨ amer, B. Normand, B. Roessli, A. M. L¨ auchli, and M. Kenzel- mann, Field-Induced Magnon Decay, Magnon Shadows, and Roton-like Excitations in the Honeycomb Antiferro- mag...

    2025

  43. [51]

    H. Jia, B. Ma, Z. D. Wang, and G. Chen, Quantum spin supersolid as a precursory Dirac spin liquid in a triangular lattice antiferromagnet, Phys. Rev. Research 6, 033031 (2024). Supplemental Material to accompany the manuscript Optimized Tensor-Network Renormalization for Quant...

    2024

  44. [52]

    to compensate for the limits inN r. By contrast, this strategy cannot be applied to the TL-XXZ model for K 2Co(SeO3)2 because the instabilities are so severe, causing significant differences between the GS- and ET- CTMRG spectra even atN r <50 (Fig. 4 of the main text). S4. SP...

  45. [53]

    Here we note that, unlike for the Goldstone-mode energy, a systematic functional form in 1/Dfor a quantity extrapolating to a finite energy is not known

    Where the Goldstone-mode energy at K shows 1/D convergence to 0, the energy retains a robust finite value at the dips characterizing the M points. Here we note that, unlike for the Goldstone-mode energy, a systematic functional form in 1/Dfor a quantity extrapolating to a fini...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.