Skyrmions of Frustrated Quantum Dimer Systems

Cristian D. Batista; David Dahlbom; Fletcher Williams; Hao Zhang; Kipton Barros; Shruti Agarwal

REVIEW 3 major objections 4 minor 1 cited by

Skyrmions of Frustrated Quantum Dimer Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Frustrated isotropic Heisenberg dimers can host CP^3 skyrmion crystals at zero temperature, with no Dzyaloshinskii-Moriya or easy-axis anisotropy needed.

desk verdict Clear and useful extension of skyrmion physics to CP^3 in frustrated spin-dimer systems, with a plausible frustration-induced easy-axis mechanism, but the numerical evidence for the headline phases rests on a single 5x5 commensurate cell and needs finite-size validation. read the letter →

arxiv 2506.22320 v1 pith:OK7UWU24 submitted 2025-06-27 cond-mat.str-el

Fletcher Williams , David Dahlbom , Hao Zhang , Shruti Agarwal , Kipton Barros , Cristian D. Batista This is my paper · ORCID

classification cond-mat.str-el

keywords skyrmioncrystalsCP^3skyrmionsfrustratedmagnetismspindimersSU(4)coherentstatestriangularlatticeisotropicHeisenbergexchangezero-temperaturephasediagram

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 claims that skyrmion crystals can form at zero temperature in a model made only of isotropic Heisenberg exchanges and a magnetic field, provided the spins are arranged as antiferromagnetic dimers on a frustrated triangular lattice. The target space of the classical theory is $\mathbb{CP}^3$, the space of single-dimer quantum states spanned by one singlet and three triplet states, rather than the ordinary sphere $S^2$ of magnetization directions. Working in the classical limit defined by SU(4) coherent states, which keeps the quantum entanglement between the two spins of each dimer, the authors compute a zero-temperature phase diagram containing two field-induced $\mathbb{CP}^3$ skyrmion crystal phases, SkX-I and SkX-II. If correct, this shows that the usual requirement of Dzyaloshinskii-Moriya or easy-axis anisotropy for skyrmion stabilization can be replaced by exchange frustration alone, and that the resulting topological textures can pass through regions that are locally singlet, hence invisible to standard dipolar probes.

What carries the argument

The machinery is the SU(4)-coherent-state classical limit of the dimer Hilbert space, in which the local state of each dimer is represented by a 15-component color field $n^\mu_j$ living on the manifold $\mathbb{CP}^3$, a complex projective space whose second homotopy group $\Pi_2(\mathbb{CP}^3)\cong\mathbb{Z}$ classifies skyrmion textures. The classical Hamiltonian is a generalized Landau-Lifshitz dynamics for the color field, and the skyrmion charge is computed as a sum of Berry phases around triangular plaquettes. The load-bearing identity is that the low-energy projected Hamiltonian becomes an XXZ pseudospin-$1/2$ model with effective exchange anisotropy $\Delta = J_+/2J_-$; $\Delta>1$ (easy-axis) selects six ordering wave vectors whose magnitude is set by the ratio $J_-^2/|J_-^1| = 2/(1+\sqrt5)$, making the ordering wave vector $Q=1/5$ commensurate with a $5\times 5$ magnetic unit cell.

What would settle it

Recompute the classical SU(4) energy minimization for the same Hamiltonian at, say, $\alpha=0.5$ and field just below saturation using magnetic unit cells of size $L=6$, $L=7$, or $L=8$; if a single-Q spiral, a multi-domain state, or a different triple-Q state is found to be lower in energy than the SkX-II skyrmion crystal for any larger cell, the claim that the phase persists beyond the perturbative regime is false. A complementary experimental falsifier would be inelastic neutron scattering on a candidate material showing elliptical, rather than circular, Goldstone cones at the six ordering wave vectors.

Watch

Extended reading notes

Core claim

The central claim is that the classical limit of a frustrated spin-dimer bilayer on a triangular lattice, with only SU(2)-invariant Heisenberg couplings, supports two thermodynamically stable zero-temperature $\mathbb{CP}^3$ skyrmion crystal phases in an applied field. The SkX-I phase consists of a triangular lattice of fully polarized cores in a singlet background, while the SkX-II phase inverts the texture, with singlet cores surrounded by a polarized background. The effective low-energy pseudospin model derived in the weak-coupling limit shows that the competition between parallel and crossed isotropic inter-dimer exchanges generates an easy-axis exchange anisotropy $\Delta>1$, which is the known criterion for stabilizing skyrmion crystals in centrosymmetric frustrated magnets. The numerical phase diagram shows the SkX-II phase persisting for inter-dimer exchange comparable to the intra-dimer exchange ($\alpha\lesssim 0.5$), well outside the perturbative regime. The paper also argues that the same mechanism should apply to other hexagonal dimer lattices such as honeycomb and kagome.

Load-bearing premise

The numerical phase diagram rests on a single magnetic unit cell of linear dimension $L=5$, chosen to match the ordering wave vector $Q=1/5$, with no finite-size scaling or larger-cell checks reported.

Editorial extensions

If this is right

Skyrmion crystals at zero temperature no longer require Dzyaloshinskii-Moriya interactions or single-ion anisotropy; isotropic exchange frustration plus a field suffices.
Because the target space is $\mathbb{CP}^3$ rather than $S^2$, skyrmion textures can interpolate between a singlet state and a polarized triplet state, so the cores or backgrounds of the textures can be locally paramagnetic.
The SkX-II phase surviving to $\alpha\approx 0.5$ means the mechanism is not confined to weak inter-dimer couplings and could be realized in strongly coupled dimer materials.
Triple-$Q$ skyrmion phases can be distinguished from multi-domain single-$Q$ spirals by inelastic neutron scattering: the Goldstone-mode cones are circular in the triple-$Q$ phases and elliptical in single-$Q$ phases.
Stacked triangular-lattice dimer compounds with strong interlayer coupling are named as candidate platforms for realizing these $\mathbb{CP}^3$ skyrmions.

Reading between the lines

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

The paper does not test this, but recomputing the phase diagram with magnetic unit cells of linear dimension $L=6$, $L=7$, and $L=8$ would show whether the SkX-II phase and the triple-$Q$ minima survive the $L=5$ commensuration, or whether they are an artifact of pinning $Q=1/5$.
The same SU($N$)-coherent-state construction should, in principle, produce $\mathbb{CP}^{N-1}$ skyrmion crystals in any dimer or cluster system whose local Hilbert space has $N$ levels and whose effective low-energy model acquires easy-axis-like anisotropy; testing this on $N>4$ cases, such as systems with multiple singlet states, would generalize the result.
Because the SkX phases carry finite scalar spin chirality on each layer, optical probes of magnetochiral or magneto-optical response might detect them even where neutron diffraction shows no dipolar long-range order; the paper suggests magnetic circular dichroism but does not compute the expected signal strength.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things. The paper is a serious, carefully derived extension of the CP^2 skyrmion framework to CP^3 for coupled spin dimers, and the mechanism—frustration between parallel and crossed isotropic exchanges generating an effective easy-axis anisotropy—is plausible and clearly explained. The catch: the actual phase diagram that establishes the skyrmion crystals is computed only in a 5×5 commensurate cell, so the headline stability of SkX-II beyond the perturbative regime is not yet backed by finite-size evidence.

What's genuinely new: a concrete microscopic Hamiltonian (bilayer triangular lattice of dimers, isotropic Heisenberg only) whose classical SU(4) limit yields two field-induced CP^3 skyrmion crystal phases. The derivation of the classical limit and the mapping to the XXZ pseudospin model (Appendices B, D, F) is standard and competently done. The expression Δ = J_+/2J_- linking frustration to easy-axis anisotropy is elegant and gives a clear materials criterion. The spin wave spectra in Fig. 5 and Appendix H offer concrete, testable distinctions between single-Q and triple-Q order, which is a genuinely useful experimental guide.

The softest spot is the numerical evidence. The whole phase diagram of Fig. 2 comes from minimizing the SU(4) Hamiltonian in a 5×5 magnetic unit cell, with Q locked to 1/5 by the parameter choice immediately below Eq. (21). There is no report of larger cells, no variation of the cell size, no off-commensuration test. Skyrmion crystals are triple-Q states selected by an energy balance between single-, double-, and triple-Q candidates, and small commensurate cells are exactly the kind of constraint that can bias that balance. The SkX-II phase at α≲0.5 sits outside the perturbative regime where the low-energy pseudospin mapping is controlled, so the mechanism is plausible but not proven there. The code availability placeholder '**' also prevents independent reproduction, which matters for a claim resting entirely on numerical minimization. The sentence in the introduction claiming the conclusions hold for honeycomb and Kagome lattices is an overstatement—no evidence is given. These are gaps in evidence, not demonstrated errors.

The central logic of the paper holds up: the low-energy mapping is derived, not fitted; the skyrmion phases are found by minimizing the full Hamiltonian, not by imposing an ansatz; and the connection to known XXZ phase diagrams provides a consistency check. The circularity burden is low.

This paper is for theorists working on frustrated magnets and topological spin textures. A careful reader will get a clear mechanism and a useful set of predictions, but should treat Fig. 2 as provisional until larger-cell results appear. It deserves a serious referee; I would send it to review with a request for finite-size scaling and better code availability.

Referee Report

3 major / 4 minor

Summary. The paper studies the classical limit of a frustrated spin-1/2 dimer bilayer on a triangular lattice, using SU(4) coherent states that retain intra-dimer entanglement. The authors derive an effective low-energy pseudospin model, compute a zero-temperature phase diagram by numerical minimization of the classical Hamiltonian, and identify two field-induced CP^3 skyrmion crystal phases (SkX-I and SkX-II) that emerge from isotropic Heisenberg exchange interactions alone. They argue that SkX-II remains stable up to inter-dimer exchange comparable to the intra-dimer exchange (alpha ~ 0.5), and they compute spin-wave spectra and neutron scattering signatures to guide experimental detection.

Significance. If the phase diagram is correct, this would be the first demonstration that zero-temperature skyrmion crystals can be stabilized by isotropic exchange frustration alone, with a CP^3 target space rather than the usual CP^1. The paper's analytical framework is a clear strength: the classical limit based on SU(4) coherent states, the derivation of the effective pseudospin model (Appendix B), the continuum Hamiltonian (Appendix D), and the topological charge formula (Eq. 17-18) are all presented carefully and connect to established results. The predicted inelastic neutron scattering distinctions between single-Q and triple-Q orders (Figs. 5 and 11) are concrete and falsifiable. However, the central numerical claim rests on a single commensurate 5x5 magnetic unit cell, without finite-size scaling or off-commensuration checks, so the phase diagram must be treated as preliminary until that evidence is strengthened.

major comments (3)

[Section V and Figure 2] The entire phase diagram is computed with a single magnetic unit cell of linear dimension L=5, made commensurate with the ordering wavevector by setting J^-_2/|J^-_1| = 2/(1+sqrt(5)) in Eq. (21). No finite-size scaling, larger-cell calculation, or off-commensuration test is reported. Because the allowed wavevectors are locked to multiples of 1/5, competing phases with incommensurate or slightly different Q cannot be represented, and skyrmion crystals are triple-Q states whose stability depends precisely on the energy balance among single-Q, double-Q, and triple-Q candidates. The claim that SkX-II persists up to alpha ~ 0.5 (Section III.A) is therefore not yet established for the infinite-lattice Hamiltonian; it may be a commensuration artifact. The authors should provide calculations for larger commensurate cells (e.g., L=6, 7, 8) and at least one off-commensuration point, or an energy comparison with incommensurate single-Q states, to support the robustness of the phase boundaries.
[Section III.A versus Section V] There is a technical inconsistency about what is actually minimized. Section III.A states that the phase diagram is obtained by 'numerically minimizing the classical spin Hamiltonian H_SU(4) given in Eq. (16)', which is the continuum Hamiltonian density, while Section V states that a 5x5 cell of SU(4) coherent states is optimized using 'the Hamiltonian Eq. (11)', which is the lattice Hamiltonian. Section III.A also mentions 'a finite lattice of up to 2L x 2L = 4L^2 dimers', which is ambiguous and does not match the 5x5 description in Section V. Please clarify which Hamiltonian and which lattice size were used, because this directly affects the interpretation of the finite-size evidence and the reproducibility of the results.
[Section VII (Code Availability)] The Code Availability section says the numerical code 'can be found at **', with a placeholder instead of a URL or repository identifier. Since the central result is a numerical phase diagram and the authors state that all data can be reproduced using that code, the placeholder blocks independent verification. The repository should be cited with a working link (e.g., DOI or persistent URL) before the manuscript is accepted.

minor comments (4)

[Section III.A] After Eq. (21), 'controls the validly of the perturbation theory' should be 'controls the validity of the perturbation theory'.
[Section III.B] Near Eq. (23), 'the dipersion of the magnon modes' should be 'the dispersion of the magnon modes'.
[Abstract] The abstract says the phase diagram is for 'weak inter-dimer coupling', but Section III.A states that SkX-II extends well beyond the perturbative regime to alpha ~ 0.5. Consider rephrasing to avoid an apparent contradiction.
[References] Reference [50] is a funding acknowledgment that appears in the reference list; it would be cleaner to place it as an unnumbered acknowledgment footnote rather than as a numbered reference.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the CP3 skyrmion crystal phases are obtained by unconstrained numerical minimization of the full SU(4) lattice Hamiltonian, not by imposing a skyrmion ansatz. The low-energy pseudospin model is derived, not fitted. Minor self-citations supply the coherent-state framework and numerical tools but are not load-bearing.

full rationale

I walked the claimed derivation chain and found no step in which a prediction reduces to its inputs by construction. The central result, the T=0 phase diagram of Fig. 2, is obtained by minimizing the lattice Hamiltonian H_SU(4) (Eq. 11) with random initial conditions and a dense grid of parameter points (Section V); the SkX-I and SkX-II phases emerge as energy minima rather than being inserted as an ansatz. The low-energy pseudospin model (Eq. 19) is derived in Appendix B by projecting the microscopic Hamiltonian onto the singlet/Sz=1 subspace, and it is used only to explain the mechanism and to guide parameter choice, not as the energy function for the phase diagram. The choice J^-2/|J^-1| = 2/(1+sqrt5) (Eq. 21) and L=5 fixes the ordering wavevector Q=1/5, but this is an explicit modeling constraint, not a fit whose output is then called a prediction; it does restrict incommensurate competitors, which is a finite-size correctness risk, not circularity. Self-citations to Zhang et al. (2023) and Dahlbom et al. (2024) provide the SU(4) coherent-state classical limit, the skyrmion-charge formula, and the numerical method, but these are not used to assert the existence of the skyrmion phases; the energy minimization is performed independently in this paper. The equivalence of the CP3 Berry flux to a solid angle on an S2 Bloch sphere (Appendix G) is a mathematical proof, not a circular reuse of the result. The placeholder code availability ('**') is a reproducibility gap but does not make the derivation circular. Overall: no load-bearing circularity; minor self-citations are present but the central claim has independent content.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the SU(4) coherent-state classical limit, the standard homotopy classification, and two hand-chosen model parameters (Delta=1.2 and the exchange ratio fixing Q=1/5). No new particles or forces are introduced. The most fragile input is the numerical assumption that a 5x5 commensurate cell captures the thermodynamic ground state.

free parameters (2)

Delta (effective exchange anisotropy) = 1.2
Chosen by hand ('we fix Delta=1.2') as a representative point in the skyrmion-stabilizing region. The central claim of two SkX phases is demonstrated only for this value, not for a general parameter range.
J_-2/|J_-1| exchange ratio = 2/(1+sqrt5)
Chosen so the ordering wavevector is Q=1/5, commensurate with the L=5 unit cell used in the numerical phase diagram. This choice may bias the stability of long-period triple-Q phases.

assumptions (5)

domain assumption The SU(4) coherent-state product ansatz |Z> = ⊗_j |Z_j> (Eq. 4) provides a valid classical limit for the quantum dimer model, so the ground state of the full quantum Hamiltonian is approximated by a product of such states.
Invoked in Section II to replace operators by expectation values and derive the classical Hamiltonian (Eq. 11). If intra-dimer entanglement beyond the coherent-state manifold or inter-dimer entanglement is essential, the phase diagram may not describe the quantum model.
standard math Second homotopy group Pi_2(CP^{N-1}) = Z for N >= 2, so mappings from the compactified plane to CP^3 carry integer topological charge.
Used in the Introduction to justify the existence of skyrmion textures in CP^3; standard algebraic topology fact.
standard math In the M -> infinity limit of completely symmetric SU(4) irreps, the generators commute to leading order and the factorization rule ⟨T^mu T^nu⟩ = ⟨T^mu⟩⟨T^nu⟩ holds.
Section II used to obtain the classical Landau-Lifshitz-like equation of motion (Eq. 14) and the color-field Hamiltonian; this is the standard large-N classical limit, but it is an approximation for finite S=1/2.
domain assumption The effective low-energy pseudospin model (Eq. 19) derived to first order in the inter-dimer couplings, with the field isolating the singlet and S_z=1 triplet states, captures the physics relevant for skyrmion stabilization.
Appendix B and Section II: used to identify Delta > 1 as a necessary condition, and to map parameters back to the microscopic Hamiltonian. The perturbative derivation assumes |J'_p,c,nu|/J << 1, but the full numerical phase diagram goes beyond this regime.
domain assumption The L=5 magnetic unit cell, made commensurate with the ordering wavevector Q=1/5 by the exchange ratio choice J_-2/|J_-1| = 2/(1+sqrt5), is large enough to capture the true ground state, including triple-Q skyrmion crystal phases.
Section V and below Eq. (21): the numerical phase diagram is computed only with this cell; larger-cell calculations are not reported, so the stability of the SkX phases, especially SkX-II at large alpha, is contingent on this assumption.

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Cite this review

Pith. "Pith review of Skyrmions of Frustrated Quantum Dimer Systems." pith.science (2026). https://pith.science/paper/OK7UWU24

@misc{pith2026250622320,
  author       = {Pith},
  title        = {Pith review of: Skyrmions of Frustrated Quantum Dimer Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OK7UWU24}},
  note         = {Machine review of arXiv:2506.22320}
}

abstract

Magnetic skyrmions are topologically protected solitons observed in various classes of real magnets. In two-dimensional systems, where the target space of local magnetization values is the two-sphere $S^2$, skyrmion textures are classified by the homotopy classes of two-loops $S^2$ in $S^2$: $\Pi_2(S^2) \cong Z$. Here, we demonstrate that more general topological skyrmion textures emerge in the classical limit of quantum dimer systems, where the phase space of the relevant classical theory is $\mathbb{CP}^{N-1}$ (with $N=4$ for the case of interest), because the relevant second homotopy group, $\Pi_2(\mathbb{CP}^{N-1}) \cong Z$ for $N\geq 2$, remains unchanged. Building on the framework established by Zhang et al. (2023), we consider a classical limit based on SU(4) coherent states, which preserve intra-dimer entanglement. We show that the zero-temperature phase diagram of frustrated spin-dimer systems on a bilayer triangular lattice with weak inter-dimer coupling includes two magnetic-field-induced $\mathbb{CP}^{3}$ skyrmion crystal phases.

Figures

Figures reproduced from arXiv: 2506.22320 by the authors.

**Figure 1.** Exchange interactions. Intra-site and inter-site exchange interactions up to second nearest neighbor on a triangular lattice. of each dimer. We begin by assuming that the many-body quantum state is a direct product of SU(4) coherent states [31–38]: |Z⟩ = ⊗𝑗 |Z𝑗⟩. (4) SU(4) coherent states are chosen since they are the simplest states capable of completely accounting for the entanglement between two 𝑆 = 1/2 spins. W… view at source ↗

**Figure 2.** 𝑻 = 0 phase diagram of the classical Hamiltonian H as a function of 𝛼 and the external field 𝐵. The ordering wave vector was fixed by setting 𝐽 − 2 /|𝐽 − 1 | = 2/(1 + √ 5). Additionally, Δ, the effective anisotropy of the low-energy pseudospin model, was set to Δ = Δ1 = Δ2 = 1.2, where Δ > 1 corresponds to an effective easy-axis anisotropy. The triplon dispersion is shown as an inset in the QPM phase, with the six … view at source ↗

**Figure 3.** Triple-𝑸 skyrmions and their single-𝑸 cross sections. a, b. Real space distribution of the dipolar sector and c. static structure factor for CP3 single-𝑸 spirals. d, e. Real space distribution of the dipolar sector and f. triple-𝑸 static structure factor for individual CP3 skyrmions from the phases SkX-I and SkX-II, respectively. Bottom Layer Top Layer Top Layer Bottom Layer SkX-I SkX-II -1.5 -1.0 -0.5 0.0 0.5 1.0 1… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Real space distribution of the dipolar sector of the CP3 skyrmion crystals SkX-I and SkX-II. The length of the arrow represents the magnitude of the dipole moment, |⟨Sˆ 𝜎 𝑗 ⟩| = √︃ (𝑛 1 𝑗 ) 2 + (𝑛 2 𝑗 ) 2 + (𝑛 3 𝑗 ) 2 . The color scale of the arrows indicates ⟨𝑆ˆ 𝑧 𝜎 𝑗…

**Figure 5.** Figure 5: Magnetic excitations of the YZ-Spiral and SkX-I Phases. a. The dynamical spin structure factor (DSSF) of a system in the YZ-Spiral phase calculated with linear spin wave theory (LSWT) at the point (𝛼, 𝐵) = (0.1,0.865) of the phase diagram of [PITH_FULL_IMAGE:figures/f…

**Figure 6.** Figure 6: Zero temperature phase diagram of low-energy Hamiltonian ℋ˜. The diagram is computed in terms of pseudo-magnetic field ℎ and effective anisotropy Δ. FM is ferromagnetic phase; SkX is the skyrmion crystal; VS is the vertical spiral; CS is the conical spiral; FL is the f…

**Figure 7.** Figure 7: Real space distribution of the top layer dipolar sector of the three CP3 single-𝑸 orderings. a. AFM-CS. b. YZ-S. c. FM-CS. The length of the arrow represents the magnitude of the dipole moment of the color field |⟨Sˆ + 𝑗 ⟩| = √︃ (𝑛 1 𝑗 ) 2 + (𝑛 2 𝑗 ) 2 + (𝑛 3 𝑗 ) 2 . T…

**Figure 8.** Figure 8: Real space distribution of the top layer dipolar sector of the three CP3 double-Q orderings. a. 2Q-I. b. 2Q-II. c. 2Q-III. The length of the arrow represents the magnitude of the dipole moment of the color field |⟨Sˆ + 𝑗 ⟩| = √︃ (𝑛 1 𝑗 ) 2 + (𝑛 2 𝑗 ) 2 + (𝑛 3 𝑗 ) 2 . T…

**Figure 9.** Figure 9: Real space distribution of the top layer dipolar sector of the two CP3 triple-𝑸 orderings. a. 3Q-I. b. 3Q-II. The length of the arrow represents the magnitude of the dipole moment of the color field |⟨Sˆ + 𝑗 ⟩| = √︃ (𝑛 1 𝑗 ) 2 + (𝑛 2 𝑗 ) 2 + (𝑛 3 𝑗 ) 2 . The color scal…

**Figure 10.** Figure 10: Magnetic excitations of the AFM-CS, FM-CS, 3Q-II, and SkX-II phases. Each pair of panels shows the 𝑞𝑧 = 0 (left) and 𝑞𝑧 = 1 channels (right) of the dynamical spin structure factor (DSSF) calculated with linear spin wave theory from different phases of the phase diagra…

**Figure 11.** Figure 11: Dispersion cross sections at 𝝎 = 0.012𝑱. a. Dynamical spin structure factor (DSSF) for (𝛼, 𝐵) = (0.1,0.8654), in the YZ-Spiral phase. Note that the cross sections at the ordering wave vectors are elliptical. Even in the case of multi-domain single-𝑸 such elliptical cr…

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This manuscript has been authored by UT-Battelle, LLC, un- der contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, ...
[52]

(19) and (20), and explicitly map its param- eters back to those of the original Hamiltonianˆℋ

Mapping the pseudospin model to the full Hamiltonian Wenowfocusonthelow-energyeffectivepseudospinmodel defined by Eqs. (19) and (20), and explicitly map its param- eters back to those of the original Hamiltonianˆℋ. This is achieved by inverting Eqs. (20) and (21), while keepin...
[53]

2cos 𝑘𝑥𝑎 2 cos √ 3𝑘𝑦𝑎 2 ! + cos(𝑘𝑥𝑎) # (B5) + 𝐽− 2

Momentum Space Hamiltonian Next,werewritethepseudospinHamiltonian ˜ℋ inmomen- tum space by Fourier transforming the pseudospin operators: ˜𝝉𝒌 = 1√ 𝑁 ∑︁ 𝑗 𝑒𝑖𝒌·𝒓 𝑗 𝝉𝑗, 𝝉𝑗 = 1√ 𝑁 ∑︁ 𝒌 𝑒−𝑖𝒌·𝒓 𝑗 ˜𝝉𝒌. (B3) Note that ˜𝝉∗ 𝒌= ˜𝝉−𝒌 because 𝝉𝑗 is real𝝉𝑗 = 𝝉∗ 𝑗. The expression for the pse...
[54]

Note that −(𝐽− 1− 27𝐽−

cos6𝜙 𝑘6 + O(𝑘8), where 𝒌 = 𝑘(cos𝜙, sin𝜙). Note that −(𝐽− 1− 27𝐽−
[55]

Fromthispointon,wesetthelatticeconstant 𝑎astheunitof length (𝑎= 1)

> 0, implying that the six vectors±𝑸𝜈 (𝜈 = 1,2,3) that mini- mize the dispersion relation have values of𝜙=±𝜙𝜈 with 𝜙𝜈=−𝜋/6+ 2𝜈𝜋/3. Fromthispointon,wesetthelatticeconstant 𝑎astheunitof length (𝑎= 1). Equation (B6) shows that the anisotropy of the hexagonal lattice first appears...
[56]

2 features three distinct single-𝑸 ordered phases

Single-Q Spiral Phases Fig. 2 features three distinct single-𝑸 ordered phases. The largest of these is the AFM-CS phase (antiferromagnetic con- ical spiral; Fig. 7a), in which the spins rotate within the𝑥𝑦- plane. In this phase, the dipole moment exhibits minimal modulation al...
[57]

2, two of which (Fig

Double-Q Phases Therearethreedifferentdouble- 𝑸 orderingsthatareshown in Fig. 2, two of which (Fig. 8a,b) only appear in the non- perturbative regime. The third (Fig. 8c) only appears in a very narrow region within the perturbative regime and has al- most zero magnitude of dip...
[58]

CP1 Skyrmion flux

Asymmetric Triple-Q Phases Two distinct asymmetric triple-𝑸 spiral phases appear in Fig. 2, both characterized by a dominant contribution from one of the three ordering wave vectors𝑸𝜈 (𝜈= 1,2,3) [see Fig. 9]. In the 3Q-I phase [Fig. 9(a)], the longitudinal spin components exhi...

2000

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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