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Emergence of correlation-driven altermagnetism in Hubbard model on geometrically frustrated lattice-clusters

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Adding a single hole or electron to a 3×3 frustrated Hubbard torus creates altermagnetic spin order that grows with on-site repulsion, without spin–orbit coupling or engineered hopping anisotropy.

desk verdict The ED work is real but the central altermagnetic diagnostic is undefined (eta_ij is never specified), so the main claim cannot be checked; with A reported near zero, the interpretation is especially shaky. read the letter →

arxiv 2603.00536 v2 pith:2GHJHIT4 submitted 2026-02-28 cond-mat.str-el

Md Fahad Equbal , M. A. H. Ahsan This is my paper · ORCID
classification cond-mat.str-el
keywords altermagnetismelectroniccorrelationsgeometricfrustrationHubbardmodelexactdiagonalizationdopedMottinsulatord-wavespinparticle-holeasymmetry
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 claims that altermagnetism—a collinear magnetic state with zero net magnetization but momentum-dependent spin splitting—can arise purely from electron–electron repulsion on a symmetric, geometrically frustrated lattice. Using exact diagonalization of the Hubbard model on a 3×3 torus, the authors find that local moment formation alone is not enough: at half-filling the average altermagnetic response ⟨Δ_spin⟩ is tiny, but doping by one hole or electron (in the S=0 sector) produces a nonzero ⟨Δ_spin⟩ that increases monotonically with U. The response has a d-wave-like sign pattern in real-space spin correlations, a broad momentum-space structure factor, and a strong electron–hole asymmetry favoring electron doping. The same correlations vanish on an unfrustrated 2×4 cylinder, establishing geometric frustration as a necessary condition, while nearest-neighbor repulsion V acts as a destabilizing control knob. If correct, this establishes a fluctuation-mediated, correlation-only route to altermagnetism on symmetric frustrated lattices, with carrier concentration and non-local Coulomb interactions as tunable parameters.

What carries the argument

The core diagnostic is the altermagnetic correlation matrix Ω_ij = ⟨S_i·S_j⟩ η_ij C_i (Eq. 2), where η_ij is a real-space staggered sign structure meant to encode altermagnetic (d-wave-like) symmetry and C_i is a local compensation factor that suppresses spin-polarized contributions. From Ω_ij the authors construct the site-resolved response Δ^spin_i, the global average ⟨Δ_spin⟩, and the anisotropy parameter A comparing x- and y-bond correlations. This projection, applied to exact equal-time spin correlators obtained in spin-adapted exact diagonalization, is the only order parameter used to claim altermagnetic order. The paper does not specify η_ij explicitly, making the projection the load-

What would settle it

Compute ⟨Δ_spin⟩ on the same 3×3 cluster with η_ij set to +1 on every bond (no staggered projection). If the doped S=0 sectors still show a U-dependent nonzero signal, the response is not an artifact of the η_ij choice; if it vanishes, the altermagnetic claim rests entirely on the hand-picked sign structure. Alternatively, check the ground-state spin by diagonalizing in all S sectors: if a higher-spin state is lower in energy for N=8 or N=10, the reported ⟨Δ_spin⟩ is computed in an excited sector.

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Extended reading notes

Core claim

In exact diagonalization of the one-band Hubbard model on a fully frustrated 3×3 torus, the spin-0 doped sectors (N=8 holes and N=10 electrons) develop an altermagnetic spin texture whose average diagnostic ⟨Δ_spin⟩ grows monotonically with U and saturates at strong coupling, while the half-filled S=1/2 sector shows large local moments but negligible ⟨Δ_spin⟩. The doped states display a d-wave-like staggered sign pattern in the real-space correlation matrix Ω_5j, a broad distribution in the altermagnetic structure factor S_alm(q), and particle–hole asymmetry with stronger response for electron doping. The correlations vanish on the unfrustrated 2×4 cylinder, indicating geometric frustration

Load-bearing premise

The paper's results rise or fall on Eq. (2), where η_ij — the real-space sign pattern that encodes altermagnetic symmetry — is never specified; if η_ij is chosen to match a d-wave form factor, then a nonzero ⟨Δ_spin⟩ partly measures that choice rather than an emergent order, and the paper also assumes the lowest-total-spin sector is the ground-state sector for each filling.

Editorial extensions

If this is right

  • At half-filling on the 3×3 torus, local moment formation alone does not produce altermagnetic correlations; a mobile carrier must be present for ⟨Δ_spin⟩ to grow with U.
  • The altermagnetic response in the doped S=0 sectors is d-wave-like in real space and appears as a broad modulation in the momentum-space structure factor S_alm(q).
  • Electron doping yields a stronger altermagnetic response than hole doping, establishing a particle–hole asymmetry in this frustrated lattice.
  • On the unfrustrated 2×4 cylinder the altermagnetic correlations are zero, so geometric frustration is a necessary ingredient; on the partially frustrated 2×3 cylinder, V can promote altermagnetic order beyond a critical threshold.
  • Nearest-neighbor repulsion V monotonically weakens the 3×3 altermagnetic correlations; the electron-doped sector undergoes a first-order-like collapse at V≈1.8 for U=4 and at V≈4.9 for U=10, while the hole-doped sector remains more robust.

Reading between the lines

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

  • Because Eq. (2) projects spin correlations through a hand-chosen sign pattern η_ij, a cleaner test of genuine altermagnetism would be to extract a d-wave form factor directly from the raw spin–spin correlators without the η_ij weighting, or to compute the spin-resolved spectral function on the same cluster.
  • The nine-site result suggests that on larger frustrated clusters (e.g., 4×4 with flux, triangular lattices, or other odd-site tori) the single-carrier doped S=0 sector may also show enhanced altermagnetic correlations; larger-scale exact diagonalization or tensor-network studies could test whether the effect survives at thermodynamic sizes.
  • The observed particle–hole asymmetry may be a finite-size artifact of the 3×3 torus's Fermi surface; a comparison on a bipartite frustrated lattice with particle–hole symmetric dispersion could separate intrinsic asymmetry from cluster-geometry effects.
  • The report that anisotropy A is nonzero only for degenerate ground states implies that, on finite clusters, altermagnetic symmetry breaking is tied to degeneracy; in the thermodynamic limit this could mean the phase is stabilized only near a frustration-driven quantum critical point, possibly observable in specific heat or magnetic susceptibility measurements.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper reports exact-diagonalization (ED) studies of the single-band and extended Hubbard model on a 3×3 periodic (toroidal) cluster at filling N=8, 9, 10, with on-site U up to 12 and nearest-neighbor V up to U/2. The authors define a site-resolved 'altermagnetic spin-response' Ω_ij = ⟨S_i·S_j⟩ η_ij C_i (Eq. 2), an averaged response ⟨Δ_spin⟩ (Eq. 4), the local moment m̄ (Eq. 5), and an anisotropy A (Eq. 6). The central claims are: (i) at half-filling the local moments are large but the altermagnetic response is small; (ii) in the S=0 doped sectors ⟨Δ_spin⟩ increases with U and with the degree of geometric frustration; (iii) a first-order-like transition occurs at V≈1.8 for N=10 at U=4; and (iv) these results establish a correlation-driven altermagnetic phase. The abstract additionally claims results for 2×3 and 2×4 cylinders, an altermagnetic structure factor S_alm(q), and a degeneracy–A relation, none of which appear in the body.

Significance. If substantiated, the claim that purely electronic correlations on a symmetric, frustrated lattice produce a compensated, anisotropic magnetic state without spin-orbit coupling or engineered hopping anisotropy would be a noteworthy contribution, especially because ED is an unbiased method for small clusters. The machine-checkable nature of ED and the explicit U- and V-dependence are strengths. However, the central diagnostic is not fully specified, the abstract promises results not contained in the text, and the ground-state sector is not verified. These issues currently prevent the reader from assessing whether the reported ⟨Δ_spin⟩ measures an emergent altermagnetic order or an artifact of the chosen projection.

major comments (4)
  1. [Eq. (2), Sec. II] The sign structure η_ij is never defined. The text only states that it is 'a real-space staggered sign structure reflecting altermagnetic symmetry.' Because ⟨Δ_spin⟩ is a linear projection of the spin-correlation matrix onto η_ij, the numerical result cannot be reproduced, and the physical interpretation is ambiguous. The manuscript must give an explicit formula (e.g., η_ij = +1 on x-directed bonds and −1 on y-directed bonds, or a translation-invariant pattern) and show that the U-dependence of ⟨Δ_spin⟩ is robust under alternative sign conventions. Without this, the central claim risks being an artifact of the projection.
  2. [Abstract vs. body] The abstract and introduction advertise results on the partially frustrated 2×3 cylinder and unfrustrated 2×4 cylinder, the altermagnetic structure factor S_alm(q), and the statement that A is nonzero only for degenerate ground states. None of these appear in the full text. In particular, the claim that geometric frustration is a 'fundamental prerequisite' relies on the 2×4 comparison, which is absent. The authors should either add the missing results or rewrite the abstract to match the body.
  3. [Sec. II, Sec. III.A] The calculations are restricted to the lowest-total-spin sector (S=1/2 for N=9, S=0 for N=8,10) with the justification that these are magnetically compensated, but the paper never demonstrates that these sectors contain the actual ground state. If the ground state has a different total S, then claims about 'ground-state' correlations and phase transitions are unsupported. The authors should report the ground-state spin quantum number for each (U,V,N) and reconcile their sector choice with it.
  4. [Eq. (6), Figs. 2 and 5, Sec. III.A] There is an apparent tension between the reported A≈0 in the doped sectors and the claimed 'd-wave-like alternating sign structure.' If η_ij were the natural d-wave choice (opposite signs on x- and y-directed bonds), then a state with A=0 would require a specific relation between x- and y-bond correlations; the text does not explain how a nonzero ⟨Δ_spin⟩ coexists with A=0 under the adopted η. The paper must specify the transformation properties of η under C4 and show explicitly how the data satisfy the relation between A, ⟨Δ_spin⟩, and η. As written, the interpretation of the doped sectors as 'rotationally symmetric altermagnetic' is unclear.
minor comments (4)
  1. [General] The manuscript contains numerous typos and stylistic issues ('Hubbar d model', 'career dopings', 'It therefore serves', 'identifies' vs 'identify' in the abstract). A careful proofreading pass is needed.
  2. [Figures] The heatmaps (Figs. 4, 8, 12) use colored matrices, but the color scales are sometimes unreadable and the sign of Ω_5j is not explained in relation to η_ij. The figure captions should state whether the plotted quantity is the raw correlation or the projected Ω_5j, and should give the explicit η pattern used.
  3. [Sec. II, Eq. (2)] The compensation factor C_i is defined as C_i=1−2|⟨n_i↑⟩−⟨n_i↓⟩|. For a singly occupied site with ⟨n_i↑⟩=1, ⟨n_i↓⟩=0, C_i=0, so the factor suppresses polarized sites. The text states C_i≈1 in the lowest spin sectors; this should be verified numerically or at least stated as an assumption.
  4. [Conclusions] The conclusion refers to 'spin-sector selectivity' as a central finding, but this is not demonstrated because only one spin sector per filling is studied. The paper would be stronger if a comparison with higher-spin sectors were shown.

Circularity Check

1 steps flagged · score 6.0 of 10

The central altermagnetic signal is generated by the unspecified η_ij projection in Eq. (2), so the claimed d-wave staggered pattern is defined into the diagnostic.

  1. self definitional [Sec. II, Eqs. (2)-(4); Sec. III.A, Fig. 4 discussion]
    "To diagnose altermagnetic correlations beyond conventional magnetic order, we introduce a site-resolved altermagnetic correlation matrix constructed from equal-time spin-spin correlation Ωij = ⟨Si · Sj⟩ηijCi ... ηij = ±1 is a real-space staggered sign structure reflecting altermagnetic symmetry ... At strong coupling (U = 12), the matrix elements Ω5j clearly distinguish between x- and y-directed bonds, forming a pattern consistent with d-wave symmetry."

    ⟨Δ_spin⟩ is defined as the average over sites of Σ_j ⟨S_i·S_j⟩ η_ij C_i. Thus the observable is a linear projection of the spin correlations onto an undisplayed sign pattern η_ij that the paper itself describes as 'reflecting altermagnetic symmetry.' The heatmaps presented as microscopic confirmation of altermagnetism are plots of Ω_5j = ⟨S_5·S_j⟩η_5j C_5, so any staggered or d-wave-looking structure they show is put in by the η factor. Since η_ij is never specified, the nonzero ⟨Δ_spin⟩ and its U-growth are not independent evidence of altermagnetism; they are consequences of having projected onto an altermagnetic-style pattern. The paper neither gives η_ij nor shows that the raw ⟨S_i·S_j⟩ matrix (without the projection) has the same alternating structure, so the central finding reduces to

full rationale

The derivation chain is short: ED ground states of the Hubbard model are computed, and Eqs. (2)-(4) convert them into ⟨Δ_spin⟩. There is no fitting of parameters to the target quantity, and self-citations (refs 6, 10, 53, 54) are methodological/contextual rather than load-bearing. However, the altermagnetic conclusion rests entirely on a projection whose sign structure η_ij is not stated. A diagnostic that multiplies correlations by a pre-chosen staggered sign structure and then reports that the correlations have that staggered sign structure is self-definitional unless η_ij is specified a priori and/or raw correlations are shown to exhibit the same pattern. The reported near-zero A in the doped S=0 sectors further compounds the ambiguity, because the role of η_ij in separating x/y anisotropy is never clarified. The ED computation itself is real, but the claim that the result is an altermagnetic phase is, as written, a property of the chosen (undisclosed) projection. This is a partial circularity of the central measure rather than a fitted prediction or self-citation chain, so a score of 6 is appropriate.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The exact ED calculation has no fitted Hubbard parameters (U,V are physical); the main hand-picked element is the undefined η_ij sign pattern. The paper also assumes spin-sector selection and finite-size representativeness without verification.

free parameters (1)
  • η_ij staggered sign pattern = unspecified (±1 per bond)
    Eq. (2) introduces η_ij as ±1 to define Ω_ij; the pattern is never specified, yet it controls ⟨Δ_spin⟩ and A.
assumptions (4)
  • domain assumption The 3x3 odd torus with PBC is a geometrically frustrated lattice suppressing bipartite Néel order.
    Section II: 'the combination of PBC and odd number of site frustrates simple bipartite Néel order'.
  • domain assumption The ground state lies in the lowest total-spin sector (S=1/2 for N=9, S=0 for N=8,10).
    Section II restricts calculations to this sector without demonstrating it is the ground-state sector.
  • ad hoc to paper Projection of equal-time spin correlations onto an unspecified sign pattern η_ij is a valid altermagnetic diagnostic.
    Eqs. (2)-(4): the central measure is defined by this projection; no symmetry derivation from altermagnetic order parameter is given.
  • domain assumption Equal-time correlations on a 9-site cluster indicate a phase relevant to the thermodynamic limit.
    All conclusions about 'altermagnetic order' are drawn from ED on a single small cluster with no finite-size scaling.

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Pith. "Pith review of Emergence of correlation-driven altermagnetism in Hubbard model on geometrically frustrated lattice-clusters." pith.science (2026). https://pith.science/paper/2GHJHIT4

@misc{pith2026260300536,
  author       = {Pith},
  title        = {Pith review of: Emergence of correlation-driven altermagnetism in Hubbard model on geometrically frustrated lattice-clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2GHJHIT4}},
  note         = {Machine review of arXiv:2603.00536}
}
read the original abstract

We investigate the emergence of correlation-driven altermagnetism in simple and extended Hubbard model on geometrically frustrated lattice-clusters using exact diagonalization. By systematically tuning the degree of geometric frustration across different cluster geometries -- including the fully frustrated 3x3 torus, the partially frustrated 2x3 cylinder and the unfrustrated 2x4 cylindrical lattice -- we isolate the necessary conditions for compensated anisotropic spin order. At half-filling on 3x3 lattice, the average altermagnetic spin-response <{\Delta}_spin> remains small despite robust local moment formation, indicating that localized moments alone are insufficient to break directional symmetry. In contrast, the introduction of a single mobile charge carrier (hole or electron) induces a finite <{\Delta}_spin> that increases monotonically with on-site interaction U . This altermagnetic phase is characterized by a d-wave-like alternating sign structure in real-space correlations, a broad momentum-space distribution in the altermagnetic structure factor S_alm(q) and a distinct particle-hole asymmetry with significantly enhanced response for one-electron-doped system. We demonstrate that these correlations remain zero on the unfrustrated 2x4 lattice, establishing geometric frustration as a fundamental prerequisite. The inclusion of nearest-neighbor Coulomb repulsion V weakens the altermagnetic correlations in 3x3 lattice through enhanced electronic localization but facilitates the onset of altermagnetic order in 2x3 lattice beyond a critical threshold. Finally, we show that the anisotropy parameter A is non-zero only for degenerate ground states, revealing that macroscopic symmetry breaking on finite clusters requires a confluence of geometric frustration and ground-state degeneracy.

Figures

Figures reproduced from arXiv: 2603.00536 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic representation of the frustrated square l [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Average altermagnetic spin-response measure [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Average local moment ¯m [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Anisotropy parameter [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Average altermagnetic spin-response measure [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Average local moment ¯m [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Anisotropy parameter [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Average local moment ¯m [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Anisotropy parameter [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]

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Reference graph

Works this paper leans on

81 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [1]

    This minimal toroidal geometry retains full quantum coherence while incorporat- ing geometric frustration and non-trivial loop connectivity. By systematically analyzing half-filling and single-carrier (hole or electron) doped sectors ( N = 8 , 9, 10), where N represents number of electrons, we isolate how carrier itin- erancy modifies local moment formation...

    arXiv 2026

  2. [2]

    In the one-hole-doped system (top row, Fig

    In contrast, the doped systems exhibit highly struc- tured and anisotropic correlation patterns with increasing U . In the one-hole-doped system (top row, Fig. 4), the cen- tral site shows strong positive self-correlation, while corre- lations with its NN along the cardinal axes display distinct variations in both magnitude and sign. The one-electron- dop...

  3. [3]

    This reflects a univer- sal crossover from a weakly correlated metallic regime to a regime of well-defined local moments as the energetic cost for double occupancy increases

    For all fillings considered ( N = 8 , 9, 10), ¯m increases monotonically with U . This reflects a univer- sal crossover from a weakly correlated metallic regime to a regime of well-defined local moments as the energetic cost for double occupancy increases. At half-filling ( N = 9), ¯m approaches its theoretical maximum most rapidly, indicat- ing a robust Mott...

  4. [4]

    At strong coupling (U = 12), the matrix elements Ω 5j clearly distinguish be- tween x- and y-directed bonds, forming a pattern consistent with d-wave symmetry

    develops an even more pronounced staggered pattern. At strong coupling (U = 12), the matrix elements Ω 5j clearly distinguish be- tween x- and y-directed bonds, forming a pattern consistent with d-wave symmetry. The spatial heatmaps thus provide microscopic confirmation of the trends observed in ⟨∆spin⟩. N=8, U=2 1 2 3 1 2 3y-direction 0 0.005 0.01 0.015 0...

  5. [5]

    U=4 case To assess the stability of the interaction-driven magnetic phases against non-local correlations, we examine the aver- age altermagnetic spin-response measure ⟨∆spin⟩ as a func- tion of the NN Coulomb interaction V at fixed intermediate coupling U = 4, shown in Fig

  6. [6]

    For the half-filled sys- tem ( N = 9, S = 1 /2), ⟨∆spin⟩ remains consistently low for all V values, indicating that NN interactions do not alleviate the geometric frustration inherent in the 3 × 3 lat- tice at half-filling; instead, V further suppress the already negligible staggered spin correlations. In contrast to the half-filled case, the doped systems (...

  7. [7]

    At half-filling, the average local moment ¯m decreases grad- ually as V increases indicating that NN repulsion does not induce any sharp magnetic transition in this sector

    This quantity allows us to distin- guish between changes in the spatial organization of spins and the actual magnitude of local magnetic polarization. At half-filling, the average local moment ¯m decreases grad- ually as V increases indicating that NN repulsion does not induce any sharp magnetic transition in this sector. In- stead, the monotonic suppressi...

  8. [8]

    At half-filling (middle-row, Fig

    These maps provide a site-resolved perspective on how the NN interaction V al- ters the staggered spin-correlation patterns established by U . At half-filling (middle-row, Fig. 8), the spatial cor- relations remain consistently fragmented and weak across the entire range of V . The matrix elements Ω 5j remain in the 10 −3 range, and the heatmaps show no di...

Show all 81 references
  1. [9]

    However, at V = 2, the heatmap undergoes a noticeable transformation

    sector displays a clear and structured altermagnetic pattern at V = 0 .2 and V = 1. However, at V = 2, the heatmap undergoes a noticeable transformation. The sharp, staggered inten- sity observed at lower V is replaced by a significantly more diffuse and lower-magnitude distribu...

  2. [10]

    U=10 case We now examine the evolution of the average altermag- netic spin-response measure ⟨∆spin⟩ as a function of NN interaction V at strong coupling U = 10, shown in Fig

  3. [11]

    At half-filling (N = 9), ⟨∆spin⟩ remains very small across the entire range 0.1 ⩽ V ⩽ 5

    In this regime, local moments are well developed and the system approaches the Heisenberg limit, allowing us to assess the robustness of carrier-mediated magnetic correla- tions against non-local Coulomb repulsion. At half-filling (N = 9), ⟨∆spin⟩ remains very small across the ...

  4. [12]

    At half-filling ( N = 9), the local moment is strongly enhanced compared to the intermediate-coupling case and remains close to saturation for small to moder- ate V

    This quantity directly probes the degree of local spin polarization and allows us to assess how robust the mag- netic moments remain when non-local Coulomb repulsion is introduced. At half-filling ( N = 9), the local moment is strongly enhanced compared to the intermediate-coup...

  5. [13]

    The central site (site 5) is used as the reference to visualize how NN repulsion alters the spin-correlation texture. At half-filling ( N = 9, mid- dle row), the correlations remain extremely small across all values of V , with magnitudes confined to the 10 −3-10−4 N=8, U=10, V=...

  6. [14]

    At half-filling (N = 9), A remains very small, of order 10 −3, across the entire V -range

    This quan- tity probes the imbalance between correlations along the two lattice directions and therefore serves as a sensitive indicator of rotational symmetry breaking. At half-filling (N = 9), A remains very small, of order 10 −3, across the entire V -range. Although a gradua...

  7. [15]

    Dagotto, Correlated electrons in high-temperature su- perconductors, Rev

    E. Dagotto, Correlated electrons in high-temperature su- perconductors, Rev. Mod. Phys. 66, 763 (1994)

    1994

  8. [16]

    Dagotto, Complexity in strongly correlated electronic systems, Science 309, 257 (2005)

    E. Dagotto, Complexity in strongly correlated electronic systems, Science 309, 257 (2005)

    2005

  9. [17]

    Imada, A

    M. Imada, A. Fujimori, and Y. Tokura, Metal-insulator transitions, Rev. Mod. Phys. 70, 1039 (1998)

    1998

  10. [18]

    H. Park, K. Haule, and G. Kotliar, Cluster dy- namical mean field theory of the mott transition, Phys. Rev. Lett. 101, 186403 (2008)

    2008

  11. [19]

    Chatzieleftheriou, S

    M. Chatzieleftheriou, S. Biermann, and E. A. Stepanov, Local and nonlocal electronic correlations at the metal- insulator transition in the two-dimensional hubbard model , Phys. Rev. Lett. 132, 236504 (2024)

    2024

  12. [20]

    M. F. Equbal and M. A. H. Ahsan, Spin-resolved mott crossover and entanglement in the half-filled hubbard model, Annals of Physics 488, 170414 (2026)

    2026

  13. [21]

    C. C. Tsuei and J. R. Kirtley, Pairing symmetry in cuprate superconductors, Rev. Mod. Phys. 72, 969 (2000)

    2000

  14. [22]

    Fradkin, S

    E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Collo- quium: Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015)

    2015

  15. [23]

    Keimer, S

    B. Keimer, S. Kivelson, M. Norman, S. Uchida, and J. Zaa- nen, From quantum matter to high-temperature supercon- ductivity in copper oxides, Nature 518, 179 (2015)

    2015

  16. [24]

    M. F. Equbal and M. A. H. Ahsan, Binding of holes and competing spin-charge order in simple and extended hub- bard model on cylindrical lattice: An exact diagonalizatio n study, arXiv preprint arXiv:2512.10577 (2025)

    2025

  17. [25]

    Savary and L

    L. Savary and L. Balents, Quantum spin liquids: a review, Reports on Progress in Physics 80, 016502 (2016)

    2016

  18. [26]

    Broholm, R

    C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quan- tum spin liquids, Science 367, eaay0668 (2020) , https://www.science.org/doi/pdf/10.1126/science.aay0668

    2020 doi

  19. [27]

    Jiang and S

    H.-C. Jiang and S. A. Kivelson, High temperature su- perconductivity in a lightly doped quantum spin liquid, Phys. Rev. Lett. 127, 097002 (2021)

    2021

  20. [28]

    S. S. Salunke, M. A. H. Ahsan, R. Nath, A. V. Ma- hajan, and I. Dasgupta, Electronic structure of spin- 1 2 heisenberg antiferromagnetic systems: Ba 2Cu(PO4)2 and Sr2Cu(PO4)2, Phys. Rev. B 76, 085104 (2007)

    2007

  21. [29]

    M. M. Haque and M. A. H. Ahsan, Sr2Cu(PO4)2 and Ba2Cu(PO4)2 as quasi-one- dimensional spin- 1 2 heisenberg antiferromagnet, Journal of Magnetism and Magnetic Materials 402, 143 (2016)

    2016

  22. [30]

    J. H. Drewes, L. A. Miller, E. Cocchi, C. F. Chan, N. Wurz, M. Gall, D. Pertot, F. Brennecke, and M. K¨ ohl, Antiferromagnetic correlations in two- dimensional fermionic mott-insulating and metallic phase s, Phys. Rev. Lett. 118, 170401 (2017)

    2017

  23. [31]

    Werner, X

    P. Werner, X. Chen, and E. Gull, Ferromagnetic spin correlations in the two-dimensional hubbard model, Phys. Rev. Res. 2, 023037 (2020)

    2020

  24. [32]

    Wrzosek and K

    P. Wrzosek and K. Wohlfeld, Hole in the two-dimensional ising antiferromagnet: Origin of the incoherent spectrum, Phys. Rev. B 103, 035113 (2021)

    2021

  25. [33]

    Hubbard, Electron correlations in narrow energy bands, Proceedings of the Royal Society A 276, 238–257 (1963)

    J. Hubbard, Electron correlations in narrow energy bands, Proceedings of the Royal Society A 276, 238–257 (1963)

    1963

  26. [34]

    M. C. Gutzwiller, Effect of correlation on the ferromagnetism of transition metals, Phys. Rev. Lett. 10, 159 (1963)

    1963

  27. [35]

    Kanamori, Electron Correlation and Ferromagnetism of Transition Metals, Progress of Theoretical Physics 30, 275 (1963)

    J. Kanamori, Electron Correlation and Ferromagnetism of Transition Metals, Progress of Theoretical Physics 30, 275 (1963)

    1963

  28. [36]

    Micnas, J

    R. Micnas, J. Ranninger, and S. Robaszkiewicz, Supercon- ductivity in narrow-band systems with local nonretarded attractive interactions, Rev. Mod. Phys. 62, 113 (1990)

    1990

  29. [37]

    Arai, Exchange interaction and heisenberg’s spin hamil- tonian, Phys

    T. Arai, Exchange interaction and heisenberg’s spin hamil- tonian, Phys. Rev. 126, 471 (1962)

    1962

  30. [38]

    M. A. Tusch, Y. H. Szczech, and D. E. Logan, Magnetism in the hubbard model: An effective spin hamiltonian ap- proach, Phys. Rev. B 53, 5505 (1996) . 11

    1996

  31. [39]

    L. F. Tocchio, A. Montorsi, and F. Becca, Magnetic and spin-liquid phases in the frustrated t − t′ hubbard model on the triangular lattice, Phys. Rev. B 102, 115150 (2020)

    2020

  32. [40]

    Huang and Z

    G.-H. Huang and Z. Wu, Magnetic correlations of a doped and frustrated hubbard model: Benchmarking the two- particle self-consistent theory against a quantum simulat or, Phys. Rev. B 110, L100406 (2024)

    2024

  33. [41]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging research landscape of altermagnetism, Phys. Rev. X 12, 040501 (2022)

    2022

  34. [42]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conven- tional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12, 031042 (2022)

    2022

  35. [43]

    Mazin (The PRX Editors), Editorial: Altermagnetism— a new punch line of fundamental magnetism, Phys

    I. Mazin (The PRX Editors), Editorial: Altermagnetism— a new punch line of fundamental magnetism, Phys. Rev. X 12, 040002 (2022)

    2022

  36. [44]

    Jungwirth, R

    T. Jungwirth, R. M. Fernandes, E. Fradkin, A. H. MacDonald, J. Sinova, and L. ˇSmejkal, Altermag- netism: An unconventional spin-ordered phase of matter, Newton 1, 100162 (2025)

    2025

  37. [45]

    P. Zhou, X. N. Peng, Y. Z. Hu, B. R. Pan, S. M. Liu, P. B. Lyu, and L. Z. Sun, Transition from antifer- romagnets to altermagnets: Symmetry-breaking theory, Phys. Rev. B 112, 144419 (2025)

    2025

  38. [46]

    Sim and J

    G. Sim and J. Knolle, Pair density waves and supercurrent diode effect in altermagnets, Phys. Rev. B 112, L020502 (2025)

    2025

  39. [47]

    Karetta, X

    B. Karetta, X. H. Verbeek, R. Jaeschke-Ubiergo, L. ˇSmejkal, and J. Sinova, Strain-controlled g- to d-wave transition in altermagnetic crsb, Phys. Rev. B 112, 094454 (2025)

    2025

  40. [48]

    Chakraborty and A

    D. Chakraborty and A. M. Black-Schaffer, Con- straints on superconducting pairing in altermagnets, Phys. Rev. B 112, 014516 (2025)

    2025

  41. [49]

    Z. Feng, X. Zhou, L. ˇSmejkal, L. Wu, Z. Zhu, H. Guo, R. Gonz´ alez-Hern´ andez, X. Wang, H. Yan, P. Qin, X. Zhang, H. Wu, H. Chen, Z. Meng, L. Liu, Z. Xia, J. Sinova, T. Jungwirth, and Z. Liu, An anomalous hall effect in altermagnetic ruthenium dioxide, Nature Electronics 5, 7...

    2022

  42. [50]

    A. Bose, N. J. Schreiber, R. Jain, D.-F. Shao, H. P. Nair, J. Sun, X. S. Zhang, D. A. Muller, E. Y. Tsymbal, D. G. Schlom, and D. C. Ralph, Tilted spin current gen- erated by the collinear antiferromagnet ruthenium dioxide , Nature Electronics 5, 267 (2022)

    2022

  43. [51]

    T. A. Maier and S. Okamoto, Weak-coupling theory of neutron scattering as a probe of altermagnetism, Phys. Rev. B 108, L100402 (2023)

    2023

  44. [52]

    L. Bai, W. Feng, S. Liu, L. ˇSmejkal, Y. Mokrousov, and Y. Yao, Altermagnetism: Exploring new frontiers in magnetism and spintronics, Advanced Functional Materials 34, 2409327 (2024)

    2024

  45. [53]

    W. Lu, S. Feng, Y. Wang, D. Chen, Z. Lin, X. Liang, S. Liu, W. Feng, K. Yamagami, J. Liu, C. Felser, Q. Wu, and J. Ma, Signature of topological surface bands in altermagnetic we yl semimetal crsb, Nano Letters 25, 7343 (2025)

    2025

  46. [54]

    C. Song, H. Bai, Z. Zhou, L. Han, H. Reichlova, J. H. Dil, J. Liu, X. Chen, and F. Pan, Al- termagnets as a new class of functional materials, Nature Reviews Materials 10, 473 (2025)

    2025

  47. [55]

    L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Giant momentum-dependent spin splitting in centrosymmetric low- z antiferromagnets, Phys. Rev. B 102, 014422 (2020)

    2020

  48. [56]

    Isidori, Altermagnetism arising from a spontaneous electronic instability, Communications Materials 5, 134 (2024)

    A. Isidori, Altermagnetism arising from a spontaneous electronic instability, Communications Materials 5, 134 (2024)

    2024

  49. [57]

    Giuli, C

    S. Giuli, C. Mejuto-Zaera, and M. Capone, Alter- magnetism from interaction-driven itinerant magnetism, Phys. Rev. B 111, L020401 (2025)

    2025

  50. [58]

    S. W. Cheong and F. T. Huang, Altermagnetism classifica- tion, npj Quantum Mater 10, 38 (2025)

    2025

  51. [59]

    V. Leeb, A. Mook, L. ˇSmejkal, and J. Knolle, Sponta- neous formation of altermagnetism from orbital ordering, Phys. Rev. Lett. 132, 236701 (2024)

    2024

  52. [60]

    S. W. Cheong and F. T. Huang, Altermagnetism with non- collinear spins, npj Quantum Mater 9, 13 (2024)

    2024

  53. [61]

    Alfonso-Moro, V

    M. Alfonso-Moro, V. Guisset, P. David, B. Canals, J. Coraux, and N. Rougemaille, Geometrical frustration, correlated disorder, and emerging order in a corrugated c60 monolayer, Phys. Rev. Lett. 131, 186201 (2023)

    2023

  54. [62]

    J. Zhou, G. Quirion, J. A. Quilliam, H. Cao, F. Ye, M. B. Stone, Q. Huang, H. Zhou, J. Cheng, X. Bai, M. Mouri- gal, Y. Wan, and Z. Dun, Anticollinear order and de- generacy lifting in square lattice antiferromagnet lasrcro 4, Phys. Rev. B 105, L180411 (2022)

    2022

  55. [63]

    D. V. Dmitriev and V. Y. Krivnov, Macroscopic degeneracy of the ground state in the frustrated heisenberg diamond chain, Phys. Rev. B 111, 064427 (2025)

    2025

  56. [64]

    Syzranov and A

    S. Syzranov and A. Ramirez, Eminuscent phase in frus- trated magnets: a challenge to quantum spin liquids, Nat Commun 13, 2993 (2022)

    2022

  57. [65]

    B.-B. Chen, Z. Chen, S.-S. Gong, D. N. Sheng, W. Li, and A. Weichselbaum, Quantum spin liquid with emer- gent chiral order in the triangular-lattice hubbard model, Phys. Rev. B 106, 094420 (2022)

    2022

  58. [66]

    Y. Liu, J. Li, P. Liu, and Q. Liu, Unconventional spin tex- tures emerging from a universal symmetry theory of spin- momentum locking, npj Quantum Mater 9, 69 (2024)

    2024

  59. [67]

    M. F. Equbal, S. R. Hassan, and M. A. H. Ahsan, Exact diagonalization study of ground state properties and level statistics in simple and extended hubbard lattice cluster, Physica Scripta 100, 035964 (2025)

    2025

  60. [68]

    M. F. Equbal, S. R. Hassan, and M. A. H. Ah- san, Principal component analysis of compet- ing correlations in quarter-filled hubbard models, arXiv preprint arXiv:2511.12551 (2025)

    2025 arXiv

  61. [69]

    M. A. H. Ahsan and C. R. Sarma, Configura- tion space truncation scheme for hubbard hamiltonian, Journal of Chemical Sciences 106, 379 (1994)

    1994

  62. [70]

    C. R. Sarma and M. A. H. Ahsan, Elec- tron correlation studies: Rumer basis approach, International Journal of Quantum Chemistry 60, 147 (1996)

    1996

  63. [71]

    S. M. Hussain and K. Son, Exploring altermagnetism in ruo2: A review of a new paradigm in spintronics, Physica B: Condensed Matter 716, 417723 (2025)

    2025

  64. [72]

    Zeng and Y.-J

    S. Zeng and Y.-J. Zhao, Description of two-dimensional altermagnetism: Categorization using spin group theory, Phys. Rev. B 110, 054406 (2024)

    2024

  65. [73]

    P. A. McClarty and J. G. Rau, Landau theory of altermag- netism, Phys. Rev. Lett. 132, 176702 (2024)

    2024

  66. [74]

    P. Das, V. Leeb, J. Knolle, and M. Knap, Realizing alter- magnetism in fermi-hubbard models with ultracold atoms, Phys. Rev. Lett. 132, 263402 (2024)

    2024

  67. [75]

    Schiff, P

    H. Schiff, P. McClarty, J. G. Rau, and J. Romh´ anyi, Collinear altermagnets and their landau theories, Phys. Rev. Res. 7, 033301 (2025)

    2025

  68. [76]

    P. M. Cˆ onsoli and M. Vojta, su(n) altermagnetism: Lattice models, magnon modes, and flavor-split bands, Phys. Rev. Lett. 134, 196701 (2025)

    2025

  69. [77]

    H. Peng, S. Zeng, J.-H. Liao, C.-C. He, X.-B. Yang, and Y.-J. Zhao, Multicomponent altermag- net: A general approach to generating multicompo- 12 nent structures with two-dimensional altermagnetism, Phys. Rev. B 111, 195123 (2025)

    2025

  70. [78]

    S. A. A. Ghorashi and Q. Li, Dynamical genera- tion of higher-order spin-orbit coupling, topology, and persistent spin texture in light-irradiated altermagnets , Phys. Rev. Lett. 135, 236702 (2025)

    2025

  71. [79]

    Z.-M. Wang, Y. Zhang, S.-B. Zhang, J.-H. Sun, E. Dagotto, D.-H. Xu, and L.-H. Hu, Spin-orbital altermagnetism, Phys. Rev. Lett. 135, 176705 (2025)

    2025

  72. [80]

    A. T. Costa, J. C. G. Henriques, and J. Fern´ andez-Rossier, Giant spatial anisotropy of magnon Landau damping in al- termagnets, SciPost Phys. 18, 125 (2025)

    2025

  73. [81]

    Kaushal and M

    N. Kaushal and M. Franz, Altermagnetism in modified lieb lattice hubbard model, Phys. Rev. Lett. 135, 156502 (2025)

    2025

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