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arxiv: 2606.30607 · v1 · pith:ZARKDX4Inew · submitted 2026-06-29 · ❄️ cond-mat.str-el

Ferromagnetic ordering in Hubbard models

Wojciech Niedzi\'o{\l}ka , Jacek Wojtkiewicz This is my paper

Pith reviewed 2026-06-30 03:09 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hubbard modelferromagnetismfrustrated latticesdensity expansionground state energycondensed matter
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The pith

Hubbard models on frustrated lattices have ferromagnetic ground states at moderate and low densities.

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

The paper examines whether the Hubbard model can produce ferromagnetic ordering on lattices with frustration. It applies the first term of a density expansion for the ground-state energy of interacting fermions, previously established for the continuum and the simple cubic lattice, to five frustrated lattices under the assumption that the expansion carries over. Calculations show that most of these models favor a fully polarized ferromagnetic state already at moderate or even low electron densities. This supports the idea that lattice geometry, specifically frustration, is key to enabling ferromagnetism in the Hubbard model where it does not appear on unfrustrated lattices. The results are not a rigorous proof but provide strong numerical indication that ferromagnetism can occur in these cases.

Core claim

Applying the first term of the density expansion for the ground-state energy shows that Hubbard models on most of the five examined frustrated lattices, including the face-centered cubic lattice, possess ferromagnetic ground states at moderate or low densities.

What carries the argument

The first term of the density expansion for the ground-state energy of interacting fermions, used to compare energies of different spin configurations on frustrated lattices.

If this is right

  • Ferromagnetism appears at low carrier densities on frustrated lattices.
  • The density expansion approach indicates but does not rigorously establish the ground state.
  • Cases with low-density ferromagnetism offer a route to prove convergence of the expansion and establish wealthy ferromagnetism rigorously.

Where Pith is reading between the lines

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

  • The contrast with simple cubic lattices suggests frustration is the structural feature enabling the ordering.
  • If the expansion converges on these lattices, it would give a controlled way to study magnetic phases at finite density.
  • The same method could be tested on other lattices known to support flat bands or geometric frustration to check generality.

Load-bearing premise

The density expansion formula for ground-state energy holds for the five frustrated lattices studied, even though it was proven only for the continuum and the simple cubic lattice.

What would settle it

An exact diagonalization or quantum Monte Carlo calculation on one of the smaller frustrated lattices at the densities in question that finds a lower energy for a non-fully-polarized state would disprove the ferromagnetic ordering.

Figures

Figures reproduced from arXiv: 2606.30607 by Jacek Wojtkiewicz, Wojciech Niedzi\'o{\l}ka.

Figure 1
Figure 1. Figure 1: Density of states for 3D, numerical results (red points) compared to exact result (blue line) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of 𝐸(𝜌↑∕𝜌) for 𝑈 = 20 corresponding to 𝑎 = 0.23 and 𝜌 = 0.2, 0.4, 0.6, 0.8. As we increase density for cubic lattice we see that the ground state has no spontaneous magnetization up to non-physical range 𝜌 = 0.8 3.3.3 Next nearest neighbors (NNN) hoppings The second lattice (more precisely, the one-parameter family of lattices) that was considered was a cubic lattice with additional hoppings 𝑡2 betwee… view at source ↗
Figure 3
Figure 3. Figure 3: Next nearest neighbors hoppings visualized in the 2D case. Thick lines represent nearest [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Density of states for 𝑡 = −1 and 𝑡2 = −0.05,−0.10,−0.15,−0.20. We can see the continuous transition from the shape for a simple cubic lattice to the shape for maximal frustration -4 -2 2 4 6 E 0.05 0.10 0.15 0.20 D(E) [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Density of states at maximal frustration [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: First order phase transition in cubic lattice with NNN at [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Diagonal nearest neighbors hoppings visualized in the 2D case. Thick lines represent nearest [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Density of states for 𝑡3 𝑡 = 0.10, 0.15, 0.20, 0.23, 0.245. We can see the continuous transition from the shape for simple cubic lattice to the shape for maximal frustration 11 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Density of states for the case of maximal frustration [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Second-order phase transition in an almost maximally frustrated ( [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Non-physical behavior for completely frustrated ( [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Single cell of FCC lattice. Solid lines correspond to hopping constant [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Density of states for the FCC lattice for [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Second-order phase transition in FCC lattice with [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Non-physical behavior for FCC lattice with [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
read the original abstract

One of the long-standing and only partially solved problems of theoretical condensed matter physics and mathematical physics is to demonstrate that ground states of some of the versions of the Hubbard model can exhibit a ferromagnetic ordering. It has long been speculated that the opportunity crucial for the occurrence of ferromagnetism is the structure of the lattice on which the Hubbard model is formulated \cite{TasakiMB}. As a consequence, while on simple cubic lattices no ferromagnetic ordering seems to be possible, it can naturally arise, even for low densities of magnetic moment carriers, on so-called frustrated lattices. We investigate the problem of ground state ferromagnetic ordering with the use of the formula for ground-state energy of interacting fermions as the first term of `density expansion', proven rigorously by Lieb, Seiringer and Solovej \cite{fermi exact} in continuum and by Giuliani \cite{hub exact} for the simple cubic lattice. Assuming that analogous expansion holds also for certain another lattices we apply this formula to five frustrated lattices -- among them to the face-centered cubic one. The hypothesis is confirmed: most of examined models formulated on frustrated lattices do indeed have ferromagnetic ground states already for densities being moderate or even low. Although the approach adopted cannot be treated as a rigorous proof that the ground state is ferromagnetic, the results obtained here strongly indicate that it can be the case. Moreover, as in some cases FM occurs at low densities, one can hope that it would be possible to prove convergence of the density expansion and prove rigorously the occurrence of `wealthy ferromagnetism' in these cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper investigates ferromagnetic ordering in the Hubbard model on frustrated lattices by applying the first term of a rigorously proven density expansion for the ground-state energy (established for the continuum and simple cubic lattice) under the explicit assumption that an analogous expansion holds for five frustrated lattices including the FCC lattice. Numerical evaluation of this leading term is used to conclude that most examined models exhibit ferromagnetic ground states at moderate or even low densities, providing a strong indication (though not a rigorous proof) of 'wealthy ferromagnetism' on frustrated lattices.

Significance. If the central assumption holds, the work supplies concrete numerical indications that lattice frustration can stabilize ferromagnetism at low carrier densities in Hubbard models, extending beyond the simple cubic case and aligning with long-standing conjectures; this could motivate targeted rigorous proofs of convergence for the density expansion on specific frustrated lattices where the indication is strongest.

major comments (1)
  1. [Abstract] The central numerical indications and the claim that 'most of examined models... do indeed have ferromagnetic ground states already for densities being moderate or even low' rest on the unverified assumption (stated in the abstract) that the first term of the density expansion proven by Lieb-Seiringer-Solovej and Giuliani extends to the five frustrated lattices; without justification, convergence control, or a check against known cases, the energy comparison between ferromagnetic and other states is an extrapolation whose validity is not demonstrated.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading and the detailed comment. We respond point-by-point below.

read point-by-point responses

  1. Referee: [Abstract] The central numerical indications and the claim that 'most of examined models... do indeed have ferromagnetic ground states already for densities being moderate or even low' rest on the unverified assumption (stated in the abstract) that the first term of the density expansion proven by Lieb-Seiringer-Solovej and Giuliani extends to the five frustrated lattices; without justification, convergence control, or a check against known cases, the energy comparison between ferromagnetic and other states is an extrapolation whose validity is not demonstrated.

    Authors: We agree that the central results rely on the assumption that the leading term of the density expansion extends to the frustrated lattices under consideration. The manuscript already states this assumption explicitly and qualifies all conclusions as 'strong indications' rather than rigorous proofs. We cannot supply a mathematical justification or convergence control for these lattices within the present work, as that would require new rigorous analysis beyond the scope of the numerical study. We will revise the abstract and the opening paragraphs of the introduction to state the assumption more prominently and to underscore that the ferromagnetic indications are conditional on the validity of the expansion. We will also add a short remark noting that the simple-cubic case serves as the only available benchmark. revision: partial

standing simulated objections not resolved
  • Rigorous proof or convergence control of the density expansion on the five frustrated lattices examined.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's central step is an explicit assumption that the first term of the density expansion (proven rigorously for the continuum by Lieb-Seiringer-Solovej and for the simple cubic lattice by Giuliani) also holds for five frustrated lattices; this assumption is stated openly and the subsequent energy comparison is performed under it. No step reduces by construction to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation chain. The cited theorems are external, the assumption is not smuggled via prior work by the same authors, and no known empirical pattern is merely renamed. The derivation therefore remains self-contained against external benchmarks once the stated assumption is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The sole load-bearing item is the domain assumption that the density expansion proven for the continuum and simple cubic lattice continues to hold on the five frustrated lattices; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption The first-term density expansion for ground-state energy proven for continuum and simple cubic lattice also holds for the five frustrated lattices studied.
    Explicitly invoked in the abstract as the step that allows application of the formula.

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Reference graph

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