The ultimate fate of the Stoner instability at a Van Hove singularity in the two-dimensional Hubbard model

Mean-field theory predicts that a two-dimensional Fermi liquid tuned to an ordinary Van Hove singularity — a saddle point of the band dispersion where the density of states diverges logarithmically — must develop itinerant (Stoner) ferromagnetism at a finite temperature for any repulsive coupling strength. Experiments (Sr2RuO4 under uniaxial strain, twisted WSe2) instead show superconductivity, and a 2026 numerically exact diagrammatic Monte Carlo study by Tupitsyn, Currie, Chubukov, Svistunov, Kozik, and Prokof'ev found that the 2D Hubbard model at a single ordinary Van Hove point avoids the Stoner instability down to $T/t=0.01$, roughly an order of magnitude below the mean-field transition temperature, through the combination of a downward-renormalized effective coupling $U^{*}(T)$ and a density of states suppressed by the loss of quasiparticle residue. Whether this avoidance persists as $T\to 0$ — or whether the Stoner condition $U^{*}|\Pi(T)|=1$ is eventually met at some lower temperature — is unknown, as are the extension to higher-order Van Hove singularities with power-law divergences, the nature of an observed finite-momentum shift of the susceptibility peak, and the connection to the superconductivity that appears instead.

Background

The Stoner criterion is the mean-field (ladder) condition for itinerant ferromagnetism in a metal with repulsive interaction: the uniform static spin susceptibility $\chi(T)$ diverges when $U\Pi_{0}(T)=-1$, where $U$ is the interaction and $\Pi_{0}(T)<0$ is the static polarization bubble of the non-interacting Fermi gas. A Van Hove (VH) point is a saddle point of the electronic dispersion at which the Fermi surface touches the Brillouin-zone boundary; there the density of states (DOS) diverges logarithmically, $\Pi_{0}(T)\sim\ln(T/E_{0})$, so mean-field theory predicts a ferromagnetic transition at a finite temperature $T_{MF}$ for arbitrarily weak $U>0$. A higher-order Van Hove singularity (HOVHS) is a flatter saddle at which the DOS diverges as a power law rather than logarithmically; such points are realized, for example, in twisted WSe$_{2}$.

The paradigmatic model is the two-dimensional (2D) Hubbard model, $H=\sum_{\mathbf{k}\sigma}\epsilon_{\mathbf{k}}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}$, with a dispersion tuned so that a single ordinary VH point sits at the Fermi level. In the reference study (Tupitsyn et al. 2026, arXiv:2605.28761) the dispersion is asymmetric, $\epsilon_{\mathbf{k}}=-2\sum_{\alpha=x,y}[t_{\alpha}\cos k_{\alpha}-t'_{\alpha}\cos 2k_{\alpha}]$ with $t_{x}=t/2$, $t_{y}=t$, $t'_{x}=0.12t$, $t'_{y}=-0.24t$, placing one VH point at $\mathbf{Q}_{VH}=(\pi/a,0)$, with $U=2t$ and $T_{MF}\approx 0.065t$.

Experiments conflict with the mean-field prediction: uniaxial compressive strain in Sr$_{2}$RuO$_{4}$ drives a Fermi-surface sheet through a Van Hove singularity and produces a peak in the superconducting $T_{c}$ (reaching about 3.5 K) together with a resistivity anomaly, but no ferromagnetism (Barber et al. 2018); superconductivity rather than Stoner magnetism is likewise observed near Van Hove physics in twisted bilayer WSe$_{2}$ (Xia et al. 2024). Renormalization-group (RG) studies — Isobe and Fu's “supermetal” analysis of power-law Van Hove divergences, Classen et al.'s study of competing orders at higher-order Van Hove points, and Ojajärvi et al.'s analysis of pairing at a single Van Hove point — found no Stoner instability at a VH point and proposed that the effective coupling is driven down by renormalization in the particle-particle (Cooper) channel; Ojajärvi et al. conjectured $U^{*}/U=(1-Ub\Pi_{0})^{-1}$ with $b\approx 1.88>1$, which can never satisfy the Stoner condition, and argued that triplet pairing wins instead.

Tupitsyn et al. 2026 settled the mechanism question with two diagrammatic Monte Carlo (DiagMC) approaches — methods that stochastically sample and sum Feynman diagram series for the Green's function and vertices. The first, Bold4+, is a four-channel self-consistent scheme including all diagrams up to fourth order plus partial higher-order resummations; the second, DiagMC-CoS (combinatorial summation), sums diagrammatic contributions up to high orders with controlled resummation and is numerically exact. Writing the spin susceptibility as $\chi(T)=-\Pi(T)/(1+U^{*}(T)\Pi(T))$, where $\Pi(T)$ is the polarization bubble built from dressed propagators and $U^{*}(T)$ absorbs the vertex corrections, they found: (i) $\chi(T)$ does not diverge down to $T/t=0.01$; (ii) $\Pi(T)$ is strongly reduced relative to the bare bubble because the quasiparticle residue $Z$ (the weight of the coherent quasiparticle pole in the Green's function) is suppressed, with a non-Fermi-liquid self-energy at the VH point consistent with $\Sigma(\mathbf{Q}_{VH},\omega_{n})\propto\omega_{n}^{1/2}$ in the CoS data; (iii) $U^{*}(T)$ decreases overall, the particle-particle downward renormalization outweighing the upward vertex corrections; and (iv) the product $U^{*}|\Pi(T)|$ stays below 1 at all temperatures studied. Notably, the two schemes disagree about the low-temperature tendencies: in Bold4+ both $U^{*}$ and $\Pi$ saturate, while in CoS $|\Pi|$ keeps growing and $U^{*}$ keeps decreasing down to the lowest temperature. They also found that the peak of the momentum-dependent susceptibility $\chi(\mathbf{Q})$ shifts from $\mathbf{Q}=0$ to finite momentum at low temperature.

Problem Statement

Determine the ultimate fate of the ferromagnetic (Stoner) instability of the two-dimensional Hubbard model tuned to a single ordinary Van Hove point: for the model studied by Tupitsyn et al. (single ordinary Van Hove point at $\mathbf{Q}_{VH}=(\pi/a,0)$, $U=2t$, $T_{MF}\approx 0.065t$), establish whether the product $U^{*}(T)|\Pi(T)|$ remains below 1 as $T\to 0$ (ferromagnetism genuinely avoided, with the ground state being a non-Fermi-liquid paramagnetic metal, a superconductor, or another ordered state), or whether the Stoner condition $U^{*}\Pi=-1$ is eventually met at some finite lower temperature. The two published DiagMC schemes disagree about the $T\to 0$ tendencies (saturation versus continued growth of $|\Pi|$ with decreasing $U^{*}$), so resolving which tendency wins is the crux.

Beyond this central open question, the source discusses as outlook directions — not additional mandatory requirements: whether the two avoidance mechanisms (suppression of the density of states by the loss of quasiparticle residue, and the net downward renormalization of $U^{*}$ by the particle-particle channel) persist at higher-order Van Hove points, where the density of states diverges as a power law rather than logarithmically (as realized in twisted WSe$_{2}$); whether the observed shift of the peak of $\chi(\mathbf{Q})$ from $\mathbf{Q}=0$ to finite momentum at low temperature is a non-analyticity of the type analyzed for 2D Fermi liquids away from a Van Hove point, or the precursor of a genuine finite-momentum (modulated) magnetic instability; and whether the mechanisms that suppress ferromagnetism at a Van Hove point underpin the superconducting pairing observed at Van Hove points in Sr$_{2}$RuO$_{4}$ under uniaxial strain and in twisted WSe$_{2}$ — that is, whether the avoided Stoner instability and the emergence of superconductivity are two faces of the same renormalization physics.

References

1. I. S. Tupitsyn, B. Currie, A. V. Chubukov, B. V. Svistunov, E. Kozik, N. V. Prokof'ev, 'Avoided Stoner instability at a single ordinary Van Hove point', arXiv:2605.28761 (2026), DOI: 10.48550/arXiv.2605.28761, https://arxiv.org/abs/2605.28761
2. R. Ojajärvi, A. V. Chubukov, Y.-C. Lee, M. Garst, J. Schmalian, 'Pairing at a single Van Hove point', npj Quantum Materials 9, 105 (2024), DOI: 10.1038/s41535-024-00717-4, arXiv:2408.05572, https://arxiv.org/abs/2408.05572
3. L. Classen, A. V. Chubukov, C. Honerkamp, M. M. Scherer, 'Competing orders at higher-order Van Hove points', Physical Review B 102, 125141 (2020), DOI: 10.1103/PhysRevB.102.125141, arXiv:2006.14729, https://arxiv.org/abs/2006.14729
4. H. Isobe, L. Fu, 'Supermetal', Physical Review Research 1, 033206 (2019), DOI: 10.1103/PhysRevResearch.1.033206, arXiv:1905.05188, https://arxiv.org/abs/1905.05188
5. M. E. Barber, A. S. Gibbs, Y. Maeno, A. P. Mackenzie, C. W. Hicks, 'Resistivity in the Vicinity of a Van Hove Singularity: Sr2RuO4 Under Uniaxial Pressure', Physical Review Letters 120, 076602 (2018), DOI: 10.1103/PhysRevLett.120.076602, arXiv:1709.06545, https://arxiv.org/abs/1709.06545
6. Y. Xia, Z. Han, K. Watanabe, T. Taniguchi, J. Shan, K. F. Mak, 'Unconventional superconductivity in twisted bilayer WSe2', arXiv:2405.14784 (2024), DOI: 10.48550/arXiv.2405.14784, https://arxiv.org/abs/2405.14784
7. Y.-C. Lee, N. V. Prokof'ev, A. V. Chubukov, 'Stoner transitions beyond mean-field in two-dimensional electronic systems: a diagrammatic Monte Carlo study', arXiv:2607.13675 (2026), DOI: 10.48550/arXiv.2607.13675, https://arxiv.org/abs/2607.13675
8. T. P. Sheerin, M. Ramirez, C. A. Hooley, L. C. Rhodes, 'Engineering correlated phases through manipulation of Van Hove singularities', arXiv:2608.07714 (2026), DOI: 10.48550/arXiv.2608.07714, https://arxiv.org/abs/2608.07714
9. V. Berger, I. S. Tupitsyn, B. Currie, B. V. Svistunov, E. Kozik, N. V. Prokof'ev, 'Emergent BEC mechanism in flat-band superconductors', arXiv:2608.19044 (2026), DOI: 10.48550/arXiv.2608.19044, https://arxiv.org/abs/2608.19044
10. Z. Wei, Y. Cai, B. Wen, T. Ma, 'Magnetic correlations and superconducting pairing near higher-order Van Hove singularities', arXiv:2505.10358 (2025), DOI: 10.48550/arXiv.2505.10358, https://arxiv.org/abs/2505.10358