Avoided Stoner instability at a single ordinary Van Hove point

Abstract: When the Fermi surface and the Brillouin zone boundary touch at a Van Hove point, mean-field analysis predicts a ferromagnetic (Stoner) instability at finite $T_{MF}$ for any coupling strength due to the divergent density of states. However, the predicted effect has not been observed experimentally. Several qualitative theoretical proposals have been put forward to explain why the mean-field prediction fails. Based on numerically exact results for the two-dimensional Hubbard model with an ordinary Van Hove singularity, we uncover the mechanisms behind the suppression of the ferromagnetic instability. We employ two diagrammatic Monte Carlo approaches: (i) the four-channel self-consistent approximation and (ii) numerically exact method of combinatorial summation of diagrams with controlled resummation of the truncated expansion. We find that the system avoids the Stoner instability down to temperatures an order of magnitude below $T_{MF}$ due to the combination of the downward renormalization of the effective coupling and the suppression of the density of states by the loss of the quasiparticle residue.

• The paper demonstrates that an ordinary two-dimensional Van Hove point does not produce a Stoner instability at weak repulsion, with susceptibility remaining finite down to temperatures about ten times below the mean-field scale.
• Controlled diagrammatic Monte Carlo identifies two cooperating mechanisms—non-Fermi-liquid residue suppression with self-energy scaling as approximately ω^{1/2}, and downward renormalization of the effective interaction—that keep U*|Π| below one.
• The results challenge selective crossed-diagram resummations, reveal a possible shift toward finite-momentum magnetic correlations, and help explain why strained Sr₂RuO₄ can favor superconductivity over ferromagnetism.

The mean-field prediction that a two-dimensional Fermi system tuned to a Van Hove point must become ferromagnetic at arbitrarily weak repulsion has long stood in tension with quantum Monte Carlo simulations and experiments on strained Sr$_2$RuO$_4$, where superconductivity rather than ferromagnetism is observed. In "Avoided Stoner instability at a single ordinary Van Hove point" (2605.28761), Tupitsyn, Currie, Chubukov, Svistunov, Kozik, and Prokof'ev resolve this tension for the ordinary (logarithmically divergent) Van Hove singularity using controlled diagrammatic Monte Carlo methods, identifying two cooperating mechanisms—suppression of the polarization bubble by a reduced quasiparticle residue and a downward renormalization of the effective coupling—that keep the Stoner criterion unfulfilled down to temperatures an order of magnitude below the mean-field scale.

Model and diagnostic framework

The authors study an anisotropic square-lattice Hubbard model with hopping amplitudes $t_x = t/2$, $t_y = t$, $t'_x = 0.12t$, $t'_y = -0.24t$, chosen to enhance the density of states $\rho_F$ at an ordinary Van Hove point where the Fermi surface touches the Brillouin zone boundary at $\mathbf{Q}_{VH} = (\pi/a, 0)$. The interaction is fixed at $U = 2t$, giving a mean-field critical temperature $T_{MF} \approx 0.065t$. A key control is that the interaction-induced deformation of the Fermi surface remains negligible even at the Van Hove point, so the instability question can be isolated from band-structure feedback.

The analysis decomposes the uniform spin susceptibility as

$_4$0

where $_4$1 is the static zero-momentum bubble built from dressed Green's functions, and all vertex corrections are absorbed into a dressed effective interaction $_4$2. This decomposition treats on equal footing the upward renormalization of $_4$3 from particle-hole vertex corrections and its downward renormalization from the particle-particle channel, avoiding the ambiguity of assigning logarithmic corrections to either factor alone. The Stoner criterion is then simply whether $_4$4 reaches unity.

Two diagrammatic Monte Carlo approaches

Two complementary schemes are employed. Bold4+ performs a fully self-consistent one-loop renormalization of the single-particle Green's function and four-point vertices in all three two-body channels, extended beyond one loop to include leading vertex corrections; it retains all diagrams up to fourth order plus selected geometric series, and is computationally efficient enough to probe low temperatures. DiagMC-CoS computes the Taylor coefficients $_4$5 of observables in powers of $_4$6 by combinatorial summation of all connected Feynman diagrams up to order $_4$7, directly in the thermodynamic limit, with reconstruction via DLog Padé and Integral Approximant methods whose spread provides a controlled estimate of extrapolation error. The only systematic errors are statistical noise in the coefficients and truncation at finite order.

Both approaches agree on the central result: no divergence of $_4$8 down to the lowest accessible temperature, roughly $_4$9–$t_x = t/2$0, about ten times below $t_x = t/2$1. They disagree, however, on how the avoidance is realized—a discrepancy that itself carries physical information about which diagram classes dominate.

Bold4+ results: saturation of both factors

Within Bold4+, both $t_x = t/2$2 and $t_x = t/2$3 saturate to finite values as $t_x = t/2$4. The bubble $t_x = t/2$5 fails to track the divergent bare $t_x = t/2$6 because the quasiparticle residue at the Van Hove point collapses: the self-energy $t_x = t/2$7 saturates to a finite value as $t_x = t/2$8, in stark contrast to the Fermi-liquid form $t_x = t/2$9 observed just away from the singularity (at a detuning of only $t_y = t$0). This frequency dependence signals non-Fermi-liquid behavior localized at the touching point. Meanwhile, $t_y = t$1 is well fit by the self-consistent equation

$t_y = t$2

an ad hoc extension of the maximally-crossed-diagram conjecture of earlier RG work, yielding $t_y = t$3; since $t_y = t$4 saturates, so does $t_y = t$5. The product $t_y = t$6 stays below unity throughout.

Controlled results: ongoing flow of both factors

The numerically exact CoS data revise this picture qualitatively. Here $t_y = t$7 continues to grow as $t_y = t$8 decreases (though more slowly than $t_y = t$9), and $t'_x = 0.12t$0 continues to decrease without saturating, following neither the logarithmic form proposed from crossed diagrams nor the self-consistent square-root form. The self-energy at the Van Hove point exhibits a clean non-Fermi-liquid power law, $t'_x = 0.12t$1 with $t'_x = 0.12t$2. Because the underlying spectral function shows no sign of recovering quasiparticle weight, the authors argue it is unlikely that $t'_x = 0.12t$3 will saturate at temperatures beyond their reach—the suppression mechanism operates through the reduced residue rather than through a hard cutoff of the density-of-states divergence.

A further unexpected feature appears in the momentum-resolved susceptibility at $t'_x = 0.12t$4: the peak shifts from $t'_x = 0.12t$5 to finite momentum, suggesting non-analyticities in $t'_x = 0.12t$6 of the type analyzed previously for 2D systems away from Van Hove points, and indicating only short-range ferromagnetic correlations persist.

Limitations and open questions

Several caveats bound the conclusions. First, the CoS analysis relies on truncation at eighth order with Padé-type resummation; while the approximant spread gives a controlled error estimate, the extrapolation to $t'_x = 0.12t$7 remains inferential, and the claim that $t'_x = 0.12t$8 diverges (rather than saturates) rests on the observed $t'_x = 0.12t$9 self-energy rather than on direct low-temperature data. Second, the results are specific to $t'_y = -0.24t$0 and to this particular dispersion; the fate at stronger coupling, or when interaction-induced Fermi-surface deformation becomes relevant, is not established. Third, the decomposition of $t'_y = -0.24t$1 into $t'_y = -0.24t$2 and $t'_y = -0.24t$3 is a convention—vertex corrections could be assigned differently—and the physical interpretation of $t'_y = -0.24t$4 as an "effective coupling" inherits that choice. Finally, the shift of the susceptibility peak to finite $t'_y = -0.24t$5 is noted but not explained; whether it reflects true non-analytic structure or a precursor to another ordered phase remains open. The extension to extended (higher-order) Van Hove singularities, where the DOS diverges as a power law—as realized in twisted WSe$t'_y = -0.24t$6—is explicitly left for future work.

Conclusion

This work establishes, with numerically exact control, that a single ordinary Van Hove point does not generically trigger a Stoner instability: the divergent bare density of states is compensated jointly by residue suppression producing non-Fermi-liquid self-energy ($t'_y = -0.24t$7) and by a net downward renormalization of the effective interaction dominated by the particle-particle channel. The disagreement between Bold4+ and CoS on the low-temperature trends of $t'_y = -0.24t$8 and $t'_y = -0.24t$9 demonstrates that the problem is genuinely non-perturbative and cannot be captured by selective resummations of crossed diagrams alone. These mechanisms provide a concrete microscopic basis for understanding why strained Sr$\rho_F$0RuO$\rho_F$1 develops superconductivity rather than ferromagnetism at a Van Hove point, and they frame the corresponding open question for higher-order Van Hove singularities in moiré materials.