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arXiv:2606.16912

Band-minimum degeneracy is not enough: density-of-states control of low-density ferromagnetism

Jinghao Wang, P. Pieri

Published Jun 15, 2026• 0 citations

Abstract

Recent ultracold-atom experiments have observed ferromagnetic correlations in a geometrically tunable Fermi--Hubbard lattice at intermediate densities. Motivated by subsequent theoretical work that connected the low-density limit of the same model to the Mueller--Hartmann mechanism for itinerant ferromagnetism, we investigate the stability of ferromagnetism across a broad density range using the T-matrix approximation, a controlled approach in the dilute limit. We reproduce the ferromagnetic regime observed experimentally at intermediate densities and systematically compare finite-size and thermodynamic-limit calculations in the dilute regime. We find that the low-density ferromagnetic phase reported for the one-diagonal hopping lattice is strongly suppressed with increasing system size and disappears in the thermodynamic limit, indicating that it originates from finite-size effects. By contrast, low-density ferromagnetism remains robust in the square-lattice Hubbard model with hopping along both diagonal directions. We show that this qualitative difference cannot be explained by the degeneracy of the band minima alone, which occurs in both models. Instead, it is controlled by the singular behavior of the density of states at the band bottom: a weak divergence is insufficient to stabilize ferromagnetism, whereas a much stronger quasi-one-dimensional singularity supports a fully polarized ground state even in the dilute limit. Our results demonstrate that band-minimum degeneracy alone is not sufficient for low-density ferromagnetism and that the nature of the band-bottom density-of-states singularity ultimately controls its stability.

DOI: 10.48550/arxiv.2606.16912

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Recent ultracold-atom experiments have observed ferromagnetic correlations in a geometrically tunable Fermi--Hubbard lattice at intermediate densities. Motivated by subsequent theoretical work that connected the low-density limit of the same model to the Mueller--Hartmann mechanism for itinerant ferromagnetism, we investigate the stability of ferromagnetism across a broad density range using the T-matrix approximation, a controlled approach in the dilute limit. We reproduce the ferromagnetic regime observed experimentally at intermediate densities and systematically compare finite-size and thermodynamic-limit calculations in the dilute regime. We find that the low-density ferromagnetic phase reported for the one-diagonal hopping lattice is strongly suppressed with increasing system size and disappears in the thermodynamic limit, indicating that it originates from finite-size effects. By contrast, low-density ferromagnetism remains robust in the square-lattice Hubbard model with hopping along both diagonal directions. We show that this qualitative difference cannot be explained by the degeneracy of the band minima alone, which occurs in both models. Instead, it is controlled by the singular behavior of the density of states at the band bottom: a weak divergence is insufficient to stabilize ferromagnetism, whereas a much stronger quasi-one-dimensional singularity supports a fully polarized ground state even in the dilute limit. Our results demonstrate that band-minimum degeneracy alone is not sufficient for low-density ferromagnetism and that the nature of the band-bottom density-of-states singularity ultimately controls its stability.

We studied ferromagnetism in the one-dimensional Hubbard model with doubly degenerate atomic orbitals by means of the density-matrix renormalization-group method and obtained the ground-state phase diagrams. It was found that ferromagnetism is stable from low to high

(0<n<1.75)

electron density when the interactions are sufficiently strong. The quasi-long-range order of triplet superconductivity coexists with the ferromagnetic order for a strong Hund coupling region, where the interorbital interaction ${U}^{\ensuremath{'}}\ensuremath{-}J$ is attractive. At quarter-filling

(n=1),

the insulating ferromagnetic state appears, accompanying orbital quasi-long-range order. For low densities

(n<1),

ferromagnetism occurs owing to the ferromagnetic exchange interaction caused by virtual hoppings of electrons, the same as in the quarter-filled system. This comes from separation of the charge and spin-orbital degrees of freedom in the strong-coupling limit. This ferromagnetism is fragile against variation of band structure. For high densities

(n>1),

the phase diagram of the ferromagnetic phase is similar to that obtained in infinite dimensions. In this case, the double exchange mechanism is operative for stabilizing the ferromagnetic order and this long-range order is robust against variation of the band dispersion. A partially polarized state appears in the density region $1.68\ensuremath{\lesssim}n\ensuremath{\lesssim}1.75$ and phase separation occurs for n just below the half-filling

(n=2).

Using exact diagonalization techniques, the Hubbard model with hopping between nearest (t) and next-nearest neighbors ${(t}^{\ensuremath{'}})$ is studied in a two-dimensional cluster with $4\ifmmode\times\else\texttimes\fi{}4$ sites. Within a range of values of ${t}^{\ensuremath{'}}<0$ and low densities, states with different total spin are degenerate in the noninteracting system. At a finite value of the Coulomb repulsion U, states with high spin are energetically favored and the ground state can be fully polarized. The behavior of the density of states differs from that obtained within the Hartree-Fock approximation. The low energy region of the density of states (DOS) for the majority spins remains weakly modified by U. Our results bring further support to the idea that itinerant ferromagnetism occurs in strongly correlated systems with high noninteracting DOS at the bottom of the band.

Ferromagnetism in the one-dimensional Hubbard model with the next-nearest-neighbor hopping is explored by using the exact-diagonalization method in a small cluster and the equation-of-motion method in the thermodynamic limit with electron density n = 3/2. With these two complementary methods, it is found that an intermediate value of the next-nearest-neighbor hopping amplitude t1 tends to stabilize the fully polarized ferromagnetic state under the condition that the on-site coulomb interaction U is sufficiently large in our model. The ground-state phase diagram of the model is presented in the t1—U plane.

Since its introduction in 1963, the Hubbard model has becomes one of the most popular models used in the literature to study cooperative phenomena in narrow-band metals (ferromagnetism, metal-insulator transitions, charge-density waves, high-Tc superconductivity). Amongst all these cooperative phenomena, the problem of itinerant ferromagnetism in the Hubbard model has the longest history. However, in spite of an impressive research activity in the past, the underlying physics (microscopic mechanisms) that leads to the stabilization of itinerant ferromagnetism in Hubbard model (narrow-band metals) is still far from being understood. In this review we present our numerical results concerning this subject, which have been reached by small cluster exact diagonalization, density matrix renormalization group and quantum Monte Carlo calculations within various extensions of the Hubbard model. Particular attention is paid to a description of crucial mechanisms (interactions) that support the stabilization of the ferromagnetic state, and namely: (i) the long-range hopping, (ii) the correlated hopping, (iii) the long-range Coulomb interaction, (iv) the flat bands and (v) the lattice structure. Most of the presented results have been obtained for the one-dimensional case, but the influence of the increasing dimension of the system on the ferromagnetic state is also intensively discussed.

The infinite-U one-dimensional Hubbard model is 'solvable' in the limit that the nearest-neighbour hopping is dominant and infinitesimal longer-range hopping lifts the one-dimensional spin degeneracy. In this limit the Hubbard model maps onto an effective spin interaction: cyclic or ring exchange with a variety of lengths and strengths. For bipartite geometries we find a region of stable ferromagnetism, which originates from the mechanism intrinsic to Nagaoka's ferromagnetism. We look at connectivities that limit exactly to the square-lattice connectivity as the range of the infinitesimal hopping diverges. When we extend this hopping range, we find that the stable region of ferromagnetism shrinks, eventually vanishing as the infinitesimal hopping range diverges and we limit to the square-lattice geometry. Since the calculation involves a non-trivial density of particles in the thermodynamic limit, it suggests how the proposed region of ferromagnetism may become lost for the case of the two-dimensional square lattice, as numerical simulations predict.

We investigate the instability of the saturated ferromagnetic ground state (Nagaoka state) in the Hubbard model on various lattices in dimensions

d=2

and

d=3

. A variational resolvent approach is developed for the Nagaoka instability both for $U=\ensuremath{\infty}$ and for $U<\ensuremath{\infty}$ which can easily be evaluated in the thermodynamic limit on all common lattices. Our results significantly improve former variational bounds for a possible Nagaoka regime in the ground-state phase diagram of the Hubbard model. We show that a pronounced particle-hole asymmetry in the density of states and a diverging density of states at the lower band edge are the most important features in order to stabilize Nagaoka ferromagnetism, particularly in the low-density limit.