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arXiv:2510.05226 · Physical review. B./Physical review. B

Spin-spiral instability of the Nagaoka ferromagnet in the crossover between square and triangular lattices

Darren Pereira, Erich J. Mueller

Published Apr 1, 2026• 0 citations
arXivHTMLPDF

Abstract

We study the hard-core Fermi-Hubbard model in the crossover between square and triangular lattices near half-filling. As was recognized by Nagaoka in the 1960s, on the square lattice the presence of a single hole leads to ferromagnetic spin ordering. On the triangular lattice, geometric frustration instead leads to a spin-singlet ground state, which can be associated with a 120-degree spiral order. On lattices which interpolate between square and triangular, there is a phase transition at which the ferromagnetic order becomes unstable to a spin spiral. We model this transition, finding the exact location of the spin-spiral instability.

DOI
10.1103/6q8x-rsqt

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We study the hard-core Fermi-Hubbard model in the crossover between square and triangular lattices near half-filling. As was recognized by Nagaoka in the 1960s, on the square lattice the presence of a single hole leads to ferromagnetic spin ordering. On the triangular lattice, geometric frustration instead leads to a spin-singlet ground state, which can be associated with a 120-degree spiral order. On lattices which interpolate between square and triangular, there is a phase transition at which the ferromagnetic order becomes unstable to a spin spiral. We model this transition, finding the exact location of the spin-spiral instability.
We study the single-hole infinite-UUU Hubbard model on the triangular lattice with nearest- and next-nearest-neighbor hoppings t1t_1t1​ and t2t_2t2​. A frustrating t2t_2t2​ destabilizes Nagaoka ferromagnetism through a Lifshitz transition to a long-wavelength coplanar spiral at t2,c/t1=−0.182t_{2,c}/t_1=-0.182t2,c​/t1​=−0.182. We determine the critical point analytically by reducing the many-body problem to an effective two-body hole--magnon scattering problem in which the hard-core hole--magnon constraint is treated exactly. Numerical calculations confirm the spiral ground state and reveal a coherent spin polaron with quasiparticle weight Z≃0.92Z\simeq0.92Z≃0.92 at the Lifshitz-shifted momentum K=−Q∗\mathbf{K}=-\mathbf{Q}^*K=−Q∗. These results establish the first quantum instability of the triangular-lattice Nagaoka ferromagnet and provide the magnetic baseline for magnon-mediated pairing at finite doping.
The ground state of the hard-core Fermi-Hubbard model with a single hole (and otherwise half-filled) is a ferromagnet on the square lattice, but an antiferromagnet on the triangular lattice, as those spin patterns minimize the kinetic energy of the hole. What is the magnetic behavior as one interpolates between these two geometries? Using a sophisticated quantum Monte Carlo algorithm, the authors determine here the magnetic correlations as the lattice is varied. Their calculations are performed in the thermodynamic limit and at temperatures comparable to current cold-atom experiments. They find a magnetic crossover between the two geometries, and calculate signatures of it that are observable in these experiments.
We investigate the Nagaoka-Thouless ferromagnetic instability in the strongly interacting t − t ′ Hubbard model by continuously breaking particle-hole symmetry on a tunable square-triangular lattice geometry. We use an analytic approach to show that the fully spin-polarized state becomes unstable to a metastable spin polaron when the kinetic frustration t ′ / t exceeds a critical, dimension-dependent value. Large-scale density matrix renormalization group simulations reveal a quantum phase transition from the Nagaoka ferromagnet to a spiral spin-density wave, which evolves continuously into the Haerter-Shastry antiferromagnet in the large-frustration limit. Remarkably, this transition remains robust at low but finite hole density, making it accessible in cold-atom and moiré Hubbard platforms under strong interactions. A variational analysis further captures the instability mechanism at finite hole density via frustration-induced magnon band deformation.
Abstract In order to analyse the lattice dependence of ferromagnetism in the two‐dimensional Hubbard model we investigate the instability of the fully polarised ferromagnetic ground state (Nagaoka state) on the triangular, honeycomb and kagome lattices. We mainly focus on the local instability, applying single spin flip variational wave functions which include majority spin correlation effects. The question of global instability and phase separation is addressed in the framework of Hartree‐Fock theory. We find a strong tendency towards Nagaoka ferromagnetism on the non‐bipartite lattices (triangular, kagome) for more than half filling. For the triangular lattice we find the Nagaoka state to be unstable above a critical density of n = 1.887 at U = ∞, thereby significantly improving former variational results. For the kagome lattice the region where ferromagnetism prevails in the phase diagram widely exceeds the flat band regime. Our results even allow the stability of the Nagaoka state in a small region below half filling. In the case of the bipartite honeycomb lattice several disconnected regions are left for a possible Nagaoka ground state.
We study the metal-insulator transition and magnetic ordering in the Hubbard model using the particle-hole mapping. The analysis simplifies near the ferromagnetic limit. We find that the two-dimensional Hubbard model has a charge excitation gap at half filling for any finite U in this region on both the bipartite square lattice and the nonbipartite triangular lattice. In some cases, the system goes through a first-order phase transition to become a paramagnetic metal as Sz{\mathit{S}}_{\mathit{z}}Sz​ is lowered. We also discuss the extension to the doped case. We find that in the large U limit, a single doped hole has a bandwidth of order of J rather than t at Sz{\mathit{S}}_{\mathit{z}}Sz​=0.
We investigate, using the density-matrix renormalization group, the evolution of the Nagaoka state with ${t}^{\ensuremath{'}}$ hopping that frustrates the hole kinetic energy in the $U=\ensuremath{\infty}$ Hubbard model on the square and anisotropic triangular lattices. We find that the Nagaoka ferromagnet survives up to a rather small ${t}_{c}^{\ensuremath{'}}/t\ensuremath{\sim}0.2.$ At this critical value, there is a transition to an antiferromagnetic phase that depends on the lattice: a Q=(Q,0)\mathbf{Q}=(Q,0)Q=(Q,0) spiral order, which continuously evolves with ${t}^{\ensuremath{'}}$, for the triangular lattice and the usual $\mathbf{Q}=(\ensuremath{\pi},\ensuremath{\pi})$ N\'eel order for the square lattice. Remarkably, the local magnetization takes its classical value for all considered ${t}^{\ensuremath{'}}$ (${t}^{\ensuremath{'}}/t\ensuremath{\le}1$). Our results show that the recently found classical kinetic antiferromagnetism, a perfect counterpart of Nagaoka ferromagnetism, is a generic phenomenon in these kinetically frustrated electronic systems.