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

TL;DR

The work tackles how Nagaoka ferromagnetism on a square lattice evolves into spiral magnetism under a square-to-triangle lattice crossover implemented via diagonal hopping $t'. The authors introduce a variational mean-field framework for a single hole moving in a static spin background and analyze the spin-spiral instability by expanding the energy in the spiral wavevector $q$, obtaining an exact critical point $t'_c = 0.24 t$. They show that spin fluctuations select a spin-spiral ground state and that the transition is continuous, connecting to the $120^ ext{o}$ order in the triangular limit; the result provides a clean, analytic determination of the phase boundary in this kinetic-magnetism problem. The findings have implications for kinetic frustration in strongly correlated systems and suggest experimental routes using cold-atom optical lattices and other quantum simulators to observe the ferromagnet-to-spiral transition and associated polaronic spin textures.

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 instability, finding the exact critical point.

Figures (2)

• Figure 1: Spin-spiral instability of ground state of half-filled hard-core Fermi-Hubbard model, with a single hole. Inset shows hopping model where $t$ corresponds to hopping matrix element along cardinal directions, and $t^\prime$ along diagonals with positive slope. This interpolates between a square lattice at $t^\prime=0$ and a triangular lattice at $t^\prime=t$. Main figure shows the variational energy of a spin-spiral configuration, as calculated from Eqs. \ref{['eq:Helements1']} and \ref{['eq:Helements2']} on a small $15\times 15$ spatial grid. Vertical axis shows $q$ where the spiral has wave-vector $\boldsymbol{Q} = (q,q)$. Horizontal axis corresponds to the hopping ratio, $t^\prime/t$. For each $t^\prime/t$ a yellow dot is placed at the $q$ which minimizes the energy. The variational calculation is exact as $q\to 0$, and hence predicts the exact location of the critical point, labeled by (1): $(t_{\rm c}^\prime)_{\rm exact} = 0.24t$. This can be contrasted to the spin-wave instability (2), occurring at $(t_{\rm c}^\prime)_{\rm SW} = 0.42t$, and the mean-field prediction (3), $(t_{\rm c}^\prime)_{\rm MF} = 0.5t$. In the lower left and upper right corner, one unit cell of the $t^\prime=0$ and $t^\prime=t$ spin patterns are depicted, corresponding to the uniform ferromagnet and the 120-degree state.
• Figure 2: Variational energies of generic spin textures, parameterized by the angles $\alpha$ and $\beta$ as described in the main text. The energies are calculated by solving Eqs. \ref{['eq:NUHElements1']} and \ref{['eq:NUHElements2']} on a small $15 \times 15$ spatial grid, as a function of $\beta$ and for different choices of $\alpha$: $\pi/6$ (red dotted line), $\pi/4$ (orange dash-dotted line), $\pi/3$ (green dashed line), and $\pi/2$ (blue solid line). The minimum-energy configuration is for $\alpha=\pi/2$ and $\beta=\pi/2$, corresponding to a spin spiral.