Evolution of Nagaoka phase with kinetic energy frustrating hopping

We investigate, using the density-matrix renormalization group, the evolution of the Nagaoka state with $t^{\prime }$ hopping that frustrates the hole kinetic energy in the $U=\infty$ Hubbard model on the square and anisotropic triangular lattices. We find that the Nagaoka ferromagnet survives up to a rather small $t_c^{\prime }/t\sim 0.2.$ At this critical value, there is a transition to an antiferromagnetic phase that depends on the lattice: a $\mathbf{Q}=(Q,0)$ spiral order, which continuously evolves with $t^{\prime }$, for the triangular lattice and the usual $\mathbf{Q}=(\pi ,\pi )$ Néel order for the square lattice. Remarkably, the local magnetization takes its classical value for all considered $t^{\prime }$ ($t^{\prime }/t\le 1$). 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.

Figure 1

Triangular lattice with spatially anisotropic hopping and square lattice with second-nearest-neighbor hopping. $t^{\prime }$ is the kinetically frustrating hopping. We take $t=1$ as the energy unit.

Figure 2

Ground-state energy $E_N\left(S^z\right)$ for the triangular lattice as a function of $t^{\prime }$ and for different $N$-site clusters with $L_y=6$. Dashed (solid) lines correspond to the $S^z=S_{\max }^z$ ($S_{\min }^z$) sector energy. For smaller $t^{\prime }$ both energies overlap. $E_{\mathrm{Nag}}^{▵}$ is the thermodynamic limit of the Nagaoka state energy. Inset: zoom of the critical region, showing the intersection of $E_N\left(S_{\min }^z\right)$ with the ferromagnetic energy $E_{\mathrm{Nag}}^{▵}$. In the critical region we have used a step of $0.02t^{\prime }$ for the energy calculation, while in the main panel the step is $0.1t^{\prime }$.

Figure 3

Ground-state energy $E_N\left(S_z\right)$ for the square lattice as a function of $t^{\prime }$ and for different $N$-site clusters with $L_y=6$. Dashed (solid) lines correspond to the $S_z=S_z^{\max }$ ($S_z=\frac{1}{2}$) sector energy. $E_{\mathrm{Nag}}^{\square }$ is the thermodynamic limit of the Nagaoka state energy. Inset: zoom of the critical region, showing the intersection of $E_N\left(\frac{1}{2}\right)$ with the ferromagnetic energy $E_{\mathrm{Nag}}^{\square }$. In the critical region we have used a step of $0.02t^{\prime }$ for the energy calculation, while in the main panel the step is $0.1t^{\prime }$.

Figure 4

Intensity plot of $\mathcal{S}\left(\mathbf{k}\right)$ for the triangular lattice with (a) $t^{\prime }=0$ and (b) $t^{\prime }=t$. The solid lines indicate the edges of the hexagonal Brillouin zones. Notice the discreteness of $k_y$ values.

Figure 5

Spiral pitch $Q$ as a function of $t^{\prime }$ for the triangular lattice. Insets: intensity plots of $\mathcal{S}\left(\mathbf{k}\right)$ for $t^{\prime }=0.25$ and 0.6. The darker regions correspond to the magnetic wave vector $\mathbf{Q}=(Q,0)$ (and other equivalent points).

Figure 6

Intensity plot of $\mathcal{S}\left(\mathbf{k}\right)$ for the square lattice with (a) $t^{\prime }=0$ and (b) $t^{\prime }=t$. Solid lines refer to the edges of the square Brillouin zones.

Figure 7

Local magnetization $M_s$ as a function of $t^{\prime }$ for (a) the $N=54$ triangular lattice without a magnetic field (open triangles) and with a magnetic field $h=0.1$ applied to one ${120}^{\circ }$ sublattice (solid triangles) and (b) the $N=60$ square lattice without a magnetic field (open squares) and with a magnetic field $h=0.1$ applied to one of the two sublattices of the $(\pi ,\pi )$ Néel order (solid squares). The dashed lines correspond to the classical local magnetization, $M_s^{\mathrm{clas}}=\frac{1}{2}-\frac{1}{2N}$.