Frustration-induced itinerant ferromagnetism of fermions in optical lattices

When the Fermi Hubbard model was first introduced 60 years ago, one of the original motivations was to understand the correlation effects in itinerant ferromagnetism. In the past two decades, ultracold Fermi gas in an optical lattice has been used to study the Fermi Hubbard model. However, the metallic ferromagnetic correlation was observed only in a recent experiment using frustrated lattices, and its underlying mechanism is not yet clear. In this paper, we point out that, under the particle-hole transformation, the single-particle ground state can exhibit double degeneracy in such a frustrated lattice. Therefore, the low-energy state exhibits valley degeneracy, reminiscent of multiorbit physics in ferromagnetic transition metals. The local repulsive interaction leads to the valley Hund's rule, responsible for the observed ferromagnetism. We generalize this mechanism to distorted honeycomb lattices and square lattices with flux. This mechanism was first discussed by Müller-Hartmann in a simpler one-dimensional model. However, this mechanism has not been widely discussed and still needs to be related to experimental observations. Hence, our study not only explains the experimental findings but also enriches our understanding of itinerant ferromagnetism.

Figure 1

(a) The tunable optical lattice realized in a recent experiment. (b) The ferromagnetic correlation found in the experiment is marked by the shaded yellow regime in the $n-t^{\prime }$ phase diagram, which is mapped to the shaded blue area under the particle-hole transformation. (c) Honeycomb lattice with nearest-neighbor and horizontal next-nearest-neighbor hopping. (d) Square lattice with nearest-neighbor hopping only, but with magnetic flux in each plaquette. The red lines in (a) and (c) denote the one-dimensional chain considered by Müller-Hartmann's paper.

Figure 2

(a) Two-body problem in a $4\times 4$ lattice with an open boundary condition. The singlet and triplet state energies as a function of $t^{\prime }$. (b) The total spin $S_{\mathrm{tot}}$ and nearest-neighbor correlation function ${\langle \mathbf{S}·\mathbf{S}\rangle }_{\mathrm{n}.\mathrm{n}}$ of the ground state calculated by the DMRG method for $N=30$ particles in a $4\times 20$ strip. Here, $S_{\mathrm{tot},\max }=N/2=15$. (c), (d) The total spin $S_{\mathrm{tot}}$ as a function of $n$ and $t^{\prime }$, normalized by the maximum total spin of the filling $S_{\mathrm{tot},\max }=N/2$, for (c) $2\times 20$ and (d) $4\times 20$ strips, respectively. (e), (f) The nearest-neighbor correlation function ${\langle \mathbf{S}·\mathbf{S}\rangle }_{\mathrm{n}.\mathrm{n}}$ as a function of $n$ and $t^{\prime }$, normalized by the particle number $N$, for $2\times 20$ and $4\times 20$ strips, respectively. All DMRG calculations have bond dimension $\chi =1400$.

Figure 3

(a) Single-particle dispersion $\mathcal{E}(k_x,k_y)$ of the Hamiltonian Eq. (1) for three different values of $t^{\prime }$. $t^{\prime }=-0.2$ (left), $-0.5$ (middle), and $-0.8$ (right). (b) The real and imaginary parts of two valley Wannier wave functions. These two wave functions are constructed by using the Bloch wave functions around each minimum of single-particle dispersion, as indicated by the solid circle in the right dispersion in (a).

Figure 4

(a) The ferromagnetic regime indicated by DMRG calculation in the $n-t^{\prime }$ phase diagram of the distorted honeycomb lattice. The model is shown in Fig. 1, in which $t^{\prime }$ denotes the next-nearest hopping along the horizontal dashed line. The DMRG calculation is performed in a $3\times 6$ zigzag cylinder, with bond dimension $\chi =1400$. The dashed line indicates the value of $t^{\prime }$ for the Lifshitz transition. (b) The single-particle dispersion of the distorted honeycomb lattice, with $t^{\prime }=-0.1$ (left), $=-0.25$ (middle), and $=-0.4$ (right).

Figure 5

(a) $S_{\mathrm{tot}}$ obtained by DMRG calculation. $S_{\max }=N/2$. Here, we consider a square lattice with nearest hopping only, but with magnetic flux $\varphi$. $\varphi =0$, $\pi$, $2\pi /3$, $\pi /2$, and $2\pi /5$ from the top to the bottom. The DMRG calculation is performed in a $4\times 10$ strip with bond dimension $\chi =1400$. (b) The single-particle dispersion around the bottom of the band for different magnetic flux corresponding to (a).