Canted antiferromagnetic order of imbalanced Fermi-Fermi mixtures in optical lattices by dynamical mean-field theory

We investigate antiferromagnetic order of repulsively interacting fermionic atoms in an optical lattice by means of dynamical mean-field theory (DMFT). Special attention is paid to the case of an imbalanced mixture. We take into account the presence of an underlying harmonic trap, both in a local-density approximation and by performing full real-space DMFT calculations. We consider the case in which the particle density in the trap center is at half-filling, leading to an antiferromagnetic region in the center, surrounded by a Fermi liquid region at the edge. In the case of an imbalanced mixture, the antiferromagnetism is directed perpendicular to the ferromagnetic polarization and canted. We pay special attention to the boundary structure between the antiferromagnetic and the Fermi liquid phase. For the moderately strong interactions considered here, no Stoner instability toward a ferromagnetic phase is found. Phase separation is only observed for strong imbalance and sufficiently large repulsion.

Figure 1

Real-space spin profile for $U/J=10$, $\Delta \mu /J=0.5$, and $V_0/J=0.05$. The vertical components of the arrows symbolize the local magnetization in the $z$ direction, $\langle {\mathrm{Ŝ}}_i^z\rangle$, while the horizontal components correspond to the (staggered) local magnetization in the $x$ direction, $\langle {\mathrm{Ŝ}}_i^x\rangle$. For all data in this paper, we took the filling at the center equal to 1, which is induced by the choice of the chemical potential equal to $\mathrm{\mu ̄}=U/2$. To obtain the ground-state expectation values, a small fictitious temperature of $T/J=0.02$ was applied.

Figure 2

Radial profiles for the ground state of two-dimensional Fermi-Fermi mixtures for different values of $U$ and $\Delta \mu$. Plotted are the total density $\langle {\mathrm{n̂}}_i\rangle =\langle {\mathrm{n̂}}_{i\uparrow }+{\mathrm{n̂}}_{i\downarrow }\rangle$ and the expectation values of the spin in the $z$ direction, $\langle {\mathrm{Ŝ}}_i^z\rangle$, and in the $x$ direction, $\langle {\mathrm{Ŝ}}_i^x\rangle$. The left column is for a balanced mixture ($\Delta \mu /J=0$), the right column is an imbalanced mixture with $\Delta \mu /J=0.5$. The other parameters are $\mathrm{\mu ̄}=U/2$ and from top to bottom: $U/J=5$, $V_0/J=0.03$; $U/J=10$, $V_0/J=0.05$; $U/J=20$, $V_0/J=0.1$. Here and in the following, spin expectation values are plotted in units of $\hslash$.

Figure 3

Radial distribution for the minority density $\langle n_{i\downarrow }\rangle$ and the in-plane staggered order $\langle {\mathrm{Ŝ}}_i^x\rangle$ (here plotted in absolute value) for large imbalance and different values of the repulsion: $U/J=10$ (left column), $U/J=20$ (right column), and $\mathrm{\mu ̄}=U/2$. Other parameters are (a) $V_0/J=0.05$, $\Delta \mu /J=1.2$; (b) $V_0/J=0.07$, $\Delta \mu /J=0.6$; (c) $V_0/J=0.05$, $\Delta \mu /J=1.6$; and (d) $V_0/J=0.07$, $\Delta \mu /J=0.9$.

Figure 4

Radial profiles for the ground state of three-dimensional Fermi-Fermi mixtures for $U/J=10$, $\mathrm{\mu ̄}=U/2$, and different values of $\Delta \mu$. Plotted are the total density $\langle {\mathrm{n̂}}_i\rangle =\langle {\mathrm{n̂}}_{i\uparrow }+{\mathrm{n̂}}_{i\downarrow }\rangle$, the expectation values of the spin in the $z$ direction, $\langle {\mathrm{Ŝ}}_i^z\rangle$, and the absolute value of the expectation value in the $x$ direction, $|\langle {\mathrm{Ŝ}}_i^x\rangle |$. The points denote results of the full R-DMFT calculation, whereas the solid lines are obtained within the LDA (TFA) approximation combined with DMFT. The values for $\Delta \mu$ are (a) $\Delta \mu /J=0$, (b) $\Delta \mu /J=0.4$, (c) $\Delta \mu /J=0.8$, (d) $\Delta \mu /J=1.2$, (e) $\Delta \mu /J=1.6$, and (f) $\Delta \mu /J=2$. The trap parameter is chosen as $V_0/J=0.15$.

Figure 5

Global observables for the three-dimensional system obtained by R-DMFT at $U/J=10$, $\mathrm{\mu ̄}=U/2$, and $V_0/J=0.15$ as a function of the chemical potential difference $\Delta \mu$. Shown are the total magnetization in the $x$ and $z$ directions, $S_{\mathrm{tot}}^{x,z}$, and the staggered magnetization in the $x$ and $z$ directions, $S_{\mathrm{stag}}^{x,z}$, all normalized to the particle number $N_{\mathrm{tot}}$.