Interaction Effects with Varying $N$ in $\mathrm{SU}(N)$ Symmetric Fermion Lattice Systems

The interaction effects in ultracold Fermi gases with $\mathrm{SU}(N)$ symmetry are studied nonperturbatively in half filled one-dimensional lattices by employing quantum Monte Carlo simulations. We find that, as $N$ increases, weak and strong interacting systems are driven to a crossover region, but from opposite directions as a convergence of itinerancy and Mottness. In the weak interaction region, particles are nearly itinerant, and interparticle collisions are enhanced by $N$, resulting in the amplification of interaction effects. In contrast, in the strong coupling region, increasing $N$ softens the Mott-insulating background through the enhanced virtual hopping processes. The crossover region exhibits nearly $N$-independent physical quantities, including the relative bandwidth, Fermi distribution, and the spin structure factor. The difference between even-$N$ and odd-$N$ systems is most prominent at small $N$’s with strong interactions, since the odd case allows local real hopping with an energy scale much larger than the virtual one. The above effects can be experimentally tested in ultracold atom experiments with alkaline-earth(-like) fermions such as ${}^{87}\mathrm{Sr}$ (${}^{173}\mathrm{Yb}$).

Figure 1

The relative bandwidth $W_R$ for even (a) and odd (b) $N$ at $\beta =30$. In both cases, the dashed lines shown as a guide from top to bottom correspond to $U/t=0.5,1.0,2.0,3.0,5.0,7.0,9.0,11.0,13.0,15.0,17.0,19.0$, respectively. The crossover lines with $U/t\approx 2$ (marked red) separating the weak and strong interaction regions are nearly $N$ independent. (a) For even $N$, $W_R$ decreases with $N$ in the weak interaction region, while it increases in the strong interaction region. (b) For odd $N$, the behavior for weak interactions is similar to (a). However, in the strong interaction regime, $W_R$ is nonmonotonic, first decreasing and then increasing with $N$.

Figure 2

Momentum distribution $n(k)$ in the strong interaction region for even (a) and odd (b) values of $N$. Parameter values are $U/t=15$ and $\beta =30$. All curves cross at $n(k_f=\pi /2)=1/2$. (a) For even $N$’s, $n(k)$ is driven towards the weak interaction distribution limit as $N$ increases. (b) For odd $N$’s, $n(k)$ exhibits nonmonotonic behavior as $N$ increases, which is consistent with that of the $W_R$ shown in Fig. 1.

Figure 3

The $N$ dependence of $S(Q=\pi )$ at $\beta =30$. (a) At $U/t=15$ (strong interaction), $S(\pi )$ for $N=2m-1$ and $2m$ are close, and the latter is slightly larger. As $N$ increases, $S(\pi )$ drops rapidly showing the suppression of AFM correlations. (b) Similar to $W_R$, $S(\pi )$ exhibits opposite $N$ dependence in the weak and strong interaction regions. The crossover occurs around $U/t\approx 2$, consistent with $W_R$ shown in Fig. 1.

Figure 4

(a) The $T-U$ relations during the adiabatic process with a fixed specific entropy $\mathcal{S}/k_B=0.3$. As $U$ increases to the strong interaction region, the isoentropy curves with even and odd $N$’s behave differently, and they merge together from opposite directions in the large-$N$ limit. (b) The dimer correlation length ${\xi }_d$ vs the specific entropy $\mathcal{S}$. The correlation length increases with $N$ in general. The odd-$N$ systems (red) overall exhibit stronger dimerization instability than the even-$N$ ones (blue). (Inset) The real space correlation function $C_d(i,j)$ is shown for $N=7$ and $N=8$ for comparison at $\mathcal{S}/k_B=0.2$.