SU$\left(N\right)$ magnetism in chains of ultracold alkaline-earth-metal atoms: Mott transitions and quantum correlations

We investigate one-dimensional SU$\left(N\right)$ Hubbard chains at zero temperature, which can be emulated with ultracold alkaline-earth-metal atoms, by using the density matrix renormalization group (DMRG), Bethe ansatz (BA), and bosonization. We compute experimental observables and use the DMRG to benchmark the accuracy of the Bethe ansatz for $N>2$ where the BA is only approximate. In the worst case, we find a relative error $\varepsilon \lesssim 4\%$ in the BA ground-state energy for $N⩽4$ at filling $1/N$, which is due to the fact that BA improperly treats the triply and higher occupied states. Using the DMRG for $N⩽4$ and the BA for large $N$, we determine the regimes of validity of strong- and weak-coupling perturbation theory for all values of $N$ and, in particular, the parameter range in which the system is well described by a SU$\left(N\right)$ Heisenberg model at filling $1/N$. We find this depends only weakly on $N$. We investigate the Berezinskii-Kosterlitz-Thouless phase transition from a Luttinger liquid to a Mott insulator by computing the fidelity susceptibility and the Luttinger parameter $K_{\rho }$ at $1/N$ filling. The numerical findings give strong evidence that the fidelity susceptibility develops a minimum at a critical interaction strength which is found to occur at a finite positive value for $N>2$.

Figure 1

Bethe ansatz results for the energies in the thermodynamic limit for $N=2,3,4,5$ for density $n=1/2$ [$1/\left(2N\right)$ filling, top] and $n=1$ [$1/N$ filling, bottom]. The dashed lines indicate the weak-coupling and the strong-coupling limits.

Figure 2

Relative error between DMRG and BA for $N=2,3,4$ at densities $n=0.5$ [$1/\left(2N\right)$ filling, top] and at $n=1$ [$1/N$ filling, bottom].

Figure 3

Triple or higher occupancy ($P_{⩾3}$) vs $N$ for various densities $n$.

Figure 4

Finite size extrapolation of $\chi (U=0)$ for $N=2,3,4$. The horizontal lines show the exact values of Table . For $N=3$ and $N=4$ we have not taken into account the results for the largest system sizes ($L=96$ and $L=36$, respectively) due to a discarded weight of similar order of magnitude as $1-\mathcal{F}\left(U\right)$. The black lines show a quadratic fit for $N=2$ and $N=3$ and a linear fit for $N=4$.

Figure 5

Fidelity susceptibility for different system sizes for $N=2$ and $N=3$. The black continuous line shows the value after extrapolating to the thermodynamic limit using the system sizes shown.

Figure 6

DMRG results for the absolute value of the ground-state energy per site $e_0$ in the thermodynamic limit. Top: weak-coupling regime. The black lines are linear fits to the energy using the first two data points. Bottom: crossover region to the strong-coupling regime (log-log scale). The black lines are fits to a function $\sim t^2/U$ using the last two data points.

Figure 7

Black lines (top two curves in each panel) indicate where relative error of the ground-state energy is within 1$\%$ (lower black line) and 3$\%$ (upper black line) of the $t^2/U$ asymptote. Blue lines (bottom two curves in each panel) indicate where the relative error of the ground-state energy is within 1$\%$ (upper blue line) and 3$\%$ (lower blue line) of the weak-coupling perturbation theory. The red curve (extra middle line in bottom panel) indicates the critical $U_c$ for the Mott transition within the Bethe ansatz [99]. Top: $1/\left(2N\right)$ filling ($n=0.5$), Bottom: $1/N$ filling ($n=1$).

Figure 8

Site averaged double, triple, quadruple, and quintuple occupancies as a function of $U/t$ for small systems at filling (a) $n=1/\left(2N\right)$, and (b) $n=1/N$. The system sizes in (a) are $L=24$ for $N=2,3,4$, and $L=20$ for $N=5$. The system sizes in (b) are $L=20$ for $N=2,4$, $L=21$ for $N=3$, and $L=10$ for $N=5$. The black straight lines are a guide to the eyes showing the power-law behavior. The insets show the double occupancy as a function of $U/t$ for the different values of $N$.

Figure 9

Comparison of spin correlation functions [Eqs. (7) and (6)] of the $1/N$ filled SU$\left(N\right)$ Hubbard and SU$\left(N\right)$ Heisenberg chains at $U=2t,8t,12t$ for $N=2$, 3, 4.

Figure 10

Distance $d$ [Eq. (25)] between the spin correlation functions of the $1/N$ filled Hubbard and of the Heisenberg chains [Eqs. (7) and (6)] as a function of $U/t$ for $N⩽4$.

Figure 11

Spin correlation functions [Eq. (7)] of SU$\left(N\right)$ Hubbard chains at half filling $n=0.5$ for the same parameters as in Fig. 9.

Figure 12

Structure factors of the various correlation functions Eqs. (7, 8, 9, 10, 11). (a) Charge structure factor [Eq. (10)] for $U=t$. (b) Spin structure factor [Eq. (7)] for $U=t$. (c) Structure factor ${\mathcal{N}}_{\alpha ,\alpha }\left(k\right)$ [Eq. (8)] for $U=t$. (d) Structure factor ${\mathcal{N}}_{\alpha ,\beta }\left(k\right)$ [Eq. (9)] for $U=t$. (e) Momentum distribution function [Eq. (11)] for $U=t$. (f)–(j): The same quantities, but for $U=15t$.

Figure 13

$K_{\rho }\left(U\right)$ as obtained for systems with $L=48$ sites via Eq. (24) for $N=2,3,4$.