Quantum magnetism in ultracold alkali and alkaline-earth fermion systems with symplectic symmetry

We numerically study the quantum magnetism of ultracold alkali and alkaline-earth fermion systems with large hyperfine spin $F=\frac{3}{2}$, which are characterized by a generic $Sp\left(N\right)$ symmetry with $N=4$. The methods of exact diagonalization (ED) and density matrix renormalization group are employed for the large size one-dimensional (1D) systems, and the ED method is applied to a two-dimensional (2D) square lattice on small sizes. We focus on the magnetic exchange models in the Mott-insulating state at quarter-filling. Both 1D and 2D systems exhibit rich phase diagrams depending on the ratio between the spin exchanges $J_0$ and $J_2$ in the bond spin singlet and quintet channels, respectively. In one dimension, the ground states exhibit a long-range-ordered dimerization with a finite spin gap at $J_0/J_2>1$ and a gapless spin-liquid state at $J_0/J_2⩽1$, respectively. In the former and latter cases, the correlation functions exhibit the two-site and four-site periodicities, respectively. In two-dimensions, various spin-correlation functions are calculated up to the size of $4\times 4$. The Néel-type spin correlation dominates at large values of $J_0/J_2$, while a $2\times 2$ plaquette correlation is prominent at small values of this ratio. Between them, a columnar spin-Peierls dimerization correlation peaks. We infer the competition among the plaquette ordering, the dimer ordering, and the Néel ordering in the 2D system.

Figure 1

Phase diagram of the 1D chain in terms of the singlet and quintet channel interaction $J_0$ and $J_2$. In this context, $\theta$ is the angle defined by $\theta ={\tan }^{-1}(J_0/J_2)$. The $SU{\left(4\right)}_A$ type ($\theta ={45}^{\circ }$) denoted by the dotted line belongs to the gapless spin-liquid state whereas $SU{\left(4\right)}_B$ is along $J_2=0$. The phase boundary separating the dimerization phase and the gapless liquid state is the $SU{\left(4\right)}_A$ line.

Figure 2

The exact diagonalization on a 1D chain with 12 sites for (a) open and (b) periodic boundary conditions. The dispersion of the (GS) and low excited states and the dimensions d of their corresponding representations of the $Sp\left(4\right)$ group are shown.

Figure 3

Exact diagonalization results on the $Sp\left(4\right)$ singlet-singlet gap with $J_0>J_2$ and periodic boundary conditions ($\theta ={60}^{\circ }$ and ${75}^{\circ }$ with $N=8,12$ and 16). Finite size scaling shows the vanishing of the singlet-singlet gap ${\Delta }_{ss}$.

Figure 4

The finite-size scaling of the spin gap ${\Delta }_{sp}$ of the $Sp\left(4\right)$ spin chain vs. $1/N$ at various values of $\theta$. $\theta$ is defined as $\theta ={\tan }^{-1}(J_0/J_2)$ and $N$ is the system size.

Figure 5

The NN correlation of $\langle L_{15}\left(i\right)L_{15}(i+1)\rangle$ with OBCs for (a) $\theta ={60}^{\circ }$ and (b) $\theta ={30}^{\circ }$, respectively. The dimer ordering is long ranged in (a). Note the two-site periodicity in (a) and the four-site periodicity in (b).

Figure 6

The finite-size scaling for the dimer order parameters (a) $D_{L_{15}}$ and (b) $D_{n_4}$ vs. $1/N$ at various $\theta$’s.

Figure 7

(a) The two point correlations $\langle L_{15}\left(i_0\right)L_{15}\left(i\right)\rangle$ at $\theta =0^{\circ }$,${15}^{\circ }$, ${30}^{\circ }$, ${45}^{\circ }$, and ${60}^{\circ }$. The dotted line is plotted by the fitting result using $\cos (x\pi /2)/x.{}^{1.52}$ The reference point $i_0$ ($=40$) is the most middle site of the chain ($N=80$). The inset indicates that all $S\left(q\right)$ for $\theta ⩽{45}^{\circ }$ have peaks located at $q=41\pi /81\sim \pi /2$ whereas $\pi$ for $\theta ={60}^{\circ }$. (b) $\langle n_4\left(i_0\right)n_4\left(i\right)\rangle$ at $\theta =0^{\circ }$, ${15}^{\circ }$, ${30}^{\circ }$, and ${60}^{\circ }$ and the fitting uses $\kappa =1.55$.

Figure 8

Speculated phase diagram of the 2D $Sp\left(4\right)$ spin-$3/2$ systems at quarter filling from Ref.4. $\theta ={\tan }^{-1}(J_0/J_2)$. The $SU{\left(4\right)}_A$ type drawn by the dotted line is at $J_0=J_2$ ($\theta ={45}^{\circ }$) whereas $SU{\left(4\right)}_B$ at $J_2=0$ ($\theta ={90}^{\circ }$). Bold letters A, B, and C represent the plaquette, columnar dimerized, and Néel-order states, respectively.

Figure 9

(a) The GS wave functions of the $2\times 2$ cluster at various $\theta$. $a$ and $b$ are coefficients depending on $\theta$ and the thick bonds denote the two-site $Sp\left(4\right)$ spin singlet states. (b) The position indices before and after the permutation $P_{2341}$.

Figure 10

(a) The low-lying states for the $4\times 4$ cluster at various values of $\theta$. The dimensions of the corresponding $Sp\left(4\right)$ representations $d$ are marked. The GS wave functions are always $Sp\left(4\right)$ singlets. (b) The zooming-in around $\theta \approx {63}^{\circ }$ exhibiting various energy level crossings.

Figure 11

The energy dispersion $E\left(\vec{K}\right)-E_0$ vs. $\theta$ for the $4\times 4$ cluster. $\Gamma$, $M,X$ are the high symmetry points for the many-body GS momenta, corresponding to $\vec{K}=(0,0)$, $(\pi ,\pi )$, and $(\pi ,0)$ respectively, in the first Brillouin zone.

Figure 12

The magnetic structure factors for the $4\times 4$ cluster. (a) The structure factors $S_L\left(\vec{q}\right)$ for the $Sp\left(4\right)$ generator sector. The inset is the comparison between $S_L(\pi ,0)$ and $S_L(\pi ,\pi )$ versus $\theta$. (b) The $Sp\left(4\right)$ vector structure factor $S_n\left(\vec{q}\right)$.

Figure 13

The susceptibilities defined in Eq. (24) with respect to the perturbations $O_{dim}$ and $O_{rot}$ for the $N=4\times 4$ cluster. (a) ${\chi }_{dim}\left(\vec{Q}\right)$ vs. $\theta$ at $\vec{Q}=(\pi ,0)$; (b) ${\chi }_{rot}$ vs. $\theta$; (c) ${\chi }_{dim}\left(\vec{Q}\right)$ vs. $\theta$ at $\vec{Q}=(\pi ,\pi )$. In both cases, a small value of $\delta =0.01$ is taken to evaluate the susceptibilities. Both susceptibilities exhibit peaks around $\theta \approx {60}^{\circ }\sim {70}^{\circ }$.

Figure 14

$C\left(\vec{r}\right)$ defined in Eq. (27) versus $\theta$. The positions of plaquette $A$, $B$, and $C$ are defined on the right schematics. $A$ (blue) is located at the corner whereas $C$ (black) in the most middle one.

Figure 15

The Young patterns of the $SU\left(4\right)$ representations 1 to 8 presented in Table .