Dimensional crossover in the SU(4) Heisenberg model in the six-dimensional antisymmetric self-conjugate representation revealed by quantum Monte Carlo and linear flavor-wave theory

Using linear flavor-wave theory (LFWT) and auxiliary field quantum Monte Carlo (QMC), we investigate the properties of the SU(4) Heisenberg model on the anisotropic square lattice in the fully antisymmetric six-dimensional irreducible representation, a model that describes interacting fermions with four flavors at half-filling. Thanks to the calculations on very large systems, we have been able to convincingly demonstrate that QMC results are consistent with a small but finite antiferromagnetic moment at the isotropic point, in qualitative agreement with LFWT obtained earlier [F. H. Kim et al., Phys. Rev. B 96, 205142 (2017)], and in quantitative agreement with results obtained previously on the Hubbard model [D. Wang et al., Phys. Rev. Lett. 112, 156403 (2014)] after extrapolation to infinite $U/t$. The presence of a long-range antiferromagnetic order has been further confirmed by showing that a phase transition takes place into a valence-bond solid (VBS) phase not too far from the isotropic point when reducing the coupling constant along one direction on the way to decoupled chains.

Figure 1

Illustrations of the (a) Néel-like pattern with an ordering vector $\mathbf{q}=(\pi ,\pi )$ and (b) VBS configuration with an ordering vector $\mathbf{q}=(\pi ,0)$ for the Mott-insulating state of SU(4) fermions with two particles per site. The flavors $A,B,C,\text{and}D$ of the fermions are represented by the colors blue, yellow, red, and green, respectively. The horizontal lines represent the intrachain coupling $J_x$ whereas the vertical dashed lines represent the interchain coupling $J_y$ that controls the dimensional crossover. The grey ellipse-shape objects in (b) indicate the strongly entangled pairs of sites.

Figure 2

Weight diagram of the six-dimensional antisymmetric $\mathrm{SU}\left(4\right)$ irrep. The flavors $A,B,C,\text{and}D$ are represented by the colors blue, yellow, red, and green, respectively, as in Fig 1. The weight diagram is in three dimensions in the Cartan-Weyl basis $\{{\mathrm{H}}_1\propto {\widehat{S}}_A^A-{\widehat{S}}_B^B,{\mathrm{H}}_2\propto {\widehat{S}}_A^A+{\widehat{S}}_B^B-2{\widehat{S}}_C^C,{\mathrm{H}}_3\propto {\widehat{S}}_A^A+{\widehat{S}}_B^B+{\widehat{S}}_C^C-3{\widehat{S}}_D^D\}$, since $\mathrm{SU}\left(4\right)$ is a group of rank 3, and its vertices form an octahedron. The plane with circles is beneath the plane with dots. The states $\overline{AB}$ and $\overline{CD}$ are antipodal and are farthest from each other.

Figure 3

Magnetization calculated from LFWT, as a function of $J_y$, while keeping $J_x=1$. The magnetization below $J_y^c=0.279$ is negative, suggesting that the order is completely destroyed below this value.

Figure 4

QMC results for the isotropic SU(4) Heisenberg model, $J_x=J_y=1$. (a) Spin-spin and (b) VBS correlation functions $S_{\text{VBS}}\left(\mathbf{q}\right)={\sum }_{\delta }{\left[S_{\text{VBS}}\left(\mathbf{q}\right)\right]}_{\delta ,\delta }$ for various lattice sizes along a high symmetry path in the Brillouin zone. The projection parameter $\mathrm{\Theta }$ grows as a function of system size $L$ so to guarantee that we have indeed converged to the ground state. (c) Spin correlation ratio. For each system size, we have checked convergence in the projection parameter. This quantity grows but shows no clear saturation to unity for lattice sizes up to $40\times 40$. The data are consistent with a small local moment. (d) Spin-spin correlation at the antiferromagnetic wave vector divided by the volume, the ordered moment corresponds to $m^2=S_{\text{Spin}}\left(\mathbf{Q}\right)/L^2$. For each system size, we have checked for convergence in $\mathrm{\Theta }$. Extrapolation of converged results support a small but finite local moment m.

Figure 5

(a) VBS and (b) spin correlation ratios as a function of $J_y$, while keeping $J_x=1$. The crossing in the spin and VBS channels are slightly shifted. From the VBS data, one would have: $J_y^c\simeq 0.76\text{–}0.78$, whereas for the spin $J_y^c\simeq 0.74\text{–}0.76$. Given the overall scatter of the crossing point, this difference is not significant enough to claim two separate transitions. (c) $\frac{1}{L^2}\frac{\partial E_0}{\partial J_y}$ shows no jump, thereby supporting a continuous transition.

Figure 6

Extrapolation of $m$ as a function of $\frac{t}{U}$ using data extracted from Fig. 4 of Ref. [37]. In the limit $U/t\to \infty$, the linear fit in $t/U$ yields the magnetization value $m=0.125\pm 0.044$. This extrapolated value compares favorably with our estimate of Eq. (41).