Ground States of the $\mathrm{SU}(N)$ Heisenberg Model

The $\mathrm{SU}(N)$ Heisenberg model with various single-row representations is investigated by quantum Monte Carlo simulations. While the zero-temperature phase boundary agrees qualitatively with the theoretical predictions based on the $1/N$ expansion, some unexpected features are also observed. For $N\ge 5$ with the fundamental representation, for example, it is suggested that the ground states possess exact or approximate U(1) degeneracy. In addition, for the representation of Young tableau with more than one column, the ground state shows no valence-bond-solid order even at $N$ greater than the threshold value.

Figure 1

Three types of VBS states: (a) columnar (${\overline{D}}_x\ne 0$ and ${\overline{D}}_y=0$), (b) nematic (${\overline{D}}_x={\overline{D}}_y=0$), and (c) plaquette (${\overline{D}}_x={\overline{D}}_y\ne 0$). The quantities ${\overline{D}}_x$ and ${\overline{D}}_y$ characterize spontaneous translational-symmetry-breaking in the $x$ and $y$ directions, respectively. [See Eq. (4) for the definition.]

Figure 2

The correlation ratio, $R_{\Sigma }(L)$, of the magnetization for the fundamental representation ($n=1$).

Figure 3

The two-point correlation function the rotational-symmetry-breaking order parameter, $A$, for the model with the fundamental representation ($n=1$).

Figure 4

The squared difference of the average bond strengths in the $x$ and $y$ directions for the fundamental representation. The straight lines that correspond to $B\propto 1/L^2$ are drawn for comparison.

Figure 5

The frequency of observing a pair of values $({\overline{D}}_x,{\overline{D}}_y)$ during the Monte Carlo simulation for $N=6$ at $L=32$ and $\beta =16$. The value at each pixel is the average over eight symmetrically equivalent points, $(\pm {\overline{D}}_x,\pm {\overline{D}}_y)$ and $(\pm {\overline{D}}_y,\pm {\overline{D}}_x)$.

Figure 6

The correlation ratio of the magnetization for the representation of two-box Young tableau ($n=2$).

Figure 7

The zero-temperature phase diagram of the $\mathrm{SU}(N)$ model on a square lattice with single-row ($m=1$) representations.

Figure 8

The correlation function of the rotational-symmetry-breaking order-parameter $A$ at the largest distance for $n=1$, 2, 3, 4. The value of $N$ for each $n$ is close to but larger than the threshold value, $N_c(n)$.