$\text{SU}\left(N\right)$ Heisenberg model on the square lattice: A continuous-$N$ quantum Monte Carlo study

A quantum phase transition is typically induced by tuning an external parameter that appears as a coupling constant in the Hamiltonian. Another route is to vary the global symmetry of the system, generalizing, e.g., SU(2) to $\text{SU}\left(N\right)$. In that case, however, the discrete nature of the control parameter prevents one from identifying and characterizing the transition. We show how this limitation can be overcome for the $\text{SU}\left(N\right)$ Heisenberg model with the help of a singlet projector algorithm that can treat $N$ continuously. On the square lattice, we find a direct, continuous phase transition between Néel-ordered and crystalline bond-ordered phases at $N_{\text{c}}=4.57\left(5\right)$ with critical exponents $z=1$ and $\beta /\nu =0.81\left(3\right)$.

Figure 1

(Color online) (a) An A-sublattice and a B-sublattice spin can pair to form a singlet, which corresponds to a column of zero (modulo $N$) boxes. (b) Update rules for the action of $H_{ij}$ (red line) on VB singlet states (black bonds).

Figure 2

(Color online) Loop diagrams contributing to two- and four-spin correlators [${\mathbf{S}}_i\cdot {\mathbf{S}}_j$ and $\left({\mathbf{S}}_i\cdot {\mathbf{S}}_j\right)\left({\mathbf{S}}_k\cdot {\mathbf{S}}_l\right)$, respectively] and their contributions. These overlap loops (schematically circular here) are obtained by superimposing the VB configurations $|v_1⟩$ and $|v_2⟩$. The straight (red) lines link the corresponding sites of the correlation function. Only diagrams with nonvanishing contributions are displayed.

Figure 3

(Top-left panel) Square of the staggered magnetization $\mathbf{M}=\frac{1}{L^2}{\sum }_{\mathbf{r}}{\left(-1\right)}^{r_x+r_y}{\mathbf{S}}_{\mathbf{r}}$ as a function of $N$ for systems up to linear size $L=48$ (lines are guide to the eyes). (Top-right panel) The Binder cumulant $U$ measures the kurtosis of the staggered magnetization with respect to a purely Gaussian distribution. It vanishes in the limit $L\to \infty$ in the absence of antiferromagnetic order. (Bottom panel) A histogram of the magnetic observable is shown for values of $N$ spanning the transition. The distribution has a single peak.

Figure 4

(Color online) Snapshots of typical VB configurations for a system of size $L=32$. Short, nearest-neighbor bonds are drawn with a line; long bonds are indicated by circles (whose area is proportional to bond length) at their end points. For $N<N_{\text{c}}$, many long bonds stretch across the system. For $N>N_{\text{c}}$, short bonds dominate. The shaded (pink) area marks a crystal domain with columnar bond order.

Figure 5

(Color online) Density plots of histograms $P\left(D_x,D_y\right)$ of the $x$ and $y$ components of the dimer order parameter $\left(L=32\right)$. On the magnetic side of the transition (e.g., $N=4.5<N_{\text{c}}$), the distribution is characterized by a central peak. On the VBC side, the distribution is ringlike but develops additional weight along the main axes as $N$ is increased.

Figure 6

(Color online) Simultaneous data collapse of the Néel and dimer correlations can be achieved with a single set of critical exponents. The left panel shows the QMC measurements plotted in rescaled coordinates with the values $\nu =0.88$, $\beta =0.71$, and $N_{\text{c}}=4.57\left(5\right)$. Other values of $\beta /\nu \sim 0.8$ produce good data collapse. The right panel shows bootstrapped results of $\beta /\nu$ versus $N_{\text{c}}$ when fits of $M^2$ and $D^2$ are performed independently. A single critical point corresponds to the crossing $\beta /\nu =0.81\left(3\right)$ and $N_{\text{c}}=4.57\left(5\right)$.

Figure 7

The spin gap ${\Delta }_{\text{s}}$ versus the inverse linear size $1/L$ close to the critical point. The dashed line indicates a linear fit for $N=4.6$ that extrapolates very close to 0 in the thermodynamic limit.