Phase diagram of the $\mathrm{SU}\text{(}N\text{)}$ antiferromagnet of spin $S$ on a square lattice

We investigate the ground-state phase diagram of an $\text{SU}\left(N\right)$-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of $\text{SU}\left(N\right)$ described by a square Young tableau of $N/2$ rows and $2S$ columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any $S$ and even values of $N$. In the large-$N$ limit, the saddle point approximation favors a fourfold degenerate valence bond solid phase. In the large $S$ limit, the semiclassical approximation points to the Néel state. On a line set by $N=8S+2$ in the $S$ versus $N$ phase diagram, we observe a variety of phases proximate to the Néel state. At $S=1/2$ and $3/2$, we observe the aforementioned fourfold degenerate valence bond solid state. At $S=1$, a twofold degenerate spin nematic state in which the $C_4$ lattice symmetry is broken down to ${\mathrm{C}}_2$ emerges. Finally, at $S=2$ we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry-protected topological state is characterized by an SU(18), $S=1/2$ boundary state that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular, one expects spin-disordered states with degeneracy set by $\text{mod}(4,2S)$.

Figure 1

Ground-state phase diagram of the $\text{SU}\left(N\right)$-antiferromagnet model Eq. (1) on the square lattice, as obtained from QMC simulations. $S$ identifies the chosen representation of the $\mathfrak{su}\left(N\right)$ algebra of SU($N$), illustrated by the Young tableau in Fig. 2. Striped regions indicate the part of the phase diagram where current QMC data do not allow an unambiguous identification of the phase; in such cases we indicate between parenthesis the most likely identified order. The insets show QMC data in the highlighted dimerized phases, obtained through a pinning-field approach (see Sec. 4a).

Figure 2

Young tableau corresponding to the irreducible representation of $\mathfrak{su}\left(N\right)$ considered here; $N$ is even and $S$ is semi-integer.

Figure 3

Decomposition of the tensor product of $2S$ antisymmetric self-adjoint representations, whose maximum Dynkin weight is given in Eq. (6), into irreducible ones.

Figure 4

Sketch of the structure of the QMC Hamiltonian for $2S=2$. ${\widehat{H}}_U$: Hubbard term for freezing out charge degrees of freedom. ${\widehat{H}}_{\text{Casimir}}$: Term for maximizing the eigenvalue of the Casimir operator. ${\widehat{H}}_J$: Antiferromagnetic interaction between elemental spins.

Figure 5

Suppression of the charge fluctuations $\langle {({\widehat{n}}_{i,1}-N/2)}^2\rangle$ as a function of $\beta U$ for $N=2$ and $S=1/2$. We consider a Hamiltonian containing the Hubbard term only and the case of a model with an antiferromagnetic interaction [Eq. (8)], with coupling constant $J=1$ on a system size $L=4$, and for different inverse temperatures $\beta$. The charge fluctuations fall off asymptotically as $exp(-\beta U)$ [Eq. (21)].

Figure 6

Difference between the Casimir eigenvalue $C(N=2,S=1)$ [Eq. (22)] of the representation $S=1$, $N=2$, and the sampled one $\langle \widehat{C}\rangle$, as a function of the effective interaction strength $\beta J_H$ [Eq. (12)] with $J=0$, and for four inverse temperatures. Data shown are obtained for a lattice of size $L=4$, with a Hubbard interaction $\beta U=6$ [Eq. (17)], vanishing nearest-neighbor antiferromagnetic interaction $J=0$, and a Trotter discretization $\mathrm{\Delta }\tau =0.1$.

Figure 7

Casimir eigenvalue for representations with $S=1$ and as a function of $N$. We compare the predicted value of Eq. (22) with the sampled Casimir eigenvalue from QMC simulations of a lattice with size $L=4$, with a Hubbard interaction $U=2$ [Eq. (17)], antiferromagnetic coupling $J=1$ [Eq. (1)], projection strength $J_H=1$ [Eq. (12)], and a Trotter discretization $\mathrm{\Delta }\tau =0.1$. In the inset, we plot the difference between the sampled and expected value.

Figure 8

Sketch of possible dimerized ground states. (a)–(c) The VBS states. Each of those states is fourfold degenerate, the corresponding states can be obtained by rotations and translations. (d) A Haldane nematic state, an equivalent state is obtained by rotations of ${90}^{\circ }$. (e) A unique ground state. States (d) and (e) are at best understood within an AKLT construction, in which, very much as done in our calculation, the spin $S$ on each site is constructed by a totally symmetric superposition of $2S$ states that are denoted by bullets around each site on the right-hand side of (d) and (e). These bullets correspond to an irreducible representation of $\mathfrak{su}\left(N\right)$, with one column ($S=1/2$) and $N/2$ rows. In the nematic state, each spin-$1/2$ forms a singlet with the nearest neighbor along the axis of the broken symmetry. The AKLT state is relevant for the $S=2$ state, where each spin-$1/2$ on a given site can be combined into a singlet with a nearest-neighbor spin-$1/2$ without breaking a lattice symmetry.

Figure 9

Correlation functions $S\left(\mathit{k}\right)$ [Eq. (25)] and $D_{ij}\left(\mathit{k}\right)$ [Eq. (25)] for the representation of Fig. 2 with $S=1/2$ and different values of $N$.

Figure 10

Correlation ratios of the staggered magnetization $m$ [Eq. (27)], $\varphi$ [Eqs. (28)] and $\psi$ [Eqs. (29)] order parameters for $S=1/2$, as a function of $N$, and for lattice sizes $L=4-16$.

Figure 11

Real-space value of bonds $B_i\left(\mathit{r}\right)$ [Eq. (33)], as measured after pinning the central bond, for $S=1/2$ and lattice size $L=18$. Due to the observed different variations in the bond strength, to better highlight the patterns of bond correlations we have used different color scales for the region close and far from the pinned bond. The biggest error of the outer bonds is indicated by two red lines on the color scale. We have symmetrized the results with regards to inversions $y\to -y$, $x\to -x$ around the pinned bond.

Figure 12

Same as Fig. 9 for $S=1$

Figure 13

Same as Fig. 10 for $S=1$.

Figure 14

Same as Fig. 11 for $S=1$.

Figure 15

Same as Fig. 9 for $S=3/2$.

Figure 16

Same as Fig. 10 for $S=3/2$.

Figure 17

Same as Fig. 11 for $S=3/2$ and lattice size $L=14(N=14,16)$, $L=12(N=18)$.

Figure 18

Same as Fig. 9 for $S=2$.

Figure 19

Same as Fig. 10 for $S=2$.

Figure 20

Real-space value of bonds $B_i\left(\mathit{r}\right)$ for $S=2$, $N=18$, lattice size $L\in \{8,10,12\}$, and open boundary conditions in $\mathit{y}$ direction. Comparison between pinning a bond at the edge (a) and a bond in the bulk (b). (c) $B_{\mathrm{x}}\left(\mathit{r}\right)$ on horizontal lines through the pinned bonds.

Figure 21

Scaling of systematic ${\mathrm{\Delta }}_{\tau }$ error for $L=4$, $\mathrm{\Theta }=2$, and $S=1/2$. For each point, we simulated with a range of different values for $\mathrm{\Delta }\tau$ and fitted the energy to $E\left({\mathrm{\Delta }}_{\tau }\right)=E_0+\alpha {{\mathrm{\Delta }}_{\tau }}^2$.