Marshall-positive $\mathrm{SU}\left(N\right)$ quantum spin systems and classical loop models: A practical strategy to design sign-problem-free spin Hamiltonians

We consider bipartite $\mathrm{SU}\left(N\right)$ spin Hamiltonians with a fundamental representation on one sublattice and a conjugate to fundamental on the other sublattice. By mapping these antiferromagnets to certain classical loop models in one higher dimension, we provide a practical strategy to write down a large family of $\mathrm{SU}\left(N\right)$ symmetric spin Hamiltonians that satisfy Marshall's sign condition. This family includes all previously known sign-free $\mathrm{SU}\left(N\right)$ spin models in this representation and in addition provides a large set of new models that are Marshall positive and can hence be studied efficiently with quantum Monte Carlo methods. As an application of our idea to the square lattice, we show that in addition to Sandvik's $Q$ term, there is an independent nontrivial four-spin $R$ term that is sign free. Using numerical simulations, we show how the $R$ term provides a new route to the study of quantum criticality of Néel order.

Figure 1

Cartoon illustration of the various Marshall-positive interaction terms studied in the paper. (a) The decomposition of the bipartite square lattice into A and B sublattices, and the groups of sites that the various interactions act on. (b)–(e) Cartoons of the different interactions acting on the basis states; the initial state is shown at the bottom and the final state is shown on top. (b) The $J_1$ interaction that acts between sites on different sublattices. (c) The $J_2$ interaction that acts on sites on the same sublattice. (d) An example of a $Q$ interaction and (e) an example of an $R$ interaction that both act on elementary plaquettes of the square lattice. Colors of the loop represent the $N$ colors of $\mathrm{SU}\left(N\right)$. Note that loops meeting at a vertex may have the same color and that the loops always travel in opposite directions on the A and B sublattices. (f) Cartoon illustration of singlet rearrangements affected by $R$ and $Q$ terms; see discussion on Eq. (5).

Figure 2

Illustration of a path-integral or SSE contribution to the partition function of an $\mathrm{SU}\left(N\right)$ antiferromagnet with only $J_1$ interactions with seven sites at a temperature $\beta$, which may be viewed as a higher dimensional closed packed loop model. The orientation of one particular loop is followed with black arrows to show how it always travels in imaginary time in opposite directions on opposite sublattices (up on A and down on B). This orientation is preserved by the $J_1$ interaction as well as all other $\mathrm{SU}\left(N\right)$ invariant interactions. An $\mathrm{SU}\left(N\right)$ antiferromagnet will have $N$ different color assignments to the loops.

Figure 3

For completeness we show here the diagrams that correspond to two more sign-free four-spin interactions, in addition to those shown in Figs. 1 and 1. Taken together, these four terms are the complete set of sign-free four-spin interactions that can be obtained from our construction.

Figure 4

Magnetic phase transitions from the crossing of ${\rho }_s$ (stiffness) data shown for SU(3) and SU(4). In the main panels data for $L{\rho }_s$ is shown for the $J_1-R$ model as a function of $R/J_1$. Clear evidence for a crossing is found which implies the existence of a critical point where magnetic order is destroyed. The insets show the magnetic phase boundaries for the $J_1-Q-R$ model inferred from the kind of data shown in the main panel with $Q\ne 0$ (not shown). We have chosen $\beta =L$ and $J_1=1$ here.

Figure 5

Representative VBS (main panels) and magnetic (insets) order parameters for SU(3) and SU(4) for $R=0$ and on either side close to the critical points (the order parameters are defined in Appendix pp3). All fits through the QMC data points are shown only as a guide to the eye. The dashed (solid) lines correspond to cases where finite (zero) order parameters are obtained in the thermodynamic limit.

Figure 6

Comparison and contrast of the VBS ordering in the $J_1-Q$ (left) and $J_1-R$ (right) model for SU(4). (a),(b) 2D histograms of the VBS order parameter, $D_y$ and $D_x$, measured at equal time on a $64\times 64$ systems at a coupling of $Q/J_1=0.5$ and $R/J_1=0.5$ (deep in the VBS phase for both models; see inset of right panel of Fig. 4 for phase diagrams). At the same couplings, the columnar nature of the ordering is much more apparent in the $Q$ model as compared to the $R$ model, where a U(1) symmetry is observed. (c),(d) The probability $P\left(\theta \right)$ of getting a particular angle for the VBS order parameters on $64\times 64$ system for different couplings. By symmetry the function repeats itself in an eightfold way and hence we show it only for $0<\theta <\pi /4$ (a half period). Note again the weak decay of $P$ with $\theta$ in the $R$ model, as compared to the $Q$ model. See text for discussion.

Figure 7

Approximate phase diagram of the SU(2) $J_1-Q-R$ model. This graph shows the location of the crossings of the size $L=16$ and $L=32$ data for $L{\rho }_s$. Similar to data shown in Fig. 4 but on smaller system sizes.