First-order superfluid to valence-bond solid phase transitions in easy-plane $\mathrm{SU}\left(N\right)$ magnets for small $N$

We consider the easy-plane limit of bipartite $\mathrm{SU}(N$) Heisenberg Hamiltonians, which have a fundamental representation on one sublattice and the conjugate to fundamental on the other sublattice. For $N=2$ the easy plane limit of the SU(2) Heisenberg model is the well-known quantum $XY$ model of a lattice superfluid. We introduce a logical method to generalize the quantum $XY$ model to arbitrary $N$, which keeps the Hamiltonian sign-free. We show that these quantum Hamiltonians have a world-line representation as the statistical mechanics of certain tightly packed loop models of $N$ colors in which neighboring loops are disallowed from having the same color. In this loop representation we design an efficient Monte Carlo cluster algorithm for our model. We present extensive numerical results for these models on the two dimensional square lattice, where we find the nearest neighbor model has superfluid order for $N\le 5$ and valence-bond order for $N>5$. By introducing $\mathrm{SU}(N$) easy-plane symmetric four-spin couplings we are able to tune across the superfluid-VBS phase boundary for all $N\le 5$. We present clear evidence that this quantum phase transition is first order for $N=2$ and $N=5$, suggesting that easy-plane deconfined criticality runs away generically to a first-order transition for small $N$.

Figure 1

A section of a stochastic series expansion configuration of the Hamiltonian, Eq. (5), which takes a simple form in terms of $N$-colored tightly packed oriented closed loops. By “oriented,” we mean an orientation of the links (e.g., shown here as arrows going up on the A sublattice and down on the B sublattice) can be assigned before any loops are grown, and the orientation remains unviolated by each of the loops in a configuration. The extreme easy-plane anisotropy results in an important constraint; i.e., loops that share a vertex (operators represented by the black bars) are restricted to have different colors.

Figure 2

Loop updating moves describing how loops pass through a vertex, depicted with a black bar. These moves cause conversions between the diagonal and off-diagonal matrix elements present in the Hamiltonian. For each update pictured here there is a reverse process that we have not drawn. (a) Moves with probability 1/2 and (b) Moves with probability 1.

Figure 3

The stiffness and VBS order parameter extrapolations as a function of $1/L$ for different $N$ in the nearest-neighbor easy-plane model $H_{J_{\perp }}^N$, defined in Eq. (4) or equivalently in Eq. (5). We find clear evidence that the system has superfluid order for all $N\le 5$ and VBS order for $N>5$. To show ground-state convergence we plot both $\beta =L,2L$ in diamond and circular points, respectively.

Figure 4

Phase diagram of the $H_{J_{\perp }{Q}_{\perp }}^N$ model as a function of the ratio $g_c=Q_{\perp }/J_{\perp }$ and $N$. Using the $H_{J_{\perp }{Q}_{\perp }}^N$ model we have access to the superfluid-VBS phase boundary for $N=2,3,4$, and 5. In this work, we provide clear evidence that the transition for $N=2$ and $N=5$ is direct but discontinuous for both order parameters, suggesting that the easy-plane-$\mathrm{SU}(N$) superfluid-VBS transition is generically first order for small $N$.

Figure 5

Crossings at the SF-VBS quantum phase transition for $N=2$ in $H_{J_{\perp }{Q}_{\perp }}^N$. The main panels shows the crossings for $L{\rho }_s$ and ${\mathcal{R}}_{\text{VBS}}$, which signal the destruction and onset of SF and VBS order, respectively. The inset shows that the SF-VBS transition is direct; i.e., the destruction of SF order is accompanied by the onset of VBS order at a coupling of $g_c=8.63\left(1\right)$.

Figure 6

Evidence for first-order behavior at the SF-VBS quantum phase transition for $N=2$. The data was collected at a coupling $g_c=8.63$. The left panel shows MC histories for both ${\rho }_s$ and ${\mathcal{O}}_{\text{VBS}}$, with clear evidence for switching behavior characteristic of a first-order transition. The right panel shows histograms of the same quantities with double-peaked structure, which gets stronger with system size, again clearly indicating a first-order transition. The bin size for the histories is 400 MC steps per point

Figure 7

Crossings at the SF-VBS quantum phase transition for $N=5$ in $H_{J_{\perp }{Q}_{\perp }}^N$. The main panels show the crossings for $L{\rho }_s$ and ${\mathcal{R}}_{\text{VBS}}$, which again show a direct transition between the superfluid and VBS states at $g_c=0.014\left(1\right)$.

Figure 8

Evidence for first-order behavior at the SF-VBS quantum phase transition for $N=5$. The data was collected at a coupling $g=0.01875$. The left panel shows MC histories for both ${\rho }_s$ and ${\mathcal{O}}_{\text{VBS}}$, with clear evidence for switching behavior characteristic of a first-order transition at the largest system size $L=48$. The right panel shows histograms of the same quantities with double-peaked structure emerging at $L=48$. Here we note that the location of the transition for each system size drifts more significantly than in the $N=2$ case. Also it can be argued that the transition shows signs of weakening. Here we use a finer bin size for the histories (100 MC steps per point).