Quantum phase transitions in bilayer SU($N$) antiferromagnets

We present a detailed study of the destruction of SU($N$) magnetic order in square lattice bilayer antiferromagnets using unbiased quantum Monte Carlo numerical simulations and field theoretic techniques. We study phase transitions from an SU($N$) Néel state into two distinct quantum disordered “valence-bond” phases: a valence-bond liquid (VBL) with no broken symmetries and a lattice-symmetry-breaking valence-bond solid (VBS) state. For finite interlayer coupling, the cancellation of Berry phases between the layers has dramatic consequences on the two phase transitions: the Néel-VBS transition is first order for all $N\ge 5$ accesible in our model, whereas the Néel-VBL transition is continuous for $N=2$ and first order for $N\ge 4$; for $N=3$ the Néel-VBL transition show no signs of first-order behavior.

Figure 1

(a) Bilayer geometry: The white (black) sites are the A(B) sublattice on which spins transform as the fundamental (conjugate) representation of SU($N$). $J_1$ connects nearest neighbors in the plane, $J_2$ connects next nearest neighbors in the plane, and $J_{\perp }$ connects sites on different layers. Panels (b) and (c) show cartoon product wave functions of local singlets for the VBL and VBS states. In reality, the ground state is a strongly interacting superposition of all valence-bond coverings. The ground state nevertheless (b) preserves all symmetries for the VBL, but (c) breaks lattice symmetry (as shown) for the VBS. In this Rapid Communication, we provide a detailed study of the Néel-VBL and Néel-VBS quantum phase transitions.

Figure 2

Phase diagram of the model $H_{\mathrm{bil}}$ defined in Eq. (1) for SU(2), SU(4), SU(6) and SU(8) symmetry in the plane of $g_2\equiv J_2/J_1$ and $g_{\perp }\equiv J_{\perp }/J_1$. The unfilled symbols are locations of first order phase transitions, Néel-VBL (diamonds), Néel-VBS (circles) and VBS-VBL (squares). The solid black circles mark continuous transitions. For SU(2), the line of Néel-VBL critical points shown are in the universality class of the O(3) non-linear $\sigma -$model. For SU(6) and SU(8) the Néel-VBS transitions shown are in the universality class of the non-compact CP${}^{N-1}$ models (with $N=6,8$ respectively). Solid lines and shaded regions are guides to the eye.

Figure 3

Néel-VBL: The spin stiffness ${\rho }_s$ close to the Néel-VBL transition for SU(2), SU(4), and SU(6). The SU(2) transition is continuous and in the O(3) universality class. The quantity ${\rho }_s$ for SU(4) and SU(6) show signs of steplike behavior. Close to the step we find double-peaked histograms (see Fig. 4) characteristic of a first-order transition. The Néel-VBL transition shows such first-order behavior for all $N\ge 4$. The parameters used are $g_2=0.8$ for SU(2), $g_2=0.4$ for SU(4), and $g_2=0.6$ for SU(6). The legend shows the value of $L$; we have set $J_1\beta =L$ everywhere.

Figure 4

Néel-VBL: Hysteresis and double-peaked histograms at a first-order Néel-VBL transition in the SU(6) bilayer. In the main frame we show a sample Monte-Carlo history of the binned squared spatial winding number, $W^2$, which shows clear evidence for metastability. The inset shows a histogram for the same quantity, with clear double-peaked structure. This behavior is found only very close to the transition and for sufficiently large volumes, providing unambiguous evidence for a first-order transition. Here it is shown for $L=32$, $g_2=0.6$, and $g_{\perp }=1.36$.

Figure 5

Néel-VBL: Crossings of the fluctuations of the spatial winding number at the Néel-VBL transition for SU(2) and SU(3). In both cases up to sizes of $L=128$ we see good evidence for a nice crossing, indicating a continuous transition. No evidence for first-order behavior was found in these two cases.

Figure 6

Néel-VBS: First-order nature of the Néel-VBS transition in the two-dimensional square lattice bilayer. Both $O_{\mathrm{VBS}}^2$ and ${\rho }_s$ show evidence for steplike behavior at the same $g_{\perp }$. Close to the jump we find the same kind of double-peaked behavior in ${\rho }_s$ that is illustrated in Fig. 4. Here we have shown sample data for $N=8$ and $g_2=0.8$. Similar behavior is found for all $N$ studied here.