Example of a first-order Néel to valence-bond-solid transition in two dimensions

We consider the $S=1/2$ Heisenberg model with nearest-neighbor interaction $J$ and an additional multispin interaction $Q_3$ on the square lattice. The $Q_3$ term consists of three bond-singlet projectors and is chosen to favor the formation of a valence-bond solid (VBS) where the valence bonds (singlet pairs) form a staggered pattern. The model exhibits a quantum phase transition from the Néel state to the VBS as a function of $Q_3/J$. We study the model using quantum Monte Carlo (stochastic series expansion) simulations. The Néel-VBS transition in this case is strongly first order in nature, in contrast to similar previously studied models with continuous transitions into columnar VBS states. The qualitatively different transitions illustrate the important role of an emerging $U\left(1\right)$ symmetry in the latter case, which is not possible in the present model due to the staggered VBS pattern (which does not allow local fluctuations necessary to rotate the local VBS order parameter).

Figure 1

(Color online) (a) The interaction term $Q_3$ involving three bond-singlet projection operators (shown with thicker lines) on the square lattice. All terms related by lattice translations and rotations of the shown instance of a product of singlet projectors are included in the Hamiltonian. Examples of VBS patterns: (b) staggered, (c) columnar, and (d) plaquette. The singlets preferentially form on the thicker (red) bonds. The arrows in (c) indicate how a local resonance of a pair of bonds between horizontal and vertical orientations corresponds to a plaquette in (d). Such resonances correspond to fluctuations of the VBS angle, which is $\varphi =n\pi /2$ $\left(n=0,1,2,3\right)$ for the columnar state and $\varphi n\pi /2+\pi /4$ for the plaquette state. In the DQC scenario (Ref. 2) they lead to an emergent $U\left(1\right)$ symmetry, i.e., a continuous circular-symmetric $\varphi$ as the transition into the Néel state is approached. The $Q_3$ term favors the staggered-type VBS (b), where such angular fluctuations should always be small.

Figure 2

(Color online) The squared staggered magnetization $⟨{\left(m_s^z\right)}^2⟩$ shown for different system sizes at inverse temperature $\beta J=L$. The vertical line at ${\left(Q_3/J\right)}_c=1.1933$ is the estimated $L\to \infty$ transition point from crossings of metastable energy branches (Fig. 7).

Figure 3

(Color online) The Binder cumulant of the staggered magnetization shown for different system sizes at inverse temperature $\beta J=L$. Note that the minimum of the Binder cumulant is negative for $L\ge 8$ and diverges to $-\infty$ as $L\to \infty$ based on these sizes.

Figure 4

(Color online) The spin stiffness ${\rho }_s$ (top panel) and ${\rho }_{s}L$ (bottom panel) for different system sizes.

Figure 5

(Color online) Size dependence of the VBS order parameter $D^2$ for different system sizes.

Figure 6

(Color online) The probability density $P\left(D_x,D_y\right)$ shown for $L=12$ at $Q_3/J=1.22,1.23,1.24$ (from top to bottom) and $\beta J=12$. The maxima present both at (0,0) and $\left(\pm D,0\right),\left(0,\pm D\right)$ at $Q_3/J=1.23$ show phase coexistence of the Neel and VBS orders.

Figure 7

(Color online) Critical coupling ${\left(Q_3/J\right)}_c\left(L\right)$ extracted from the crossing of the energy branches (top panel) of the Neel phase (labeled as fw) and the VBS phase (labeled as rev) for a system of size $L=36$ at $\beta J=36$. The bottom panel shows a fit to the form ${\left(Q_3/J\right)}_c\left(L\right)={\left(Q_3/J\right)}_c+A/L^3$ to extract the thermodynamic value of ${\left(Q_3/J\right)}_c=1.1933\left(1\right)$.