Exactly solvable topological phase transition in a quantum dimer model

Abstract

We introduce a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We then focus on a quantum dimer model on the triangular lattice, with doubly-periodic edge weights. For simplicity we consider a $2\times1$ periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled $α$. We analytically show that the model exhibits a continuous quantum phase transition at $α=3$, changing from a topological $\mathbb{Z}_2$ quantum spin liquid ($α<3$) to a columnar ordered state ($α>3$). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length $ξ\propto1/|α-3|$ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase. We explain the constant vison correlator in terms of loops statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class.

Figures (12)

• Figure 1: The quantum dimer model on the weighted triangular lattice as given in Eq. \ref{['eqn:Htri']} exhibits a continuous quantum phase transition as an edge weight parameter $\alpha$ varies. Statistical behavior in the different phases and at the critical point can be seen through the behavior of random double-dimer coverings in the corresponding classical model, shown here for the $2\times1$ periodic triangular lattice (defined in Fig. \ref{['fig:triangle']}) with $\alpha=1$ (left, quantum spin liquid), $\alpha=3$ (center, critical), and $\alpha=5$ (right, columnar order), on an $81\times81$ grid. Double-edges are shown in gray, while loops are shown in black. In the spin liquid phase $\alpha<3$, the double-dimer covering forms many large macroscopic loops, while in the ordered phase $\alpha>3$, there are only small loops and double edges. The ordered phase for $\alpha>3$ is a columnar phase with a nonzero fraction of defects. This loop behavior can be used to intuitively explain the observed vison correlator behavior in the quantum dimer model. The double-dimer coverings are generated using the Kasteleyn matrix method to compute conditional edge probabilities.
• Figure 2: A plaquette $p$ with surrounding weights $w_{p_1}$, $w_{p_2}$, $w_{p_3}$, and $w_{p_4}$ in a clockwise order.
• Figure 3: Fundamental domain and Kasteleyn orientation for the $2\times1$ periodic triangular lattice with one horizontal edge weight $\alpha$ and all five other weights equal to 1. Horizontal and vertical translates of the fundamental domain are indexed by variables $w,z\in\mathbb{C}$.
• Figure 4: Path in the triangular lattice for the dimer-dimer and vison correlator calculations. For the vison correlator, the starting face is shaded, with the "face distance" for faces in the path labeled. We count the number of dimers crossed by the dashed path from the starting face to the ending face. The relevant edges to count are shown in bold. For the lattice sizes used in the numerics, this path goes through the horizontal edges with weight $1$. The dimer-dimer and vison correlators behave similarly on the adjacent vertical path which goes through the edges with weight $\alpha$, as well on horizontal paths.
• Figure 5: Dimer-dimer correlator for different values of $\alpha$, calculated along the path shown in Fig. \ref{['fig:path']} on a $303\times303$ size grid. The path starts in the center and ends roughly midway to the boundary. As we show analytically, the dimer-dimer correlator exhibits exponential decay for all values of $\alpha\ne3$.

Theorems & Definitions (1)

• Lemma 1