Topologically ordered phases of matter are quantum liquids with a non-local quantum order. Because of their unique properties due to the non-local quantum entanglement present in these phases, topological phases have been proposed as the basis of a physically fault-tolerant quantum computer. The formation of such topological order is well understood in terms of the mechanism of loop condensation in systems with loop-like degrees of freedom. Certain quantum dimer models posses topologically ordered dimer liquid ground states and can be mapped to loop models. In this dissertation we present a study of the geometric properties of the loop condensates of quantum dimer models and related models using classical Monte Carlo as well as ground state quantum Monte Carlo calculations. Additionally, we present an approach for the robust experimental generation of a topologically ordered phase in a system of neutral atoms trapped in an optical lattice.
We consider 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 focus on a quantum dimer model on the triangular lattice, with doubly periodic edge weights. For simplicity we consider a
2×1 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
Z2 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∝1/∣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, which we explain in terms of loop 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. Additionally, we analytically show that the topological Rényi entropy of order
∞ (topological min-entropy) changes from
log2 for the quantum spin liquid phase
α<3, to
0 for the ordered phase
α>3, thereby analytically confirming the topological nature of the phase transition.
The plaquette phase of the square-lattice quantum dimer model is studied using a continuous-time reptation quantum Monte Carlo method for lattices of sizes up to $48\ifmmode\times\else\texttimes\fi{}48$ sites. We determine the location of the phase transition between the columnar and plaquette phases to occur at ${V}_{c}∕J=0.60\ifmmode\pm\else\textpm\fi{}0.05$ which is significantly larger than inferred from previous exact diagonalization studies on smaller lattices. Off diagonal correlation functions are obtained. They exhibit long-range order in the plaquette phase but not at the Rokhsar-Kivelson point. We also observe significant finite-size corrections to scaling for the transition between the plaquette phase and the critical resonating valence bond liquid. This study demonstrates the importance of understanding finite-size effects when considering critical properties of the square-lattice quantum dimer model.
We study the mechanism of loop condensation in the quantum dimer model on the triangular lattice. The triangular lattice quantum dimer model displays a topologically ordered quantum liquid phase in addition to conventionally ordered phases with broken symmetry. In the context of systems with extended loop-like degrees of freedom, the formation of such topological order can be described in terms of loop condensation. Using Monte Carlo calculations with local and directed-loop updates, we compute geometric properties of the transition graph loop distributions of several triangular lattice quantum dimer wavefunctions that display dimer-liquid to dimer-crystal transitions and characterize these in terms of loop condensation.
The quantum dimer and loop models attract great attention, partially because the fundamental importance in the phases and phase transitions emerging in these prototypical constrained systems, and partially due to their intimate relevance toward the on-going experiments on Rydberg atom arrays in which the blockade mechanism naturally enforces the local constraint. Here we show, by means of the sweeping cluster quantum Monte Carlo method, the complete ground state phase diagram of the fully packed quantum loop model on the square lattice. We find between the lattice nematic (LN) phase with strong dimer attraction and the staggered phase (SP) with strong dimer repulsion, there emerges a resonating plaquette (RP) phase with off-diagonal translational symmetry breaking. Such a quantum phase is separated from the LN via a first order transition and from the SP by the famous Rokhsar-Kivelson point. Our renormalization group analysis reveals the different flow directions, fully consistent with the order parameter histogram in Monte Carlo simulations. The realization and implication of our phase diagram in Rydberg experiments are proposed.
We introduce the quantum dimer-pentamer model (QDPM) on the square lattice. This model is a generalization of the square-lattice quantum dimer model as its configuration space comprises fully packed hard-core dimer coverings as well as dimer configurations containing pentamers, where four dimers touch a vertex. Thus in the QDPM, the fully packed, hard-core constraint of the quantum dimer model is relaxed such that the local dimer number at each vertex is fixed modulo 3, resulting in an exact local
Z3 gauge symmetry. We construct a local Hamiltonian for which the Rokhsar-Kivelson (RK) equal superposition state is the exact ground state and has a ninefold topological degeneracy on the torus. Using Monte Carlo calculations, we find no spontaneous symmetry breaking in the RK wave function and that its dimer-dimer correlation function decays exponentially. By doping the QDPM RK state with a pair of monomers, we demonstrate that
Z3 electric charges are deconfined. Additionally, we introduce a
Z3 magnetic string operator that we find decays exponentially and shows no signatures of magnetic vortex condensation. These numerical results suggest that the ground state of the QDPM is a dimer liquid with
Z3 topological order.