We study a model of close-packed dimers on the square lattice with a nearest neighbor interaction between parallel dimers. This model corresponds to the classical limit of quantum dimer models [D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988)]. By means of Monte Carlo and transfer matrix calculations, we show that this system undergoes a Kosterlitz-Thouless transition separating a low temperature ordered phase where dimers are aligned in columns from a high temperature critical phase with continuously varying exponents. This is understood by constructing the corresponding Coulomb gas, whose coupling constant is computed numerically. We also discuss doped models and implications on the finite-temperature phase diagram of quantum dimer models.
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 a model of two-dimensional classical dimers on the square lattice with strong geometric constraints (there is exactly one bond with the nearest point for every point in the lattice). This model corresponds to the quantum dimer model suggested by D.S. Rokhsar and S.A. Kivelson (1988). We use the directed-loop algorithm to show the system undergoes a Berezinskii-Kostelitz Thousless transition (BKT transition) in finite temperatures. After that, if we destroy the geometric constraint of dimers, the topological transition will transfer to a quasi one-order transition. For the dimer updates, we also introduce a new cluster updating algorithm called the edged cluster algorithm. By this method, we succeed in rapidly traversing the winding (topological) sections uniformly and widening the effective matrical ensemble to include more topological sections.
The quantum dimer model has been continuously discussed both theoretically and numerically.Numerical methods have been developed on the quantum dimer model, but efficiency has been an essential problem due to the strong geometry constraint of the model.Recently, sweeping cluster update based on stochastic series expansion has shown power on quantum Monte Carlo on quantum dimer model.In this thesis, we propose improvements to the sweeping cluster update algorithm, which can reduce the time complexity by smartly choosing the next operator to update on.We also show a general framework to extend the current method to a broader range of models, including dimermonomer and monomer-monomer operators.As a benchmark of our algorithm, we simulate the quantum dimer model at finite temperatures in a wide region of parameters.By measuring several order parameters by the quantum Monte Carlo and using the finitescaling analysis, the phase boundary of the quantum dimer model on the square lattice can be estimated numerically, and the phase diagram in V /J < 1 region is determined accurately.We find evidence to support the expectation of previous research that the phase transition at finite temperature of quantum dimer model shows similar behavior to the Kosterlitz-Thouless transition in the classical dimer model.The intermediate plaquette phase may not exist, and the columnar phase in the classical model extends up to the Rokhsar-Kivelson point.iii This thesis is supported by Prof. Synge Todo and Prof. Hidemaro Suwa.I greatly appreciate the discussions we had during my doctoral years with them.Many ideas have been discussed in the meetings and then applied to the model for interesting numerical results.I also appreciated
We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size L\ifmmode\times\else\texttimes\fi{}L, where L\ensuremath{\leqslant}8, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of V/J. This state is consistent with the plaquette resonating-valence-bond state. The energy gap of this state vanishes as a power law of L. It exhibits columnar dimer order, but has disorder within the dimer columns. This state persists up to V/J1, and the system changes to a dimer liquid state only at V/J=1. \textcopyright{} 1996 The American Physical Society.
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.