Short-ranged RVB physics, quantum dimer models and Ising gauge theories

Quantum dimer models are believed to capture the essential physics of antiferromagnetic phases dominated by short-ranged valence bond configurations. We show that these models arise as particular limits of Ising (Z_2) gauge theories, but that in these limits the system develops a larger local U(1) invariance that has different consequences on different lattices. Conversely, we note that the standard Z_2 gauge theory is a generalised quantum dimer model, in which the particular relaxation of the hardcore constraint for the dimers breaks the U(1) down to Z_2. These mappings indicate that at least one realization of the Senthil-Fisher proposal for fractionalization is exactly the short ranged resonating valence bond (RVB) scenario of Anderson and of Kivelson, Rokhsar and Sethna. They also suggest that other realizations will require the identification of a local low energy, Ising link variable {\it and} a natural constraint. We also discuss the notion of topological order in Z_2 gauge theories and its connection to earlier ideas in RVB theory. We note that this notion is not central to the experiment proposed by Senthil and Fisher to detect vortices in the conjectured Z_2 gauge field.Comment: 17 pages, 4 postscript figures automatically include

"The Ising model on spherical lattices: dimer versus Monte Carlo approach"

We study, using dimer and Monte Carlo approaches, the critical properties and finite size effects of the Ising model on honeycomb lattices folded on the tetrahedron. We show that the main critical exponents are not affected by the presence of conical singularities. The finite size scaling of the position of the maxima of the specific heat does not match, however, with the scaling of the correlation length, and the thermodynamic limit is attained faster on the spherical surface than in corresponding lattices on the torus.Comment: 25 pages + 6 figures not included. Latex file. FTUAM 93-2

Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density

The classical monomer-dimer model in two-dimensional lattices has been shown to belong to the \emph{``#P-complete''} class, which indicates the problem is computationally ``intractable''. We use exact computational method to investigate the number of ways to arrange dimers on $m \times n$ two-dimensional rectangular lattice strips with fixed dimer density $\rho$. For any dimer density $0 < \rho < 1$, we find a logarithmic correction term in the finite-size correction of the free energy per lattice site. The coefficient of the logarithmic correction term is exactly -1/2. This logarithmic correction term is explained by the newly developed asymptotic theory of Pemantle and Wilson. The sequence of the free energy of lattice strips with cylinder boundary condition converges so fast that very accurate free energy $f_2(\rho)$ for large lattices can be obtained. For example, for a half-filled lattice, $f_2(1/2) = 0.633195588930$, while $f_2(1/4) = 0.4413453753046$ and $f_2(3/4) = 0.64039026$. For $\rho < 0.65$, $f_2(\rho)$ is accurate at least to 10 decimal digits. The function $f_2(\rho)$ reaches the maximum value $f_2(\rho^*) = 0.662798972834$ at $\rho^* = 0.6381231$, with 11 correct digits. This is also the \md constant for two-dimensional rectangular lattices. The asymptotic expressions of free energy near close packing are investigated for finite and infinite lattice widths. For lattices with finite width, dependence on the parity of the lattice width is found. For infinite lattices, the data support the functional form obtained previously through series expansions.Comment: 15 pages, 5 figures, 5 table

Exact solution of the 2d2d dimer model: Corner free energy, correlation functions and combinatorics

In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model {\it via} the so-called height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predict particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms.Comment: Final version to be published in Nuclear Physics B (53 pages and a lot of figures

Dimers and the Critical Ising Model on Lattices of genus>1

We study the partition function of both Close-Packed Dimers and the Critical Ising Model on a square lattice embedded on a genus two surface. Using numerical and analytical methods we show that the determinants of the Kasteleyn adjacency matrices have a dependence on the boundary conditions that, for large lattice size, can be expressed in terms of genus two theta functions. The period matrix characterizing the continuum limit of the lattice is computed using a discrete holomorphic structure. These results relate in a direct way the lattice combinatorics with conformal field theory, providing new insight to the lattice regularization of conformal field theories on higher genus Riemann Surfaces.Comment: 44 pages, eps figures included; typos corrected, figure and comments added to section