Yang-Baxter integrable dimers and fused restricted-solid-on-solid lattice models

Abstract

The main objects of investigation in this thesis are two Yang-Baxter integrable lattice models of statistical mechanics in two dimensions: nonunitary RSOS models and dimers. At criticality they admit continuum descriptions with nonunitary conformal field theories (CFTs) in (1+1) dimensions. CFTs are quantum field theory invariant under conformal transformations. They play a major role in the theory of phase transition and critical phenomena. In quantum field theory unitarity is the requirement that the probability is conserved, hence realistic physical problems are associated with unitary quantum field theories. Nevertheless, in statistical mechanics this property loses a physical meaning and statistical systems like polymers and percolations, which model physical problems with long-range interactions, in the continuum scaling limit give rise to nonunitary conformal field theories.

Both the nonunitary RSOS models and dimers are defined on a two-dimensional square lattice. Restricted solid-on-solid (RSOS) models are so called because their degrees of freedom are in the form of a finite (therefore restricted) set of heights which live on the sites of the lattice and their interactions take place between the four sites around each face of the lattice (solid-on-solid). Each allowed configuration of heights maps to a specific Boltzmann weight. RSOS are integrable in the sense that their Boltzmann weights and transfer matrices satisfy the Yang-Baxter equation. The CFTs associated to critical RSOS models are minimal models, the simplest family of rational conformal field theories. The process of fusion on elementary RSOS models has a different outcome on the CFT side depending on both the level of fusion and the value of their crossing parameter λ. Precisely, in the interval 0 < λ < π/2, the 2x2 fused RSOS models correspond to higher-level conformal cosets with integer level of fusion equal to two. Instead in the complementary interval π/n < λ < π the 2x2 fused RSOS models are related to minimal models with integer level of fusion equal to one. To prove this conjecture one-dimensional sums, deriving from the well-known Yang-Baxter corner transfer matrix method, have been calculated, extended in the continuum limit and ultimately compared to the conformal characters of the related minimal models.

The dimer model has been for a long time object of various scientific studies, firstly as simple prototype of diatomic molecules and then as equivalent formulation to the well-known domino tilings problem of statistical mechanics. However, only more recently it has attracted attention as conformal field theory thanks to its relation with another famous integrable lattice model, the six-vertex model. What is particularly interesting of dimers is the property of being a free-fermion model and at the same time showing non-local properties due to the long-range steric effects propagating from the boundaries. This non-locality translates then in the dependance of their bulk free energy on the boundary conditions. We formulate the dimer model as a Yang-Baxter integrable free-fermion six-vertex model. This model is integrable in different geometries (cylinder, torus and strip) and with a variety of different integrable boundary conditions. The exact solution for the partition function arises from the complete spectra of eigenvalues of the transfer matrix. This is obtained by solving some functional equations, in the form of inversion identities, usually associated to the transfer matrix of the free-fermion six-vertex model, and using the physical combinatorics of the pattern of zeros of the transfer matrix eigenvalues to classify the eigenvalues according to their degeneracies. In the case of the cylinder and torus, the transfer matrix can be diagonalized, while, in the other cases, we observe that in a certain representation the double row transfer matrix exhibits non trivial Jordan-cells. Remarkably, the spectrum of eigenvalues of dimers and critical dense polymers agree sectors by sectors. The similarity with critical dense polymers, which is a logarithmic field theory, raises the question whether also the free-fermion dimer model manifests a logarithmic behaviour in the continuum scaling limit. The debate is still open. However, in our papers we provide a final answer and argue that the type of conformal field theory which best describe dimers is a logarithmic field theory, as it results by looking at the numerically estimate of the finite size corrections to the critical free energy of the free-fermion six-vertex-equivalent dimer model.

The thesis is organized as follows. The first chapter is an introduction which has the purpose to inform the reader about the basics of statistical mechanics, from one side, and CFTs, on the other side, with a specific focus on the two lattice models that have been studied (nonunitary RSOS and dimers) and the theories associated to the their continuum description at criticality (minimal models and logarithmic CFTs). The second chapter considers the family of non-unitary RSOS models with π/n < λ < π and brings forward the discussion around the one-dimensional sums of the elementary and fused models, and the associated conformal characters in the continuum scaling limit.

The third and fourth chapters are dedicated to dimers, starting with periodic conditions on a cylinder and torus, and then more general integrable boundary conditions on a strip. In each case, a combinatorial analysis of the pattern of zeros of the transfer matrix eigenvalues is presented and extensively treated. It follows then the analysis of the finite-size corrections to the critical free energy. Finally, the central charges and minimal conformal dimensions of critical dimers are discussed in depth with concluding remarks about the logarithmic hypothesis. Next, there is a conclusion where the main results of these studies are summarized and put into perspective with possible future research goals.