School of Mathematics and Statistics - Theses
Permanent URI for this collection
Search Results
1 - 2 of 2
-
ItemCritical dense polymers on the strip and cylinder( 2013)This thesis is concerned with the study of solvable critical dense polymers using Yang-Baxter techniques in two distinct topologies, namely the strip and the cylinder. Critical dense polymers is the first member ${\cal LM}(1,2)$ of the Yang-Baxter integrable series of logarithmic minimal models ${\cal LM}(p,p')$. The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels $r,s=1,2,\ldots$. We begin by explicitly constructing the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion \mbox{labels} $(r,s)=(r,1)\otimes (1,s)$ and involve a boundary field $\xi$. Tuning the field $\xi$ appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized $q$-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge $c=-2$ and the Kac formula for the conformal weights in the infinitely extended Kac table $\Delta_{r,s}=\frac{(2r-s)^2-1}{8}$ for $r,s=1,2,3,\ldots$. The cylinder topology allows for non-contractible loops with fugacity $\alpha$ that wind around the cylinder or for an arbitrary number $\ell$ of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley-Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even $N$, this involves a non-diagonalizable braid operator $\vec J$. The number of defects $\ell$ thus separates the theory into Ramond ($\ell/2$ even), Neveu-Schwarz ($\ell/2$ odd) and $\mathbb{Z}_4$ ($\ell$ odd) sectors. For the case of loop fugacity $\alpha=2$, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge $c=-2$ and conformal weights $\Delta,\bar\Delta=\Delta_t=(t^2-1)/8$. Here $t=\ell/2$ and $t=2r-s\in\mathbb{N}$ in the $\ell$ even sectors with Kac labels $r=1,2,3,\ldots; s=1,2$ while $t\in\mathbb{Z}-\half$ in the $\ell$ odd sectors. Strikingly, the $\ell/2$ odd sectors exhibit a ${\cal W}$-extended symmetry but the $\ell/2$ even sectors do not. Moreover, the naive trace summing over all $\ell$ even sectors does not yield a modular invariant.
-
ItemCombinatorics of lattice paths and polygons( 2012)We consider the enumeration of self-avoiding walks and polygons on regular lattices. Such objects are connected with many other problems in combinatorics, as well as in fields as diverse as physics and chemistry. We examine the general models of walks and polygons and methods we can use to study them; subclasses whose properties enable a rather deeper analysis; and extensions of these models which allow us to model physical phenomena like polymer collapse and adsorption. While the general models of self-avoiding walks and polygons are certainly not considered to be ‘solved’, recently a great deal of progress has been made in developing new methods for studying these objects and proving rigorous results about their enumerative properties, particularly on the honeycomb lattice. We consider these recent results and show that in some cases they can be extended or generalised so as to enable further proofs, conjectures and estimates. In particular, we find that properties shared by all two-dimensional lattices allow us to develop new methods for estimating the growth constants and certain amplitudes for the square and triangular lattices. The subclasses of self-avoiding walks and polygons that we consider are typically defined by imposing restrictions on the way in which a walk or polygon can be constructed. Ideally the restrictions should be as weak as possible, so as to result in a model that closely resembles the unrestricted case, while still enabling some manner of simple recursive construction. These recursions can sometimes lead to solutions for generating functions or other quantities of interest. Some of the models we consider display quite unusual asymptotic properties despite the relatively simple restrictions which lead to their construction. We approach the modelling of polymer adsorption in several ways. Firstly, we adapt some of the new methods for studying self-avoiding walks on the honeycomb lattice to account for interactions with an impenetrable surface. In this way we are able to prove the exact value of the critical surface fugacity for adsorbing walks, confirming an existing conjecture. Then, we show that some key identities for the honeycomb lattice model lead to a new method for estimating the critical surface fugacities for adsorption models on the square and triangular lattices. Many of the estimates we obtain in this way are new; for the cases where previous estimates did already exist, our results are several orders of magnitude more precise. Finally, we define some new solvable models of polymer adsorption which generalise existing models, and find that some of these models exhibit interesting and unexpected critical behaviour.