Jané Kondev

Field Theory of Compact Polymers on the Square Lattice

Exact results for conformational statistics of compact polymers are derived from the two-flavour fully packed loop model on the square lattice. This loop model exhibits a two-dimensional manifold of critical fixed points each one characterised by an infinite set of geometrical scaling dimensions. We calculate these dimensions exactly by mapping the loop model to an interface model whose scaling limit is described by a Liouville field theory. The formulae for the central charge and the first few scaling dimensions are compared to numerical transfer matrix results and excellent agreement is found. Compact polymers are identified with a particular point in the phase diagram of the loop model, and the non-mean field value of the conformational exponent γ = 117112 is calculated for the first time. Interacting compact polymers are described by a line of fixed points along which γ varies continuously.

Nonlinear measures for characterizing rough surface morphologies

We develop an approach for characterizing the morphology of rough surfaces based on the analysis of the scaling properties of contour loops, i.e., loops of constant height. Given a height profile of the surface we perform independent measurements of the fractal dimension of contour loops, and the exponent that characterizes their size distribution. Scaling formulas are derived, and used to relate these two geometrical exponents to the roughness exponent of a self-affine surface, thus providing independent measurements of this important quantity. Furthermore, we define the scale-dependent curvature, and demonstrate that by measuring its third moment departures of the height fluctuations from Gaussian behavior can be ascertained. These nonlinear measures are used to characterize the morphology of computer generated Gaussian rough surfaces, surfaces obtained in numerical simulations of a simple growth model, and surfaces observed by scanning-tunneling microscopes. For experimentally realized surfaces the self-affine scaling is cut off by a correlation length, and we generalize our theory of contour loops to take this into account.

Kac-Moody symmetries of critical ground states

Abstract The symmetries of critical ground states of two-dimensional lattice models are investigated. We show how mapping a critical ground state to a model of a rough interface can be used to identify the chiral symmetry algebra of the conformal field theory that describes its scaling limit. This is demonstrated in the case of the six-vertex model, the three-coloring model on the honeycomb lattice, and the four-coloring model on the square lattice. These models are critical and they are described in the continuum by conformal field theories whose symmetry algebras are the su (2) k =1 , su (3) k =1 , and the su (4) k =1 Kac-Moody algebra, respectively. Our approach is based on the Frenkel-Kac-Segal vertex operator construction of level-one Kac-Moody algebras.

Supersymmetry and localization in the quantum Hall effect

Abstract We study the localization transition in the integer quantum Hall effect as described by the network model of quantum percolation. Starting from a path integral representation of transport Green functions for the network model, which employs both complex (bosonic) and Grassmann (fermionic) fields, we map the problem of localization to the problem of diagonalizing a one-dimensional non-hermitian Hamiltonian of interacting bosons and fermions. An exact solution is obtained in a restricted subspace of the Hilbert space which preserves boson-fermion supersymmetry. The physically relevant regime is investigated using the density matrix renormalization group (DMRG) method, and critical behavior is found at the plateau transition.

Liouville Field Theory of Fluctuating Loops

Effective field theories of two-dimensional lattice models of fluctuating loops are constructed by mapping them onto random surfaces whose large scale fluctuations are described by a Liouville field theory. This provides a geometrical view of conformal invariance in two-dimensional critical phenomena and a method for calculating critical properties of loop models exactly. As an application of the method, the conformal charge and critical exponents for two mutually excluding Hamiltonian walks on the square lattice are calculated.

Conformational Entropy of Compact Polymers

Exact results for the scaling properties of compact polymers on the square lattice are obtained from an effective field theory. The entropic exponent \gamma=117/112 is calculated, and a line of fixed points associated with interacting chains is identified; along this line \gamma varies continuously. Theoretical results are checked against detailed numerical transfer matrix calculations, which also yield a precise estimate for the connective constant \kappa=1.47280(1).

Loop Models, Marginally Rough Interfaces, and the Coulomb Gas

We develop a coarse-graining procedure for two-dimensional models of fluctuating loops by mapping them to interface models. The result is an effective field theory for the scaling limit of loop models, which is found to be a Liouville theory with imaginary couplings. This field theory is completely specified by geometry and conformal invariance alone, and it leads to exact results for the critical exponents and the conformal charge of loop models. A physical interpretation of the Dotsenko-Fateev screening charge is found.