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Register a new userCorrelation functions of twist operators applied to single self-avoiding loops
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The O(n) spin model in two dimensions may equivalently be formulated as a loop model, and then mapped to a height model which is conjectured to flow under the renormalization group to a conformal field theory (CFT). At the critical point, the order n terms in the partition function and correlation functions describe single self-avoiding loops. We investigate the ensemble of these self-avoiding loops using twist operators, which count loops which wind non-trivially around them with a factor -1. These turn out to have level two null states and hence their correlators satisfy a set of partial differential equations. We show that partly-connected parts of the four point function count the expected number of loops which separate one pair of points from the other pair, and find an explicit expression for this. We argue that the differential equation satisfied by these expectation values should have an interpretation in terms of a stochastic(Schramm)-Loewner evolution (SLE_kappa) process with kappa=6. The two point function in a simply connected domain satisfies a closely related set of equations. We solve these and hence calculate the expected number of single loops which separate both points from the boundary.
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Using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the O(n) model at the critical point. We focus on correlation functions containing twist operators, combining these with anchored loops, boundaries with SLE processes and with double SLE processes. We focus further upon n=0, representing self-avoiding loops, which corresponds to a logarithmic conformal field theory (LCFT) with c=0. In this limit the twist operator plays the role of a zero weight indicator operator, which we verify by comparison with known examples. Using the additional conditions imposed by the twist operator null-states, we derive a new explicit result for the probabilities that an SLE_{8/3} wind in various ways about two points in the upper half plane, e.g. that the SLE passes to the left of both points. The collection of c=0 logarithmic CFT operators that we use deriving the winding probabilities is novel, highlighting a potential incompatibility caused by the presence of two distinct logarithmic partners to the stress tensor within the theory. We provide evidence that both partners do appear in the theory, one in the bulk and one on the boundary and that the incompatibility is resolved by restrictive bulk-boundary fusion rules.
We consider a self-avoiding walk model of polymer adsorption where the adsorbed polymer can be desorbed by the application of a force. In this paper the force is applied normal to the surface at the last vertex of the walk. We prove that the appropriate limiting free energy exists where there is an applied force and a surface potential term, and prove that this free energy is convex in appropriate variables. We then derive an expression for the limiting free energy in terms of the free energy without a force and the free energy with no surface interaction. Finally we show that there is a phase boundary between the adsorbed phase and the desorbed phase in the presence of a force, prove some qualitative properties of this boundary and derive bounds on the location of the boundary.
Quantum mechanics can be formulated in terms of phase-space functions, according to Wigners approach. A generalization of this approach consists in replacing the density operators of the standard formulation with suitable functions, the so-called generalized Wigner functions or (group-covariant) tomograms, obtained by means of group-theoretical methods. A typical problem arising in this context is to express the evolution of a quantum system in terms of tomograms. In the case of a (suitable) open quantum system, the dynamics can be described by means of a quantum dynamical semigroup in disguise, namely, by a semigroup of operators acting on tomograms rather than on density operators. We focus on a special class of quantum dynamical semigroups, the twirling semigroups, that have interesting applications, e.g., in quantum information science. The disguised counterparts of the twirling semigroups, i.e., the corresponding semigroups acting on tomograms, form a class of semigroups of operators that we call tomographic semigroups. We show that the twirling semigroups and the tomographic semigroups can be encompassed in a unique theoretical framework, a class of semigroups of operators including also the probability semigroups of classical probability theory, so achieving a deeper insight into both the mathematical and the physical aspects of the problem.
This is the second of two papers on the end-to-end distance of a weakly self-repelling walk on a four dimensional hierarchical lattice. It completes the proof that the expected value grows as a constant times sqrt{T} log^{1/8}T (1+O((log log T)/log T)), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z^4. - Apart from completing the program in the first paper, the main result is that the Greens function is almost equal to the Greens function for the Markov process with no self-repulsion, but at a different value of the killing rate beta which can be accurately calculated when the interaction is small. Furthermore, the Greens function is analytic in beta in a sector in the complex plane with opening angle greater than pi.
By using the self-dual Yang-Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by a non-auto-Backlund transformation. We prove the Lie-algebra isomorphism between the symmetries of the SDYM equation and those of the potential SDYM (PSDYM) equation, and we describe the construction of the recursion operators for these two systems. Using certain known aspects of the PSDYM symmetry algebra, we draw conclusions regarding the Lie algebraic structure of the potential symmetries of the SDYM equation.
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