arXiv subjects

Peter Kleban

Publications and source records attributed to Peter Kleban.

26 records · Page 2Linked to original sources

arXiv2005

Generalized Number Theoretic Spin Chain-Connections to Dynamical Systems and Expectation Values

We generalize the number theoretic spin chain, a one-dimensional statistical model based on the Farey fractions, by introducing a new parameter x>=0. This allows us to write recursion relations in the length of the chain. These relations are closely related to the Lewis three-term equation, which is useful in the study of the Selberg ζ-function. We then make use of these relations and spin orientation transformations. We find a simple connection with the transfer operator of a model of intermittency in dynamical systems. In addition, we are able to calculate certain spin expectation values explicitly in terms of the free energy or correlation length. Some of these expectation values appear to be directly connected with the mechanism of the phase transition.

math-ph↗

arXiv2004

Thermodynamics of the Farey Fraction Spin Chain

We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field $h$. From rigorous and renormalization group arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a "marginal" field is present) despite the unusual nature of the transition for h=0.

math-ph↗

arXiv2003

Crossing Probabilities and Modular Forms

We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic Löwner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension 1/3), follows from a simple modular argument. A new type of "higher-order modular form" arises and its properties are discussed briefly.

math-ph↗

arXiv2002

The Phase Transition in Statistical Models Defined on Farey Fractions

We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as "spin chains", with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-Frobenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form [t (ln t)^2]^(-1). The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.

math-ph↗

arXiv1999

Crossing Probabilities in Critical 2-D Percolation and Modular Forms

Crossing probabilities for critical 2-D percolation on large but finite lattices have been derived via boundary conformal field theory. These predictions agree very well with numerical results. However, their derivation is heuristic and there is evidence of additional symmetries in the problem. This contribution gives a preliminary examination some unusual modular behavior of these quantities. In particular, the derivatives of the "horizontal" and "horizontal-vertical" crossing probabilities transform as a vector modular form, one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form. We include consideration of the interplay between conformal and modular invariance that arises.

cond-mat.stat-mech↗

arXiv1998

A Fully Magnetizing Phase Transition

We analyze the Farey spin chain, a one dimensional spin system with effective interaction decaying like the squared inverse distance. Using a polymer model technique, we show that when the temperature is decreased below the (single) critical temperature T_c=1/2, the magnetization jumps from zero to one.

math-ph↗

arXiv1998

Shape-dependent universality in percolation

The shape-dependent universality of the excess percolation cluster number and cross-configuration probability on a torus is discussed. Besides the aspect ratio of the torus, the universality class depends upon the twist in the periodic boundary conditions, which for example are generally introduced when triangular lattices are used in simulations.

cond-mat.dis-nn↗

arXiv1996

Casimir Terms and Shape Instabilities for Two-Dimensional Critical Systems

We calculate the universal part of the free energy of certain finite two- dimensional regions at criticality by use of conformal field theory. Two geometries are considered: a section of a circle ("pie slice") of angle ϕand a helical staircase of finite angular (and radial) extent. We derive some consequences for certain matrix elements of the transfer matrix and corner transfer matrix. We examine the total free energy, including non- universal edge free energy terms, in both cases. A new, general, Casimir instability toward sharp corners on the boundary is found; other new instability behavior is investigated. We show that at constant area and edge length, the rectangle is unstable against small curvature.

cond-mat↗