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 \phi and 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.Comment: 15 pages PostScript, accepted for publication in Z. Phys.

Exact Results in Model Statistical Systems

This research focuses on the exact study of the statistical mechanics of model systems. Research concentrates on critical percolation in two-dimensions, and important and extensively studied statistical system, and the phase transition of the farey fraction spin chain, a new and interesting one-dimensional model with connections to multifractals and Monte Carol simulations, mathematically exact solutions of model statistical mechanical systems, and techniques from number theory. The goal is to gain new insights into, and understanding of, these systems. An important feature of the project is the use of techniques from pure mathematics and close collaboration with mathematicians

Exact Results in Model Statistical Systems

Intellectual merit: This project focuses on continued research on the exact study of the statistical mechanics of model systems. The research concentrates on two areas:1) critical percolation in two dimensions, an important and very extensively studied model system, to which we are bringing new and unexpected approaches, and2) the thermodynamics of the Farey fraction spin chain, a set of one dimensional models with interesting phase transition behavior and connections to multifractals, and dynamical systems.This project aims at new results and insights in both these areas. Research on the Farey models illuminates an interesting borderline case in the theory of phase transitions and has inspired several publications in number theory. The percolation research has already led to some surprising features, especially the applicability of modular forms. Ongoing work aims at extending and deepening these connections.An important feature of the research is the relation to pure mathematics, including close collaboration with mathematicians. This has already led to some new developments in number theory in both areas mentioned, and to synergistic developments, where the number theory provides new tools to gain insight into physical systems.We employ a variety of theoretical methods including conformal field theory, Monte Carlo simulations, mathematically exact solutions of model statistical mechanical systems, and techniques from number theory. This research extends and develops the PI\u27s program of basic research in statistical mechanics in two dimensions and related topics.Broader impacts of this project include the synergy with number theory already mentioned, training of graduate students in these research areas, and impacts on the PI\u27s teaching methods. There is also ongoing significant spinoff on a very practical level, involving instrumentation for mass spectrometers (and similar instruments). This has led, in particular, to the formation of Stillwater Scientific Instruments, Inc. Our main role in this company exploits our familiarity with conformal methods.Non-technical Abstract:In this research funded by the Divisions of Materials Research, Physics, and Mathematical Sciences, the study of the behavior of particles and small magnets moving on surfaces will be connected to the study of pure numbers. This connection between surface physics and pure mathematics provides a fertile area for the education of students and the discovery of unexpected mathematics and physics

New percolation crossing formulas and second-order modular forms

We consider the three new crossing probabilities for percolation recently found via conformal field theory by Simmons, Kleban and Ziff. We prove that all three of them (i) may be simply expressed in terms of Cardy's and Watts' crossing probabilities, (ii) are (weakly holomorphic) second-order modular forms of weight 0 (and a single particular type) on the congruence group $\Gamma(2)$, and (iii) under some technical assumptions (similar to those used by Kleban and Zagier, are completely determined by their transformation laws. The only physical input in (iii) is Cardy's crossing formula, which suggests an unknown connection between all crossing-type formulas.Comment: 15 page

Development of a Fourier Transform-Based Time-of-Flight Electron Spectrometer with Ultra-High Resolution

This project, funded by the Major Research Instrumentation program, will develop a time-of-flight electron velocity analyzer using advanced modulation and Fourier deconvolution techniques with a throughput advantage on the order of 1000 over existing instruments. The new spectrometer will operate with ultra-high resolution in the energy range 1-1000 electron volts. It will be useful for the investigation of surface properties under ultra-high vacuum and a variety of other scientific and commercial applications. The device utilizes secondary chopping of the electron beam in the nanosecond or sub-nanosecond time regime, and state-of-the-art Fourier transform-based digital signal recovery methods. Additionally, there is potential for several orders of magnitude more throughput by using array detectors. Other substantial performance advantages arise when it is used with synchrotron sources. The ultimate result will be a new generation of spectrometers allowing a wide range of new applications, both applied and fundamental, to research, research training, and analytical work. This sophisticated instrumentation is of wide applicability to vibrational electron energy loss spectroscopy, e.g. high resolution electron energy loss spectroscopy (HREELS), higher energy spectroscopies (X-ray photoelectron (XPS), Auger electron (AES), and ultraviolet photoemission (UPS) spectroscopy), and gas phase ion (mass) spectrometry and gas phase photoemission. The new device allows a number of new experiments and analytical techniques. By reducing acquisition time from hours to seconds, surface reactions can be followed on a much faster time-scale, for example during thermal processing. The several orders of magnitude improvement in throughput has a dramatic affect on the sensitivity to low intensity processes, such as trace analysis in analytical surface chemistry applications, non-dipolar scattering in HREELS and inelastic diffraction. Broader applications include practical surface analysis of soft and rough materials, e.g. HREELS of polymers, as well as greatly improved analytical capabilities in more standard types of measurements, such as XPS. Development requires optimizing the design of charged particle beam chopping devices. The new instrument requires the adaptation of deconvolution software for optimum performance. In collaboration with several experienced private sector software companies, sophisticated numerical algorithms for signal recovery will be implemented. Development of the spectrometer includes extensive participation by faculty and students, involving both experimental and theoretical work. A new time-of-flight electron spectrometer, developed with funding from the Major Research Instrumentation program, will be useful for the investigation of surface properties under ultra-high vacuum and a variety of other scientific and commercial applications. The scientific and engineering infrastructure at the University of Maine will be strengthened not only by the research enabled by this new instrument, but also by the education and career development of participants in the project, since students as well as faculty will be involved