Basis-conjugating automorphisms of a free group and associated Lie algebras

Let F_n = denote the free group with generators {x_1,...,x_n}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F_n to the general linear group over the integers. In particular among them are the automorphisms chi_{k,i} which conjugate the generator x_k by the generator x_i leaving the x_j fixed for j not k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by chi_{k,i} for i<k is given here. Partial results are obtained for the group generated by all chi_{k,i}.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

Orbital ice: an exact Coulomb phase on the diamond lattice

We demonstrate the existence of orbital Coulomb phase as the exact ground state of p-orbital exchange Hamiltonian on the diamond lattice. The Coulomb phase is an emergent state characterized by algebraic dipolar correlations and a gauge structure resulting from local constraints (ice rules) of the underlying lattice models. For most ice models on the pyrochlore lattice, these local constraints are a direct consequence of minimizing the energy of each individual tetrahedron. On the contrary, the orbital ice rules are emergent phenomena resulting from the quantum orbital dynamics. We show that the orbital ice model exhibits an emergent geometrical frustration by mapping the degenerate quantum orbital ground states to the spin-ice states obeying the 2-in-2-out constraints on the pyrochlore lattice. We also discuss possible realization of the orbital ice model in optical lattices with p-band fermionic cold atoms.Comment: 6 pages, 5 figure

Lattice Statistics in Three Dimensions: Exact Solution of Layered Dimer and Layered Domain Wall Models

Exact analyses are given for two three-dimensional lattice systems: A system of close-packed dimers placed in layers of honeycomb lattices and a layered triangular-lattice interacting domain wall model, both with nontrivial interlayer interactions. We show that both models are equivalent to a 5-vertex model on the square lattice with interlayer vertex-vertex interactions. Using the method of Bethe ansatz, a closed-form expression for the free energy is obtained and analyzed. We deduce the exact phase diagram and determine the nature of the phase transitions as a function of the strength of the interlayer interaction.Comment: 22 pages in Revtex, 6 PS files, submitted to PR

On the duality relation for correlation functions of the Potts model

We prove a recent conjecture on the duality relation for correlation functions of the Potts model for boundary spins of a planar lattice. Specifically, we deduce the explicit expression for the duality of the n-site correlation functions, and establish sum rule identities in the form of the M\"obius inversion of a partially ordered set. The strategy of the proof is by first formulating the problem for the more general chiral Potts model. The extension of our consideration to the many-component Potts models is also given.Comment: 17 pages in RevTex, 5 figures, submitted to J. Phys.

Exact Solution of a Three-Dimensional Dimer System

We consider a three-dimensional lattice model consisting of layers of vertex models coupled with interlayer interactions. For a particular non-trivial interlayer interaction between charge-conserving vertex models and using a transfer matrix approach, we show that the eigenvalues and eigenvectors of the transfer matrix are related to those of the two-dimensional vertex model. The result is applied to analyze the phase transitions in a realistic three-dimensional dimer system.Comment: 11 pages in REVTex with 2 PS figure

Understanding I=2 pi-pi Interaction

A correct understanding and description of the I=2 pi-pi S-wave interaction is important for the extraction of the I=0 pi-pi S-wave interaction from experimental data and for understanding the I=0 pi-pi S-wave interaction theoretically. With t-channel rho, f2(1270) exchange and the pi pi -> rho rho -> pi pi box diagram contribution, we reproduce the pi-pi isotensor S-wave and D-wave scattering phase shifts and inelasticities up to 2.2 GeV quite well in a K-matrix formalism.Comment: Talk given at Hadron 03: 10th International Conference on Hadron Spectroscopy, Aschaffenburg, Germany, 31 Aug - 6 Sep 200

Introduction to Graphene Electronics -- A New Era of Digital Transistors and Devices

The speed of silicon-based transistors has reached an impasse in the recent decade, primarily due to scaling techniques and the short-channel effect. Conversely, graphene (a revolutionary new material possessing an atomic thickness) has been shown to exhibit a promising value for electrical conductivity. Graphene would thus appear to alleviate some of the drawbacks associated with silicon-based transistors. It is for this reason why such a material is considered one of the most prominent candidates to replace silicon within nano-scale transistors. The major crux here, is that graphene is intrinsically gapless, and yet, transistors require a band-gap pertaining to a well-defined ON/OFF logical state. Therefore, exactly as to how one would create this band-gap in graphene allotropes is an intensive area of growing research. Existing methods include nano-ribbons, bilayer and multi-layer structures, carbon nanotubes, as well as the usage of the graphene substrates. Graphene transistors can generally be classified according to two working principles. The first is that a single graphene layer, nanoribbon or carbon nanotube can act as a transistor channel, with current being transported along the horizontal axis. The second mechanism is regarded as tunneling, whether this be band-to-band on a single graphene layer, or vertically between adjacent graphene layers. The high-frequency graphene amplifier is another talking point in recent research, since it does not require a clear ON/OFF state, as with logical electronics. This paper reviews both the physical properties and manufacturing methodologies of graphene, as well as graphene-based electronic devices, transistors, and high-frequency amplifiers from past to present studies. Finally, we provide possible perspectives with regards to future developments.Comment: This is an updated version of our review article, due to be published in Contemporary Physics (Sept 2013). Included are updated references, along with a few minor corrections. (45 pages, 19 figures

The Emergence of Superconducting Systems in Anti-de Sitter Space

In this article, we investigate the mathematical relationship between a (3+1) dimensional gravity model inside Anti-de Sitter space $\rm AdS_4$, and a (2+1) dimensional superconducting system on the asymptotically flat boundary of $\rm AdS_4$ (in the absence of gravity). We consider a simple case of the Type II superconducting model (in terms of Ginzburg-Landau theory) with an external perpendicular magnetic field ${\bf H}$. An interaction potential $V(r,\psi) = \alpha(T)|\psi|^2/r^2+\chi|\psi|^2/L^2+\beta|\psi|^4/(2 r^k )$ is introduced within the Lagrangian system. This provides more flexibility within the model, when the superconducting system is close to the transition temperature $T_c$. Overall, our result demonstrates that the two Ginzburg-Landau differential equations can be directly deduced from Einstein's theory of general relativity.Comment: 10 pages, 2 figure