Groupes et algébres de lie

Introduction to lie algebras and representation theory

Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems; (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals; and (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and nonequilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. A self-contained derivation of the exact flow equations is presented and frequently used truncation schemes are described. Reviewing selected applications it is shown how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.

Functional renormalization group approach to correlated fermion systems

"Extended Ensemble Monte Carlo" is a generic term that indicates a set of algorithms, which are now popular in a variety of fields in physics and statistical information processing. Exchange Monte Carlo (Metropolis-Coupled Chain, Parallel Tempering), Simulated Tempering (Expanded Ensemble Monte Carlo) and Multicanonical Monte Carlo (Adaptive Umbrella Sampling) are typical members of this family. Here, we give a cross-disciplinary survey of these algorithms with special emphasis on the great flexibility of the underlying idea. In Sec. 2, we discuss the background of Extended Ensemble Monte Carlo. In Secs. 3, 4 and 5, three types of the algorithms, i.e., Exchange Monte Carlo, Simulated Tempering, Multicanonical Monte Carlo, are introduced. In Sec. 6, we give an introduction to Replica Monte Carlo algorithm by Swendsen and Wang. Strategies for the construction of special-purpose extended ensembles are discussed in Sec. 7. We stress that an extension is not necessary restricted to the space of energy or tempe...

Extended ensemble monte carlo

The classical Yang–Baxter equation(CYBE) is an algebraic equation central in the theory of integrable systems. Its nondegenerate solutions were classified by Belavin and Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric interpretation of CYBE was given by Drinfeld and gave rise to the theory of Poisson–Lie groups.

Geometry and Classificatin of Solutions of the Classical Dynamical Yang–Baxter Equation

We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as a subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculations are possible and confirm the conjecture. The Friedan-Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik-Zamolodchikov equation.

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Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard equations

There are 880 magic squares of size 4 by 4, and 275 305 224 of size 5 by 5. It seems very difficult if not impossible to count exactly the number of higher order magic squares. We propose a method to estimate these numbers by Monte Carlo simulating magic squares at finite temperature. One is led to perform low temperature simulations of a system with many ground states that are separated by energy barriers. The Parallel Tempering Monte Carlo method turns out to be of great help here. Our estimate for the number of 6 by 6 magic squares is (0.17745± 0.00016)×1020.

Number of Magic Squares from Parallel Tempering Monte Carlo

We define a topological action of the quantum groupUq(sl2) on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc. This action commutes with the monodromy action of the braid groupoid, which is given by theR-matrix ofUq(sl2).

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Topological representations of the quantum group U q ( sl 2 )

We derive norm bounds that imply the convergence of perturbation theory in fermionic quantum field theory if the propagator is summable and has a finite Gram constant. These bounds are sufficient for an application in renormalization group studies. Our proof is conceptually simple and technically elementary; it clarifies how the applicability of Gram bounds with uniform constants is related to positivity properties of matrices associated to the procedure of taking connected parts of Gaussian convolutions. This positivity is preserved in the decouplings that also preserve stability in the case of two-body interactions.

Positivity and Convergence in Fermionic Quantum Field Theory

We investigate Symanzik's improvement program in a four-dimensional Euclidean scalar field theory with smooth momentum space cutoff. We use Wilson's renormalization group transformation to define the improved actions as a sequence of initial data for the effective action at the fundamental cutoff. This leads to a sequence of solutions to the renormalization group equation. We define the parameters of the improved actions implicitly by conditions on the effective action at a renormalization scale. The improved actions are close approximations to the continuum effective action. We prove their existence to every order of improvement and to every order of renormalized perturbation theory.

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Christian Wieczerkowski

Papers

Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard equations

Number of Magic Squares from Parallel Tempering Monte Carlo

Topological representations of the quantum group U q ( sl 2 )

Positivity and Convergence in Fermionic Quantum Field Theory

Symanzik's improved actions from the viewpoint of the renormalization group