Scaling in Condensed Matter Physics: the Kondo Problem, Aggregation, and the Random-Field Ising Model.

Ian K. Affleck

Impurities are ubiquitous in condensed matter. Boundary Conformal Field Theory (BCFT) provides a powerful method to study a localized quantum impurity interacting with a gapless continuum of excitations. The results can also be implied to nanoscopic devices like quantum dots. In these lecture notes, I review this field, including the following topics: I. General Renormalization Group (RG) framework for quantum impurity problems: example of simplest Kondo model II. Multi-channel Kondo model III. Quantum Dots: experimental realizations of one and two channel Kondo models IV. Impurities in Luttinger liquids: point contact in a quantum wire V. Quantum impurity entanglement entropy VI. Y-junctions of Luttinger liquids VII. Boundary condition changing operators and the X-ray edge problem.

Philip Warren Anderson, G. Yuval, Donald R. Hamann

Nikolaos G. Fytas, Anastasios Malakis

Manoj Kumar, Martin Weigel

Spin systems exposed to the influence of random magnetic fields are paradigmatic examples for studying the effect of quenched disorder on condensed-matter systems. In this context, previous studies have almost exclusively focused on systems with Ising or continuous symmetries, while the Potts symmetry, albeit being of fundamental importance also for the description of realistic physical systems, has received very little attention. In the present study, we use a recently developed quasiexact method for determining ground states in the random-field Potts model to study the problem with four states. Extending the protocol applied for the three-state model, we use extensive finite-size scaling analyses of the magnetization, Binder parameter, energy cumulant, specific heat, and the connected as well as disconnected susceptibilities to study the magnetic ordering transition of the model. In contrast to the system in the absence of disorder, we find compelling evidence for a continuous transition, and we precisely determine the critical point as well as the critical exponents, which are found to differ from the exponents of the three-state system as well as from those of the random-field Ising model.

Laurence A Jacobs, Michael Nauenberg

R. Shankar

Providing a broad review of many techniques and their application to condensed matter systems, this book begins with a review of thermodynamics and statistical mechanics, before moving onto real and imaginary time path integrals and the link between Euclidean quantum mechanics and statistical mechanics. A detailed study of the Ising, gauge-Ising and XY models is included. The renormalization group is developed and applied to critical phenomena, Fermi liquid theory and the renormalization of field theories. Next, the book explores bosonization and its applications to one-dimensional fermionic systems and the correlation functions of homogeneous and random-bond Ising models. It concludes with Bohm–Pines and Chern–Simons theories applied to the quantum Hall effect. Introducing the reader to a variety of techniques, it opens up vast areas of condensed matter theory for both graduate students and researchers in theoretical, statistical and condensed matter physics.