Renormalization plays an important role in the theoretically and mathematically careful analysis of models in condensed-matter physics. I review selected results about correlated-fermion systems, ranging from mathematical theorems to applications in models relevant for materials science, such as the prediction of equilibrium phases of systems with competing ordering tendencies, and quantum criticality.
Renormalization plays an important role in the theoretically and mathematically careful analysis of models in condensed-matter physics. I review selected results about correlated-fermion systems, ranging from mathematical theorems to applications in models relevant for materials science, such as the prediction of equilibrium phases of systems with competing ordering tendencies, and quantum criticality.
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.
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.
In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d -wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high-level theoretical presentations and research papers specialized in certain classes of materials. It will give an overview on a variety of different quantum transitions, critically discuss open theoretical questions, and frequently make contact with recent experiments in condensed matter physics.
We investigate various topics in theoretical quantum condensed matter physics.First, we show that a one-dimensional quantum wire with as few as 2 channels of interacting fermions can host metallic Luttinger liquid states of matter that are stable against all perturbations up to $q^\\text{th}$-order in fermion creation/annihilation operators for any fixed finite q.The leading relevant perturbations are thus complicated operators that are expected to modify the physics only at very low energies, below accessible temperatures. The stability of these Luttinger liquid fixed points is due to strong interactions between the channels, which can (but need not) be chosen to be purely repulsive. Our results might enable elementary physical realizations of these phases, and may also serve as a useful paradigm for thinking about higher-dimensional non-Fermi liquids. Separately, we present an elementary but general description of relaxation to gaussian and equilibrium generalized Gibbs states in lattice models of fermions or bosons with quadratic hamiltonians. Our analysis applies to arbitrary initial states that satisfy a mild condition on clustering of correlations.We obtain quantitive, model-independent predictions for how quickly local quantities relax in such systems.These predictions can be tested in near-term quantum gas experiments.Finally, we study chaotic many-body quantum systems that obey the eigenstate thermalization hypothesis (ETH).We show that a known bound on the growth rate of the out-of-time-order four-point correlator in such systems follows directly from the general ETH structure of operator matrix elements.This ties together two key paradigms of thermal behavior in isolated many-body quantum systems.We also consider a bipartition of the system, and study the entanglement properties of an energy eigenstate with nonzero energy density.When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy).
The Density Matrix Renormalization Group (DMRG) has become a powerful numerical method that can be applied to low-dimensional strongly correlated fermionic and bosonic systems. It allows for a very precise calculation of static, dynamical and thermodynamical properties. Its field of applicability has now extended beyond Condensed Matter, and is successfully used in Statistical Mechanics and High Energy Physics as well. In this article, we briefly review the main aspects of the method. We also comment on some of the most relevant applications so as to give an overview on the scope and possibilities of DMRG and mention the most important extensions of the method such as the calculation of dynamical properties, the application to classical systems, inclusion of temperature, phonons and disorder, field theory, time-dependent properties and the ab initio calculation of electronic states in molecules.