Quantum Theory of Many-Body Systems: Techniques and Applications

Alexandre M. Zagoskin

Henrik Bruus, Karsten Flensberg

Abstract This book is an introduction to the techniques of many-body quantum theory with a large number of applications to condensed matter physics. The basic idea of the book is to provide a self-contained formulation of the theoretical framework without losing mathematical rigor, while at the same time providing physical motivation and examples. The examples are taken from applications in electron systems and transport theory. On the formal side, the book covers an introduction to second quantization, many-body Green's function, finite temperature Feynman diagrams and bosonization. The applications include traditional transport theory in bulk as well as mesoscopic systems, where both the Landau-Büttiker formalism and recent developments in correlated transport phenomena in mesoscopic systems and nano-structures are covered. Other topics include interacting electron gases, plasmons, electron-phonon interactions, superconductivity and a final chapter on one-dimensional systems where a detailed treatment of Luttinger liquid theory and bosonization techniques is given. Having grown out of a set of lecture notes, and containing many pedagogical exercises, this book is designed as a textbook for an advanced undergraduate or graduate course, and is also well suited for self-study.

Jincheng Zhang

This paper explores the application of topological quasiparticle theory to the simulation of quantum many-body systems. The core claim is that this approach offers a pathway to significantly reduce computational complexity in simulating these systems. The mechanism involves mapping the constituent particles of a quantum many-body system into a topological quasiparticle space, leveraging the inherent properties of topological quasiparticles for efficient simulation and calculation. This work presents a novel methodology for tackling complex quantum many-body problems by exploiting the robust and computationally advantageous characteristics of topological order. The focus is on the theoretical framework and the potential for improved simulation techniques, without empirical experimental validation.

Alexandre M. Zagoskin

J A Sheikh, J. Dobaczewski, P Ring +2 more

Abstract The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. In this approach, the linear Schrödinger equation for the exact many-body wave function is mapped onto a non-linear one-body potential problem. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. In this way, additional correlations are subsumed in the system. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional (EDF) theory, as well as in other mesoscopic systems. We also discuss applications of projection techniques to quantum statistics in order to treat the averaging over restricted ensembles with fixed quantum numbers. Further, unresolved problems in the application of the symmetry restoration methods to the EDF theories are highlighted in the present work.

Hal Tasaki