This chapter discusses phase transitions and symmetry breaking and the mean field approach to condensation in Bose systems and looks at strong interactions in ‘normal’ Fermi systems. ‘Statistical field theory’ encompasses all the applications of quantum field theory to statistical physics. In the context of these notes, it is interpreted as the study of ‘statistical mechanics of quantum many particle systems, employing the methods of field theory’. The identification of the symmetry broken in a phase with long range order allows to construct simple ‘mean field’ theories which provide a qualitative description of the system. Field theoretic methods were those developed by Feynman, Dyson, Schwinger, and others for understanding QED. They were ‘field theoretic’ in that one had to go beyond quantum mechanics, and the Schrodinger equation, to set up a theory. Unlike quantum field theory, there is a Schrodinger equation governing many particle phenomena in condensed matter.
This chapter discusses phase transitions and symmetry breaking and the mean field approach to condensation in Bose systems and looks at strong interactions in ‘normal’ Fermi systems. ‘Statistical field theory’ encompasses all the applications of quantum field theory to statistical physics. In the context of these notes, it is interpreted as the study of ‘statistical mechanics of quantum many particle systems, employing the methods of field theory’. The identification of the symmetry broken in a phase with long range order allows to construct simple ‘mean field’ theories which provide a qualitative description of the system. Field theoretic methods were those developed by Feynman, Dyson, Schwinger, and others for understanding QED. They were ‘field theoretic’ in that one had to go beyond quantum mechanics, and the Schrodinger equation, to set up a theory. Unlike quantum field theory, there is a Schrodinger equation governing many particle phenomena in condensed matter.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.
The theory of fermion phase transitions is reviewed from a unified field theoretic standpoint, based on the diagrammatic perturbation expansion of a generalized matrix propagator. Transitions from a normal to a condensed phase are characterized by the spontaneous appearance of long-range order and (in the presence of a suitable infinitesimal external field) broken symmetry. This is illustrated by the ferromagnetic, solid, superconducting and spindensity wave ground states. The phenomenon is explained qualitatively as caused by the creation of a long-range internal field, F, due to the interactions between particles. This field establishes long-range order in the system, and is in turn itself established by the long-range order, in a self-consistent fashion. The mechanism here is expressed quantitatively in terms of a self-consistent Dyson equation relating a generalized matrix propagator, G, to a proper self-energy matrix, Σ. The off-diagonal elements of G describe ‘anomalous’ propagation processes which are characteristic for the condensed phase, and they yield directly the long-range order parameters. The Σ-matrix is just the potential of the internal field. The method is illustrated by applying it to the ferromagnetic phase of a system with δ-function interaction between particles. Finally, the technique is used to derive the vertex part equation for the transition point.
Abstract This book provides a thorough introduction to the fascinating world of phase transitions as well as many related topics, including random walks, combinatorial problems, quantum field theory and S-matrix. Fundamental concepts of phase transitions, such as order parameters, spontaneous symmetry breaking, scaling transformations, conformal symmetry, and anomalous dimensions, have deeply changed the modern vision of many areas of physics, leading to remarkable developments in statistical mechanics, elementary particle theory, condensed matter physics and string theory. This self-contained book provides an excellent introduction to frontier topics of exactly solved models in statistical mechanics and quantum field theory, renormalization group, conformal models, quantum integrable systems, duality, elastic S-matrix, thermodynamics Bethe ansatz and form factor theory. The clear discussion of physical principles is accompanied by a detailed analysis of several branches of mathematics, distinguished for their elegance and beauty, such as infinite dimensional algebras, conformal mappings, integral equations or modular functions. Besides advanced research themes, the book also covers many basic topics in statistical mechanics, quantum field theory and theoretical physics. Each argument is discussed in great detail, paying attention to an overall coherent understanding of physical phenomena. Mathematical background is provided in supplements at the end of each chapter, when appropriate. The chapters are also followed by problems of different levels of difficulty. Advanced undergraduate and graduate students will find a rich and challenging source for improving their skills and for accomplishing a comprehensive learning of the many facets of the subject.
The development of methods of quantum statistical mechanics is considered in light of their applications to quantum solid-state theory. We discuss fundamental problems of the physics of magnetic materials and the methods of the quantum theory of magnetism, including the method of two-time temperature Green’s functions, which is widely used in various physical problems of many-particle systems with interaction. Quantum cooperative effects and quasi-particle dynamics in the basic microscopic models of quantum theory of magnetism: the Heisenberg model, the Hubbard model, the Anderson Model, and the spinfermion model are considered in the framework of novel self-consistent-field approximation. We present a comparative analysis of these models; in particular, we compare their applicability for description of complex magnetic materials. The concepts of broken symmetry, quantum protectorate, and quasi-averages are analyzed in the context of quantum theory of magnetism and theory of superconductivity. The notion of broken symmetry is presented within the nonequilibrium statistical operator approach developed by D.N. Zubarev. In the framework of the latter approach we discuss the derivation of kinetic equations for a system in a thermal bath. Finally, the results of investigation of the dynamic behavior of a particle in an environment, taking into account dissipative effects, are presented.
This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book highlights quantum field theory techniques that were developed in parallel in high energy physics and condensed matter physics. It explains the relevance of quantum field theory methods developed in the context of high energy physics, to many particle systems, even though, in principle, quantum mechanics may suffice to understand these systems. The book discusses the modern perspective of phase transitions and use a length scale based approach. It deals with the application of the ideas of bosonization in simple one-dimensional models such as spin models and the Hubbard model, where the calculational tools of bosonization and Renormalisation group are used to compute physically relevant quantities like conductances. The book also highlights the idea that topological concepts play an important role in a wide range of phenomena.