Central to our understanding of quantum many particle physics are two ideas due to Landau. The first is the notion of the electron as a well-defined quasiparticle excitation in the many body state. The second is that of the order parameter to distinguish different states of matter. Experiments in a number of correlated materials raise serious suspicions about the general validity of either notion. A growing body of theoretical work has confirmed these suspicions, and explored physics beyond Landau's paradigms. This article provides an overview of some of these theoretical developments.
Central to our understanding of quantum many particle physics are two ideas due to Landau. The first is the notion of the electron as a well-defined quasiparticle excitation in the many body state. The second is that of the order parameter to distinguish different states of matter. Experiments in a number of correlated materials raise serious suspicions about the general validity of either notion. A growing body of theoretical work has confirmed these suspicions, and explored physics beyond Landau's paradigms. This article provides an overview of some of these theoretical developments.
Central to our understanding of quantum many particle physics are two ideas due to Landau. The first is the notion of the electron as a well-defined quasiparticle excitation in the many body state. The second is that of the order parameter to distinguish different states of matter. Experiments in a number of correlated materials raise serious suspicions about the general validity of either notion. A growing body of theoretical work has confirmed these suspicions, and explored physics beyond Landau's paradigms. This article provides an overview of some of these theoretical developments.
I give a pedagogical introduction to some of the many particles and gauge fields that can emerge in correlated matter. The standard model of materials is built on Landau's foundational principles: adiabatic continuity and spontaneous symmetry breaking. These ideas lead to quasiparticles that inherit their quantum numbers from fundamental particles, Nambu-Goldstone bosons, the Anderson-Higgs mechanism, and topological defects in order parameters. I then describe the modern discovery of physics beyond the standard model. Here, quantum correlations (entanglement) and topology play key roles in defining the properties of matter. This can lead to fractionalised quasiparticles that carry only a fraction of the quantum numbers that define fundamental particles. These particles can have exotic properties: for example Majorana fermions are their own antiparticles, anyons have exchange statistics that are neither bosonic nor fermionic, and magnetic monopoles do not occur in the vacuum. Gauge fields emerge naturally in the description of highly correlated matter and can lead to gauge bosons. Relationships to the standard model of particle physics are discussed.
The Landau paradigm is a central dogma for understanding phase and phase transitions in condensed matter systems, yet for decades it has been known that a variety of quantum phases exist beyond the framework. Is there a more general framework that provides a systematic understanding of phase and phase transitions in quantum many-body systems? Recent developments of the notion of generalized symmetry and generalized gauging seem to point to a way to generalize the Landau Paradigm. In this essay, we discuss how `beyond Landau' phases and phase transitions can be captured by a generalized Landau paradigm in terms of the breaking of generalized symmetries, often induced by the generalized gauging procedure facilitated through the topological holography formalism. We also discuss what needs to be understood to make the generalized Landau paradigm useful in the study of quantum phase and phase transitions.
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.
Condensed matter physics is a rich field with a wide variety of phenomena. In condensed matter, we consider systems composed of many electrons and nuclei, with the only fundamental interaction being electromagnetism. Nuclei of a given charge and mass are considered as point particles. The relevant fundamental physics that we use as our foundations is the quantum many-body problem. That is, just standard many-body quantum mechanics, but formulated in terms of a useful second quantized field theory, which can deal with an arbitrary number of particles within a single Hilbert space. In general, this many-body problem is essentially intractable for numbers of particles as large as those we must consider for materials. Thus we must look for simplifications and look at easily measurable aspects which allow us to make predictions still. We employ the trick of phrasing the questions differently. Rather than look at the time evolution of an arbitrary many-body wavefunction for the fundamental degrees of freedom, we look at transport in scenarios where the starting point is a relatively simple assumed equilibrium state of the material, consisting of a ground state with postulated quasiparticle excitations.
The Landau theory of phase and phase transition, which describes the state of matter by the spontaneous symmetry breaking, and the Fermi liquid theory of interacting Fermi systems are the two corner stones of contemporary condensed matter physics. They act as our starting points to understand the structure, excitations and phase transitions of any condensed matter system. According to such a Landau paradigm, an interacting Fermi system will behave just as non-interacting Fermi gas system at sufficient low energy, if the interaction is not strong enough to induce a symmetry breaking phase transition in the system. In such a case, the perturbation expansion in the interaction strength is assumed to converge and the low energy excitation can be connected to that of a Fermi gas system adiabatically. In strongly correlated electron systems, such as the high temperature superconductors and quantum antiferromagnets, a pertubative picture on the interaction effect usually fails. The correlation effect between the electrons can induce new forms of order that are beyond the Landau symmetry breaking paradigm. Non-Fermi liquid behavior can emerge in the fully symmetric correlated phases. The Mott insulating phase and the quantum spin liquid phase are the most prominent example of such exotic state of matter. After about three decades of intensive studies, people find that the quantum non-locality effect is at the heart of such new forms of order. The quantum non-locality effect manifests itself in the emergence of gauge degree of freedom in the low energy physics of such strongly correlated system. Such emergent gauge filed is not only important for the characterization and classification of the ground state structure, but is also directly responsible for the non-Fermi liquid behavior of the system. The Landau paradigm should be extended to accommodate such new forms of order and the related new form of excitations and new universal class of phase transitions. Such an extension will elevate our understanding of the structure of matter to the level of quantum wave function. Unlike the conventional symmetry breaking order, which is characterized by the correlation functions between local physical observables, the new form of orders, which are commonly called quantum order (with the topological order as one of its typical example), can be better characterized by the entanglement property in the quantum wave function. From a mathematical point of view, the new form of order is closely related to the projective representation theory of the symmetry group, while the conventional symmetry breaking order is well described by the linear representation theory of the symmetry group.