Condensed matter is the area of physics which deals with the phases of matter in condensed form with an aim to understand the properties of these phases and the transition between them based on the known laws of physics. This is usually done by figuring out some characteristic properties of the phase which sets it apart from the others, e.g., symmetry, topological order, entanglement, etc. The most common way of characterizing the transition between these phases is the broken symmetry. In this classification method, one finds the symmetry which phase breaks spontaneously during the transition and develops a local long-range order. This is the famous Landau’s symmetry breaking paradigm [ 1 ]. Landau’s symmetry breaking theory is a phenomenological theory which describes the transition from a symmetric phase (high-temperature phases) to a less symmetric/broken symmetry phase (phases at low temperature). A zero-temperature phase transition is also possible (by tuning the parameters of the Hamiltonian) and is known as quantum phase transition [ 2 ]. Although there were other paradigms of phase transition available, like Anderson localization [ 3 ], Berezinsky—Kosterlitz—Thouless transition [ 4 ] (broken symmetry but no long-range order), etc., Landau’s spontaneous symmetry breaking dominated over the others.
Condensed matter is the area of physics which deals with the phases of matter in condensed form with an aim to understand the properties of these phases and the transition between them based on the known laws of physics. This is usually done by figuring out some characteristic properties of the phase which sets it apart from the others, e.g., symmetry, topological order, entanglement, etc. The most common way of characterizing the transition between these phases is the broken symmetry. In this classification method, one finds the symmetry which phase breaks spontaneously during the transition and develops a local long-range order. This is the famous Landau’s symmetry breaking paradigm [ 1 ]. Landau’s symmetry breaking theory is a phenomenological theory which describes the transition from a symmetric phase (high-temperature phases) to a less symmetric/broken symmetry phase (phases at low temperature). A zero-temperature phase transition is also possible (by tuning the parameters of the Hamiltonian) and is known as quantum phase transition [ 2 ]. Although there were other paradigms of phase transition available, like Anderson localization [ 3 ], Berezinsky—Kosterlitz—Thouless transition [ 4 ] (broken symmetry but no long-range order), etc., Landau’s spontaneous symmetry breaking dominated over the others.
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.
The quantum phase of a physical system is defined as its ground state. As examples of quantum phases we mention super uids, superconductors, Bose- Einstein condensates and quantum Hall states. Different phases of a system originate from the different ways a systems' constituents can be organised or, in other words, the different orders of the system. Landau had the insight that the difference between two phases can be attributed to the corresponding orders having different symmetries [1]. Also, the transition between phases can be associated with the breaking of symmetries. This insight led to Ginzburg and Landau describing orders using local order parameters and developing a general theory of phase transitions based on symmetry breaking. This theory was successful in describing a large number of observed phases of matter. However, phases of matter have been discovered which cannot be described using a local order parameter and Landau's theory of symmetry breaking known as topological phases.
One problem of paramount importance in physics over various fields such as elementary particle physics, condensed matter physics, and statistical mechanics has been to make classification of phases of matter, including nontrivial structures of the vacuum. The triumphant history of theory on the phase classification and the phase transition can be traced back to Landau, which is based on the spontaneous breaking of symmetry. It is known, however, that other kinds of phases exist that cannot be characterized by conventional arguments of symmetry breaking, namely, topological phases. Topological phases are gapped phases at zero temperature without symmetry breaking. Examples of the topological phases include a quantum phase in a fractional quantum Hall system, where the ground state exhibits degeneracy, and an anyon, a quasi-particle with nontrivial statistical nature, neither boson nor fermion, emerges as a low-energy excitation. In contemporary physics of the 21st century, it has been revealed that, among various topological states, there exists a nontrivial ground state that does not have ground state degeneracy and yet can be distinguished from a trivial state. This nontrivial ground state, i.e., a topological insulator, has been recognized first in free electron systems [1–3] and later extended to more general quantum many-body systems with interactions including bosonic (spin) systems. Now, such a generalized situation is called a symmetry protected topological (SPT) phase [4–7]. From the point of a modern view of the SPT phase, nontrivial states, such as the Haldane state proposed in antiferromagnetic quantum spin systems in the 1980’s [8] and the ground state in the Affleck-Kennedy-Lieb-Tasaki model [9], belong to the SPT phase. A remarkable feature of the SPT phase is that, if the system has a surface, gapless surface states appear and they are stable against disturbances which do not break the symmetry. One way to understand these surface states is the quantum anomaly on the surface. That is, as long as the surface is concerned, the quantum anomaly arises, but the whole system has no quantum anomaly, and the symmetry is preserved due to cancellation from the bulk contribution. This canceling mechanism of the quantum anomaly has been long known as the anomaly inflow in elementary particle physics [10]. In fact, the classification of the SPT phases can be identified as the classification of the quantum anomaly through the bulk-boundary correspondence. This type of the quantum anomaly closely related to the SPT phase can provide us with a useful theoretical device to constrain phases that can or cannot be realized under the ’t Hooft anomaly matching condition [11–13]. It is quite difficult to scrutinize nonperturbatively interacting systems, but the ’t Hooft anomaly matching condition can be applied to those systems, and possible phases can be constrained. Nowadays, our understanding of the SPT phases has been so developed that applications of the anomaly matching condition can cover systems with finite groups and/or extending objects [14–16]. In this way, the concept of the topological phases has been essential by now in a wide range of physics, particularly in elementary particle physics, condensed matter physics, and statistical mechanics, as mentioned in the beginning. This special section is a collection of papers related to the topological aspects of physics including the topological phases. We would like to express our sincere gratitude to all the authors who made excellent contributions.
The Landau-Ginzburg theory has been immensely successful to model different states of matter. In this theory, order is associated with spontaneous symmetry breaking described using group theory. For many decades now, new states of matter have been experimentally produced (e.g. Hall fluids). In these liquids, matter is strongly correlated and has an internal structures, like in a solid, however the implied order cannot be characterized by symmetry breaking of local order parameters. In other words, to understand these states of matter, we have to go beyond the Landau-Ginzburg approach and, as sustain by Xiao- Gang Wen (2004), develop a new classification of orders, especially what he called topological orders. Moreover this notion of order should be introduced via quantities that can be measured. This talk sets out to achieve three primary objectives: 1. Evaluate Different Notions of Order: This involves a critical examination of various proposed notions of order to discern whether they represent an evolution or a transformation from the concept as envisaged in the Landau-Ginzburg theory. The development of topological order, particularly noted for its adherence to measurable quantities, provides an intriguing backdrop for this analysis. The theories of Nussinov and Ortiz (2009a, 2009b), which continue to honor symmetry principles, will be scrutinized in detail to facilitate this comparison. 2. Explore the Concept of Topological Transition: Inspired by Continentino (2017), this section will delve into the implications of topological order on our understanding of phase transitions. The inquiry will center on whether topological transitions introduce an entirely new paradigm or if they are fundamentally analogous to traditional notions of phase transition. 3. Investigate the Notion of Transition as Ontological Emergence: The final aim is to discuss how this novel perspective on transitions might exemplify a case of diachronic ontological emergence, as suggested by Humphreys (2016) and Guay and Sartenaer (2016). The hypothesis posits that quantum fluid transitions offer the most compelling evidence yet of such emergence.
Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classification of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal fluctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal fluctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into different phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-field treatments and ordinary perturbation theory are known to fail due to the enhancement of fluctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an effective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles confined to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the effects of the interplay between kinetic fluctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are efficient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to efficiently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a final remark pointing out the relevance of the research effort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.
It has long been thought that all different phases of matter arise from symmetry breaking. Without symmetry breaking, there would be no pattern, and matter would be featureless. However, it is now clear that for quantum matter at zero temperature, even symmetric disordered liquids can have features, giving rise to topological phases of quantum matter. Some of the topological phases are highly entangled (that is, have topological order), whereas others are weakly entangled (that is, have symmetry-protected trivial order). This Review provides a brief summary of these zero-temperature states of matter and their emergent properties, as well as their importance in unifying some of the most basic concepts in nature.