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.
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.
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.
According to the principle of emergence, the properties of material are mainly determined by how the atoms are organized in the material. Such organization is formally called order. The vast range of materials is a result of the rich variety of orders that atoms can have. For a long time, it has been believed that all orders are described as symmetry breaking. A comprehensive theory for phases and phase transitions is developed here based on the symmetry breaking picture. The existence of FQH states (and superconducting states) indicates that there are new states of matter that cannot be described as symmetry breaking. Completely new theory is needed to describe those new states of matter. This chapter outlines the theory of topological order and theories of quantum order for the new states of matter, such as FQH states. Many new concepts and new language, such as topology-dependent degeneracy, fractional statistics, edge states, etc, are introduced to describe new states of matter.
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.
A recurring pattern in recent condensed matter theory is the inadequacy of the Landau paradigm—spontaneous breaking of invertible symmetry characterized by a local order parameter—to describe a growing class of quantum phase transitions. This synthesis examines a cluster of eight recent preprints (primarily from cond-mat.str-el, with supporting results from cond-mat.mes-hall and cond-mat.dis-nn) that collectively develop the microscopic underpinnings of beyond-Landau transitions. The central thesis, offered as a **heuristic reading rather than a derivation**, is the following structural pattern: when the symmetry protecting or separating phases is non-invertible—meaning defect fusion does not close in a group—the conventional Landau machinery (local order parameter, symmetry-breaking pattern, Ginzburg-Landau effective field theory) must be replaced by categorical data encoding fusion, F-symbols, and tube-algebra modules. This replacement is not merely formal: it has concrete microscopic consequences for which transitions are allowed, what order parameters signal them, and whether a conventional hidden symmetry-breaking description is even possible after gauging. The corpus sources span deconfined quantum critical points (DQCPs), twin phases, non-invertible symmetry-enriched topological orders (NI-SETOs), Rydberg-chain phase diagrams, topological edge memory under quenches, and the algebraic structure of string order parameters. We identify a falsification path for the central claim: if a numerical (DMRG/tensor-network) study of any of the proposed Rep(H₈)- or Rep(D₈)-type lattice models finds a conventional Landau description after an exhaustive gauging search, the categorical necessity argument collapses. The Limitations section acknowledges abstract-only reading and the absence of full proofs from several sources. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.31601, 2605.31602, 2606.05856, 2606.06343, 2606.06466, 2606.08558, 2606.11303, 2606.12089, 2606.18529, 2606.19437, 2606.20403, 2606.20522
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.
The Landau Fermi liquid theory and the Ginzburg-Landau phase transition theory stand as two pivotal cornerstones in traditional condensed matter physics, achieving significant success in addressing crucial physical phenomena such as BCS superconductors and liquid helium superfluids. However, marked by the discoveries of the quantum Hall effect and high-temperature superconductivity in the 1980s, it gradually became evident that for a broad class of novel quantum states, such as fractional quantum Hall states and quantum spin liquids, their properties transcend the Landau Fermi liquid theory and Ginzburg-Landau phase transition theory. Topological order and its related concepts of long-range many-body quantum entanglement and fractionalized excitation have become the key concepts to understand these exotic quantum states. Designing and identifying topologically ordered states of matter in quantum materials and quantum simulation systems, and probing and manipulating their fractionalized excitations, are important research directions in modern condensed matter physics. In recent years, great progress has been made in the quantum simulation and manipulation of topological order on highly controllable quantum simulation platforms, such as Rydberg atomic systems, superconducting quantum processors, and two-dimensional moiré superlattices. This article provides a brief overview of recent research advances and challenges in the study of topological order in traditional condensed matter systems and quantum simulation experimental platforms. It also provides prospects for the future developments of this field.