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The Kitaev Model on Surfaceswith Lattice Defects and ofHigher Genus

John K. Brennan

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

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.

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Recently, a generalized Landau paradigm has been proposed, in which a broad class of "unconventional" phase transitions lying beyond the original symmetry-breaking framework of Landau are understood as the symmetry-breaking transitions of generalized symmetries. In particular, in (1+1)D, such "topological" or "deconfined" transitions can be mapped to the symmetry-breaking transitions of non-invertible symmetries. How to extract the universal dynamical properties of the transitions from this understanding? In this paper, we follow the Ginzburg-Landau philosophy and develop a field-theoretic description of such critical points in terms of the fluctuations of local order parameters, now of non-invertible symmetries. The Symmetry Topological Field Theory formalism plays a central role in our analysis, allowing us to identify the algebra of local order parameters from the Lagrangian algebra of a (2+1)D topological order. We illustrate the key ideas and the full steps of this generalized Ginzburg-Landau procedure using the transitions between gapped phases with a simple non-invertible symmetry --- the Rep($S_3$) symmetry.

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Topological Transition in Quantum fluids

Alexandre Guay, Reductionism in Transitions

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.

•Jan 1, 2024•0 citations
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.
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.
Matter exists in many different phases, for example in solid state or in liquid phase. There are also phases in which the ordering of atoms is the same, but which differ in some other respect, for example ferromagnetic and paramagnetic states. According to Landau's symmetry breaking theory every phase transition is connected to a symmetry breaking process. A solid material has discrete translational symmetry, while liquid phase has continuous translational symmetry. But it has turned out that there also exist phase transitions that can occur without a symmetry breaking. This phenomenon is called topological order. In this thesis we consider one example of a theoretical model constructed on a two dimensional lattice in which one obtains topological order.
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.
Recently, a generalized Landau paradigm has been proposed, in which a broad class of "unconventional" phase transitions lying beyond the original symmetry-breaking framework of Landau are understood as the symmetry-breaking transitions of generalized symmetries. In particular, in (1+1)D, such "topological" or "deconfined" transitions can be mapped to the symmetry-breaking transitions of non-invertible symmetries. How to extract the universal dynamical properties of the transitions from this understanding? In this paper, we follow the Ginzburg-Landau philosophy and develop a field-theoretic description of such critical points in terms of the fluctuations of local order parameters, now of non-invertible symmetries. The Symmetry Topological Field Theory formalism plays a central role in our analysis, allowing us to identify the algebra of local order parameters from the Lagrangian algebra of a (2+1)D topological order. We illustrate the key ideas and the full steps of this generalized Ginzburg-Landau procedure using the transitions between gapped phases with a simple non-invertible symmetry --- the Rep(S3S_3S3​) symmetry.
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.