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Characterizing Gapless Phases and Phase Transitions with Non-Invertible Symmetries


Core Concepts
The authors provide a generalized framework to characterize gapless phases and phase transitions in the presence of categorical (non-invertible) symmetries, extending the Symmetry Topological Field Theory (SymTFT) approach.
Abstract

The paper presents a generalization of the SymTFT framework to study phase transitions and gapless phases with categorical symmetries. The central concept is the "club sandwich", which extends the SymTFT setup to include an interface between two topological orders.

The key points are:

  1. Gapped boundary phases with non-invertible symmetries are characterized by the "club quiche", which couples a (d+1)-dimensional TQFT to the SymTFT via a topological interface.

  2. Phase transitions between gapped phases with categorical symmetries are described by the "club sandwich", which provides a map between two SymTFTs via a non-maximal condensable algebra interface. This generalizes the Kennedy-Tasaki (KT) transformation.

  3. The authors construct new gapless phases and phase transitions by applying suitable KT transformations on known phase transitions, such as the critical Ising model and 3-state Potts model. The order parameters in these gapless theories are generally mixtures of conventional and string-type order parameters.

  4. Removing the physical boundary from the club sandwiches results in "club quiches", which characterize all possible gapped boundary phases with (possibly non-invertible) symmetries that can arise on the boundary of a bulk gapped phase.

  5. The authors provide a mathematical characterization of gapped boundary phases with symmetries as pivotal tensor functors whose targets are pivotal multi-fusion categories.

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Deeper Inquiries

How can the club sandwich framework be extended to study gapless phases and phase transitions in higher dimensions?

The club sandwich framework, initially designed to analyze gapless phases and phase transitions in two-dimensional systems, can be extended to higher dimensions by generalizing the construction of the Symmetry Topological Field Theory (SymTFT). In higher dimensions, the club sandwich can be represented as a configuration of multiple topological orders separated by interfaces, with each layer corresponding to a different symmetry or phase. This involves considering a (d + 1)-dimensional TQFT that incorporates not only the symmetry boundaries but also physical boundaries that may be gapless. To achieve this, one can introduce additional layers of topological orders and interfaces, allowing for a richer structure that captures the interactions between various categorical symmetries. The mathematical framework of pivotal tensor categories can be employed to characterize the gapped and gapless phases, as well as the phase transitions that occur between them. By applying Kennedy-Tasaki (KT) transformations in this extended setup, one can explore the implications of non-invertible symmetries in higher-dimensional systems, leading to a comprehensive understanding of the resulting gapless phases and their associated order parameters.

What are the potential applications of the club sandwich construction in other areas of physics beyond condensed matter, such as high-energy physics or quantum information?

The club sandwich construction has significant potential applications in various fields of physics beyond condensed matter. In high-energy physics, it can be utilized to study phase transitions in quantum field theories (QFTs) that exhibit non-invertible symmetries, providing insights into the behavior of gauge theories and their dualities. The framework can help in understanding the dynamics of critical phenomena in QFTs, particularly in the context of conformal field theories (CFTs) and their associated symmetry structures. In quantum information, the club sandwich construction can be applied to analyze topological quantum computing, where the manipulation of anyonic excitations and their braiding properties are crucial. The framework can aid in the classification of quantum states and the design of fault-tolerant quantum gates based on topological phases. Furthermore, the insights gained from the club sandwich approach can inform the development of new quantum error-correcting codes that leverage the properties of non-invertible symmetries, enhancing the robustness of quantum information processing.

Can the insights from this work lead to the discovery of new exotic phases of matter or novel phase transition phenomena in experimental systems?

Yes, the insights derived from the club sandwich framework can indeed pave the way for the discovery of new exotic phases of matter and novel phase transition phenomena in experimental systems. By characterizing gapless phases and phase transitions with non-invertible symmetries, researchers can identify previously unrecognized phases that may arise in materials with complex symmetry structures. The framework's ability to blend conventional and string-type order parameters allows for a more nuanced understanding of phase transitions, particularly in systems where traditional symmetry breaking approaches fall short. This could lead to the realization of exotic phases such as fractional quantum Hall states, topological insulators, or even new forms of quantum liquid states that exhibit non-trivial topological order. Moreover, the application of the club sandwich construction in experimental setups can facilitate the exploration of multi-critical points and higher-order phase transitions, which are often overlooked in standard phase transition theories. As experimental techniques advance, the ability to manipulate and probe these exotic phases could lead to groundbreaking discoveries in both fundamental physics and practical applications, such as in the development of next-generation quantum materials and devices.
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