Generalized Symmetries in Condensed Matter (2204.03045v2)

Abstract: Recent advances in our understanding of symmetry in quantum many-body systems offer the possibility of a generalized Landau paradigm that encompasses all equilibrium phases of matter. This is a brief and elementary review of some of these developments.

• The paper extends the Landau paradigm by incorporating generalized symmetries to classify quantum phases beyond conventional symmetry breaking.
• It introduces higher-form symmetries that illuminate the behavior of topological order, fracton phases, and deconfined states in physical systems.
• The framework provides actionable insights for discovering new materials and refining quantum models by addressing symmetry anomalies.

Overview of "Generalized Symmetries in Condensed Matter" by John McGreevy

This paper provides a thorough exploration of the concept of generalized symmetries within the framework of condensed matter physics. The research navigates beyond the traditional Landau paradigm, which primarily classifies phases of matter based on the symmetries they break, by accommodating modern advancements that reveal a more expansive range of equilibrium phases characterized by generalized symmetries. The essay captures the nuances in understanding quantum phases of matter through modifications in established paradigms, utilizing both theoretical and practical insights on symmetry.

Key Concepts and Claims

1. Generalization of Landau Paradigm:
   • The paper posits an extended Landau paradigm that not only incorporates traditional symmetry considerations but also includes generalized symmetries to classify phases of matter.
   • These generalized symmetries account for various known equilibrium phases beyond the Landau's original scope.
2. Apparent Exceptions to the Traditional Paradigm:
   • The paper discusses several phenomena and states of matter that do not conform to the conventional transformation under the Landau paradigm, such as topologically-ordered states, deconfined phases of gauge theories, fracton phases, and topological insulators.
   • It shows that these exceptions can be understood within this generalized framework.
3. Higher-form Symmetries:
   • McGreevy introduces higher-form symmetries, which involve conserved currents of forms beyond zero, expanding the scope of how symmetries can manifest in physical systems.
   • Constructs such as 1-form symmetries and their associated conserved currents are introduced to account for states of matter and their transitions.
4. Topological and Non-topological Order:
   • Topological order is redefined in light of higher-form symmetries. Such orders often require a more complex symmetry-breaking framework than simple flaw corrections.
   • Gapped and gapless phases are analyzed, illustrating how symmetries play a crucial role in defining the properties of various quantum phases.
5. Implications of Anomalies:
   • Anomalies, traditionally discussed in context with gauge invariance and conservation laws, are reconsidered in the framework of generalized symmetry.
   • An understanding of how anomalies can shape phase structure and constraint symmetry provides a deeper theoretical toolset for exploring phase transitions and state formations.

Numerical Results and Theoretical Implications

The paper primarily deals with theoretical frameworks rather than specific numerical results. It, however, implies important mathematical relationships and constraints that generalized symmetries hold in quantum many-body systems, influencing the countability and classification of such states. Furthermore, theoretical implications extend into practical realms, suggesting that a robust comprehension of generalized symmetries could reveal new materials and phenomena in condensed matter systems, advance quantum computing, and refine models for quantum fields.

Speculative Future Developments

The paper hints at several future avenues for research in both theoretical and applied physics. By articulating a pathway for integrating generalized symmetries and exhibiting how these could broaden the scope of condensed matter physics, McGreevy sets the stage for further research into:

• The discovery of new phases of matter enabled by these symmetries.
• Applications in designing quantum materials with specific symmetry properties.
• Enhancing theoretical models that are currently bounded by traditional frameworks.

In conclusion, McGreevy's overview of generalized symmetries is a significant stride in understanding the full landscape of quantum phases. It proposes a substantial shift from classical paradigms towards embracing the complexity and richness of symmetries that dictate the properties of novel quantum states. Future research efforts leveraging this perspective could lead to groundbreaking findings in both fundamental physics and material sciences.