Overview
This research addresses the characterization and diagnosis of symmetries in mixed states of quantum systems, specifically focusing on quantum many-body systems. Symmetries in such systems are categorized into two types: strong and weak. A previously proposed phenomenon, 'strong-to-weak symmetry breaking,' describes the spontaneous reduction of strong symmetries to weak symmetries. The challenge associated with this concept, as previously formulated, is its apparent non-local nature.
The core contribution of this work is the development of a methodology to understand and diagnose both strong symmetries and the process of strong-to-weak symmetry breaking through an explicitly local description. This local framework is achieved by introducing a rigorous definition of strong symmetry applicable in the infinite-volume limit, which generalizes existing definitions for finite volumes. This generalized definition is presented with several equivalent formulations, one of which incorporates the concept of 'local charge coherence.'
Additionally, the study introduces 'von Neumann systems' as a new classification for infinite-volume symmetries. These von Neumann symmetries are positioned as intermediate states between strong and weak symmetries. The research further establishes a Lieb-Schultz-Mattis type anomaly constraint. This constraint is derived for von Neumann symmetries, and by extension, for strong symmetries, specifically within the context of quantum spin chains.
Research Context
The foundational context for this research lies in the distinction between strong and weak symmetries present in mixed states of quantum systems. Prior discussions in the field have posited that strong symmetries within quantum many-body systems can undergo 'spontaneous breaking,' resulting in their reduction to weak symmetries. This process, termed 'strong-to-weak symmetry breaking,' has been understood as having a non-local character in its earlier formulations.
Approach
The primary methodological strategy employed in this research involves developing an explicitly local framework to address symmetries and their breaking. This framework is built upon a redefined understanding of strong symmetry in an infinite-volume context. Key aspects of the approach include:
- Infinite-volume strong symmetry definition: The research establishes a rigorous definition for strong symmetry in the limit of infinite volume. This definition is presented as a generalization of conventional definitions that apply to finite-volume systems.
- Equivalent formulations: Multiple equivalent formulations of this generalized strong symmetry definition are provided. One of these formulations integrates the concept of 'local charge coherence.' This concept serves as a technical tool within the local description.
- Introduction of von Neumann systems: The study introduces a new category of symmetries, termed 'von Neumann systems.' These systems are defined as infinite-volume symmetries that are intermediate between strong and weak symmetries.
- Anomaly constraint derivation: A Lieb-Schultz-Mattis type anomaly constraint is derived. This derivation specifically applies to von Neumann symmetries and, consequently, to strong symmetries when considered within quantum spin chains.
Findings
- The research demonstrates a method for understanding and diagnosing strong symmetries and strong-to-weak symmetry breaking using an explicitly local approach.
- A rigorous definition of strong symmetry in the infinite-volume limit is presented, generalizing conventional finite-volume definitions.
- Several equivalent formulations of the infinite-volume strong symmetry definition are provided, including one that incorporates the concept of 'local charge coherence'.
- Von Neumann systems are introduced as infinite-volume symmetries that occupy an intermediate position between strong and weak symmetries.
- A Lieb-Schultz-Mattis type anomaly constraint is derived for von Neumann symmetries and, by extension, for strong symmetries within quantum spin chains.
Why This Matters
This work offers a refined theoretical framework for understanding fundamental aspects of quantum many-body systems. By providing a local description for strong symmetries and their breaking, it addresses a previously identified non-local characteristic, potentially simplifying future analyses in theoretical physics. The introduction of von Neumann systems and the derivation of anomaly constraints contribute to the foundational understanding of symmetry properties in complex quantum states, particularly in the infinite-volume limit relevant to theoretical models.