Advances in Gapless Phase Interfaces Enable Classification of Symmetry-Enriched Systems, Including 1+1d CFTs

The subtle distinctions between seemingly identical critical phases challenge physicists to understand how symmetry impacts their behaviour, and a team led by Saranesh Prembabu, Shu-Heng Shao from the Massachusetts Institute of Technology, and Ruben Verresen from the University of Chicago now demonstrates a novel way to characterise these phases. Researchers traditionally seek protected edge states to reveal differences between phases, but this approach doesn’t always work for gapless systems, prompting the team to investigate the interfaces between these phases instead. They reveal that when two critical phases exhibit differing symmetry properties, any interface connecting them inevitably becomes a ‘non-invertible defect’, a characteristic that serves as a robust fingerprint of the underlying symmetry enrichment. This breakthrough provides a new physical indicator for understanding the interplay between topology and critical behaviour, offering a powerful tool for classifying and distinguishing complex quantum systems.

Gapless quantum phases become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological phases. Researchers propose that the spatial interface between gapless phases, rather than their individual properties, defines the emergent behaviour. This work investigates the conditions under which these interfaces exhibit robust, symmetry-protected states, extending the established framework of topological phases to include gapless systems and their boundaries.

Ising Model Interfaces and Defect Behaviour

The team investigated the properties of interfaces between two variations of the Ising model, differing in their symmetry properties. A key focus was understanding the degeneracy of these interfaces, whether they have multiple equally likely configurations, and how this degeneracy is protected by symmetry. The authors explored connections to quantum critical points and symmetry-enriched phases of matter. They utilized anomaly arguments to constrain the allowed defects at the interface, finding that only a specific defect configuration is permitted, potentially with a highly constrained and degenerate interface.

This analysis involved establishing a dictionary connecting microscopic spin configurations to effective low-energy operators at the interface, allowing the team to analyze symmetry properties. The researchers predicted the existence of degenerate edge modes at the interface, indicative of a symmetry-enriched deconfined quantum critical point. Numerical calculations, using exact diagonalization, supported predictions about the scaling of energy splitting between these degenerate modes, demonstrating robust symmetry protection. The interface between these Ising models exhibits robust degeneracy of edge modes, protected by enhanced symmetry, and represents a signature of a symmetry-enriched deconfined quantum critical point. In simpler terms, the strict rules governing how two different magnets connect result in special properties, like multiple possible configurations, related to a complex quantum phase transition, as demonstrated through computer simulations.

Non-Invertible Defects Signal Symmetry-Enriched Criticality

Scientists established a novel method for identifying symmetry-enriched criticality in gapless phases of matter, focusing on the properties of interfaces between these phases. The work demonstrates that whenever two one-dimensional conformal field theories exhibit differing symmetry charge assignments for local operators, any symmetry-preserving spatial interface between them must develop into a non-invertible defect. This finding provides a new physical indicator for distinguishing between these complex quantum states, moving beyond reliance on edge modes. Experiments revealed that interfaces between Ising conformal field theories host one-dimensional symmetry-breaking phases, exhibiting finite-size splittings that scale as 1/L³, where L represents the system size.

Researchers mapped interfaces between gapless phases differing by a symmetry-protected topological entangler to conformal defects possessing a specific defect anomaly, solidifying the connection between interfacial properties and underlying topological order. This classification extends to higher-dimensional systems, including symmetry-enriched variants of the two-dimensional Ising conformal field theory. The breakthrough delivers a powerful tool for characterizing symmetry-enriched criticality through symmetry-protected interfaces, offering a new approach to understanding the interplay between topology and gapless phases. Data shows that the non-invertible nature of these defects manifests in observable signatures, including unique correlation functions and distinct finite-size scaling of energy spectra, providing concrete experimental handles for detecting these phases. The research establishes that interfaces, rather than boundaries, serve as the fundamental objects for diagnosing distinctions between critical chains with internal symmetry.

Symmetry Breaking Defines Phase Boundary Properties

The research establishes a novel connection between symmetry, critical behaviour, and the properties of interfaces separating different phases. Scientists demonstrate that when two gapless systems possess differing symmetry characteristics, any interface between them necessarily develops non-invertible properties, fundamentally altering how symmetry is expressed. The team showed this principle applies to conformal field theories and successfully classified the allowed interfaces for different versions of the Ising model. This work provides a new way to identify and characterise symmetry-enriched criticality, where symmetry plays a crucial role in defining the system’s behaviour. The researchers highlight that the behaviour of interfaces offers a robust indicator of underlying symmetry structure, with mismatched symmetry charges leading to symmetry breaking at the interface, detectable through measurable changes in energy levels. While the analysis primarily focuses on one and two-dimensional systems, extending these findings to higher dimensions presents a significant challenge, and further investigation is needed to fully understand the nature of the non-invertible defects.