Global symmetries of quantum lattice models under non-invertible dualities (2501.12514v2)

Abstract: Non-invertible dualities/symmetries have become an important tool in the study of quantum field theories and quantum lattice models in recent years. One of the most studied examples is non-invertible dualities obtained by gauging a discrete group. When the physical system has more global symmetries than the gauged symmetry, it has not been thoroughly investigated how those global symmetries transform under non-invertible duality. In this paper, we study the change of global symmetries under non-invertible duality of gauging a discrete group $G$ in the context of (1+1)-dimensional quantum lattice models. We obtain the global symmetries of the dual model by focusing on different Hilbert space sectors determined by the $\mathrm{Rep}(G)$ symmetry. We provide general conjectures of global symmetries of the dual model forming an algebraic ring of the double cosets. We present concrete examples of the XXZ models and the duals, providing strong evidence for the conjectures.

• The paper demonstrates that gauging discrete groups transforms global symmetries in quantum lattice models into non-invertible algebraic structures, exemplified by rings of double cosets.
• It employs matrix product operators to systematically decompose Hilbert spaces and illustrate symmetry evolution in models like the XXZ and Ising zig-zag.
• The study’s insights pave the way for future research in quantum field theory and condensed matter, unifying duality concepts across various systems.

Overview of "Global Symmetries of Quantum Lattice Models Under Non-invertible Dualities"

The paper authored by Weiguang Cao, Yuan Miao, and Masahito Yamazaki addresses an intricate yet significant area within quantum lattice models—how global symmetries transform under non-invertible dualities, specifically when executing the duality via gauging a discrete group. This paper is particularly relevant in the context of (1+1)-dimensional models, with implications extending across theoretical physics disciplines, including quantum field theories and condensed matter physics.

The authors investigate the non-invertible transformations that occur by engaging with discrete groups in the symmetry structure of quantum models. This investigation particularly focuses on how residual symmetries manifest post-transformation, often culminating in non-trivial algebraic structures such as rings of double cosets. The work expands on previously established duality concepts and scrutinizes their broader implications in quantum lattice frameworks.

Theoretical Framework and Methodology

1. Dualities in Quantum Systems: Dualities in quantum systems have seen significant examination due to their capability to simplify complex problems by transforming them to equivalently solvable forms. The traditional Kramers-Wannier duality is revisited in this paper, elevating its interpretation to modern paradigms, such as non-invertible symmetries and duality defects.
2. Transformations and Symmetries: The authors aim to resolve how global symmetries of original models translate post-duality. They examine sectors determined by representations of gauged groups and propose that these transformations culminate in algebraic rings forming double cosets. Through explicit examples such as XXZ and Ising zig-zag models, the paper demonstrates these transformations and how they manifest in practice.
3. Quantum Symmetry Analysis: The paper's methodology includes decomposing the Hilbert spaces of quantum lattice models both prior to and after transformations. It details the use of duality operators, expressed via matrix product operators (MPOs), to perform these transformations systematically. Their treatment of symmetry involves both theoretical speculation and rigorous demonstration through concrete examples.

Key Contributions and Findings

• The paper presents a profound assertion that global symmetries can transition into non-invertible forms characterized by a novel algebraic structure. Examples provided, notably involving the XXZ and its duals, illustrate scenarios wherein gauging leads to unexpected symmetries, such as cosine-type symmetries in certain model sectors.
• Symmetry transformations under dualities reflect an intricate interplay between original and gauged symmetries, often resulting in new, emergent symmetries. The paper postulates the dual model’s symmetry might equate to a representation of a ring of double cosets derived from the gauged symmetries.
• Through systematic sector analysis, the paper identifies and verifies conjectures regarding Hilbert space decompositions post-duality. Each sector displays different symmetries, often adhering to known algebraic structures such as those derived from non-Abelian groups.

Implications and Future Directions

The paper's implications extend beyond specific models, rendering the theoretical propositions applicable to a range of quantum systems involving discrete symmetries. It lays foundational thoughts for understanding how non-invertible symmetries might further interlace with topological phases and anomalies in higher-dimensional physics.

Future research could explore broadening the scope of symmetries under variant conditions, such as considering continuum quantum field theories or exploring richer fusion categories beyond group-based symmetries. Additionally, the insights gained could refine quantum information theory, particularly regarding error-correcting codes predicated on symmetry principles.

This work contributes substantially toward demystifying the roles of symmetries under non-invertible dualities in condensed matter and theoretical physics, offering a framework for future exploration and unification of dualities across theoretical paradigms.