[Paper Review] Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections

Salvatore D. Pace, Guilherme Delfino+2 more|arXiv (Cornell University)|Jun 18, 2024

Molecular spectroscopy and chirality6 citations

TL;DR

The paper studies gauging finite Abelian modulated symmetries in 1+1D spin chains, derives dualities implemented by reflections, and constructs non-invertible reflection operators via sequential quantum circuits.

ABSTRACT

Modulated symmetries are internal symmetries that act in a non-uniform, spatially modulated way and are generalizations of, for example, dipole symmetries. In this paper, we systematically study the gauging of finite Abelian modulated symmetries in ${1+1}$ dimensions. Working with local Hamiltonians of spin chains, we explore the dual symmetries after gauging and their potential new spatial modulations. We establish sufficient conditions for the existence of an isomorphism between the modulated symmetries and their dual, naturally implemented by lattice reflections. For instance, in systems of prime qudits, translation invariance guarantees this isomorphism. For non-prime qudits, we show using techniques from ring theory that this isomorphism can also exist, although it is not guaranteed by lattice translation symmetry alone. From this isomorphism, we identify new Kramers-Wannier dualities and construct related non-invertible reflection symmetry operators using sequential quantum circuits. Notably, this non-invertible reflection symmetry exists even when the system lacks ordinary reflection symmetry. Throughout the paper, we illustrate these results using various simple toy models.

Motivation & Objective

• Characterize gauging of finite Abelian modulated symmetries in 1+1 dimensions.
• Understand the dual symmetry structure and possible new spatial modulations after gauging.
• Identify conditions for isomorphisms between original and dual modulated symmetries via lattice reflections.
• Construct Kramers-Wannier dualities and non-invertible reflection operators in this setting.
• Illustrate the framework with explicit toy models and examples.

Proposed method

• Define modulated symmetries as semidirect products with a space action via a homomorphism varphi.
• Use bond algebras to describe local symmetric operator structure and derive Gauss’s law gauging maps.
• Show dual symmetry generators U^vee as products of link operators and relate them to reflected originals.
• Prove an isomorphism B ≃ M B^vee M^{-1} implemented by a reflection M under broad conditions.
• Construct KW duality operators D_M and D_KW via sequential quantum circuits in representative models.
• Develop a general self-duality analysis for Z_N dipole, quadrupole, and multipole symmetries.

Experimental results

Research questions

• RQ1 When gauging finite Abelian modulated symmetries in 1+1D, what is the structure of the dual symmetry and its spatial modulation?
• RQ2 Under what conditions is there a canonical isomorphism between original and dual bond algebras implemented by lattice reflections?
• RQ3 How do Kramers-Wannier dualities manifest as non-invertible reflection symmetries in modulated symmetry systems?
• RQ4 Can one realize self-dual gauging and non-invertible reflections in explicit toy models such as dipole, quadrupole, and exponential modulated symmetries?
• RQ5 What is the role of ring-theoretic techniques in extending gauging to non-prime N cases?

Key findings

• For prime p, the dual symmetry after gauging an exponential modulated Z_p symmetry remains exponential with the inverse exponent a^{-1} = a^{p-2}.
• There exists a canonical isomorphism between the original and dual bond algebras implemented by a reflection, yielding self-duality when reflection symmetry is present.
• The dual modulated symmetry is generally not isomorphic to the original if the spatial action varphi differs between the two, leading to new modulations.
• A non-invertible reflection operator D_M can be constructed, with a fusion rule D_M^2 proportional to a projector, and it generates a KW-type duality at a self-dual point.
• Sequential gauging can realize gauging of non-prime N modulated symmetries via a step-by-step process that preserves a bond-algebra form similar to the prime case.
• The framework yields explicit non-invertible KW operators D_KW and demonstrates their action in Z_N dipole and exponential symmetry models.