Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective

Abstract

The Kramers-Wannier self-duality of critical quantum chains is examined from the perspective of model wave functions. We demonstrate, using the transverse-field Ising chain and the $3$-state Potts chain as examples, that the symmetry operator for the Kramers-Wannier self-duality follows in a simple and direct way from a `generalised' translation symmetry of the model wave function in the anyonic fusion basis. This translation operation, in turn, comprises a sequence of $F$-moves in the underlying fusion category. The symmetry operator thus obtained naturally admits the form of a matrix product operator and obeys non-invertible fusion rules. The findings reveal an intriguing connection between the (non-invertible) translation symmetry on the lattice and topological aspects of the conformal field theory describing the scaling limit.

31 pages, 3 figures, published version

References in corpus (26)

• Non-Abelian Anyons and Topological Quantum Computation
• Generalized Global Symmetries
• Interacting anyons in topological quantum liquids: The golden chain
• Duality and defects in rational conformal field theory
• Generalized Symmetries in Condensed Matter
• Construction of two-dimensional topological field theories with non-invertible symmetries
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Lecture Notes on Generalized Symmetries and Applications
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Non-Invertible Duality Transformation Between SPT and SSB Phases
• On lattice models of gapped phases with fusion category symmetries
• Dualities in one-dimensional quantum lattice models: topological sectors
• Cluster state as a non-invertible symmetry protected topological phase
• Subsystem Non-Invertible Symmetry Operators and Defects
• Discrete torsion defects
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• Topological Disorder Parameter
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• From the XXZ chain to the integrable Rydberg-blockade ladder via non-invertible duality defects
• Non-invertible symmetries act locally by quantum operations
• Topological aspects of the critical three-state Potts model
• Generating Lattice Non-invertible Symmetries
• Chiral correlators of the Ising conformal field theory
• Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains
• The BCS wave function, matrix product states, and the Ising conformal field theory

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• Isolated zero mode in a quantum computer from a duality twist
• The Yang-Baxter integrability of the critical Ising chain
• Noninvertible Kramers-Wannier duality symmetries for the discrete-time quantum Ising chain