Lattice T-duality from non-invertible symmetries in quantum spin chains

Summary

The paper discusses the concept of T-duality in quantum field theories and its realization in lattice models, specifically in the XX model. It explores the symmetry operators and anomalies in the XX model, including the momentum and winding symmetries, and the non-invertible symmetry operator D. The paper also discusses the Onsager algebra and its relation to the lattice T-duality.

Highlights

• T-duality in quantum field theories and lattice models is discussed.
• The XX model is used as an example to realize T-duality on the lattice.
• Symmetry operators and anomalies in the XX model are explored.
• The non-invertible symmetry operator D is introduced.
• The Onsager algebra is related to the lattice T-duality.
• The XYZ model is discussed as a deformation of the XX model.
• The U(1)M and U(1)W symmetric spin chains are constructed.

Key Insights

• The XX model has a lattice T-duality that exchanges the momentum and winding symmetries, which is implemented by the unitary operator U_T.
• The non-invertible symmetry operator D satisfies the operator algebra D^2 = (1+η)T e^{-iπQM}, Dη = ηD = D, CD = DC, TDT^{-1} = eiπQMeiπQWD, and D† = DT^{-1} eiπQM.
• The Onsager charges Q_n are quantized and satisfy the Onsager algebra [Q_n,Q_m] = iG_{n-m}, [G_n,G_m] = 0, and [Q_n,G_m] = 2i(Q_{n-m} - Q_{n+m}).
• The XYZ model has three gapped phases, all of which have two ground states, and the phase transitions between them are described by the compact free boson CFT at various radii 1 ≤ R ≤ √2.
• Any Hamiltonian that commutes with both QM and QW has the non-invertible symmetry formed by D, and any QM and QW symmetric Hamiltonian has lattice T-duality that exchanges the momentum and winding charges.
• The bosonization map transforms the fermionic model to the XX model, and the fermionization map transforms the XX model to the two-Majorana chain.
• The Onsager algebra is generated by the Onsager charges Q_n, which can be expressed in terms of the Pauli operators using the bosonization map.

Mindmap

Citation

Pace, S. D., Chatterjee, A., & Shao, S.-H. (2024). Lattice T-duality from non-invertible symmetries in quantum spin chains (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18606