Invertible Symmetry and Spontaneous Duality Breaking in the Transverse-Field Ising Model

TL;DR

This paper demonstrates that adjusting boundary conditions in the transverse-field Ising model allows for an exact, invertible duality, revealing new insights into symmetry, topology, and boundary effects at quantum critical points.

Contribution

It introduces a model with open boundaries that admits an exact duality implemented by an invertible operator, clarifying the nature of duality and symmetry breaking in quantum systems.

Findings

Exact duality is achieved with open boundary conditions.

Anomalous edge degrees of freedom are necessary for the duality.

Spontaneous duality breaking can be influenced by environmental perturbations.

Abstract

The self-duality of the transverse-field Ising model is an archetype for dualities that, alongside symmetry and topology, are used as an organizing principle throughout modern physics. This duality, however, is not exact. The original and dual models have different symmetries and numbers of ground states, and the duality is implemented by a non-invertible operator giving rise to a non-invertible symmetry at the quantum critical point. Here, we show that by adjusting the model to accommodate open rather than periodic boundary conditions, it allows for an exact duality implemented by a unique invertible operator. In the model with exact duality, the symmetry at the quantum critical point is also exact, and hence invertible. Moreover, we find that the exact duality necessitates the presence of an anomalous edge degree of freedom, thus realizing a duality rather than topology based bulk-boundary correspondence. Finally, the exactness of the duality implies that the spontaneous breakdown of a global symmetry in terms of the original model can equivalently be described as spontaneously breaking a local symmetry in the dual system. We show that this seeming contradiction of Elitzur's theorem can be explained by the original and dual models obtaining different sensitivities to spatially local perturbations in any physical implementation of the Hamiltonian. Although the dual partners are mathematically equivalent, their physical implementations therefore are not. In analogy to the spontaneous breakdown of symmetries, we term this emergent distinction due to arbitrarily small environmental influences spontaneous duality breaking.