Spontaneous symmetry breaking without ground state degeneracy in generalized $N$-state clock model

Abstract

Spontaneous symmetry breaking is ubiquitous phenomenon in nature. One of the defining features of symmetry broken phases is that the large system size limit and the vanishing external field limit do not commute. In this work, we study a family of extensions of the $N$-state clock model. We find that the exact symmetry and the ground state degeneracy under the periodic boundary condition heavily depend on the system size, although the model has the manifest translation symmetry. In particular, the ground state can be unique and all excitations are gapped even when the phase exhibits non-commutativity of the two limits. Our model hence poses a question on the standard understanding of spontaneous symmetry breaking.

arXiv comment10 pages, 6 figures; v2: Fig 1 is added. Minor changes throughout the manuscript

References in corpus (11)

• Fault-tolerant quantum computation by anyons
• Probing many-body dynamics on a 51-atom quantum simulator
• Topological quantum memory
• The ITensor Software Library for Tensor Network Calculations
• Dualities and the phase diagram of the $p$-clock model
• Quantum field theory for the chiral clock transition in one spatial dimension
• Parafermionic clock models and quantum resonance
• The A-Cycle Problem for Transverse Ising Ring
• Ground state degeneracy on torus in a family of ${\mathbb{Z}}_N$ toric code
• Finite-size scaling at first-order quantum transitions when boundary conditions favor one of the two phases
• Phase transitions in the ${\mathbb{Z}}_p$ and U(1) clock models

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• ${\mathbb{Z}}_N$ generalizations of three-dimensional stabilizer codes
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