Spontaneous symmetry breaking without ground state degeneracy in generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> -state clock model

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