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The renormalization group and the scaling theories that result from its use
have become essential to our understanding of singular behavior in
thermodynamic functions at continuous phase transitions. Despite this, they
have long been viewed as irrelevant to the quantitative study of abrupt phase
transitions, and likewise the presence of an abrupt phase transition has
typically been considered irrelevant to scaling analysis. Our work
demonstrates, for one common model with both a continuous and a
line of abrupt transitions, that well-known singularities in thermodynamic
functions at the abrupt transition long assumed to be unobservable are in fact
part of the scaling theory of the critical point, and make explicit
predictions confirmed by numerics. This result indicates that many published
statistical models whose abrupt transitions were not incorporated into scaling
theories should be revisited, and introduces a novel new method for
determining new universal scaling properties in previously poorly-understood
regimes.
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