  Critical phase transitions contain a variety of deep and universal physics,
and are intimately tied to thermodynamic quantities through scaling relations.
Yet, these notions are challenged in the context of non-Hermiticity, where
spatial or temporal divergences render the thermodynamic limit ill-defined. In
this work, we show that a thermodynamic grand potential can still be defined in
pseudo-Hermitian Hamiltonians, and can be used to characterize aspects of
criticality unique to non-Hermitian systems. Using the non-Hermitian
Su-Schrieffer-Heeger (SSH) model as a paradigmatic example, we demonstrate the
fractional order of topological phase transitions in the complex energy plane.
These fractional orders add up to the integer order expected of a Hermitian
phase transition when the model is doubled and Hermitianized. More
spectacularly, gap preserving highly degenerate critical points known as
non-Bloch band collapses possess fractional order that are not constrained by
conventional scaling relations, testimony to the emergent extra length scale
from the skin mode accumulation. Our work showcases that a thermodynamic
approach can prove fruitful in revealing unconventional properties of
non-Hermitian critical points.
