  Symmetry breaking plays a pivotal role in modern physics. Although
self-similarity is also a symmetry and appears ubiquitously in nature, a
fundamental question is whether self-similarity breaking makes sense or not.
Here, by identifying the most important kind of critical fluctuations dubbed as
phases fluctuations and comparing the consequences of having self-similarity
with those of lacking self-similarity in the phases fluctuations, we show that
self-similarity can indeed be broken with significant consequences at least in
nonequilibrium situations. We find that the breaking of self-similarity results
in new critical exponents which give rise to violation of the well-known
finite-size scaling or the less-known finite-time scaling and different leading
exponents in the ordered and the disordered phases of the paradigmatic Ising
model on two- or three-dimensional finite lattices when it subjects to the
simplest nonequilibrium driving of linear heating or cooling through its
critical point, in stark contrast to identical exponents and different
amplitudes in usual critical phenomena. Our results demonstrate how surprising
driven nonequilibrium critical phenomena can be. Application to other classical
and quantum phase transitions is highly expected.
