  Spontaneous symmetry breaking is a foundational concept in physics. In
condensed matter, it characterizes conventional continuous phase transitions
but is absent at topological phase transitions such as the
Berezinskii-Kosterlitz-Thouless (BKT) transition - as in the BKT case the
expected norm (i.e., the magnitude) of the $U(1)$ order parameter vanishes in
the thermodynamic limit at all nonzero temperatures. Phenomena consistent with
low-temperature broken symmetry have been observed, however, in many different
BKT experiments. Examples include recent experiments on superconducting films
and the seminal work on two-dimensional arrays of Josephson junctions. While
the inaccessibility of the above thermodynamic limit partially explains this
paradox in finite systems, the full dynamical framework of symmetry breaking at
the BKT transition remains unresolved. Here we provide this by introducing the
broader concept of general symmetry breaking. This encompasses both spontaneous
symmetry breaking and the BKT case by allowing the expected norm of the order
parameter to go to zero in the thermodynamic limit, provided its directional
phase fluctuations are asymptotically smaller. We demonstrate this
asymptotically slow directional mixing in the low-temperature BKT phase. This
explicitly shows that the order parameter arbitrarily chooses some well-defined
direction in the thermodynamic limit, predicting negligible phase fluctuations
compared to the expected norm in arbitrarily large experimental BKT systems.
Our results provide a model for directional mixing timescales across the
diverse array of experimental BKT systems. We suggest various experiments.
