  Quantum adiabatic evolution, an important fundamental concept inphysics,
describes the dynamical evolution arbitrarily close to the instantaneous
eigenstate of a slowly driven Hamiltonian. In most systems undergoing
spontaneous symmetry-breaking transitions, their two lowest eigenstates change
from non-degenerate to degenerate. Therefore, due to the corresponding
energy-gap vanishes, the conventional adiabatic condition becomes invalid. Here
we explore the existence of quantum adiabatic evolutions in spontaneous
symmetry-breaking transitions and derive a symmetry-dependent adiabatic
condition. Because the driven Hamiltonian conserves the symmetry in the whole
process, the transition between different instantaneous eigenstates with
different symmetries is forbidden. Therefore, even if the minimum energy-gap
vanishes, symmetry-protected quantum adiabatic evolutioncan still appear when
the driven system varies according to the symmetry-dependent adiabatic
condition. This study not only advances our understandings of quantum adiabatic
evolution and spontaneous symmetry-breaking transitions, but also provides
extensive applications ranging from quantum state engineering, topological
Thouless pumping to quantum computing.
