Phase transition in space: how far does a symmetry bend before it breaks?

We extend the theory of symmetry-breaking dynamics in non-equilibrium second-order phase transitions known as the Kibble–Zurek mechanism (KZM) to transitions where the change of phase occurs not in time but in space. This can be due to a time-independent spatial variation of a field that imposes a phase with one symmetry to the left of where it attains critical value, while allowing spontaneous symmetry breaking to the right of that critical borderline. Topological defects need not form in such a situation. We show, however, that the size, in space, of the ‘scar’ over which the order parameter adjusts as it ‘bends’ interpolating between the phases with different symmetries follows from a KZM-like approach. As we illustrate on the example of a transverse quantum Ising model, in quantum phase transitions this spatial scale—the size of the scar—is directly reflected in the energy spectrum of the system: in particular, it determines the size of the energy gap.