Emergence of mesoscale quantum phase transitions in a ferromagnet

Mesoscale patterns as observed, e.g., in ferromagnets, ferroelectrics, superconductors, mono-molecular films, or block-copolymers, reflect spatial variations of a pertinent order parameter at length- and time-scales that may be described classically. This raises the question for the relevance of mesoscale patterns near zero temperature phase transitions, also known as quantum phase transitions (QPTs). Here we report the magnetic susceptibility of LiHoF$_4$ -- a dipolar Ising ferromagnet -- near a well-understood transverse-field quantum critical point (TF-QCP). When tilting the magnetic field away from the hard axis such that the Ising symmetry is always broken, a line of well-defined phase transitions emerges from the TF-QCP characteristic of an additional symmetry breaking, in stark contrast to a crossover expected microscopically. We show that a continuous suppression of ferromagnetic domains, representing a breaking of translation symmetry on mesoscopic scales in an environment of broken magnetic Ising symmetry on microscopic scales, is in excellent qualitative and quantitative agreement with the field- and temperature dependence of the susceptibility and the magnetic phase diagram of LiHoF$_4$ under tilted field. This identifies a new type of phase transition that may be referred to as mesocale quantum criticality, which emanates from the text-book example of a microscopic ferromagnetic TF-QCP. Our results establish the surroundings of QPTs as a regime of mesoscale pattern formation, where non-analytical quantum dynamics and materials properties without classical analogue may be expected.