Ruling out spontaneous symmetry breaking on a line defect in the 3d Ising conformal field theory

It is well-known that one-dimensional systems at finite temperature, such as the classical Ising model, cannot spontaneously break a discrete symmetry due to the proliferation of domain walls. The validity of this statement rests on a few assumptions, including spatial locality of interactions. In a situation where a one-dimensional system is a defect in a critical, higher-dimensional bulk system, the coupling between defect and bulk can induce an effective long-range interaction on the defect. It is thus natural to ask if long-range order can be stabilized on a defect in a critical bulk, which amounts to asking whether domain walls on the defect are relevant or not in the renormalization group sense. We explore this question in the Ising conformal field theory in two and higher dimensions by studying a conformal line defect with a symmetry-breaking field localized on it. With both perturbative techniques and numerical conformal bootstrap, we provide evidence that indeed the defect domain wall must be relevant when 2 < d < 4. Additionally, we give tight estimates of various additional quantities characterizing this defect, including critical exponents and the defect entropy.