Supersymmetry plays a crucial role in superstring theory and high-energy physics, but has never been observed in experiments. Recently, an effective space–time supersymmetry (SUSY) was argued to emerge in the low-energy region by tuning Dirac or Weyl semimetal to approach a superconducting (SC) quantum critical point (QCP), at which the Dirac or Weyl fermion and the bosonic order parameter are both massless. Here, we study under what circumstances can SUSY be realized at a quantum critical point. We demonstrate that the Yukawa-type coupling between the massless fermion and massless boson can dynamically generate an infinite number of non-supersymmetric terms in the effective field theory of the boson. Owing to these terms, no space–time supersymmetry emerges at the SC QCPs. The results provide important constraint on the exploration of emergent SUSY in condensed matter systems.

超对称性在超弦理论和高能物理中起着至关重要的作用，但从未在实验中观察到。最近，有人争辩说，通过调节Dirac或Weyl半金属以逼近超导（SC）量子临界点（QCP），在低能量区域出现有效的时空超对称（SUSY），在该临界点处Dirac或Weyl费米子与玻序参数都是无质量的。在这里，我们研究在什么情况下可以在量子临界点实现SUSY。我们证明，在玻色子有效场理论中，无质量费米子与无质量玻色子之间的Yukawa型耦合可以动态生成无限数量的非超对称项。由于这些术语，SC QCP上没有出现时空超对称性。