<p>Scale invariance in quantum mechanics can be broken in several ways. A well-known example is the breakdown of continuous scale invariance to discrete scale invariance, whose typical realization is the Efimov effect of three-body problems. Here we discuss yet another discrete symmetry to which continuous scale invariance can be broken: <i>discrete phase invariance</i>. We first revisit the one-body problem on the half line in the presence of an inverse-square potential—the simplest example of nontrivial scale-invariant quantum mechanics—and show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant. We also show that discrete phase invariance manifests itself as circularly distributed simple poles on Riemann sheets of the S-matrix. We then present three examples of few-body problems that exhibit discrete phase invariance. These examples are the one-body Aharonov-Bohm problem, a two-body problem of nonidentical particles in two dimensions, and a three-body problem of nonidentical particles in one dimension, all of which contain a codimension-two “magnetic” flux in configuration spaces.</p>