  The parity-time (PT) symmetry of a non-Hermitian Hamiltonian leads to real
(complex) energy spectrum when the non-Hermiticity is below (above) a
threshold. Recently, it has been demonstrated that the non-Hermitian skin
effect generates a new type of PT symmetry, dubbed the non-Bloch PT symmetry,
featuring unique properties such as high sensitivity to the boundary condition.
Despite its relevance to a wide range of non-Hermitian lattice systems, a
general theory is still lacking for this generic phenomenon even in one spatial
dimension. Here, we uncover the geometric mechanism of non-Bloch PT symmetry
and its breaking. We find that non-Bloch PT symmetry breaking occurs by the
formation of cusps in the generalized Brillouin zone (GBZ). Based on this
geometric understanding, we propose an exact formula that efficiently
determines the breaking threshold. Moreover, we predict a new type of spectral
singularities associated with the symmetry breaking, dubbed non-Bloch van Hove
singularity, whose physical mechanism fundamentally differs from their
Hermitian counterparts. This singularity is experimentally observable in linear
responses.
