Non-Hermitian free-fermion critical systems and logarithmic conformal field theory

Conformal invariance often accompanies criticality in Hermitian systems. However, its fate in non-Hermitian settings is less clear, especially near exceptional points where the Hamiltonian becomes non-diagonalizable. Here we investigate whether a 1+1-dimensional gapless non-Hermitian system can admit a conformal description, focusing on a PT-symmetric free-fermion field theory. Working in the biorthogonal formalism, we identify the conformal structure of this theory by constructing a traceless energy-momentum tensor whose Fourier modes generate a Virasoro algebra with central charge $c=-2$. This yields a non-Hermitian, biorthogonal realization of a logarithmic conformal field theory, in which correlation functions exhibit logarithmic scaling and the spectrum forms Virasoro staggered modules that are characterized by universal indecomposability parameters. We further present a microscopic construction and show how the same conformal data (with finite-size corrections) can be extracted from the lattice model at exceptional-point criticality, thereby supporting the field-theory prediction.