Exceptional-point criticality: Conformal and non-conformal field theories

We study free-fermion chains with an exceptional point and ask which field-theoretic structures describe their critical behaviour. The fractional-power dispersion is the single-band remnant of a local multiband model with a Jordan block, making the results exact for the free theory. We determine the symmetry, scaling dimensions, correlators, entanglement entropy, and finite-size spectrum separately, since a single central charge does not fix them. The dynamical exponent is set by the Puiseux exponent, while the Jordan structure determines whether logarithmic correlators arise. The linear dispersion gives a relativistic-like logarithmic theory. In the simplest case, this is a symplectic-fermion theory with a complex velocity, whose two Virasoro algebras contract to a sector of a Galilean conformal algebra with vanishing rapidity. The square-root dispersion has Schrödinger symmetry with position as time, whereas higher Puiseux orders exhibit scale invariance as their only manifest symmetry. These theories have no Virasoro algebra. The entanglement entropy depends on the regularisation of the unbounded correlation matrix, while the Casimir energy of a ring yields a universal complex amplitude rather than a central charge. The non-analytic dispersion encodes nonlocality, whereas the Jordan structure determines logarithms and chirality.