Conformally invariant free-parafermionic quantum chains with multispin interactions

We calculate the spectral properties of two related families of non-Hermitian free-particle quantum chains with $N$-multispin interactions ($N=2,3,...$). The first family have a $Z\left(N\right)$ symmetry and are described by free parafermions. The second have a $U\left(1\right)$ symmetry and are generalizations of $XX$ quantum chains described by free fermions. The eigenspectra of both free-particle families are formed by the combination of the same pseudo-energies. The models have a multicritical point with dynamical critical exponent $z=1$. The finite-size behavior of their eigenspectra, as well as the entanglement properties of their ground-state wave function, indicate the models are conformally invariant. The models with open and periodic boundary conditions show quite distinct physics due to their non-Hermiticity. The models defined with open boundaries have a single conformal invariant phase, while the $XX$ multispin models show multiple phases with distinct conformal central charges in the periodic case.