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New families of non-parity-time-symmetric complex potentials with all-realspectra are derived by the supersymmetry method and the pseudo-Hermiticitymethod. With the supersymmetry method, we find families ofnon-parity-time-symmetric complex partner potentials which share the samespectrum as base potentials with known real spectra, such as the (complex)Wadati potentials. Different from previous supersymmetry derivations ofpotentials with real spectra, our derivation does not utilize discreteeigenmodes of base potentials. As a result, our partner potentials featureexplicit analytical expressions which contain free functions. With thepseudo-Hermiticity method, we derive a new class of non-parity-time-symmetriccomplex potentials with free functions and constants, whose eigenvalues appearas conjugate pairs. This eigenvalue symmetry forces the spectrum to be all-realfor a wide range of choices of these functions and constants in the potential.Tuning these free functions and constants, phase transition can also beinduced, where conjugate pairs of complex eigenvalues emerge in the spectrum.

New families of non-parity-time-symmetric complex potentials with all-real spectra