Enumerating cat scars analytically in clean discrete time crystals

We construct an analytical theory to enumerate and characterize the scar eigenstates responsible for non-prethermal clean discrete time crystal (DTC) dynamics. The results include two parts. First, we introduce a projective symmetry indicator method to quickly identify all possible scars leading to clean DTC phenomena. Importantly, the symmetry indicator predicts the coexistence of ferromagnetic and antiferromagnetic scars, with the latter unnoticed before in clean systems. The inhomogeneous anti-ferromagnetic DTC patterns offer a valuable opportunity to distinguish local clean DTC oscillations from certain short-lived homogeneous global oscillations based on early-time data accessible to experiments. Second, we prove three analytical scaling relations characterizing the DTC amplitudes, Fock space localization, and lifetime, where the bounds for scaling exponents are analytically given by simply counting how many operators are multiplied in perturbation Hamiltonians. These two results pave the way to a systematic understanding and classification of clean DTCs based on spatial symmetries, and also to future connections of clean and disordered DTCs in a unified scheme.