Analytical quantification of strongly disordered discrete time crystals

We introduce an analytical framework to calculate the values of key observables in a strongly disordered discrete time crystal (DTC) without fitting parameter. The perturbatively obtained closed-form formulae show quantitative agreement with numerical simulations of inverse participation ratios for eigenstate localization in Fock space, Edwards-Anderson parameters for spin-glass orders, mutual information for long-range entanglement, and the steady-state amplitudes of autocorrelators for period-doubled oscillations. Meanwhile, we demonstrate that eigenstate resonances render the scaling for the deviation of physical observables from their unperturbed values as $O(\lambda)$, in contrast to non-resonant situations with suppressed deviation $O(\lambda^2)$. Our scheme is based on the resolvent perturbation method that can directly prescribe arbitrarily higher-order corrections without iterations. With such advantages, we analytically prove that quasienergy corrections for pairwise cat eigenstates are identical up to order $O(\lambda^{(L/n_{\text{op}})-1})$, where perturbations of strength $\lambda$ involve at most $n_{\text{op}}$-spin terms. Such spectral pairing deviations quantify the DTC lifetime as $\tau_* \sim (1/\lambda)^{L/n_{\text{op}}}$. Our analytical scheme applies to generic DTC models with dominant Ising interaction and a given number of qubits, which allows for independent quantification of physical observables beyond the system size accessible to numerical simulations.