Analytical quantification of strongly disordered discrete time crystals

Abstract

Discrete time crystals (DTCs) have emerged as a novel non-equilibrium phenomenon. Nevertheless, analytical quantification for disordered DTCs remains challenging, leaving uncertainties about their behaviors in large scale systems amid current debates on the stability of (Floquet-)many-body localization. In this work, we introduce an analytical framework to calculate the values of key observables in a strongly disordered DTC without fitting parameter. The perturbatively obtained closed-form formulae show quantitative agreement with numerical simulations for 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 newly developed scheme adopts resolvent perturbation formalism, which 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 ${\tau }_{\ast }\sim (1/\lambda {)}^{L/n_{\text{op}}}$. We further show that the analytical framework applies to generic DTC models with dominant Ising interaction and a given number of qubits, offering a unified scheme to quantify physical observables for systems beyond numerically accessible sizes, with or without disorders.