Analytical quantification of strongly disordered discrete time crystals

TL;DR

This paper develops an analytical method to precisely calculate key observables in strongly disordered discrete time crystals, revealing the effects of eigenstate resonances on their stability and lifetime.

Contribution

The authors introduce a resolvent perturbation framework that provides parameter-free, higher-order analytical predictions for DTC properties, surpassing numerical limitations.

Findings

Eigenstate resonances cause linear scaling deviations in observables.

Analytical formulas match numerical results for localization and entanglement measures.

DTC lifetime scales as (1/λ)^{L/n_{op}} based on spectral pairing deviations.

Abstract

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.