Analytical quantification of strongly disordered discrete time crystals

TL;DR

The paper develops a Floquet resolvent perturbation theory for strongly disordered discrete time crystals, enabling analytic, parameter-free predictions of key observables such as IPR, Edwards-Anderson parameter, mutual information, and autocorrelator plateaus. It reveals that resonances among pairwise cat eigenstates induce $O(λ)$ deviations and derives a lifetime scaling $τ_* \,\sim (1/λ)^{L/n_{op}}$ driven by spectral pairing, with $n_{op}$ counting the perturbation order. The framework delivers closed-form, higher-order corrections without iterative calculations and applies to generic DTC models with Ising interactions, validated by numerical benchmarks and extended to both single- and two-spin perturbations. It also contrasts disordered DTCs with clean cat scars, illustrating resonance-driven distinctions in observable scaling, and outlines implications for experiments on large-scale quantum devices and for foundational discussions of localization and avalanche phenomena.

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 $O(λ)$, in contrast to non-resonant situations with suppressed deviation $O(λ^2)$. 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 $O(λ^{(L/n_{\text{op}})-1})$, where perturbations of strength $λ$ involve at most $n_{\text{op}}$-spin terms. Such spectral pairing deviations quantify the DTC lifetime as $τ_* \sim (1/λ)^{L/n_{\text{op}}}$. 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.

Figures (4)

• Figure 1: Comparison of numerical results (dots) with analytical predictions in Eqs. \ref{['eq:ipr']}, \ref{['eq:EA_ana']}, \ref{['eq:MI_def']}--\ref{['eq:S2']} and \ref{['eq:corr']} (lines) for physical observables in a strongly disordered DTC. (a) (e) Inverse participation ratio (IPR) and its deviation from the unperturbed values $0.5$. (b) (f) Edwards-Anderson parameter and its deviation from the unperturbed values $L$ for a spin chain of $L$ sites. (c) (g) Mutual information for two distant sites $1$ and $L/2$ in a chain with periodic boundary condition, and its deviation from the unperturbed value $\ln(2)\approx0.6931$. (d) (h) Evolution of autocorrelator, and the deviation of its steady state plateau from the unperturbed value $1$. $C(t)$ in (d) illustrates an example with $\lambda=0.08$, while the value of $\langle C\rangle_t$ in (h) is extracted by averaging $C(t)$ over the periods $t\in[10^2, 10^3]T$ at each $\lambda$ and $L$. In all cases, $J=1, W=1.6,$. Results in (a)--(c) and (e)--(g) are averaged over all eigenstates, while (h) is sampled over all initial Fock states. Error bars, if invisible, are within the size of data points. In all panels except (d), we average over $10^6$, $10^6$, $10^5$, $10^4$, $0.3\sim1.4\times10^4$ samples for $L=6,8,10,12,14$ at each $\lambda$, respectively. In panel (d), we average over $10^5$, $10^5$, $10^4$, $10^4$, $2100$ samples for $L=6,8,10,12,14$.
• Figure 2: Scaling of spectral pairing deviation with different operator product order $n_{\text{op}}$ for the perturbation. Here we extract $\omega_{\boldsymbol{s},0}- \omega_{\boldsymbol{s},1}$ by checking the quasienergy difference between the $m$-th and $(m+2^{L-1})$-th eigenstate, where quasienergies obtained numerically are arranged as $\omega_m < \omega_{m+1}$. (a) Single-spin perturbation with $n_{\text{op}}=1$. (b) Single- and two-spin perturbation with $n_{\text{op}}=2$. (c) (d) Extraction of scaling exponent for (a) and (b), respectively, showing that the deviation $\delta_{\boldsymbol{s}} \sim \lambda^{\gamma L}$ from a perfect spectral pairing $|\omega_{\boldsymbol{s},1} - \omega_{\boldsymbol{s},0}|=\pi$ scales as $\gamma \rightarrow n_{\text{op}}^{-1}$ in the strong disorder (weak perturbation) limit, consistent with Eq. \ref{['eq:sp']}. Here for each numerical data point, we averaged over all eigenstates of different dominant spin patterns $\boldsymbol{s}$ and disorder samples as described in Fig. \ref{['fig1']}. In all panels, machine number precision is limited to $\sim 10^{-15}$. We average over $10^6$, $10^6$, $10^5$, $10^4$ samples for $L=6,8,10,12$, while for $L=14$, we average over $0.3\sim1.4\times10^4$ samples for $n_\mathrm{op}=1$ and $2000$ samples for $n_\mathrm{op}=2$.
• Figure 3: IPR for the generalized model with both single- and two-spin perturbations in Eq. \ref{['eq:UF_nop2']}. (a) IPRs averaged over eigenstates and disorder realizations. (b) Deviation of IPRs from the unperturbed value $0.5$, for the model. Legends are the same as in Fig. \ref{['fig1']} with colors denoting different system sizes. The analytical results, denoted by lines in panel (b), corresponds to Eq. \ref{['eq:ipr_n2']}. For $L=6,8,10,12,14$, we sample over $10^6, 10^6, 10^5, 10^4, 2\times 10^3$ disorder realizations at each $\lambda$, respectively.
• Figure 4: (a) Eigenstate IPRs and (b) their deviation from the unperturbed value $0.5$ for translation-invariant models hosting rare cat scar eigenstates huang2023PRBa. (c) Edwards-Anderson parameter and (d) their deviation from the unperturbed value $L$ for cat scars. Notably, in panels (b) and (d) for the data collapse, we see that since cat scars are free from resonance, the deviation $0.5-\text{IPR}, L-\chi_{\text{EA}} \sim \lambda^2 L$ is suppressed at the same strength of perturbation $\lambda\ll1$, compared with the majority eigenstates in disordered situation showing stronger deviations $\sim \lambda L$ in Fig. \ref{['fig1']}(a), (b), (e), and (f). Here we take $J_j=J=1, h_j=h=0.3$ for the model in Eq. \ref{['eq:UF_general']} and \ref{['eq:UF_nop1']}.