Information scrambling and butterfly velocity in quantum spin glass chains

We make lattice generalization of two well-known zero dimensional models ofquantum spin glass, Sachdev-Ye (SY) and spherical quantum $p$-spin glass model,to one dimension for studying crossovers in non-local scrambling dynamics dueto glass transition, complex dynamics, and quantum and thermal fluctuations inparamagnetic (PM) and spin glass (SG) phases. In the SY chain of quantum dots,each described by infinite-range random Heisenberg model of $N$ spin-$S$$SU(M)$ spins, we obtain the quantum Lyapunov exponent $\lambda_\mathrm{L}$ andbutterfly velocity $v_B$ as a function of temperature $T$ and the quantumparameter $S$ across the PM-SG phase boundary using a bosonic spinonrepresentation in the large-$N,M$ limit. In particular, we extract asymptotic$T$ and $S$ dependence, e.g., power laws, for $\lambda_\mathrm{L}$ and $v_B$ indifferent regions deep inside the phases and near the replica symmetry breakingSG transition. We find the chaos to be non-maximal almost over the entire phasediagram. Very similar results for chaos indicators are found for the $p$-spinglass chain as a function of temperature and a suitable quantum parameter$\Gamma$, with some important qualitative differences. In particular,$\lambda_\mathrm{L}$ and $v_B$ exhibit a maximum, coinciding with onset ofcomplex glassy relaxation, above the glass transition as a function of $T$ and$\Gamma$ in the PM phase of the $p$-spin glass model. In contrast, the maximumis only observed as a function of $S$, but not with temperature, in the PMphase of SY model. The maximum originates from enhanced chaos due to maximalcomplexity in the glassy landscape. Thus, the results in the SY model indicatevery different evolution of glassy complexity with quantum and thermalfluctuations.