Unitary Long-Time Evolution with Quantum Renormalization Groups and Artificial Neural Networks

In this work, we combine quantum renormalization group approaches with deep artificial neural networks for the description of the real-time evolution in strongly disordered quantum matter. We find that this allows us to accurately compute the long-time coherent dynamics of large many-body localized systems in nonperturbative regimes including the effects of many-body resonances. Concretely, we use this approach to describe the spatiotemporal buildup of many-body localized spin-glass order in random Ising chains. We observe a fundamental difference to a noninteracting Anderson insulating Ising chain, where the order only develops over a finite spatial range. We further apply the approach to strongly disordered two-dimensional Ising models, highlighting that our method can be used also for the description of the real-time dynamics of nonergodic quantum matter in a general context.

Figure 1

(a) Illustration of random quantum circuit built up from local unitary RG transformations. In the course of the RG, long-distance and higher-order couplings emerge. Adding time dependence leads to further broadening with increasing time. (b) Spatiotemporal buildup of MBL spin-glass order in a random quantum Ising chain with 64 lattice sites after quenching a paramagnetic initial condition into the symmetry-broken phase at $J=5h$ and $J^{(x)}=h/5$ (interacting case). Dashed lines indicate emergence of light cones, except for noninteracting case $J^{(x)}=0$, where the order stops developing at a finite distance. Numerical data obtained from average over 25 disorder realizations.

Figure 2

(a) Comparing dynamics of transverse magnetization to exact diagonalization, averaged over lattice sites and 250 disorder realizations, with and without treatment of many-body resonances. Here, $L=12$ and $h=\{4,1,1/4\}J$ from top to bottom, and $J^{(x)}=J/8$. (b) Lower bound $F^{*}$ on many-body fidelity, using $L=64$, of trained ANN state with state given by hypothetical, exact application of quantum circuit for different numbers of hidden units $M$ in each layer of deep forward ANN. System parameters are $J=5h$ and $J^{(x)}=h/5$ (interacting case). Shaded areas indicate uncertainties due to finite disorder ensemble of 25 realizations. Result from first-order cumulant expansion of quantum circuit shown for comparison.

Figure 3

Quench dynamics on $L=12\times 12$ periodic simple square lattice with disordered next-neighbor Ising couplings and disordered transverse field. Local magnetization along initial direction of polarization plotted as average over lattice sites and 25 disorder realizations for various bounds of external field strengths. Inset shows lower bound of many-body fidelity as stated in Fig. 2.