Learning the non-Markovian features of subsystem dynamics

TL;DR

This work investigates non-Markovian features in the dynamics of local subsystems embedded in an infinite quantum Ising chain after a global quench. Using transverse folding tensor networks and machine-learning–driven optimization, the authors extract time-dependent dynamical maps for single- and two-site subsystems across critical, integrable non-critical, and chaotic regimes, and learn time-local generators $\\Lambda_t$ that approximate the reduced dynamics. They show the reduced maps are non-divisible, yet the learned time-local forms reproduce local observables and can forecast long-time behavior, with a new non-Markovianity measure based on the distance to the closest CP-divisible form aligning with standard criteria such as LPP and BLP. The critical regime tends toward a Markovian representation at large times, whereas the non-critical integrable and chaotic cases retain non-Markovian features, highlighting regime-dependent memory effects. The results offer a practical route to extend subsystem dynamics beyond direct tensor-network simulation and deepen understanding of memory in strongly coupled many-body systems.

Abstract

The dynamics of local observables in a quantum many-body system can be formally described in the language of open systems. The problem is that the bath representing the complement of the local subsystem generally does not allow the common simplifications often crucial for such a framework. Leveraging tensor network calculations and optimization tools from machine learning, we extract and characterize the dynamical maps for single- and two-site subsystems embedded in an infinite quantum Ising chain after a global quench. We consider three paradigmatic regimes: integrable critical, integrable non-critical, and chaotic. For each we find the optimal time-local representation of the subsystem dynamics at different times. We explore the properties of the learned time-dependent Liouvillians and whether they can be used to forecast the long-time dynamics of local observables beyond the times accessible through direct quantum many-body numerical simulation. Our procedure naturally suggests a novel measure of non-Markovianity based on the distance between the quasi-exact dynamical map and the closest CP-divisible form and reveals that criticality leads to the closest Markovian representation at large times.

Figures (11)

• Figure 1: Pictorial representation of the transverse folding tensor network which evolves the subsystem density matrix (spanning between L and R site) up to a given time. Environment is initialized in a product state, with a local state denoted with symbol $\raisebox{0.55ex}{$\longrightarrow$} \space\raisebox{0.65ex}{$\hbox{$\bigcirc$}$}\;\,$ .
• Figure 2: Modulus of the eigenvalues $e_i(t)$ of $\epsilon_{t,0}$ as a function of time in a log-log scale. Here, we present the results for single-site subsystems ($l=1$) and the (a) critical $(g^x = 1.0,\,g^z=0.0)$, (b) integrable non-critical $(g^x = 1.5,\,g^z=0.0)$ and (c) chaotic $(g^x = -1.05,\,g^z=0.5)$ parameters. In the insets, the same quantities in a semi-log scale. Since $\abs{e_1(t)}=1$ by construction, $\abs{e_1(t)}$ is omitted.
• Figure 3: Error function $\mathcal{E}(t)$, quantifying the non-divisibility at different times $t$ for single-site $l=1$ (first row) and two-site $l=2$ (second row) subsystems, across the three case studies: (a1,a2) critical $(g^x = 1.0,\,g^z=0.0)$, (b1,b2) integrable non-critical $(g^x = 1.5,\, g^z = 0.0)$ and (c1,c2) chaotic $(g^x = -1.05,\, g^z = 0.5)$ quantum Ising chains. In red, we plot the absolute value of the determinant of $\epsilon_{t,0}$.
• Figure 4: We compare the expectation values $\langle\sigma_L^x\rangle$ and $\langle\sigma_L^x\sigma_R^x\rangle$ obtained from TN simulations (red and blue curves) with those reconstructed by the optimal time-local representations $\Lambda_t^{\rm OPT}$ (black dashed lines). We present results for two-site subsystems ($l=2$) across the three case studies: (a) critical $(g^x = 1.0,\,g^z=0.0)$, (b) integrable non-critical $(g^x = 1.5,\, g^z = 0.0)$ and (c) chaotic $(g^x = -1.05,\, g^z = 0.5)$ quantum Ising chains. The subsystem is prepared into $\hat{\rho}_{\rm S}(0)=\ket{X^+_L\,X^+_R}\bra{X^+_L\,X^+_R}$ at time $t=0$.
• Figure 5: Time evolution of learned Hamiltonian parameters $h_i(t)$ and eigenvalues $\gamma_i(t)$ for subsystems of different support $l=1$ (first two rows) and $l=2$ (last two rows) and for different Ising parameters. Left column: critical parameters ($g^x=1.0,\,g^z=0.0$). Central column: integrable non-critical parameters ($g^x=1.5,\,g^z=0.0$). Right column: chaotic parameters ($g^x=-1.05,\,g^z=0.5$). In the plots for the coefficients $h_i(t)$ of the effective Hamiltonian, we explicitly highlight the most significant terms, along with their corresponding elements of the Liouville basis, given by tensor products of normalized strings $2^{-1/2}(\hat{\mathbb{I}}_k,\,\hat{\sigma}^x_k,\,\hat{\sigma}^y_k,\,\hat{\sigma}^z_k)$.