Memory hierarchy for many-body localization: Emulating the thermodynamic limit

Local memory—the ability to extract information from a subsystem about its initial state—is a central feature of many-body localization. We introduce, investigate, and compare several information-theoretic quantifications of memory and discover a hierarchical relationship among them. We also find that while the Holevo quantity is the most complete quantifier of memory, vastly outperforming the imbalance, its decohered counterpart is significantly better at capturing the critical properties of the many-body localization transition at small system sizes. This motivates our suggestion that one can emulate the thermodynamic limit by artificially decohering otherwise quantum quantities. Applying this method to the von Neumann entropy results in critical exponents consistent with analytic predictions, a feature missing from similar small finite-size system treatments. In addition, the decohering process makes experiments significantly simpler by avoiding quantum state tomography.

Figure 1

Hierarchy of disorder-averaged informational quantities against time (normalized between 0 and 1) for disorder strength $h=4$ in the middle of the MBLT. The schematic shows the action of a single channel realization on a single contiguous message state initialized in the state ${\rho }^{\left(k\right)}=|k\rangle \langle k|$. As each channel realization has a fixed disorder profile and environment state, we average over channels when constructing the quantities. The peak at $Jt\approx 8$ occurs when the information carried away from the message first interacts with itself, causing transient revival. Error bars are shown where visible. Results for both sides of the MBLT, $h=0.1$ and $h=16$, are given in Appendix pp1. All results are for $L=16$ and $l=4$.

Figure 2

Disorder-averaged steady-state values of (normalized) informational quantities as a function of disorder strength. The message-to-system size ratio for all panels is $l/L=1/3$. The panels show (a) von Neumann entropy, (b) Holevo quantity, (c) CMI, and (d) SSMI. Insets show the corresponding data collapse for the three largest systems. Every fifth point is marked for legibility, and error bars are shown where visible.

Figure 3

Panel (a) shows the optimal critical values $h_c$ and the exponents $\nu$ for all memory quantifiers and the DVNE. Panels (b)–(d) compare the Holevo quantity and its decohered counterpart, the CMI, in the XX and Heisenberg (XXX) models. Artificial dephasing is indicated by the arrows.

Figure 4

Dynamics of the four informational quantities for $L=16$ and $l=4$ on extreme sides of the MBLT: (a) $h=0.1$ and (b) $h=16$. Results for the middle of the MBLT ($h=4$) are given in the main text. In all cases, a strict hierarchy is seen, and the system convincingly relaxes to a steady state at exponential times.

Figure 5

Scaling of the DVNE discussed in the main text (message to system size ratio $l/L=1/3$). Every fifth point is marked for readability, and error bars are shown where visible. The inset shows the corresponding data collapse of the largest three systems.

Figure 6

Steady-state values of the four informational quantities, (a) von Neumann, (b) Holevo quantity, (c) CMI, and (d) SSMI, against disorder for the message to system ratio $l/L=1/4$. Inserts show corresponding data collapses. Gray regions depict the extracted critical values and their errors.

Figure 7

Extracted critical values and exponents for the memory quantifiers for the message to system size ratio $l/L=1/4$. Black arrows show artificial decohering [see Eq (A6) and main text] which systematically increases the extracted critical values to be closer to large-scale phenomenological studies and analytic bounds. The DVNE is even consistent with the Harris bound $\nu =1.98\pm 0.17\ge 2$.