Information-theoretic memory scaling in the many-body localization transition

A key feature of many-body localization (MBL) is the breaking of ergodicity and consequently the emergence of local memory, revealed as the local preservation of information over time. As memory is necessarily a time-dependent concept, it has been partially captured by a few extant studies of dynamical quantities. However, these quantities suffer from a variety of issues which limit their value as true quantifiers of memory; thus, a fundamental and complete information-theoretic understanding of local memory in the context of MBL remains elusive. We outline these issues in detail and introduce the dynamical Holevo quantity to address them. We find that it shows clear scaling behavior across the MBL transition, and we determine a family of two-parameter scaling Ansätze which capture this behavior. We perform a comprehensive finite-sized scaling analysis to extract the transition point and scaling exponents.

Figure 1

Schematic diagram of the procedure by which individual messages ${\varrho }^{\left(k\right)}$ are transmitted via the map $\mathcal{E}$. Information initially localized within the message may bleed out into the environment during transmission.

Figure 2

(a)–(c) The disorder-averaged Holevo rate $\overline{R}(L,l,h,t)$ against time for a fixed message length $l=4$. (d)–(f) The time-averaged Holevo rate ${\overline{R}}_{SS}(L,l,h)$ against disorder strength for a variety of message lengths. Each row contains results for a single environment type: (a) and (d) Néel state, (b) and (e) evolved Néel state, and (c) and (f) eigenstate environments, respectively. We take $L=16$ for all above figures.

Figure 3

Time-averaged Holevo rate ${\overline{R}}_{SS}(L,l,h)$ for (a) Néel state, (b) evolved Néel state, and (c) eigenstate environments, respectively. The main figures show results for the fixed ratio $l/L=\frac{1}{3}$ and insets for the fixed ratio $l/L=\frac{1}{4}$. Dashed red lines indicate the region in which the data collapse was carried out, and the gray regions indicate the standard error on $h_c$.

Figure 4

The optimal data collapse of each of the results shown in Fig. 3 using the Ansätze of Eq. (8) for sizes up to $L=16$. The critical values $h_c$ and exponents $\nu$ and $\beta$ of each collapse are summarized in Table 1.