Total correlations of the diagonal ensemble herald the many-body localization transition

The intriguing phenomenon of many-body localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. In this work we introduce the total correlations, a concept from quantum information theory capturing multipartite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pin-down, probe, and better understand the MBL transition and ergodicity breaking in quantum systems. In particular, we show that the total correlations has sublinear dependence on the system size in delocalized, ergodic phases, whereas we find that it scales extensively in the localized phase developing a pronounced peak at the transition. We exemplify the power of our approach by means of an exact diagonalization study of a Heisenberg spin chain in a disordered field. By a finite size scaling analysis of the peak position and crossover point from log to linear scaling we collect evidence that ergodicity is broken before the MBL transition in this model.

Figure 1

For high $h, \overline{T}$ decays as a power law. The dashed lines are fits to the data points with $h\ge 10$ yielding exponents of $-0.9\left(1\right)$ (consistent with the expectation of $h^{-1}$ corrections). The inset shows the position of the peaks from Fig. 2 (see also Fig. (1) in the Supplemental Material [32]). The extrapolated position of the peak indicates the onset of MBL at around $h=3.8$. The fact that the different $\overline{T}/N$ curves overlap for large enough $h$ indicates that $\overline{T}$ scales essentially linearly with $N$ in this regime.

Figure 2

Subtracting the values of $\overline{T}$ for $h=0.2$ (which is just outside the integrable region around $h=0$) from $\overline{T}$ one can see that for increasing system sizes the individual curves fall on top of each other in an increasingly large region of $h$ values (shaded regions) during the approach to the peak. This increase is well captured by a power law with exponent $2.7\left(2\right)$ (dotted line guide to the eye $\propto h^{2.7}$).

Figure 3

The difference of the average total correlations $\overline{T}$ and the best possible linear fit for different values of $h$. The crossover from a nearly linear scaling to a sublinear scaling clearly happens between $h=2.8$ and $h=2.4$. This result is robust against omitting data points for large or small values of $N$ and equally holds for affine fits instead of linear ones. This gives us high confidence in this result, which indicates the existence of an extended nonergodic but not yet many-body localized region. The inset shows the data before subtracting and the fits.