Many-body localization phase transition: A simplified strong-randomness approximate renormalization group

We present a simplified strong-randomness renormalization group (RG) that captures some aspects of the many-body localization (MBL) phase transition in generic disordered one-dimensional systems. This RG can be formulated analytically and is mathematically equivalent to a domain coarsening model that has been previously solved. The critical fixed-point distribution and critical exponents (that satisfy the Chayes inequality) are thus obtained analytically or to numerical precision. This reproduces some, but not all, of the qualitative features of the MBL phase transition that are indicated by previous numerical work and approximate RG studies: our RG might serve as a “zeroth-order” approximation for future RG studies. One interesting feature that we highlight is that the rare Griffiths regions are fractal. For thermal Griffiths regions within the MBL phase, this feature might be qualitatively correctly captured by our RG. If this is correct beyond our approximations, then these Griffiths effects are stronger than has been previously assumed.

Figure 1

A sketch of the RG rules. Panel (a) shows the TIT rule where a central I (insulating) block is surrounded by two longer T (thermal) blocks, and after merging the new longer block is thermal. Panel (b) shows the ITI move where a T block is surrounded by two longer I blocks and the resulting new block is insulating.

Figure 2

The scaled fixed point length distribution $Q^{*}\left(\eta \right)$.

Figure 3

The one relevant eigenfunction $f_{-}\left(\eta \right)$ at the critical fixed point (see main text).

Figure 4

A sketch of part of the fractal structure of a rare large Griffiths T (locally thermalizing) region in the MBL phase.

Figure 5

The numerical results of $L_{\mathrm{T}}$ as a function of $L$ in the insulating phase to obtain the fractal dimension $d_f$. Nearby points have been binned and (their logarithm) averaged. Error bars show the standard error of points in each bin, assuming they are independent.