Analytically Solvable Renormalization Group for the Many-Body Localization Transition

We introduce a simple, exactly solvable strong-randomness renormalization group (RG) model for the many-body localization (MBL) transition in one dimension. Our approach relies on a family of RG flows parametrized by the asymmetry between thermal and localized phases. We identify the physical MBL transition in the limit of maximal asymmetry, reflecting the instability of MBL against rare thermal inclusions. We find a critical point that is localized with power-law distributed thermal inclusions. The typical size of critical inclusions remains finite at the transition, while the average size is logarithmically diverging. We propose a two-parameter scaling theory for the many-body localization transition that falls into the Kosterlitz-Thouless universality class, with the MBL phase corresponding to a stable line of fixed points with multifractal behavior.

Figure 1

Illustration of RG rules for the decimation of thermal (a) and insulating segments (b). (c) The length of the central thermal block ${\ell }_2^T$ is recovered with unit prefactor after two decimation steps if $\alpha \beta =1$.

Figure 2

(a) For small $\alpha ={\beta }^{-1}=1/10$ the fixed point distribution for thermal $Q_T^{*}(\eta )$ blocks (red line) behaves as a power law for $\eta \le {\alpha }^{-1}+1$, and decays exponentially for larger values of $\eta$. Insulating blocks are approximately distributed exponentially for all $\eta$ (blue line). (b) Slow decay of inverse critical exponent ${\nu }^{-1}$ with ${\alpha }^{-1}$ is well approximated by the analytical asymptotic. The dots marked with red have been used for the extrapolation of ${\nu }^{-1}(\alpha )$ for smaller values of $\alpha$. Value of ${\nu }^{-1}(1/30)$ collapses well onto the numerical fit. (c) Fractal dimension $d_I$ of insulating regions rapidly approaches one when $\alpha \to 0$, whereas fractal dimension of thermal inclusions slowly decays to zero.

Figure 3

Two-parameter RG flows in the limit $\alpha \to 0$ has the half line of stable fixed points $\gamma =0$, $\kappa >0$ describing a multifractal MBL phase. For $\gamma =0$, $\kappa <0$, the line of fixed points is unstable and gives rise to a flow to strong coupling which corresponds to thermalization. The black line separates the set of initial conditions that flow to the MBL and thermal phases.