Multifractal scaling and functional renormalization group in disordered electron systems

Taiki Haga

Serafim S. Babkin, I. S. Burmistrov

Generalized multifractality characterizes system size dependence of pure scaling local observables at Anderson transitions in all 10 symmetry classes of disordered systems. Recently, the concept of generalized multifractality has been extended to boundaries of critical disordered noninteracting systems. Here we study the generalized boundary multifractality in the presence of electron-electron interaction, focusing on the spin quantum Hall symmetry class (class C). Employing the two-loop renormalization group analysis within the Finkel'stein nonlinear sigma model, we compute the anomalous dimensions of the pure scaling operators located at the boundary of the system. We find that generalized boundary multifractal exponents are twice larger than their bulk counterparts. Exact symmetry relations between generalized boundary multifractal exponents in the case of noninteracting systems are explicitly broken by the interaction.

Ignacio García-Mata, Weitao Chen, John Martin +2 more

We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions $D_q$. The RG flows recover the qualitative structure previously reported for the information dimension $D_1$, while their extension to other $D_q$ with smaller moment orders $q<1/2$ confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension $D_1$ need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large $q>1/2$ and small $q<1/2$ eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.

Weitao Chen

Zhizhong Zhang

Éric Brillaux, David Carpentier, Andrei A. Fedorenko

A Weyl semimetal is a three-dimensional topological gapless phase. In the presence of strong enough disorder it undergoes a quantum transition towards a diffusive metal phase whose universality class depends on the range of disorder correlations. Similar to other quantum transitions driven by disorder, the critical wave functions at the semimetal--diffusive-metal transition exhibit multifractality. Using renormalization group methods we study the corresponding multifractal spectrum as a function of the range of disorder correlations for generic disorder including random scalar and vector potentials. We also discuss the relation between the geometric fluctuations of critical wave functions and the broad distribution of the local density of states (DOS) at the transition. We derive a new scaling relation for the typical local DOS and argue that it holds for other disorder-driven transitions in which both the average and typical local DOS vanish on one side of the transition. As an illustration we apply it to the recently discussed unconventional quantum transition in disordered semiconductors with power-law dispersion relation near the band edge.