Universal Properties of Many-Body Localization Transitions in Quasiperiodic Systems

The precise nature of many-body localization (MBL) transitions in both random and quasiperiodic (QP) systems remains elusive so far. In particular, whether MBL transitions in QP and random systems belong to the same universality class or two distinct ones has not been decisively resolved. Here, we investigate MBL transitions in one-dimensional ($d=1$) QP systems as well as in random systems by state-of-the-art real-space renormalization group (RG) calculation. Our real-space RG shows that MBL transitions in 1D QP systems are characterized by the critical exponent $\nu \approx 2.4$, which respects the Harris-Luck bound ($\nu >1/d$) for QP systems. Note that $\nu \approx 2.4$ for QP systems also satisfies the Harris-Chayes-Chayes-Fisher-Spencer bound ($\nu >2/d$) for random systems, which implies that MBL transitions in 1D QP systems are stable against weak quenched disorder since randomness is Harris irrelevant at the transition. We shall briefly discuss experimental means to measure $\nu$ of QP-induced MBL transitions.

Figure 1

The schematic RG flow of 1D interacting systems with both QP potential and quenched randomness. Our real-space RG analysis shows that MBL transitions induced by QP potential is Harris stable against weak randomness.

Figure 2

Finite-size scaling analysis of entanglement entropy of 1D systems with quasiperiodic potentials $W_i=W\cos (2\pi \alpha i+\varphi )$. The MBL transition is identified as the crossing point of entanglement entropy $s$ for different sizes $W_c=3.36\pm 0.01$. The data collapse shown in the inset gives rise to $\nu =2.4\pm 0.3$. Results are obtained by averaging over ${10}^5–{10}^6$ QP configurations (namely ${10}^5–{10}^6$ choices of $\varphi$).

Figure 3

The finite-size scaling for entanglement entropy $s$ of 1D system with random potentials $W_i\in [0,W]$. The MBL transition occurs at $W_c=1.78\pm 0.01$. The data collapse shown in the inset gives rise to $\nu =3.1\pm 0.3$. Results are obtained by averaging over ${10}^5–{10}^6$ disorder configurations.