Solvable Sachdev-Ye-Kitaev Models in Higher Dimensions: From Diffusion to Many-Body Localization

Many aspects of many-body localization (MBL) transitions remain elusive so far. Here, we propose a higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model and show that it exhibits a MBL transition. The model on a bipartite lattice has $N$ Majorana fermions with SYK interactions on each site of the $A$ sublattice and $M$ free Majorana fermions on each site of the $B$ sublattice, where $N$ and $M$ are large and finite. For $r\equiv M/N<r_c=1$, it describes a diffusive metal exhibiting maximal chaos. Remarkably, its diffusive constant $D$ vanishes [$D\propto (r_c-r{)}^{1/2}$] as $r\to r_c$, implying a dynamical transition to a MBL phase. It is further supported by numerical calculations of level statistics which changes from Wigner-Dyson ($r<r_c$) to Poisson ($r>r_c$) distributions. Note that no subdiffusive phase intervenes between diffusive and MBL phases. Moreover, the critical exponent $\nu =0$, violating the Harris criterion. Our higher-dimensional SYK model may provide a promising arena to explore exotic MBL transitions.

Figure 1

(a) The 1D generalization of the SYK model consists of $N$ SYK Majorana fermions ${\psi }_i$ on each site of the $A$ sublattice and $M$ free Majorana fermions ${\eta }_{\alpha }$ on each site of the $B$ sublattice. The hopping between two types of fermions is represented by $t_{i\alpha ,x}$ and $t_{i\alpha ,x}^{\prime }$. (b) The phase diagram of the 1D model in Eq. (1) as a function of $r=M/N$.

Figure 2

The distribution of level-spacing ratios for the cases of $(N,M)=(6,4)$, (5,5), and (4,6) are shown in (a), (b), and (c), respectively. The results (red solid line) are obtained by exactly diagonalizing the generalized SYK model on the six-site chain with $N+M=10$ Majorana fermions in each unit cell and with $J=t=1$, $t^{\prime }=0.5$. The Wigner-Dyson distribution (dashed line) implies thermalization while Poisson distribution (dotted line) implies MBL.

Figure 3

(a) The generalized SYK model on the square lattice. Each unit cell consists of two sites represented by a square and a disk, where $N$ SYK Majorana fermions and $M$ free Majorana fermions reside, respectively. $t$ denotes the variance of random hopping within a unit cell, while $t^{\prime }$ denotes that between neighboring unit cells. (b) The energy diffusive constant $D$ along the ${\mathbf{e}}_1$ or ${\mathbf{e}}_2$ direction as a function of $r$. We use the parameter $J=t=1$, $t_1^{\prime }=t_2^{\prime }=0.1$, $t_3^{\prime }=0$.