Solvable Sachdev-Ye-Kitaev models in higher dimensions: from diffusion to many-body localization

TL;DR

The work introduces a solvable, higher-dimensional generalization of the SYK model on a bipartite lattice that exhibits a dynamical transition from a diffusive thermal metal to a many-body localized phase. Using a replica-derived, large-$N$ framework and an emergent reparametrization symmetry, it derives IR solutions, the zero-temperature entropy, and an effective action for reparametrization modes that yields a diffusion constant $D$ and chaotic properties. The transition occurs at a critical ratio $r_c=1$, with $D o0$ as $r o1$ following $D\, extpropto\,(1-r)^{1/2}$, maximal chaos with $oldsymbol{ extlambda_L = 2oldsymbol{ extpi/eta}}$, and a universal relation $oldsymbol{D = v_B^2/oldsymbol{ extlambda_L}}$ linking diffusion, chaos, and butterfly spreading; level statistics change from Wigner-Dyson to Poisson, indicating MBL, with a peculiar $ u=0$ critical exponent. The model thus provides a controlled arena to study exotic MBL transitions in higher dimensions and their connections to holography and chaotic dynamics.

Abstract

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 the of $B$ 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$$\rightarrow$$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 $ν$=0, violating the Harris criterion. Our higher-dimensional SYK model may provide a promising arena to explore exotic MBL transitions.

Figures (4)

• 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 $t'_{i\alpha,x}$. (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, $t'$=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 $t'$ denotes that between neighboring unit cells. (b) The energy diffusive constant $D$ along the ${\bf e}_1$ or ${\bf e}_2$ direction as a function of $r$. We use the parameter $J$=$t$=1, $t_1'$=$t_2'$=0.1, $t_3'$=0.
• Figure S1: 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 $J=0.8$, $t=1.2$, $t'=0.1$, $V=0.2$. The Wigner-Dyson distribution (dashed line) implies thermalization while Poisson distribution (dotted line) implies MBL.