Localization-protected order in spin chains with non-Abelian discrete symmetries

We study the nonequilibrium phase structure of the three-state random quantum Potts model in one dimension. This spin chain is characterized by a non-Abelian $D_3$ symmetry recently argued to be incompatible with the existence of a symmetry-preserving many-body localized (MBL) phase. Using exact diagonalization and a finite-size scaling analysis, we find that the model supports two distinct broken-symmetry MBL phases at strong disorder that either break the ${\mathbb{Z}}_3$ clock symmetry or a ${\mathbb{Z}}_2$ chiral symmetry. In a dual formulation, our results indicate the existence of a stable finite-temperature topological phase with MBL-protected parafermionic end zero modes. While we find a thermal symmetry-preserving regime for weak disorder, scaling analysis at strong disorder points to an infinite-randomness critical point between two distinct broken-symmetry MBL phases.

Figure 1

Random $D_3$ chain at weak disorder $W=0.5$. (Top) Level statistics measured by $r$ ratio display two transitions: for $|{\delta }_c|\lesssim 0.5$, $r$ tends to the ETH value $r\approx 0.53$ characteristic of the Gaussian orthogonal ensemble with increasing $L$, whereas outside this region $r\to 0.38$, indicating Poisson statistics of MBL. (Center) Half-chain entanglement entropy density $S_{\text{E}}\left(L\right)/L$ is consistent with volume (area) law scaling in the ETH (MBL) regions. (Bottom) Spin-glass order parameters of ${\mathbb{Z}}_3$ and chiral symmetry (scaled by $L$, see text) also show crossings at $|{\delta }_c|\approx 0.5$, showing that MBL coincides with the onset of symmetry breaking.

Figure 2

Random $D_3$ chain at strong disorder $W=2.0$. (Top) Since $r\approx 0.38$ for all values of $\delta$, we infer that the system is always MBL. (Center) Entanglement entropy density is consistent with area-law scaling as $L\to \infty$, again consistent with MBL. (Bottom) Scaling collapses of $m_3$ ($▵$) and $m_{\chi }$ ($\diamond$), both consistent with a direct transition at ${\delta }_c=0$ between distinct broken-symmetry MBL phases. Here, $\nu =2$, and $\mathrm{\Delta }=\frac{\overline{lnJ}-\overline{lnf}}{\text{var}J+\text{var}f}=\frac{1}{2W^2}ln\frac{1+\delta }{1-\delta }$ is a rescaled tuning parameter. (The point where the two collapsed curves cross has no physical significance.)

Figure 3

Nonequilibrium global phase diagram of random $D_3$ chain. The MBL-ETH boundary (red line) is an estimate based on crossings in level statistics and scaled entanglement entropy (denoted $\times$, $†$, respectively). For weak disorder we also indicate crossings in the ${\mathbb{Z}}_3$ ($▵$) and chiral ($\diamond$) order parameters. At strong disorder, we find scaling collapse consistent with an infinite-randomness critical point at ${\delta }_c=0$; however, we cannot conclusively rule out a nonergodic quantum critical glass in the transition region (hatched). (Inset) schematic of the symmetry-breaking pattern and the topological/trivial phases of dual parafermions.

Figure 4

Bond decimation for ${\mathbb{Z}}_3$. The two parafermion modes corresponding to the bond drop out of the chain, and the resulting couplings leave the Hamiltonian self-similar, with one fewer site to consider in subsequent steps of the RG.

Figure 5

$W=0.5$ (Fig. 1 of main text).

Figure 6

Additional results for $W=0.6.$

Figure 7

Additional results for $W=0.7.$

Figure 8

Additional results for $W=0.8.$

Figure 9

Additional results for $W=0.9.$

Figure 10

$W=1.0$ for comparison.

Figure 11

“Strong disorder” plot for $W=1.0.$

Figure 12

Additional results for $W=1.5.$

Figure 13

$W=2.0$ for comparison.

Figure 14

Additional results for $W=2.5.$