Many-body localization of zero modes

The celebrated Dyson singularity signals the relative delocalization of single-particle wave functions at the zero-energy symmetry point of disordered systems with a chiral symmetry. Here we show that analogous zero modes in interacting quantum systems can fully localize at sufficiently large disorder, but do so less strongly than nonzero modes, as signified by their real-space and Fock-space localization characteristics. We demonstrate this effect in a spin-1 Ising chain, which naturally provides a chiral symmetry in an odd-dimensional Hilbert space, thereby guaranteeing the existence of a many-body zero mode at all disorder strengths. In the localized phase, the bipartite entanglement entropy of the zero mode follows an area law, but is enhanced by a system-size-independent factor of order unity when compared to the nonzero modes. Analytically, this feature can be attributed to a specific zero-mode hybridization pattern on neighboring spins. The zero mode also displays a symmetry-induced even-odd and spin-orientation fragmentation of excitations, characterized by real-space spin-correlation functions, which generalizes the sublattice polarization of topological zero modes in noninteracting systems, and holds at any disorder strength.

Figure 1

Spin correlations in a zero mode of the spin-1 Ising chain, as quantified by the spin-correlation matrix $\mathcal{C}$ in an individual realization of the disorder with strength (a) $W=1$, (b) $W=8$, and (c) $W=20$. At any disorder, the correlation matrix fragments into five sectors. Subpanels (i)–(v) show the correlation eigenvectors with the largest and smallest correlation eigenvalue in each sector. The corresponding correlation eigenvalue spectra are shown in the adjacent subpanels $\left(\alpha \text{−}\gamma \right)$. For weak disorder, these eigenvalues lie around the ergodic value $2/3$, while for strong disorder they approach quantized values 1 (for $\mathrm{\Delta }$) and 0 (for $Z$).

Figure 2

Spin correlations for a nonzero mode close to the band center, in analogy to Fig. 1. Note that the correlation eigenvectors displayed in subpanels (i) and (ii) now belong to the same correlation eigenvalues; hence, they represent their $x$ and $y$ components, as these no longer separate, also not with respect to the sublattice. Therefore, only two types of elementary spin correlations exist for such nonzero modes. As shown in subpanels $\left(\alpha \right)$ and ($\beta )$, the correlation spectra again become quantized for strong disorder, reflecting the number of spins with finite $s_z$ in the approached basis state $|\mathbf{s}\rangle$.

Figure 3

Disorder-averaged measures of localization for the zero mode (light red) and nonzero modes (dark blue) as a function of disorder strength $W$. (a) Overlap $I_k$ with the basis state $|\mathbf{0}\rangle$, as defined in Eq. (25). For the zero mode, $\overline{I_0}$ increases with increasing disorder strength, reaching $\overline{I_0}\sim 1/2$ at around $W=20$. For nonzero modes, the corresponding average $\overline{I_{\ne 0}}=O\left({\mathcal{N}}^{-1}\right)$ remains negligible at all disorder strengths. As shown in (b), the nonzero modes nonetheless have a larger extent of Fock-space localization, as quantified by the inverse participation ratio IPR [see Eq. (24)], and hence approach an eigenstate $|\mathbf{s}\rangle$ with $\mathbf{s}\ne \mathbf{0}$ more quickly than the zero mode approaches $|\mathbf{0}\rangle$. This relative delocalization of the zero mode is confirmed in panel (c) by the bipartite entanglement entropy, Eq. (23), which is enhanced for the zero mode. Panel (d) shows the maximal $Z$ spin-correlation eigenvalue $Z_k^{\max }$, which quantifies the residual hybridization of these states in the strongly localized regime, as further discussed in Sec. 5.

Figure 4

Averaged entanglement entropy of energy eigenstates ordered by their energy, with the resulting mode index centered at the zero mode, for disorder strengths (a) $W=1$, (b) $W=8$, and (c) $W=20$ and system sizes $N=4,6,8$. The entanglement entropy of the zero mode is enhanced in the localized regime, by an amount that is independent of the accessible system sizes.

Figure 5

Distributions of the entanglement entropy for zero modes (top panels) and nonzero modes (bottom panels), for parameters as in Fig. 4. From moderate disorder, the zero mode shows a significant enhancement of entropies $S\gtrsim 1$, with the indicated value $ln3$ identified in Sec. 5. Nonzero modes display less pronounced features at smaller characteristic values $ln2$ and $(ln8)/2$.

Figure 6

Analytical predictions for the localization characteristics of zero modes (top) and nonzero modes (bottom) as a function of the hybridization parameter $\delta$ from quasidegenerate perturbation theory (see text). (a) Bipartite entanglement entropy, (b) inverse participation ratio, and (c) leading $Z$ spin-correlation eigenvalue $Z_k^{\max }$. Quantities arising from the zero-mode hybridizations (36) and (37) are given in medium-dark red. In the case of nonzero modes, these hybridizations can still appear when embedded into a state in which an even number of the remaining spins have a finite $S^z$ component. For the nonzero modes, further configuration scenarios appear from the hybridizations (43) (again embedded into states with an even number of remaining nonzero spins, light yellow) and (46) (embedded into states with an odd number of remaining nonzero spins, dark blue). Examples of these hybridization patterns are shown at the bottom of the figure.

Figure 7

Analytical predictions for correlations between the localization characteristics of zero modes (top) and nonzero modes (bottom), following from the results shown in Fig. 6.

Figure 8

Scatter plot of localization characteristics of the zero mode for disorder strengths (a) $W=1$, (b) $W=8$, and (c) $W=20$. In panel (c), the dashed lines indicate the predicted analytical bounds; see the top panels in Figs. 7 and 7. Note the different scale on the horizontal axis in panel (a).

Figure 9

Scatter plot of localization characteristics of nonzero modes in analogy to Fig. 8, with the analytical bounds in (c) taken from the bottom panels of Figs. 7 and 7.

Figure 10

Disorder-averaged level-spacing ratio (A1) as a function of disorder strength $W$, for (a) $r_1$ (including the spacing of the zero mode to its nearest neighbor) and (b) $r_k$ with $k>1$ (only involving spacing between nonzero modes, which are furthermore restricted to the middle 10% of the spectrum). The dotted line indicates the value expected for an ergodic system modeled by the Gaussian orthogonal ensemble of random matrix theory, while the dashed line indicates the value for a localized system with Poissonian level statistics. For the three system sizes $N=6,8,10$, the data shown are obtained from $\approx {10}^4,{10}^3,{10}^2$ realizations, respectively.

Figure 11

Distribution of level-spacing ratios corresponding to Fig. 10, with the system size fixed at $N=6$ and the disorder strength set to (a) $W=1$, (b) $W=5$, and (c) $W=20$, using $2\times {10}^5$ disorder realizations. The curves indicate the expected distribution for an ergodic system modeled by the Gaussian orthogonal ensemble of random matrix theory (dotted), as well as a localized system with Poissonian level statistics (dashed).