Structural properties of local integrals of motion across the many-body localization transition via a fast and efficient method for their construction

Many-body localization (MBL) is a novel prototype of ergodicity breaking due to the emergence of local integrals of motion (LIOMs) in a disordered interacting quantum system. To better understand the role played by the existence of such LIOMs, we explore and study some of their structural properties across the MBL transition. We first consider a one-dimensional XXZ spin chain in a disordered magnetic field and introduce and implement a nonperturbative, fast, and accurate method of constructing LIOMs. In contrast to already existing methods, our scheme allows obtaining LIOMs not only in the deep MBL phase but, rather, near the transition point too. Then, we take the matrix representation of LIOM operators as an adjacency matrix of a directed graph whose elements describe the connectivity of ordered eigenbasis in the Hilbert space. Our cluster-size analysis for this graph shows that the MBL transition coincides with a percolation transition in the Hilbert space. By performing finite-size scaling, we compare the critical disorder and correlation exponent $\nu$ both in the presence and absence of interactions. Finally, we also discuss how the distribution of diagonal elements of LIOM operators in a typical cluster signals the transition.

Figure 1

Graphical representation of the procedure outlined in Sec. 2b to arrange the eigenstates of the system in the unitary matrix $U$ which is obtained after diagonalization. The left side shows the matrix of eigenvectors obtained via the exact diagonalization procedure in which usually the eigenstates are ordered according to their corresponding eigenvalues. The right side is the matrix of eigenstates after rearranging them using our algorithm.

Figure 2

Decay of two-point correlator [$log\left(\langle {\tau }_j^z{\sigma }_k^z\rangle \right)$] for the LIOM operators localized near the chain center versus $|j-k|$ for different system sizes $L=8-16$ and disorder intensities in both (a) interacting ($\mathrm{\Delta }=1$) and (b) noninteracting ($\mathrm{\Delta }=0$) regimes. Insets show the divergence of length scale $\zeta$ as a function of $(W-W_c)$ in the presence and absence of interaction, respectively

Figure 3

The left side shows the graphical representation of matrix elements of a typical LIOM operator obtained using Eq. (6) and the ordered set of eigenvectors in unitary matrix $U$. The right side is the corresponding adjacent graph of the same LIOM operator in the basis of product states in the $S_z$ basis for a spin chain with length $L=10$ spins for two different disorder intensities $W=2$ (upper panel) and $W=6$ (lower panel) in the MBL regime ($\mathrm{\Delta }=1$).

Figure 4

Behavior of the average largest and second-largest cluster sizes versus disorder intensity $W$ for a spin chain with $L=16$ spins in both (a) the noninteracting $(\mathrm{\Delta }=0)$ and (b) interacting $(\mathrm{\Delta }=1)$ regimes. It is obvious one needs a stronger disorder to reach the nonpercolating regime in the presence of interaction.

Figure 5

The resulting data collapse of $S_{\text{typ}}/N_H$ onto a scaling function of $(W-W_c)L^{1/\nu }$ which is obtained for spin chains with different lengths $(L=8-16)$ both in the (a) absence $\mathrm{\Delta }=0$ and (b) presence $\mathrm{\Delta }=1$ of interaction in the localized phase. The critical parameters $W_c=3.0,0.0$, and $\nu =2.0,2.5$ is obtained for the case of MBL and Anderson transition, respectively. Insets show the corresponding raw data.

Figure 6

Distribution function $P\left(m_l\right)$ of the average local magnetization of cluster $m_l$, which is the average expectation values of LIOM operator ${\tau }_l^z$ over the nodes of this cluster both for $\mathrm{\Delta }=0$ and $\mathrm{\Delta }=1$.