Exploring one-particle orbitals in large many-body localized systems

Strong disorder in interacting quantum systems can give rise to the phenomenon of many-body localization (MBL), which defies thermalization due to the formation of an extensive number of quasilocal integrals of motion. The one-particle operator content of these integrals of motion is related to the one-particle orbitals (OPOs) of the one-particle density matrix and shows a strong signature across the MBL transition as recently pointed out by Bera et al. [Phys. Rev. Lett. 115, 046603 (2015); Ann. Phys. 529, 1600356 (2017)]. We study the properties of the OPOs of many-body eigenstates of an MBL system in one dimension. Using shift-and-invert MPS, a matrix product state method to target highly excited many-body eigenstates introduced previously [Phys. Rev. Lett. 118, 017201 (2017)], we are able to obtain accurate results for large systems of sizes up to $L=64$. We find that the OPOs drawn from eigenstates at different energy densities have high overlap and their occupations are correlated with the energy of the eigenstates. Moreover, the standard deviation of the inverse participation ratio of these orbitals is maximal at the nose of the mobility edge. Also, the OPOs decay exponentially in real space, with a correlation length that increases at low disorder. In addition, we find that the probability distribution of the strength of the large-range coupling constants of the number operators generated by the OPOs approach a log-uniform distribution at strong disorder.

Figure 1

Phase diagram in the disorder strength $W$ and energy density $\epsilon$ plane for the model in Eq. (1) with $t=V=1$. The mobility edge is plotted from the results of Ref. [19]. We numerically access eigenstates at the depicted points.

Figure 2

Top: Percentage of eigenstates accessed by SIMPS that pass our filter for the standard deviation of the energy, $\sigma \left(E\right)<{10}^{-3}$. Eigenstates in the MBL region are accessed successfully through SIMPS with a low value of $\sigma \left(E\right)$, whereas eigenstates in the ergodic region (see Fig. 1) fail to be represented accurately by the low bond dimension MPS ansatz. We neglect the eigenstates in the gray area due to the bias the strong filtering might introduce. Bottom: Average bond dimension at half-cut of the eigenstates kept after filtering. As expected, the bond dimension diverges close to the transition, where it also becomes strongly system size dependent and it is eventually cutoff by the finite bond dimension used in SIMPS.

Figure 3

Top: Probability density $|U_{ki}^{†}{|}^2$ in real space ($i$) of each OPO ($k$) of a particular eigenstate at $\epsilon =0.1$ of a system of size $L=32$ and $W=3.0,10.0$. Bottom: Profile of the random chemical potential ${\mu }_i$ at $W=3.0,10.0$. At strong disorder (right), the OPOs are highly localized on one site. As the disorder is lowered, the OPOs start delocalizing, mixing over small nonoverlapping subsystems of the chain. There is a high probability of mixing along sites with a similar ${\mu }_i$, which occasionally gives rise to tunneling OPOs (see sites 14 and 16 at $W=3.0$ for an example).

Figure 4

Top: Total contribution $F_R$ from string operators of range $R$ to the definition of the OPO number operator $a_k^{†}{a}_k$ (logarithmically) averaged over OPOs ($L=32$). The average ${\overline{F}}_R$ decays exponentially with range $R$. Bottom: Correlation length $\xi$ extracted from the exponential decay of ${\overline{F}}_R$. Insets: System size and $\epsilon$ dependence of $\xi$ ($L=16$).

Figure 5

Probability distribution of the coupling constants $\left|f_R\right|$ divided by the typical coupling constant ${\left|f_R\right|}^{*}\equiv {10}^{\text{Mo}\left[{log}_{10}\left(\right|f_R\left|\right)\right]}$ (where the mode $\text{Mo}\left[{log}_{10}\left(\right|f_R\left|\right)\right]\equiv argmax\left\{p\left[{log}_{10}\left(\right|f_R\left|\right)\right]\right\}$), $p\left(|f_R|/|f_R{|}^{*}\right)$, of the number operators of the OPOs at $\epsilon =0.1$ for fixed $W$ and range $R$, for systems of size $L=64$. All curves, except for $W=2$, have been shifted in the $y$ axis for clarity; they would otherwise lay on top of each other and meet approximately at $|f_R|=|f_R{|}^{*}$ (where they are parallel to the $\propto 1/|f_R|$ reference line) and $p\left(|f_R|/|f_R{|}^{*}\right)\approx {10}^{-1}$.

Figure 6

Top: Distribution of the support of the OPOs for different energy densities as a function of $W$ for $L=32$. The $suppor{t}_{90}$ is computed as the size of the region of that contains $90\%$ of the norm $L_2$ of the OPOs. Bottom: correlation length ${\xi }_{{\mathrm{support}}_{90}}$ corresponding to the exponential decay of the distributions in the top panel. Insets: System size and energy density dependence of ${\xi }_{{\mathrm{support}}_{90}}$ ($L=16$).

Figure 7

Top: IPR as a function of $W$ for $L=32$ averaged over OPOs and disorder. Inset: Average IPR as a function of $L$. Bottom: Standard deviation of the IPR of the OPOs for $L=32$. Inset: Standard deviation of the IPR as a function of $L$.

Figure 8

Distribution of IPR for $L=32$.

Figure 9

IPR as a function of $k$, i.e., as a function of OPOs ordered by occupation, averaged over OPOs. The shapes of the curves are characteristic of, respectively, strong disorder eigenstates, eigenstates around the critical disorder strength $W_c$, and weak disorder, independent of $\epsilon$.

Figure 10

Two-dimensional histogram of the IPR of the OPOs vs $k$ for $L=64,\epsilon =0.1$. It is easy to see the emergence of the characteristic curves presented in Fig. 9. At strong disorder, intermediate values of $k$ have an IPR close to 0.5 (see Fig. 8).

Figure 11

Left: Matrix of overlaps $M_{\mathrm{overlap}}=\left|\langle {\varphi }_k|{\psi }_l\rangle \right|$ between the OPOs of two different eigenstates with $W=2$ at energy densities ${\epsilon }_1$ and ${\epsilon }_2$, for a system of size $L=32$. Middle: Permutation between OPOs of the two eigenstates, $k\mapsto l$. The coloring on the left column is red for the half of the OPOs that have highest occupation and blue for the half with lowest occupation. The coloring on the right is inherited from the color of the OPO on the left to which it is mapped. Right: Matrix of overlaps $M_{\mathrm{overlap}}$ with the columns ordered after the permutation shown by the middle diagrams.

Figure 12

The coloring on the left column is red for the half of the OPOs that have highest occupation and blue for the half with lowest occupation. The coloring on the right is the disordered average of all colors inherited from the OPO on the left to which it is mapped (see Fig. 11 (middle) for a nonaveraged version of this). The closer the eigenstate energies are to each other, the more likely the occupations of the OPOs of two eigenstates will be preserved, ranging from a few swaps in occupation when the energies are close in the spectrum to almost all swaps when the energies are in opposite sides of the spectrum.

Figure 13

Top: Distribution of overlaps $|\langle {\varphi }_k\left({\epsilon }_1\right)|{\psi }_{l\left(k\right)}\left({\epsilon }_2\right)\rangle |$ between corresponding OPOs of eigenstates at energy densities ${\epsilon }_1$ and ${\epsilon }_2$ ($L=64$). Bottom: Mode of the distribution of overlaps of corresponding OPOs.

Figure 14

Top: Average occupations $n_k$ of the OPOs. The error bars have been removed for clarity; they are of the order of those of the gap $\mathrm{\Delta }n$ (bottom). All eigenstates accessed by SIMPS are in the MBL phase, and so the occupation spectrum of the OPOs shows a finite gap [52, 53]. Bottom: Scaling of the gap $\mathrm{\Delta }n\equiv n_{L/2}-n_{L/2-1}$ as a function of $1/L$.

Figure 15

Standard deviation of the entanglement entropy of the half system $\sigma \left(S_{L/2}\right)$ as a function of disorder strength $W$ for different system sizes and energy densities. $\sigma \left(S_{L/2}\right)$ exhibits a maximum at each energy density close to the transition point [24]. Our results for the finite energy eigenstates considered in this article accessed by SIMPS show only the approach to this maximum.

Figure 16

Top: exponential decay of ${\overline{F}}_R$ for ground states. Bottom: correlation length $\xi$ for ground states. $\xi$ is a monotonically increasing function of the system size $L$ at small disorder.

Figure 17

Histogram of $F_R$ vs $R$ for systems of size $L=64$ at different values of $W$ and eigenstates at $\epsilon =0.0,0.1$.

Figure 18

Exponent $\alpha$ of the scaling law $\xi =log\left(\beta L^{\alpha }\right)$ for the correlation length of the OPOs.

Figure 19

Average decay of the OPOs' tails ($L=32$). The asymptotic behavior of the tails is equal to the one of ${\overline{F}}_R$ presented in Fig. 4.

Figure 20

Distribution of the $suppor{t}_{90}$ of the OPOs of eigenstates at $\epsilon =0.0$ (top) and $\epsilon =0.1$ (bottom) of systems of different sizes $L$. At strong disorder the distribution decays exponentially and is largely system size independent, while it collapses to the system size at weak disorder, where the exponential decay is lost.

Figure 21

Equivalent to Fig. 6 ($L=32$ is considered in the top panel). The support ($support$) is now computed for the number operator of the OPO $a_k^{†}{a}_k$ as the average range $R$ weighted by $F_R$. The phenomenology is extremely similar to the one found for the $suppor{t}_{90}$ in Fig. 6.

Figure 22

Distribution of the $support$ of the OPOs of systems of different size $L$ at $\epsilon =0.0$ (top) and $\epsilon =0.1$ (bottom). The phenomenology is extremely similar to the one found in Fig. 20.

Figure 23

Histogram of the distribution of IPR for different system sizes at $\epsilon =0.1$. The distribution is system size independent for almost all values of $W$, with only a slight drift towards high IPR for small systems at $W=2$.

Figure 24

Top: Distribution of the biggest one site contribution to the overlaps between corresponding OPOs $(k\leftrightarrow l(k\left)\right)$ obtained from pairs of eigenstates of the same Hamiltonian at two particular energy densities ${\epsilon }_1$ and ${\epsilon }_2$ ($L=64$). Bottom: Mode of the distribution of the one site contributions to the overlaps for different pairs of energy densities as a function of $W$. The one site overlap is lower than the total overlap between the OPOs.

Figure 25

Matrix $\left|f_{ij}^k\right|=\left|U_{ki}^{†}{U}_{jk}\right|$ for a generic OPO with exponentially decaying $\left|U_{ki}^{†}\right|\propto e^{-|i-m|/\xi }$ centered at site $m$. Only the elements for which the range $R(i,j)=\text{const.}$ are represented, where $R(i,j)\equiv max\left(\left|i-m\right|,\left|j-m\right|\right)$ (centered) applies to the top panel, and $R(i,j)\equiv \left|i-j\right|$ (uncentered) applies to the bottom panel. In both examples, $m=10$ and $R=6$ for a system of $L=32$.

Figure 26

Equivalent to Fig. 5, but computed using the uncentered range. Contrary to Fig. 5, the distributions of $|f_R|$ are broader and flatter at small ranges, as discussed in Appendix pp6.

Figure 27

Equivalent to Fig. 4, although the uncentered definition of the range $R$ is used ($L=32$ is considered in the top panel).

Figure 28

Equivalent to Fig. 21, although using the uncentered definition for the range $R$ ($L=32$ is considered in the top panel).

Figure 29

Equivalent to Fig. 22, but using the uncentered definition of the range $R$. $\epsilon =0.0$ (top) and $\epsilon =0.1$ (bottom) are considered.