Quantum nonergodicity and fermion localization in a system with a single-particle mobility edge

We study the many-body localization aspects of single-particle mobility edges in fermionic systems. We investigate incommensurate lattices and random disorder Anderson models. Many-body localization and quantum nonergodic properties are studied by comparing entanglement and thermal entropy, and by calculating the scaling of subsystem particle-number fluctuations, respectively. We establish a nonergodic extended phase as a generic intermediate phase (between purely ergodic extended and nonergodic localized phases) for the many-body localization transition of noninteracting fermions where the entanglement entropy manifests a volume law (hence, “extended”), but there are large fluctuations in the subsystem particle numbers (hence, “nonergodic”). Based on the numerical results, we expect such an intermediate phase scenario may continue to hold even for the many-body localization in the presence of interactions as well. We find for many-body fermionic states in noninteracting one-dimensional Aubry-André and three-dimensional Anderson models that the entanglement entropy density and the normalized particle-number fluctuation have discontinuous jumps at the localization transition where the entanglement entropy is subthermal but obeys the “volume law.” In the vicinity of the localization transition, we find that both the entanglement entropy and the particle-number fluctuations obey a single parameter scaling based on the diverging localization length. We argue using numerical and theoretical results that such a critical scaling behavior should persist for the interacting many-body localization problem with important observable consequences. Our work provides persuasive evidence in favor of there being two transitions in many-body systems with single-particle mobility edges, the first one indicating a transition from the purely localized nonergodic many-body localized phase to a nonergodic extended many-body metallic phase, and the second one being a transition eventually to the usual ergodic many-body extended phase.

Figure 1

Many-body states in the 3D Anderson model. Here, we randomly sample ${10}^5$ many-body eigenstates of fermions at half-filling. In (a), (c), and (e), we choose a disorder strength $W/t=3$, and sample extended states. In (b), (d), and (f), we choose $W/t=20$ with all states localized. (a), (b) Show the occupation number $m_0$ of the eigenmode at the single-particle spectra center. The occupation numbers for other orbitals behave similarly. For both extended and localized systems, we have strong fluctuations in $m_0$. (c), (d) Show the bipartite $\left(\gamma =\frac{1}{2}\right)$ density ${\rho }_{\mathrm{bipart}}=\frac{N_{\mathrm{bipart}}}{D/2}$, with $N_{\mathrm{bipart}}$ the particle number in one-half of the system. For localized states, the bipartite density exhibits strong fluctuation as shown in (d), whereas it is suppressed for extended states as shown in (c). (e), (f) Show the scaling of the fluctuation with increasing system size. In (e) and (f) we average over energy since the energy dependence of the fluctuation is negligible. The numerical results agree with our analytical analysis [Eq. (16)].

Figure 2

Eigenstate thermalization hypothesis and its violation in interacting Aubry-André model. (a), (b) Show the occupation number $m_{\mathrm{low}}$ of the lowest-energy orbital. (c), (d) Show scaling of the fluctuation of the bipartite $\left(\gamma =\frac{1}{2}\right)$ occupation number versus the total particle number. The numerical results for this interacting case are consistent with the $\sqrt{N}$ scaling as we find for the noninteracting case. The interaction strength is fixed to be $V/t=2$ here. (a), (c) Correspond to the delocalized phase where we choose $\lambda /t=0.25$. (b), (d) Correspond to the localized phase with $\lambda /t=2$.

Figure 3

ETH violation in the many-body states of 1D Anderson model. (a), (c), (e) Show the thermal and entanglement entropy per lattice site $\left(s\right)$ versus the energy density $\left(e\right)$ for a subsystem. The average density is fixed to be at half-filling. (b), (d), (f) Show the particle density in the subsystem $({\rho }_{\mathrm{sub}})$. In this plot we choose a subsystem of 32 sites within a global system with size $L=2048$. The calculations are carried out using the stochastic sampling method illustrated in Sec. 3b, where the total particle number is not strictly fixed. In (a)–(f), the disorder strengths $\left(W/t\right)$ are, respectively, $0,2$, and 10. For the clean case $\left(W/t=0\right)$, the entanglement entropy is equal to the thermal entropy and the subsystem density does not exhibit strong fluctuations, i.e., these observables look thermal despite the model being noninteracting. For the disorder case, the entanglement entropy is smaller than the thermal entropy and the subsystem density shows strong fluctuations.

Figure 4

Thermal and entanglement entropy across the localization transition. Here, we take the Aubry-André model for illustration. We calculate the thermal and entanglement entropy per lattice site $\left(s\right)$ versus the energy density $\left(e\right)$ for a subsystem. We use the stochastic method described in Sec. 3b. The average density is fixed to be at half-filling. In (a), we choose a subset of four lattice sites $\left(L_{\mathrm{sub}}=4\right)$ within a system of size $L=24$. For this small system, we calculate $s$ with both methods of using the exact spectra and the sampling method (see Sec. 3b). The difference between the results from the two methods is due to finite-size effect and approaches zero as we increase the system size. We choose system sizes with $L_{\mathrm{sub}}=128$ and $L=2048$ in (b)–(d), which correspond to $\lambda /t=0$ (delocalized phase), $\lambda /t=1$ (critical), and $\lambda /t=2$ (localized phases), respectively.

Figure 5

Phase transitions of noninteracting fermions in the Aubry-André model at half-filling. (a), (b) Show entanglement entropy density in one-half of the system. (c), (d) Show the bipartite $\left(\gamma =\frac{1}{2}\right)$ particle-number fluctuation. For both quantities, we average over all many-body eigenstates by stochastic sampling. (b), (d) Show the data collapse, where we take $\delta =(\lambda -{\lambda }_c)/t,{\lambda }_c=t$, and the scaling exponent $\nu =1$. From the system size dependence, we find the entanglement entropy is extensive (intensive) for $\lambda <t$ $\left(\lambda >t\right)$. The bipartite $\left(\gamma =\frac{1}{2}\right)$ particle-number fluctuation shows $\sqrt{N}$ scaling for $\lambda >t$, implying nonthermal behavior, whereas for $\lambda <t$, the fluctuation is greatly suppressed. Across the transition, both of the two quantities shown here, $s$ and $\mathrm{var}\left[\overline{N_{\mathrm{sub}}}\right]/\sqrt{N_{\mathrm{sub}}}$, develop a discontinuous jump as approaching the thermodynamic limit. In our calculation, we randomly sample ${10}^4$ $\left({10}^5\right)$ many-body states for (a)–(d), respectively.

Figure 6

Entanglement scaling as a function of the subsystem size $l$ in localized, extended, and critical phases of the Aubry-André model at half-filling. For $\lambda /t=0.5$, the system is in an extended phase, the entanglement entropy shows volume-law scaling, whereas for the localized phase with $\lambda /t=1.5$, the entanglement entropy shows area-law scaling. At the critical point $\lambda /t=1$, the entanglement entropy exhibits volume-law scaling, with slope lower than the fully extended phase. We randomly average over ${10}^3$ many-body states here. We choose a system size $L=1000$.

Figure 7

Phase transitions of noninteracting fermions in the GAA model [Eq. (3)] at half-filling. (a) Shows entanglement entropy density in one-half of the system. (b) Shows the bipartite $\left(\gamma =\frac{1}{2}\right)$ particle-number fluctuation. The tuning parameter $\alpha$ is fixed to be $-0.2$. We find the extensive-to-intensive transition for entanglement entropy is located at ${\lambda }_L\approx 1.3t$, and the thermal-to-nonthermal transition is located at ${\lambda }_T\approx 0.75t$, i.e., the bipartite particle-number fluctuation is strong (suppressed) above (below) ${\lambda }_T$. As we increase the system size, the two quantities shown here exhibit singular behavior at ${\lambda }_T$ and ${\lambda }_L$. The steplike structures in-between root in the spectra gaps of the model. In our calculation, we randomly sample ${10}^4$ $\left({10}^5\right)$ many-body states for (a) and (b), respectively.

Figure 8

Entanglement scaling as a function of the subsystem size $l$ in localized, extended, and partially extended phases of the generalized Aubry-André model [Eq. (3)] at half-filling. Like the Aubry-André model $\left(\alpha =0\right)$, the entanglement entropy shows volume law scaling for $\lambda /t=0.5$ and area law for $\lambda /t=1.5$. For $\lambda /t=1$, the GAA model is in a partially extended phase where the entanglement entropy shows volume-law scaling. The subthermal partially extended states are composed of both localized and delocalized single-particle orbitals. We randomly average over ${10}^3$ many-body states here. We choose a system size $L=1000$.

Figure 9

Thermal and entanglement entropies as a function of the energy density $e$ for the generalized Aubry-André model [Eq. (3)] at half-filling. In this plot, we take the tuning parameter $\alpha =-0.2$ [Eq. (3)]. We calculate the thermal and entanglement entropy per lattice site $\left(s\right)$ versus the energy density $\left(e\right)$ for a subsystem. The average density is fixed to be at half-filling. We choose a subset of $L_{\mathrm{sub}}$ lattice sites $\left(L_{\mathrm{sub}}=128\right)$ within a system of size $L=2048$. (a), (b) Correspond to $\lambda /t=0.5$ (fully extended phase), and $\lambda /t=1$ (partially extended phase), respectively. In the fully extended phase shown in (a), the entanglement entropy is approximately equal to the thermal entropy. We expect the deviation is due to finite-size effects. In the partially extended phase shown in (b), the entanglement entropy strongly deviates from the thermal entropy, the reason being partially extended states are composed of both localized and delocalized single-particle orbitals.

Figure 10

Phase transition of noninteracting fermions in the 1D nonlocal tunneling model at half-filling [Eq. (4)]. (a) Shows entanglement entropy density in one-half of the system. (b) Shows the bipartite $\left(\gamma =\frac{1}{2}\right)$ particle-number fluctuation. For this model, we find the extensive-to-intensive transition for entanglement entropy locates at ${\lambda }_L\approx 3t$. The thermal-to-nonthermal transition locates at ${\lambda }_T\approx 0.3t$. In our calculation, we randomly sample ${10}^4$ $\left({10}^5\right)$ many-body states for (a) and (b), respectively. The coefficient $p$ controlling the nonlocality is fixed to be 1 here.

Figure 11

Phase transition of noninteracting fermions in the 3D Anderson model at half-filling. We take a 3D system of size $L^3$ and calculate entanglement entropy per site [shown in (a) and (b)] and particle-number fluctuation [shown in (c) and (d)], in a subsystem (with size ${(L/2)}^3)$. (b), (d) Show the data collapse, where we take $\delta =(W-W_c)/t,W_c/t=16$, and scaling exponent $\nu =1.57$. For entanglement entropy, we randomly sample ${10}^2$ and ${10}^3$ many-body states, respectively, for large $\left(L=24,32\right)$ and small $\left(L=8,16,20\right)$ systems. For the particle-number fluctuation, we sample ${10}^4$ and ${10}^3$ states for small and large systems. We average over 100 (10) disorder realizations for $L=8,16$, and 20 (for $L=24$ and 32). With finite disorder strength, the particle-number fluctuation saturates to a finite value as we increase the system size.

Figure 12

A scenario for many-body localization transition. For noninteracting fermions in incommensurate lattice models, we have confirmed this phase transition scenario. Here, $E$ is the many-body energy. The intermediate “nonergodic extended” phase shows up in the presence of a single-particle mobility edge. For interacting systems, i.e., the many-body localization case, if we assume an intermediate regime with a subextensive number of local integrals of motion, the scenario shown here also applies.

Figure 13

Local integrals of motion in the GAA model. (a) The numerical energy spectrum $E/\left|t\right|$ as a function of $\lambda /\left|t\right|$ for $L=200$ sites tight-binding model for $\alpha =0.3$. Pure cyan denotes IPR $=0$ and pure black denotes IPR $=1$. Red line is a plot of the analytically obtained mobility edge. Convergence of LIOMs: plots of $ln|{\eta }_{l,m+l}^m|$ $\left(l=0,...,199\right)$ for different values of $\lambda [\lambda /|t|=0.6,0.8,1.0,1.4$ for (b)–(e)]. (f) Shows comparison between the number of localized states obtained by exact diagonalization to the number of convergent local charges.

Figure 14

Pictorial illustration of three different possibilities to construct many-body states of noninteracting fermions in the presence of a single-particle mobility edge. Here, $\epsilon$ is the single-particle energy axis. The “dashed” line marks the single-particle mobility edge. In this plot, we assume the single-particle states below (above) the mobility edge are localized (delocalized). With all particles put below the mobility edge, the many-body state is localized (shown on the left) with area-law entanglement entropy. With all particles put above the mobility edge, the many-body state is extended (shown on the right) with volume-law entanglement entropy, which is equal to the thermal entropy. Another scenario is to put some particles below the mobility edge and some above it, which leads to a partially extended state (shown in the middle). This state has volume-law entanglement entropy, yet smaller than the thermal entropy.

Figure 15

Level statistics of the AA model. We average over all single-particle eigenstates here. In (a), we average over $\varphi$. The black dashed line in (a) marks the position of $2\ln 2-1$, an expected value from Poisson statistics. As approaching the thermodynamic limit, $\langle r\rangle$ approaches to zero in the delocalized phase $\left(\lambda /t<1\right)$ and to $2\ln 2-1$ in the localized phase $\left(\lambda /t>1\right)$. In (b), we average over $\phi$ (flux), and the system size is $L=987$. In both (a) and (b), $Q$ is set from the Fibonacci sequence as described in the main text. In (c), $Q$ is set from the Pell sequence and we average over $\varphi$ as in (a). The results imply that $\langle r\rangle$ value does not depend on the choice of irrational numbers for the wave number $Q$. (d) Shows the probability distribution (with double-logarithmic scale) of the level spacing of the AA model at the self-dual point. In (d), we use $\lambda /t=1,L=4181,Q/2\pi =2584/4181$, the shift $\varphi$ is averaged over. The distribution $P\left(s\right)$ fits well to a power-law function $\sim s^{-1.68}$.

Figure 16

Inverse participation ratio near the localization transition for $\alpha =0$ in (a) and (b), and for $\lambda ={\lambda }_c$ in (c) and (d). The labels for each system size are shared across each figure. We find the IPR goes like $1/L$ in the delocalized phase and saturates to an $L$-independent constant in the localized phase. Applying scaling collapse in the vicinity of the critical point yields the critical exponents given in Table 1.

Figure 17

Energy ranges averaged over when computing the IPR. The numerical single-particle energy spectra are depicted with its corresponding IPR value that is not averaged over energy, black denotes an IPR $=0$, cyan denotes an IPR $=1$, and each red line marks the single-particle mobility edge ${\epsilon }_{ME}$. (a) $\lambda =t$, we average over all energies $\epsilon <0$ (the light tan region) giving rise to a localization transition at $\alpha =0$ shown in Figs. 16 and 16. (b) $\lambda =0.9t$ for energies $\epsilon <{\epsilon }_{ME}$ we average over the light tan region in energy whereas for $\epsilon >{\epsilon }_{ME}$ we average over the light blue region of energy, resulting in the average IPR in Fig. 18. The localization transition in each case occurs where the single-particle spectrum intersects the mobility edge which yields ${\alpha }_c^{-}=0.07286$ and ${\alpha }_c^{+}=0.07631$.

Figure 18

Inverse participation ratio near the localization transition for $\lambda =0.9t$, and averaging over energies $\epsilon >{\epsilon }_{ME}$ in (a) and (b), and averaging over energies $\epsilon <{\epsilon }_{ME}$ in (c) and (d). The labels for each system size are shared across each figure. The results for the critical exponents are given in Table 1.

Figure 19

Many-body density of states (MBDOS) of the one-dimensional Anderson model at half-filling. We show the MBDOS for the four different cases: (a) without disorder or interaction, (b) with disorder but no interaction, (c) without disorder but with interaction, and (d) with both disorder and interaction. We choose $W/t=V/t=0,W/t=2,V/t=0,W/t=0,V/t=2$, and $W/t=V/t=2$, in (a)–(d), respectively. For the disordered cases, we average over 100 realizations. The asymmetric shape of the distribution in (c) and (d) is due to interactions (for $V\to -V$ the MBDOS is peaked at $E<0)$. For the interacting case, we use the kernel polynomial method [90].