The single-particle eigenstates of the Aubry-Andr\'e-Harper model are known to show a delocalization-localization transition at a finite strength of the quasiperiodic disorder. In this work, we point out that an intimate relationship exists between the subband structure of the spectrum and transport properties of the model. To capture the transport properties, we have not only used a variety of single-particle measures like inverse participation ratio, and von Neumann entropy, but also many-particle measures such as persistent current and its variance, and many-body entanglement entropy. The many-particle measures are very sensitive to the subband structure of the spectrum. Even in the delocalized phase, surprisingly the entanglement entropy is substantially suppressed when the Fermi level is in the band gaps, whereas the persistent current is vanishingly small for the same locations of the Fermi level. The entanglement entropy seems to follow area law exclusively for these special locations of Fermi level or filling fractions of free fermions. A study of the standard deviation of persistent current offers further distinguishing features for the special fillings. In the delocalized phase, the standard deviation vs mean persistent current curves are discontinuous for the nonspecial values of filling fractions and continuous (closed) for the special values of filling fractions, whereas in the localized phase, these curves become straight lines for both types of filling fractions. We have also discussed how the results depend on the system size. Our results, especially on the persistent current, can potentially be tested experimentally using the present day setups based on ultracold atoms.
We investigate the nonequilibrium dynamics of entanglement entropy and out-of-time-order correlator (OTOC) of noninteracting fermions at half-filling starting from a product state to distinguish the delocalized, multifractal (in the limit of nearest neighbor hopping), localized and mixed phases hosted by the quasiperiodic Aubry-André-Harper (AAH) model in the presence of long-range hopping. For sufficiently long-range hopping strength a secondary logarithmic behavior in the entanglement entropy is found in the mixed phases whereas the primary behavior is a power-law the exponent of which is different in different phases. The saturation value of entanglement entropy in the delocalized, multifractal and mixed phases depends linearly on system size whereas in the localized phase (in the short-range regime) it is independent of system size. The early-time growth of OTOC shows very different power-law behaviors in the presence of nearest neighbor hopping and long-range hopping. The late time decay of OTOC leads to noticeably different power-law exponents in different phases. The spatial profile of OTOC and its system-size dependence also provide distinct features to distinguish phases. In the mixed phases the spatial profile of OTOC shows two different dependences on space for small and large distances respectively. Interestingly the spatial profile contains large fluctuations at the special locations related to the quasiperiodicity parameter in the presence of multifractal states.
We numerically investigate the dynamics of entanglement in a chain of spinless fermions with nonrandom but long-range hopping and interactions, and with random on-site energies. For moderate disorder in the absence of interactions, the chain hosts delocalized states at the top of the band which undergo a delocalization-localization transition with increasing disorder. We find an interesting regime in this noninteracting disordered chain where the long-time entanglement entropy scales as $S(t)\ensuremath{\sim}lnt$ and the saturated entanglement entropy scales with system size
L as $S(L,t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\sim}lnL$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.
We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent
α. For
α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size
L is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small
α≲1/2. For
α>3/2, the EE is in the area-law phase, namely, no scaling with
L, for any monitoring strength. For
1<α≲3/2, we identify an
α-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.
We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of long-range interactions and long-range hopping. Based on perturbative arguments there is a common belief that MBL can exist only in systems with short-range interactions and short-range hopping. We analyze effects of power-law interactions and power-law hopping, separately, on a system which has all the single-particle states localized in the absence of interactions. Since delocalization is driven by proliferation of resonances in the Fock space, we mapped this model to an effective Anderson model on a complex graph in the Fock space and calculated the probability distribution of the number of resonances up to third order. Though the most-probable value of the number of resonances diverge for the system with long-range hopping [$t(r)\ensuremath{\sim}{t}_{0}/{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}<2$], there is no enhancement of the number of resonances as the range of power-law interactions increases. This indicates that the long-range hopping delocalizes the many-body localized system but in contrast to this, there is no signature of delocalization in the presence of long-range interactions. We further provide support in favor of this analysis based on dynamics of the system after a quench starting from a charge density wave ordered state, level spacing statistics, return probability, participation ratio and Shannon entropy in the Fock space. We demonstrate that MBL persists in the presence of long-range interactions though long-range hopping with $1<\ensuremath{\alpha}<2$ delocalizes the system partially, with almost all the states extended for $\ensuremath{\alpha}\ensuremath{\le}1$. Even in a system which has single-particle mobility edges in the noninteracting limit, turning on long-range interactions does not cause delocalization.
On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations (chains up to
L=96 and up to 128 Born-rule trajectories per parameter point), whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation
FA=∑kνk(1−νk) and a participation-style effective number of entangling modes,
MA=exp[−∑kwklnwk], with
wk∝νk(1−νk). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values,
M is nearly a deterministic function of
F: a pooled curve explains 99.7\% of its variance across all rates, so
M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of
∼2 smaller than that of the entropy or
F under strong monitoring. Subsystem-scaling comparisons at
L=48--
96 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus,
F remains the natural experimentally motivated companion to the entropy, whereas
M is best viewed as a variance-reduced numerical summary of essentially the same information.