Having outlined the experimental procedure used by the Gadway research group to investigate the GPD model, we proceed to enumerate some observations from it [161]. Discussion and results presented in this Section stem from a collaboration with this group as well as with Profs. J. H. Pixley at Rutgers Univeristy and S. Ganeshan at City College of New York. At the time of writing of this thesis, a manuscript of a publication is being prepared together with these co-authors.
Main results of the experiment are summarized by Figs. 11.2 (c) and 11.3. Here, we use the participation ratio (PR) in order to distinguish localized and delocalized states. More explicitly,
is calculated from experimental data and we extract it from numerical simulations as well. For a finite- sized system having N sites, the PR takes values ranging from 1 for fully localized states to N for fully extended states. When an eigenstate of the system is tightly localized to a single site, n = n0, the sum in
Eq. (11.13) becomes Σn|ψn|4 = |ψn0|4 = 1 and the PR is equal to unity. In contrast, for a wavefunction
evenly distributed across all sites in the system i.e. ψn = 1/N for any n, it follows that Σn|ψn|4= 1/N and
the participation ratio is equal to system size N . Panel (c) of Fig. 11.2 shows the measured values of this parameter for the ground state and the highest excited state of the GPD model together with theoretically obtained predictions (using exact numerical diagonalization and the imaginary time propagation technique discussed in more detail in Sec. 11.4.1 below). Here, we recall that in the experimental realization of the AAH limit (α = 0) of the GPD model, the localization-delocalization transition takes place for λe/t = 2
rather than λ/t = 1 as in Chapter 10.
While all three parts of this plot show both states localizing as λe/tf becomes larger, we highlight that
the localization transition happens at approximately the same quasiperiodicity amplitude for the ground state and the highest excited state only when α = 0 (we attribute the small discrepancy between the two to the effect of nonlinear inter-atomic interactions and discuss it further in Sec. 11.4.2). At this point, the experimental system is equivalent to the AAH model of Eq. (10.1) where mobility edges are effectively absent and all states presumed to exhibit the same localization-delocalization physics. For α = −0.5, the ground state localizes for smaller λe/tf than the highest excited state. The opposite is true for α = 0.5, in
accordance with our discussion of the α → −α symmetry in Sec. 10.2.2.
Considering the PR of either state as function of λe/t, as shown in Fig. 11.2 (b), we see that for
α = −0.5 the ground state (localizing near λe/t = 1 or λ/t = 1/2) and the highest excited state (localizing
near λe/t = 3 or λ/t = 3/2) of the system eventually cross the mobility edge. These two plots therefore
provide experimental support for a parameter-tuned mobility edge. This conclusion is further strengthened by Fig. 11.3 which shows a localization-delocalization phase diagram with both α and λe/t varied. To
summarize, experimental data shows that for negative values of α this systems features a mobility edge and as α is increased, the mobility edge moves closer to the zero energy eigenvalue. For positive α, the mobility edge inverts so that high energy states localize at smaller quasiperiodicity strengths λ/t than the low energy states.
To gain insight into different localization behaviors for positive and negative α, we consider the distri- bution of lattice site energies in either case, as depicted in Fig. 11.4 (a). In this figure, the GPD on-site quasiperiodic term of Eq. (11.12) is shown overlaid on top of the regular one-dimensional lattice. It exhibits a more peaked character for α < 0. Lattice site energies in this case are mostly high, in a heuristic picture
of the quasiperiodic lattice being a sequence of potential wells, most states “sit” on top of the wells rather than at their bottoms. The opposite is true for α > 0 where the distribution of lattice site energies is such that most states are characterized by low energy eigenvalues. Consequently, for negative α, a high quasiperi- odicity strength is required to achieve localization as there are many nearby wavefunctions (having similar, high energies) that overlap thus being likely to produce an extended state. Low energy states, in contrast, are fewer and further apart thus leading to localization even for weaker quasiperiodicity. Since changing the sign of α inverts the distribution of low and high energy states, we expect the opposite localization trends, as in Fig. 11.2 (c), by the same argument. We have previously exhibited this inverse localization-delocalization behavior when comparing the two states in Chapter 10 by using the transfer matrix formalism; Fig. 10.6 illustrates it in the language of Lyapunov exponents.
We proceed to discuss the effects of interactions between condensed atoms that are relevant for localization- delocalization physics as most clearly evidenced by the experimental observations in the α = 0 limit. To pin-point these effects, we turn to numerical simulations.