Quantum level engineering for Aharonov-Bohm caging in the presence of electron-electron interactions

Aharonov-Bohm (AB) cages are finite regions of a quantum network where single-particle eigenstates remain extremely localized for certain values of an applied magnetic field. Electron-electron interactions partially destroy the single-particle localization by the generation of extended many-particle states. We address the effects of on-site potentials on the AB localization phenomenon in a diamond-shaped quasi-one-dimensional bipartite chain, and show that field-induced AB cages can effectively persist in the presence of electron-electron interactions.

Figure 1

(a) Schematic arrangement of the diamond chain, with two types of inequivalent sites: spinal sites (labeled as $s$) and edge sites (labeled as $e$). The flux threading each constituent rhombus alternates between $\Phi$ and $\Phi /2$. (b) Possible configurations that the electron pair can adopt in its propagation through the chain: doubly occupied edge sites (DE), simultaneously singly occupied edge sites (SSE), doubly occupied spinal sites (DS), and singly occupied neighbor sites (SN).

Figure 2

Time evolution of the system for an initial state which consists of both electrons occupying the central ($s$3) site. $\Phi$ is set to ${\Phi }_0/2$, so the flux threading the rhombi alternates between ${\Phi }_0/4$ (first and third rhombi) and ${\Phi }_0/2$ (second and fourth rhombi). The occupation number of each spinal site is calculated as a function of time for (a) $U=0$ (noninteracting electrons), (b) $U=25\lambda$ (doublon regime), and (c) $U=2\lambda$. (d1) and (d2) sketch the Aharonov-Bohm cages produced for a noninteracting electron pair and for a doublon, respectively, both initialized on the central site (empty circle). The circled sites form the limited set of sites that the wave function can visit in each case.

Figure 3

(a) Schematics of the system employed to analyze the interplay between pair hopping mechanisms: (a1) Energy diagram of the states involved in both processes. (a2) and (a3) Schematic representation of, respectively, DE and SSE pair hopping mechanisms. (b) Mean $s$2 occupation probability as a function of the reduced flux $\Phi /{\Phi }_0$ threading the rhombus for the time evolution of a two-electron state initialized on $s$1. The Hubbard potential $U$ has been set to $10\lambda$ (doublon domain, red circles), $0.1\lambda$ (weakly interacting electrons, green triangles), and 0 (noninteracting electrons, blue squares). Inset: Minimum, over $\Phi /{\Phi }_0$, of the mean $s$2 occupation as a function of $U$. (c) Time evolution for the same system as in Fig. 2c ($U=2\lambda$), but forcing artificially the degeneracy of SSE and DE states (see the text for details). Note that in this case AB caging is preserved.

Figure 4

(a) Schematics of the system employed to analyze the effects of on-site potentials on the tunneling rates of pair and single hopping processes. (b) Mean $s$2 (red dashed lines) and $s$4 (blue solid lines) occupation probabilities for the time evolution of a state initialized on $s$3, as a function of the on-site potential $\delta \epsilon$ introduced on edge sites. In the four upper panels, a two-electron state evolves in time for, respectively, $U=0,0.1\lambda ,\lambda$, and $25\lambda$ interaction potentials. In the lower panel, the initial state is a single electron. The occupation probability has been renormalized to one in order to enable the comparison between one- and two-electron systems. Note that for the selected configuration of the magnetic flux, propagation to $s$2 ($s$4) relies on pair (single) hopping mechanisms. Inset: Same as in the corresponding panel but for a wider range of $\delta \epsilon$.