Exact localized eigenstates for an extended Bose-Hubbard model with pair-correlated hopping

We show that a Bose-Hubbard model extended with pair-correlated hopping has exact eigenstates, quantum lattice compactons, with complete single-site localization. These appear at parameter values where the one-particle tunneling is exactly canceled by nonlocal pair correlations, and correspond in a classical limit to compact solutions of an extended discrete nonlinear Schrödinger model. Classical compactons at other parameter values, as well as multisite compactons, generically get delocalized by quantum effects, but strong localization appears asymptotically for increasing particle number.

Figure 1

$|c_{1\mathrm{site}}{|}^2$ for the highest-energy eigenstate as a function of $Q_2/N$ and $Q_5$ with $Q_4=0$ for $f=4$ and (a) $N=16$ and (b) $N=26$ particles. ${\tau }_k=1$.

Figure 2

$|c_{1\mathrm{site}}{|}^2$ for the highest eigenstate when condition (5a) is fulfilled, for $N=16$ and $f=4$. Black regions have $|c_{1\mathrm{site}}{|}^2<0.9$. ${\tau }_k=1$.

Figure 3

Maximum values of $|c_{2\mathrm{site}}{|}^2$ for the highest-energy state. Blue line (stars): Global max under independent parameter variations; red line (circles): max when $Q_2=-4Q_5(N/2-1)$. $f=4$, ${\tau }_k=1$.

Figure 4

$|c_{2\mathrm{site}}{|}^2$ for the highest-energy state with $N=16$ and (a) $Q_4=0$; (b) $Q_4=0.2$; (c) $Q_4=0.3$; (d) $Q_4=0.4$. $f=4$, ${\tau }_k=1$.

Figure 5

Probability distributions of two-site basis states (stars), matched with Gaussian functions (line), in the highest-energy eigenstates when $f=4$, ${\tau }_k=1$, $Q_2=-5$, $Q_5=0.08$, $N=30$, and (a) $Q_4=0.4$ ($|c_{2\mathrm{site}}{|}^2=0.9760$); (b) $Q_4=0.6$ ($|c_{2\mathrm{site}}{|}^2=0.9645$).