  We explore the effects of geometric frustration within a one-dimensional
Bose-Hubbard model using a chain of rhombi subject to a magnetic flux. The
competition of tunnelling, self-interaction and magnetic flux gives rise to the
emergence of a pair-superfluid (pair-Luttinger liquid) phase besides the more
conventional Mott-insulator and superfluid (Luttinger liquid) phases. We
compute the complete phase diagram of the model by identifying characteristic
properties of the pair-Luttinger liquid phase such as pair correlation
functions and structure factors and find that the pair-Luttinger liquid phase
is very sensitive to changes away from perfect frustration (half-flux). We
provide some proposals to make the model more resilient to variants away from
perfect frustration. We also study the bipartite entanglement properties of the
chain. We discover that, while the scaling of the block entropy pair-superfluid
and of the single-particle superfluid leads to the same central charge, the
properties of the low-lying entanglement spectrum levels reveal their
fundamental difference.
