  We present a numerical study of quasi-1D frustrated Josephson junction
ladders with diagonal couplings and open boundary conditions, in the large
capacitance limit. We derive a correspondence between the energy of this
Josephson junction ladder and the expectation value of the Hamiltonian of an
analogous tight-binding model, and show how the overall superconducting state
of the chain is equivalent to the minimum energy state of the tight-binding
model in the subspace of one-particle states with uniform density. To satisfy
the constraint of uniform density, the superconducting state of the ladder is
written as a linear combination of the allowed k-states of the tight-binding
model with open boundaries. Above a critical value of the parameter t (ratio
between the intra-rung and inter-rung Josephson couplings), the ladder
spontaneously develop currents at the edges which spread to the bulk as t is
increased until complete coverage is reached. Above a certain value of t, which
varies with ladder size (t = 1 for an infinite-sized ladder), the edge currents
are destroyed. The value t = 1 corresponds, in the tight-binding model, to the
opening of a gap between two bands. We argue that the disappearance of the edge
currents with this gap opening is not coincidental, and that this points to a
topological origin for these edge current states.
