  The geometric phase can act as a signature for critical regions of
interacting spin chains in the limit where the corresponding circuit in
parameter space is shrunk to a point and the number of spins is extended to
infinity; for finite circuit radii or finite spin chain lengths, the geometric
phase is always trivial (a multiple of 2pi). In this work, by contrast, two
related signatures of criticality are proposed which obey finite-size scaling
and which circumvent the need for assuming any unphysical limits. They are
based on the notion of the Bargmann invariant whose phase may be regarded as a
discretized version of Berry's phase. As circuits are considered which are
composed of a discrete, finite set of vertices in parameter space, they are
able to pass directly through a critical point, rather than having to
circumnavigate it. The proposed mechanism is shown to provide a diagnostic tool
for criticality in the case of a given non-solvable one-dimensional spin chain
with nearest-neighbour interactions in the presence of an external magnetic
field.
