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${\bf \frac{h}{e}}$ flux quantization in metals due to Berry phase coherence

Berry curvature does not show itself in the relative phase correlation of wave-functions at different spatial points in a metal unless the fermions have closed trajectories in momentum space, for example those around isolated impurities. But these, just as the Bloch phase correlations, disappear at lengths larger than the diffusion length. If a quasi-two dimensional metal with Berry curvature has a set of domains, their boundaries necessarily carry chiral currents precluding back-ward scattering. The Berry induced phase coherence then persists over length scales of order the scale at which the chiral one-dimensional states scatter into the bulk states, which can be macroscopic. The conditions for their occurrence and the lengths and the orientation of such states are derived. These calculations are used to understand the remarkable aspects of a recent experiment in an anisotropic metal, reported to have loop-current order, with mean-free path of about 0.01 $μ$m which exhibits flux quantization in some transport properties over lengths of several $μ$m. %It is also shown that generating the appropriately oriented channels in the plane by the field applied is plausible.