We consider a domain wall in the mesoscopic quasi-one-dimensional sample(wire or stripe) of weakly anisotropic two-sublattice antiferromagnet, andestimate the probability of tunneling between two domain wall states withdifferent chirality. Topological effects forbid tunneling for the systems withhalf-integer spin S of magnetic atoms which consist of odd number of chains N.External magnetic field yields an additional contribution to the Berry phase,resulting in the appearance of two different tunnel splittings in anyexperimental setup involving a mixture of odd and even N, and in oscillatingfield dependence of the tunneling rate with the period proportional to 1/N.
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