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Chiral and continuous symmetry of an XY spin glass on a tube lattice

We analyse the chiral symmetry in the random $\pm J$ $XY$ model on a $N\times 2$ square lattice with periodic boundary conditions in the transverse direction. This ``tube" lattice may be seen as a two-dimensional lattice of which one dimension has been compactified. In the Villain formulation the discrete-valued {\em chiralities}\/ or {\em charges}\/ associated with the plaquettes of the lattice decouple from the continuous degrees of freedom. The difficulty of the problem lies in the fact that the chiralities interact through the long range ``strong" one-dimensional Coulomb potential - which increases linearly with distance - as well as through an exponentially decaying ``weak" interaction. By comparing the ground state energies for periodic, antiperiodic, and reflecting boundary conditions in the longitudinal direction, we show that the chiralities and the $XY$ spins have the {\em same}\/ zero-$T$ correlation length exponent, whose exact value $ν_c = 0.5564\ldots$ we determine. The equality of these correlation lengths even in the presence of long range chirality-chirality interactions lends support to the view that chiral glass order cannot be sustained without simultaneous spin glass order.