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This is an analytic study of the two-dimensional XY spin glass with $\pm J$disorder. The Hamiltonian has a continuous spin symmetry and a discrete chiralsymmetry, and therefore possesses, potentially, two different order parametersand correlation lengths. The cost of breaking the symmetries is probed bycomparing the ground state energy under periodic (P) boundary conditions withthe one under antiperiodic (AP) and under reflecting (R) boundary conditions.Two energy differences (``domain wall energies'') appear, $\Delta E^{AP}$ and$\Delta E^R$, whose scaling behavior with system size is nontrivially relatedto the correlation length exponents. For a specific distribution of the $\pm J$disorder we show that the chiral and spin correlation lengths diverge with thesame exponent as $T\downarrow 0$. The common exponent has a common cause, viz.the low reversal energy of domains of chiral variables. For general disorder wegive a heuristic argumentation in terms of droplet excitations that leads tospin ordering on a longer, or equal, scale than chiral ordering. These resultsare in contrast with interpretations of Monte Carlo simulations.Comment: 18 pages in revtex, LPTHE Orsay 94/2

Chiral and spin order in the two-dimensional ±JXYspin glass: Domain-wall scaling analysis