  We study theoretically the spatial distribution of spontaneous magnetic
fields near a (110) surface of cuprate high-$T_C$ superconductor YBCO by
treating a bilayer $t-J$ model with the Bogoliubov-de Gennes method. Near a
(110) surface where $d_{x^2-y^2}$-wave superconductivity is strongly
suppressed, flux phase order occurs locally leading to a time-reversal symmetry
broken surface state. Since the spontaneous currents flowing in the flux phase
in two layers are antiparallele, resulting spontaneous magnetic fields are
quite small. They exist in a narrow region inside the superconductor, and
essentially vanish outside irrespective of the temperature. This result,
together with the splitting of zero-energy peak of the local density of states,
can provide consistent explanations for several experiments on YBCO.
