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We study theoretically the spatial distribution of spontaneous magneticfields near a (110) surface of cuprate high-$T_C$ superconductor YBCO bytreating a bilayer $t-J$ model with the Bogoliubov-de Gennes method. Near a(110) surface where $d_{x^2-y^2}$-wave superconductivity is stronglysuppressed, flux phase order occurs locally leading to a time-reversal symmetrybroken surface state. Since the spontaneous currents flowing in the flux phasein two layers are antiparallele, resulting spontaneous magnetic fields arequite small. They exist in a narrow region inside the superconductor, andessentially 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.

Spontaneous Magnetic Field and Local Density of States near a Time-Reversal Symmetry Broken Surface State of YBCO