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

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.