We investigate subdominant order parameters stabilizing at low temperatures in nanoscale high-T c cuprate islands, motivated by the recent observation of a fully gapped state in nanosized YBa 2 Cu 3 O 7À [D. Gustafsson et al., Nature Nanotech. 8, 25 (2013)]. Using complementary quasiclassical and tight-binding Bogoliubov-de Gennes methods, we show on distinctly different properties dependent on the symmetry being d x 2 Ày 2 þ is or d x 2 Ày 2 þ id xy . We find that a surface-induced d x 2 Ày 2 þ is phase creates a global spectroscopic gap which increases with an applied magnetic field, consistent with experimental observation. DOI: 10.1103/PhysRevLett.110.197001 PACS numbers: 74.78.Àw, 74.20.Rp, 74.50.+r, 74.72.Àh It is well established that high-temperature cuprate superconductors have a dominantly d x 2 Ày 2 -wave order parameter symmetry [1,2], but the existence of a subdominant symmetry has also long been considered. Of special interest are order parameters such as d x 2 Ày 2 þ id xy (d 1 þ id 2 ) and d 1 þ is, which fully gap the Fermi surface and break time-reversal (T ) symmetry [3][4][5][6]. Experimental data have been contradictory, invoking large imaginary subdominant orders to explain tunneling experiments in YBa 2 Cu 3 O 7À (YBCO) [7,8] and La 2Àx Sr x CuO 4 [9] or thermal conductivity in Bi 2 Sr 2 CaCu 2 O 8 [10] while, on the other hand, only very small imaginary components seem compatible with the absence of any measured spontaneous magnetization [2,[11][12][13][14].The possibility to find a T -symmetry breaking state is enhanced if the dominant d 1 -wave order parameter is locally reduced by, e.g., surface scattering. A nanoscale island of a curate superconductor is, by virtue of its large surface-to-area ratio, thus an ideal candidate to search for this elusive state. Very recently, a parity effect was reported in YBCO single-electron transistors (SETs), signalling a fully gapped low-temperature superconducting phase in nanoscale YBCO [15]. For a pure d 1 -wave superconductor there are always low-energy quasiparticle states available at the nodal points, into which the added charge in a SET can relax. Unless the nodal quasiparticles states are lifted by a spectroscopic energy gap E g , a parity effect should not be present in YBCO SETs. The experimental data show E g % 20-40 eV, making E g 3 orders of magnitude smaller than the gap Á d 1 % 20 meV in YBCO [15].The YBCO SETs studied in Ref.[15] have sizes of order 0:2 Â 0:2 Â 0:1 m 3 , making the energy-level spacing s of the nodal quasiparticles a candidate for the observed gap E g . While s $ 10 eV for an infinite mean-free path, surface disorder gives a reduced s $ 0:01 eV ( E g [16]. However, energy level spacing due to finite size is incompatible with the finite onset voltages of the SET current measured over a full gate charge period [17]. The measured E g is thus of superconducting origin and gaps the whole Fermi surface. Given the parent d 1 -wave order parameter of cuprates, finding E g demonstrates the presence of a complex order par...