We calculate the conductance as a function of temperature GðTÞ through Au monatomic chains containing one Co atom as a magnetic impurity, and connected to two conducting leads with a fourfold symmetry axis. Using the information derived from ab initio calculations, we construct an effective model H eff that hybridizes a 3d 7 quadruplet at the Co site with two 3d 8 triplets through the hopping of 5d xz and 5d yz electrons of Au. The quadruplet is split by spin anisotropy due to spin-orbit coupling. SolvingĤ eff with the numerical renormalization group we find that at low temperatures GðTÞ ¼ a À b ffiffiffi ffi T p and the ground state impurity entropy is lnð2Þ=2, a behavior similar to the two-channel Kondo model. Stretching the chain leads to a non-Kondo phase, with the physics of the underscreened Kondo model at the quantum critical point. DOI: 10.1103/PhysRevLett.110.196402 PACS numbers: 71.10.Hf, 73.23.Àb, 75.20.Hr Transport properties in nanoscopic systems are being extensively studied due to potential applications and their role in basic research. In particular, the Kondo effect, in which the spin 1=2 of a molecule or a quantum dot (QD) is screened by one channel of conduction electrons below a characteristic temperature T K , has been experimentally observed and the conductance as a function of temperature GðTÞ is in excellent agreement with theory [1,2]. In spite of the large effect of correlations, these systems are Fermi liquids where, for example, the conductance has the expected behavior GðTÞ ' Gð0Þ À aT 2 for T ( T K , with T K the Kondo temperature and a some constant. More recently, the underscreened Kondo (UK) effect, leading to singular Fermi liquid physics [3,4] has been observed, and quantum phase transitions (QPTs) involving partially Kondo screened spin-1 molecular states were induced by externally controlled parameters [5][6][7].The overscreened Kondo effect, the simplest manifestation of which is the two-channel Kondo (2CK) model, is even more interesting because inelastic scattering persists even at vanishing temperatures and excitation energies. Therefore, the system is a non-Fermi liquid, exhibiting fascinating low-energy properties [8][9][10]. In particular, the impurity contribution to the entropy is lnð2Þ=2 and the conductance per channel at low T has the form GðTÞ ' G 0 =2 AE a ffiffiffi ffi T p , where G 0 is the conductance at zero temperature in the one-channel case. However, experimental observations of 2CK physics have been elusive, particularly due to its instability against the asymmetry of the channels [10]. For example, while 2CK physics is expected to take place at the QPT in the model of two spin-1/2 Kondo impurities [9,10], interchannel charge transfer spoils the quantum critical point (QCP) [10] and its observation in double QD systems [11].Oreg and Goldhaber-Gordon [12] proposed to modify the simplest setup of the one-electron transistor (a QD between two conducting leads) by adding a second large QD with level spacing < T K , which acts as a second channel of conduc...