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We study the low-temperature properties of a one-dimensional spin-1/2 chainof magnetic impurities coupled to a (normal) metal environment by means ofanisotropic Kondo exchange. In the case of easy-plane anisotropy, we obtain thephase diagram of this system at T=0. We show that the in-plane Kondo couplingdestabilizes the Tomonaga-Luttinger phase of the spin-chain, and leads to twodifferent phases: i) At strong Kondo coupling, the spins in the chain formKondo singlets and become screened by the metallic environment, and ii) At weakand intermediate Kondo coupling, we find a novel dissipative phasecharacterized by diffusive gapless spin excitations. The two phases areseparated by a quantum critical point of the Wilson-Fisher universality classwith dynamical exponent $z\simeq2$.

Magnetic phases in the one-dimensional Kondo chain on a metallic surface