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

We study the low-temperature properties of a one-dimensional spin-$1/2$ chain of magnetic impurities coupled to a (normal) metal environment by means of anisotropic Kondo exchange. In the case of easy-plane anisotropy, we obtain the phase diagram of this system at $T=0$. We show that the in-plane Kondo coupling destabilizes the Tomonaga-Luttinger phase of the spin chain, and leads to two different phases: (i) At strong Kondo coupling, the spins in the chain form Kondo singlets and become screened by the metallic environment, and (ii) at weak and intermediate Kondo coupling, we find a novel dissipative phase characterized by diffusive gapless spin excitations. The two phases are separated by a quantum critical point of the Wilson-Fisher universality class with dynamical exponent z2.