Atom assemblies on surfaces represent the ultimate lower size limit for electronic circuits, and their conduction properties are governed by quantum phenomena. A fundamental prediction for a line of atoms confining the electrons to one dimension is the Tomonaga-Luttinger liquid 1. Yet, astonishingly, this has not been observed in surface systems so far. Here we scrutinize self-organized chains of single-atom width by scanning tunnelling spectroscopy and photoemission. The lowenergy spectra univocally show power-law behaviour. Even more, the density of states obeys universal scaling with energy and temperature. This demonstrates paradigmatic Tomonaga-Luttinger liquid properties 2,3 encountered at the atomic scale, with bearing for the conductivity of wires and junctions. Local control enables us to study modified interactions due to defects or bridging atoms not previously possible. Arrays of single atoms on surfaces provide an environment for a rich variety of quantum phenomena, especially regarding the electron states responsible for conduction. Their properties can be probed locally with scanning tunnelling microscopy (STM). Key examples include the superposition of electron waves in quantum corrals, leading to new coherent states 4. A challenge remains the exotic correlated state predicted to occur when the electrons are squeezed into one dimension, as in a linear chain of atoms. Quantum theory describes this regime as a Tomonaga-Luttinger liquid 1 (TLL) with collective excitations of spin and charge. It reveals itself in characteristic power-law behaviour of the excitation spectra 2,3 , with markedly depressed density of states at the chemical potential (where conduction takes place). This state is highly fragile and collapses on slight coupling to the second dimension. Experimental indications of TLL low-energy spectra are scarce, consisting of one-dimensional (1D) crystals 5-7 , carbon nanotubes 8,9 and GaAs channels 10,11 (compiled in Supplementary Table S1). Surprisingly, and contrary to expectations, this phenomenon has not been found so far in atom chains at surfaces, although such behaviour will dramatically affect atomic leads and junctions 12. Creation of a TLL state in such chains would be highly intriguing because this promises local atom manipulations that tune the interactions. Concerning approaches to build suitable chains, artificial atom placement by an STM tip 13 leads to short arrays, which do not suffice for an extended 1D regime. Hence, our approach is to use self-organized chains of large extent formed by noble-metal atoms on the semiconducting Ge(001) surface 14,15. Here Au-induced chains show metallic tunnelling conductivity at room temperature 15. Self-organized formation of Au atom wires along the Ge(001) dimer rows leads to c(8 × 2) long-range order covering the

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