We present Angle-Resolved Photoemission Spectroscopy measurements of the quasi-one dimensional superconductor K$_2$Cr$_3$As$_3$. We find that the Fermi surface contains two Fermi surface sheets, with linearly dispersing bands not displaying any significant band renormalizations. The one-dimensional band dispersions display a suppression of spectral intensity approaching the Fermi level according to a linear power law, over an energy range of ~200 meV. This is interpreted as a signature of Tomonoga-Luttinger liquid physics, which provides a new perspective on the possibly unconventional superconductivity in this family of compounds.
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