  Dimensionless ratios of physical properties can characterize low-temperature
phases in a wide variety of materials. As such, the Wilson ratio (WR), the
Kadowaki-Woods ratio and the Wiedemann\--Franz law capture essential features
of Fermi liquids in metals, heavy fermions, etc. Here we prove that the phases
of many-body interacting multi-component quantum liquids in one dimension (1D)
can be described by WRs based on the compressibility, susceptibility and
specific heat associated with each component. These WRs arise due to additivity
rules within subsystems reminiscent of the rules for multi-resistor networks in
series and parallel --- a novel and useful characteristic of multi-component
Tomonaga-Luttinger liquids (TLL) independent of microscopic details of the
systems. Using experimentally realised multi-species cold atomic gases as
examples, we prove that the Wilson ratios uniquely identify phases of TLL,
while providing universal scaling relations at the boundaries between phases.
Their values within a phase are solely determined by the stiffnesses and sound
velocities of subsystems and identify the internal degrees of freedom of said
phase such as its spin-degeneracy. This finding can be directly applied to a
wide range of 1D many-body systems and reveals deep physical insights into
recent experimental measurements of the universal thermodynamics in ultracold
atoms and spins.
