Universal description of dissipative Tomonaga-Luttinger liquids with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>SU</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> spin symmetry: Exact spectrum and critical exponents

Universal scaling relations for dissipative Tomonaga-Luttinger (TL) liquids with SU($N$) spin symmetry are obtained both fermions and bosons, by using asymptotic Bethe-ansatz solutions conformal field theory (CFT) in one-dimensional non-Hermitian quantum many-body systems symmetry. We uncover that the spectrum of TL is described sum one charge mode characterized a complex generalization $c=1$ U(1) Gaussian CFT, $N-1$ modes level-$1$ Kac-Moody algebra anomaly $c=N-1$, thereby dissipation only affects as result spin-charge separation systems. The derivation based on Haldane's ideal-gas description, which implemented Calogero-Sutherland model inverse-square long-range interactions.