Strongly correlated quantum systems often display universal behavior as, in certain regimes, their properties are found to be independent of the microscopic details of the underlying system. An example of such a situation is the Tomonaga-Luttinger liquid description of one-dimensional strongly correlated bosonic or fermionic systems. Here we investigate how such a quantum liquid responds under dissipative dephasing dynamics and, in particular, we identify how the universal Tomonaga-Luttinger liquid properties melt away. Our study, based on adiabatic elimination, shows that dephasing first translates into the damping of the oscillations present in the density-density correlations, a behavior accompanied by a change of the Tomonaga-Luttinger liquid exponent. This first regime is followed by a second one characterized by the diffusive propagation of featureless correlations as expected for an infinite temperature state. We support these analytical predictions by numerically exact simulations carried out using a number-conserving implementation of the matrix product states algorithm adapted to open systems.
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