  Motivated by advances in the manipulation and detection of ultracold atoms
with multiple internal degrees of freedom, we present a finite-temperature
lattice Monte Carlo calculation of the density and pressure equations of state,
as well as Tan's contact, of attractively interacting SU(4)- and
SU(6)-symmetric fermion systems in one spatial dimension. We also furnish a
nonperturbative proof of a universal relation whereby quantities computable in
the SU(2) case completely determine the virial coefficients of the SU($N_f$)
case. These one-dimensional systems are appealing because they can be
experimentally realized in highly constrained traps and because of the dominant
role played by correlations. The latter are typically nonperturbative and are
crucial for understanding ground states and quantum phase transitions. While
quantum fluctuations are typically overpowered by thermal ones in one and two
dimensions at any finite temperature, we find that quantum effects do leave
their imprint in thermodynamic quantities. Our calculations show that the
additional degrees of freedom, relative to the SU(2) case, provide a dramatic
enhancement of the density and pressure (in units of their noninteracting
counterparts) in a wide region around vanishing $\beta\mu$, where $\beta$ is
the inverse temperature and $\mu$ the chemical potential. As shown recently in
experiments, the thermodynamics we explore here can be measured in a controlled
and precise fashion in highly constrained traps and optical lattices. Our
results are a prediction for such experiments in one dimension with atoms of
high nuclear spin.
