  We present results of a lattice calculation of tetraquark states with quark
contents $q_1q_2\bar{Q}\bar{Q}, \, q_1,q_2 \subset u,d,s,c$ and $Q \equiv b,c$
in both spin zero ($J=0$) and spin one ($J=1$) sectors. These calculations are
performed on three dynamical $N_f = 2 + 1 + 1$ highly improved staggered quark
ensembles at lattice spacings of about 0.12, 0.09 and 0.06 fm. We use the
overlap action for light to charm quarks while a non-relativistic action with
non-perturbatively improved coefficients with terms up to $\mathcal{O}(\alpha_s
v^4)$ is employed for the bottom quark. While considering two heavy quarks as
charm or bottom, we calculate the energy levels of various four-quark
configurations with light quark masses ranging from the physical strange quark
mass to that of the corresponding physical pion mass. Results for the spin one
states show the presence of ground state energy levels which are below their
respective thresholds for all the light flavor combinations with both doubly
heavy quarks and particularly for the bottom quarks. Further, we identify a
trend that the energy splittings, defined as the energy difference between the
ground state energy levels and their respective thresholds, increase with
decreasing the light quark masses and are maximum at the physical point for all
the spin one states. The rate of increase is however dependent on the light
quark configuration of the particular spin one state. We also present a study
of hadron mass relations involving tetraquarks, baryons and mesons arising in
the limit of infinitely heavy quark and find that these relations are more
compatible with the heavy quark limit in the bottom sector but deviate
substantially in the charm sector. The ground state spectra of the spin zero
tetraquark states with various flavor combinations are seen to lie above their
respective thresholds.
