Resonances of $A=4$ $T=1$ isospin triplet states within the $\mathit{\text{ab}}\mathit{\text{initio}}$ no-core Gamow shell model

The $A=4$ nuclei, i.e., ${}^4$H, ${}^4$He and ${}^4$Li, establish an interesting isospin $T=1$ isobaric system. ${}^4$H and ${}^4$Li are unbound broad resonances, whereas ${}^4$He is deeply bound in its ground state but unbound in all its excited states. The present situation is that experiments so far have not given consistent data on the resonances. Few-body calculations have well studied the scatterings of the $4N$ systems. In the present work, we provide many-body calculations of the broad resonance structures, in an $\mathit{\text{ab}}\mathit{\text{initio}}$ framework with modern realistic interactions. It occurs that, indeed, ${}^4$H, ${}^4$Li and excited ${}^4$He are broad resonances, which is in accordance with experimental observations. The calculations also show that the first $1^{-}$ excited state almost degenerates with the $2^{-}$ ground state in the pair of mirror isobars of ${}^4$H and ${}^4$Li, which may suggest that the experimental data on energy and width are the mixture of the ground state and the first excited state. The $T=1$ isospin triplet formed with an excited state of ${}^4$He and ground states of ${}^4$H and ${}^4$Li is studied, focusing on the effect of isospin symmetry breaking.