I am wondering what is the phase diagram of the transverse-field Ising model in the presence of a longitudinal field, in particular, a one-dimensional spin-1/2 chain with ferromagnetic interactions. Something like
$$ H = -\frac{J}{2}\sum_{\langle i,j \rangle} S_i^z S_j^z - h_z \sum_i S_i^z - h_x \sum_i S_i^x. $$ where $J>0$ and $\langle i,j \rangle$ stands for nearest neighbor, and $S_i^z/S_i^x$ are spin-1/2 operators at site $i$ within an infinite chain.
I found countless discussions on the random-field-transverse Ising model, in which the longitudinal field is considered within a probability distribution, e.g., https://link.springer.com/book/10.1007/978-3-642-33039-1 and https://www.sciencedirect.com/science/article/pii/S0375960105005694 and https://iopscience.iop.org/article/10.1088/0953-8984/6/46/023
Very similar to what I am looking for, I could find the phase diagram considering antiferromagnetic interactions, e.g, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.68.214406 and https://www.nature.com/articles/nature09994 and https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.012122
The closest I could get to answer my question was Fig. 1 in this paper https://iopscience.iop.org/article/10.1088/1367-2630/aab2db but it feels weird that this recent paper is the only ref to it, and the references they provided do not show a clear phase diagram.