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.


1 Answer 1


Reference 23 of the reference you cited might also be useful. Furthermore, I wanted to add the following: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.22.436 https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.031030 https://www.nature.com/articles/ncomms14926

Don’t forget that you can also take advantage of the quantum to classical duality to check for any studies for the classical version of the model in 2D. See for instance http://www.phy.olemiss.edu/~luca/Topics/i/ising_2D.html for various references.

A possibly useful, related post Exact solution of the 2D Ising model in an external magnetic field?

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    $\begingroup$ I strongly suggest you provide, at least, a brief explanation of the (possible) solution(s) contained in the linked resources. In my opinion, your post does not constitute an answer to the question as it only provides some links and there’s no information regarding a possible solution. $\endgroup$
    – ZaellixA
    Oct 4, 2023 at 13:01
  • $\begingroup$ I perfectly agree, it’s not a complete answer. It’s just that I couldn’t write a comment… $\endgroup$
    – Kostas
    Oct 4, 2023 at 14:17
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    $\begingroup$ I understand that you don’t have enough reputation to write comments yet (you are really close though). But still I urge you to refrain from posting comments as answers. I believe that you could put some more information in your answer which summarises the content of the resources and this would, most probably get some upvotes which will provide the needed reputation to post comments on other people’s posts. In general, “link-only” answers are considered “off-topic” here (my opinion is that this is good but this is personal opinion) and yours seems to be (cont.) $\endgroup$
    – ZaellixA
    Oct 4, 2023 at 14:50
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    $\begingroup$ (cont’ed) very close to a “link-only” answer. I suggest you do add some info on the suggested solution in the resources but it is up to you whether you decide to do it or not. $\endgroup$
    – ZaellixA
    Oct 4, 2023 at 14:51

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