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Translationally invariant one-dimensional models, with interactions of finite range and a finite number of states at the site, don't allow phase transitions at positive temperatures. This fact is a simple consequence of the Perron-Frobenius theorem and relations between partition functions and transfer matrices.

For the random-field Ising model (RFIM), the partition function is not equal to the trace of some degree of the transfer matrix. So the simple argument about the non-existence of phase transition, which was valid in a spatially uniform situation, doesn't apply in the case of the random field. My question is the following. Are there any known non-existence theorems for the phase transitions at a non-zero temperature in the RFIM in one dimension?

This question is due to the rich diversity of interesting properties of the RFIM at zero temperature, even in one dimension. For some distributions of a random field, the ground state of 1d RFIM can have macroscopic degeneracy. There also can be discontinuities in the magnetization.

Update. I want to try to clarify my question more because of comments by Connor Behan and Roger Vadim. I know about the Imry-Ma argument. As far as I understand, this argument is about ferromagnetic ordering. My question is more general. In the ground state of 1d RFIM, spins tend to make clusters of different sizes and probabilities. Changes in model parameters lead to changes in cluster formation. Changes in cluster formation lead to discontinuities in some physical quantities which look like they can exist at finite temperatures. Hence my question. It is not about ferromagnetic transition, it is about the principal possibility of different phase transitions of the first kind. And my question is not about phase transitions in translationally invariant Ising model with long-range interaction like in this topic.

The energy of 1d RFIM is $$ E(\{\sigma\}) = -J\sum_{j} \sigma_j \sigma_{j+1} - \sum_j h_j \sigma_j. $$ Here $h_j$ are frozen (quenched) fields generated according to some distribution. It is improbable for any nontrivial distribution that $h_j = h_{j+1}$ for all $j$. Hence RFIM is not translationally invariant. Configurations $\{\sigma\}$ and $\{\sigma'\}$, where $\sigma_j' = \sigma_{j+1}$, have different energies.

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  • $\begingroup$ I checked the paper arxiv.org/abs/1912.01617 and the introduction says there is no phase transition for $d \leq 2$. It cites the Imry-Ma criterion for this. $\endgroup$
    – Connor Behan
    Commented Nov 30, 2021 at 12:15
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    $\begingroup$ Does this answer your question? Finite range 1D Ising model vs. infinite range Ising model $\endgroup$
    – Roger V.
    Commented Nov 30, 2021 at 12:20
  • $\begingroup$ @ConnorBehan Thank you. I've updated my quaestion. $\endgroup$
    – Gec
    Commented Nov 30, 2021 at 13:10
  • $\begingroup$ @RogerVadim No, it does not. I tried to clarify my question. $\endgroup$
    – Gec
    Commented Nov 30, 2021 at 13:11
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    $\begingroup$ @RogerVadim IvanVilenik gave a good answer to your question. But your question and his answer are about translationally invariant models. My question is about models with random fields. Such models do not possess translational invariance in the strong sense that energy doesn't change when the spin configuration is changed like $\sigma_j' = \sigma_{j+1}$. $\endgroup$
    – Gec
    Commented Nov 30, 2021 at 13:24

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