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I wrote a Mathematica code to test out the 2d Ising Gauge Theory (by computing the exact partition fucntion) on a $3\times 3$ lattice (so that there are ($4\times 3$ spins on the horizontal bonds and $3 \times 4$ spins on the vertical bonds). I put an external magnetic field \begin{equation} H_\text{ext} = h\sum_{\text{bonds}} \sigma_3, \end{equation} and plotted the magnetization per 'bond' \begin{equation} M = \frac{1}{2n(n+1)}\frac{\partial \log Z(\beta,h)}{\partial \beta h}. \end{equation} I obtained the following plot enter image description here

It seems to me that there is some residual magnetization as $T\to 0$. But this is strange to me as this model does not have a phase transition? Shouldn't this hint towards a phase transition which does not happen in the case of the Ising Gauge theory? Another plot which I made was $M$ vs. $T$ for different values of $h$ and I got this. Is this consistent with theory? enter image description here

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    $\begingroup$ I might be completely missing the point but: the 1D ising model also have a non-zero magnetization for any temperature as long as there is a magnetic field? See for example: researchgate.net/figure/… . This does not mean that there is a phase transition (there is no critical behavior at $T_c$ as expected for a continuous transition or a discontinuity of the magnetization at $h = 0^+$ and $h = 0^-$ as expected for a first order phase transition) $\endgroup$
    – Syrocco
    Commented Nov 20, 2023 at 14:32

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