# Latent heat in the Ising model

I am confused about the (non)-existence of latent heat in the first order phase transition of the Ising model. Most textbooks talk about latent heat as a signature sign of a first order transition, but I was not able to find any discussion about it in relation to ferromagnetism (other than here). The first order phase transition of the Ising model is effected by a change of sign in the applied magnetic field $$h$$ below the critical temperature $$T_c$$. This results in a discontinuous change in the magnetisation $$m$$. The latent heat $$L$$ associated with the transition depends on difference in entropy $$\Delta s = s_+ - s_-$$ between the up and down phases, and the temperature $$T$$ of the transition: $$L = T \Delta s \, .$$ However, as the up and down phases are completely symmetric, they have the same entropy and therefore $$\Delta s = L = 0 \,.$$

This can also be seen from the Clausius-Clapeyron relation, $$\frac{dh}{dT} = \frac{L}{T \Delta m} \, .$$

From the phase diagram, it is obvious that $$\frac{dh}{dT} = 0$$, leading to $$L = 0$$.

I have a bunch of questions about all this:

• Is there really no latent heat?
• When using the Ising model to model a lattice gas, how does a nonzero latent heat emerge?
• Is there a general relationship between the slope of the coexistence line and the symmetry between two phases?
• Are there any other pedagogically relevant systems undergoing a first order transition with zero latent heat?
• You are linking to another question on Physics.SE. There are good answers there (for instance, the answer by @knzhou is short and seems to be addressing exactly your question. Could you explain what is missing from his (and the other) answer? Concerning the Ising lattice gas, notice that, in the chemical potential-temperature plane, the phase coexistence line is not horizontal anymore. In particular, you can cross the phase coexistence line by varying the temperature now. This changes everything. Nov 24, 2021 at 19:10
• @knzhou's answer is insightful, but only talks about the absence of latent heat as a strange edge case in 'weird systems'. I am curious if the absence of latent heat is a general feature that can tell us something about the properties or symmetries of the system we are studying. I am also not sure if it is correct that the latent heat depends on the way in which one crosses the coexistence line. Your point about the lattice gas is very interesting. Is the change in behaviour due to the extra constraint that the number of 'up' (filled with molecule) cells has to be conserved in the gas model? Nov 25, 2021 at 9:51

Latent heat is associated with a transition in which there is a change in temperature, keeping the other fields constant. Furthermore, the transition is from an ordered configuration to a disorder. Examples with lattice models are the q=10 Potts model, Baxter-Wu 3D and others. In the Ising model, the first-order phase transition occurs with the variation of the external field, keeping the temperature constant. In this case the transition occurs between two ordered phases. In the first-order transition with temperature, the energy probability distribution has a double peak, the energy difference of the distribution maxima is the latent heat. In the transition with an external field, the probability distribution of the order parameter has a double peak. In both we find discontinuities in the thermodynamic quantities. An article where you can find more details of first-order transitions is Phys. Rev. B 26, 2507 (1982) - Scaling for first-order phase transitions in thermodynamic and finite systems.

Below I suggest some articles where the probability distribution of energy (PDE) and the probability distribution of the order parameter are used. Few articles calcule the last distribution. In the article "Phase transition properties of the Bell-Lavis model" you can see the behavior of the PDE for first and second order phase transition. In “On the order of the phase transition in the spin-1 Baxter–Wu model” you find a detailed study of the PDE, where it includes the configurations corresponding to each maximum of the PDE and the probability distribution of the order parameter.

Phys. Rev. B 30, 1477 (1984) - Finite-size scaling at first-order phase transitions

Phys. Rev. B 44, 5081 (1991) - Critical behavior of the three-dimensional Ising model: A high-resolution Monte Carlo study

Phys. Rev. B 43, 3265 (1991) - Finite-size scaling and Monte Carlo simulations of first-order phase transitions

Phys. Rev. B 34, 1841 (1986) - Finite-size effects at temperature-driven first-order transitions

Phys. Rev. E 90, 042124 (2014) - Phase transition properties of the Bell-Lavis model

On the order of the phase transition in the spin-1 Baxter–Wu model - ScienceDirect

Phys. Rev. E 100, 032141 (2019) - Three-dimensional Baxter-Wu model

Probability distribution of magnetization in the one-dimensional Ising model: effects of boundary conditions - IOPscience

• Thanks a lot for your answer! The paper you linked is very insightful. Do you have a reference that goes into more detail about the energy and order parameter distribution functions you mentioned? Feb 9, 2022 at 16:47
• I edit the post Feb 10, 2022 at 19:04
• @leapsheep I added, in the post, the information contending the references. Feb 10, 2022 at 19:14