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One may consider a scenario in which a ferromagnetic material prepared in an ordered state (eg., with $ \langle m \rangle = \uparrow $) is suddenly subjected to a finite “opposite” external magnetic field (ie., antiparallel to the initial magnetisation) while the temperature is kept fixed below the critical temperature. The question is about the latent heat involved in this “phase transition of 1st order”.

Firstly, I think this abrupt scenario cannot be called a phase transition in the common sense. Such a procedure will induce an abrupt change in the properties of the system (esp. its ground-state). Free energy will be discontinuous itself, so it is a “zeroth-order/discontinuous phase transition”, if you wish. Phase transitions are usually defined as processes in which a slow variation of a thermodynamic variable (usually, temperature) leads to drastic changes in the thermodynamic phase of a system. Abrupt changes are trivial in this regard: we know a priori that they induce qualitative changes in the system. In our case, if one changes the external magnetic field abruptly, and waits long enough to let the system relax to its equilibrium, one will see that the magnetisation of the system is reversed to be parallel to the final external field; the system adapts itself to the applied field if we allow it to relax (see diagram below).

Let's consider a quasi-static scenario. But, to be more quantitative, we need a simple mean-field analysis.

In our scenario, when the magnetic field is changed slowly from $ h_\ast - \delta h $ to $ h_\ast + \delta h $, with a small $ \delta h > 0 $, then the internal energy of the system suddenly changes by a finite value $ \sim m \, \delta h + h_\ast \, \delta M $, due to the discontinuity of $ m $ at $ h_\ast $. This amount of energy is absorbed from/released to the external field. This is similar to the usual case when temperature is varied near the transition point and a latent heat $ \Delta q $ is absorbed from/released to the reservoir.

(I) Is this scenario a first-order phase transition Finally, the fact that this scenario is indeed a first-order phase transition, can be seen by an analysis of singularities in thermodynamic quantities. First let's consider the magnetisation. The mean-field equation can be expanded around $ h = 0 $ in an ordered phase where $ \frac{T}{T_c} \lesssim 1 $ and $ m \sim \mathcal{O}(1) $ is finite. Then we obtain the approximate mean-field equation,

$$ \tanh(m/T) \sim \text{sign}(m) \sim \text{sign}(h) = \pm 1 ~; $$ The second approximation is valid since the sign of $ h $ determines the sign of $ m $. From this, we readily get the non-analyticity in $ m(h) $ near $ h = 0 $ (in the ordered phase when $ T < T_c $):

One may consider a scenario in which a ferromagnetic material prepared in an ordered state (eg., with $ \langle m \rangle = \uparrow $) is suddenly subjected to a finite “opposite” external magnetic field (ie., antiparallel to the initial magnetisation) while the temperature is kept fixed below the critical temperature.

I think this abrupt scenario cannot be called a phase transition in the common sense. Such a procedure will induce an abrupt change in the properties of the system (esp. its ground-state). Free energy will be discontinuous itself, so it is a “zeroth-order/discontinuous phase transition”, if you wish. Phase transitions are usually defined as processes in which a slow variation of a thermodynamic variable (usually, temperature) leads to drastic changes in the thermodynamic phase of a system. Abrupt changes are trivial in this regard: we know a priori that they induce qualitative changes in the system. In our case, if one changes the external magnetic field abruptly, and waits long enough to let the system relax to its equilibrium, one will see that the magnetisation of the system is reversed to be parallel to the final external field; the system adapts itself to the applied field if we allow it to relax (see diagram below).

So let's consider a quasi-static (slow) scenario. But, to be more quantitative, we need a simple mean-field analysis.

In our scenario, when the magnetic field is changed slowly from $ h_\ast - \delta h $ to $ h_\ast + \delta h $, with a small $ \delta h > 0 $, then the internal energy of the system suddenly changes by a finite value $ \sim m \, \delta h + h_\ast \, \delta M $, due to the discontinuity of $ m $ at $ h_\ast $. This amount of energy is absorbed from/released to the external field -- this is essentially the work performed by the external field on the system which leads to a change in the internal energy as $ du = đ q + dw $. This is similar to the usual case when temperature is varied near the transition point and a latent heat $ \Delta q $ is absorbed from/released to the reservoir.

(I) Is this scenario truly a first-order phase transition?

The fact that this scenario is indeed a first-order phase transition, can be seen by an analysis of singularities in thermodynamic quantities. First let's consider the magnetisation. The mean-field equation can be expanded around $ h = 0 $ in an ordered phase where $ \frac{T}{T_c} \lesssim 1 $ and $ m \sim \mathcal{O}(1) $ is finite. Then we obtain the approximate mean-field equation,

$$ \tanh(m/T) \sim \text{sign}(m) \sim \text{sign}(h) = \pm 1 ~; $$ The second approximation is valid since the sign of $ h $ determines the sign of $ m $ (see above). From this, we readily get the non-analyticity in $ m(h) $ near $ h = 0 $ (in the ordered phase when $ T < T_c $):

In our scenario, when the magnetic field is changed slowly from $ h_\ast - \delta h $ to $ h_\ast + \delta h $, with a small $ \delta h > 0 $, then the internal energy of the system suddenly changes by a finite value $ \sim m \, \delta h $, due to the discontinuity of $ m $ at $ h_\ast $. This amount of energy is absorbed from/released to the external field. This is similar to the usual case when temperature is varied near the transition point and a latent heat $ \Delta q $ is absorbed from/released to the reservoir.

In our scenario, when the magnetic field is changed slowly from $ h_\ast - \delta h $ to $ h_\ast + \delta h $, with a small $ \delta h > 0 $, then the internal energy of the system suddenly changes by a finite value $ \sim m \, \delta h + h_\ast \, \delta M $, due to the discontinuity of $ m $ at $ h_\ast $. This amount of energy is absorbed from/released to the external field. This is similar to the usual case when temperature is varied near the transition point and a latent heat $ \Delta q $ is absorbed from/released to the reservoir.

First order phase transitions are always associated with a latent heat?enter preformatted text here

First order phase transitions are always associated with a latent heat?