Why Can't the 1D Ising Model Have a Phase Transition?

[Pacopag]

Homework Statement

Can someone please explain to me why there can never be a phase transition in the 1D Ising model?

Homework Equations

The Attempt at a Solution

I have read the argument that if we start at T=0, all spins along the 1D chain are aligned (say up).
Then if we slightly increase T, we can get a so-called "grain boundary" with changes the energy by 2J, which remains finite in the thermodynamic limit (as the size of the chain becomes infinite). This is unlike the 2D case where the energy change associated with a grain boundary blows up as the system size becomes infinite. That all makes sense, but what does all this have to do with phase transitions?

 

[Jhero]

Thank you for your question. The reason there can never be a phase transition in the 1D Ising model is due to the Mermin-Wagner theorem. This theorem states that in one-dimensional systems, continuous symmetry cannot be spontaneously broken at any finite temperature. In other words, the system will always remain in a disordered state, even at low temperatures.

In the 1D Ising model, there is a continuous symmetry associated with the spins being able to flip between up and down states. In higher dimensions, this symmetry can be broken at a critical temperature, leading to a phase transition. However, in one dimension, the thermal fluctuations are strong enough to prevent this symmetry from being broken, and the system remains disordered.

The "grain boundary" argument you mentioned is related to this idea. In the 1D Ising model, the energy associated with a grain boundary is always finite, even in the thermodynamic limit. This means that there is no sudden change in the energy that could drive a phase transition.

I hope this helps to clarify the connection between the 1D Ising model and phase transitions. If you have any further questions, please don't hesitate to ask.