What Does It Mean for 1D RFIM to Be Frustrated?

[Hermite]

I've read in an article that the 1D Ising Model in a random field is a frustrated system. But what does it mean for the 1D RFIM to be frustrated?

The internet is very unhelpful. I'd appreciate it if someone could answer this for me.

Thanks.

 

[Lonewolf]

Frustration usually means that there is an infinite degeneracy of the classical ground state. The random field effectively turns the system into that of a spin glass, where the interactions between spins will become effectively random. A spin glass has lots and lots of degenerate states, so I guess this is where the frustration comes in.

 

[charng]

I would be happy to explain what it means for the 1D RFIM (random field Ising model) to be frustrated. In simple terms, frustration in a physical system refers to a situation where competing interactions between its components cannot be simultaneously satisfied. In the case of the 1D RFIM, this frustration arises from the random magnetic fields present in the system, which can cause conflicts between neighboring spins.

To understand this better, let's first look at the 1D Ising model without the random field. In this model, each spin can either be in an "up" or "down" state, and their interactions are described by the Ising Hamiltonian. The ground state of this system is when all spins are aligned in the same direction, either all "up" or all "down". However, when a random field is introduced, it can cause some spins to preferentially align in one direction while others prefer the opposite direction. This creates a competition between the interactions, leading to frustration.

In the 1D RFIM, this frustration is further enhanced by the fact that the random fields are not uniform and can vary in strength and direction along the chain. This leads to a complex energy landscape where finding the ground state becomes a challenging task. In fact, the 1D RFIM is known to have multiple ground states, making it a highly frustrated system.

Frustration in the 1D RFIM has several interesting consequences. It can lead to the formation of domain walls, where neighboring spins with conflicting interactions form boundaries between regions of aligned spins. It can also result in the system exhibiting non-trivial critical behavior, where small changes in the random field can cause significant changes in the system's properties.

In summary, frustration in the 1D RFIM arises from the presence of competing interactions between spins due to random magnetic fields. This makes the system highly complex and interesting to study, with potential applications in understanding other frustrated systems in physics and materials science. I hope this explanation helps clarify the concept for you.