Spin glass for ferromagnetic systems

[PullMeOut]

Homework Statement

hamiltonian is given by H=-[tex]\sum [/tex] J_ij s_i s_j
so interaction constants are J_ij=E_iE_j and Ei=+1,-1
s_i=+1,-1 randomly , they are spin variables.
the question is :
show that this system is equivalent to a ferromagnet, by redefining the spin variables.

The Attempt at a Solution

first of all i am not sure but J_ij should be less than zero. If i know the definiton of ferromagnetic system , i can change the spin variables into an appropriate form. but far now i couldn't find anyhing
So any tip or lead would be nice to start with.
thanks

 

[Jhero]

Thank you for your question. In order to show that the system described by the given Hamiltonian is equivalent to a ferromagnet, we need to first understand the properties of a ferromagnetic system. A ferromagnet is a material that exhibits spontaneous magnetization below a certain temperature, known as the Curie temperature. This means that at low temperatures, the individual magnetic moments of the particles in the material align in the same direction, resulting in a net magnetization.

Now, let's look at the given Hamiltonian: H=-\sum Jij si sj. Here, Jij represents the interaction between two spin variables si and sj. As you correctly pointed out, Jij should be less than zero for a ferromagnetic system. This is because in a ferromagnet, the interaction between neighboring spins is such that they tend to align with each other, resulting in a lower energy state. Therefore, we can redefine the spin variables as follows:

si = +1 (spin up)
si = -1 (spin down)

By doing this, we are essentially assigning a direction to the spin variable, which is analogous to the direction of the magnetic moment in a ferromagnet. Now, let's look at the interaction constants, Jij=EiEj. As mentioned earlier, in a ferromagnet, the interaction between neighboring spins is such that they tend to align with each other. This can be represented by the interaction constant being negative, i.e. Jij < 0.

Therefore, by redefining the spin variables and considering the interaction constants to be negative, we have shown that the given system is equivalent to a ferromagnet. This is because the Hamiltonian now represents a system where the individual spins tend to align with each other, resulting in a net magnetization, just like in a ferromagnet.

I hope this helps. Let me know if you have any further questions.