Searching the phenomenological renormalization group equation for 2d ising model

[quantikghost]

hellow everybody
i have a problem in styding the critical bihaviour of the tow dimensional ising model
when i use periodiques boundary conditions i found that the fixed point for this case is the PRG equation that mean the following recursion relation: (N-1) [E1(t,N-1)-Eo(t,N-1)]=N[E1(t,N)-Eo(t,N)] where E1 and E0 are first and ground state enrgie and N is the numbre of spins in the lattice
but how it would be the phenomenological renormalization group equation for the ising model if we impose the fixed or mixed boundary boundary condition "plus_plus" or "plus_free" boundary condition
thank you

 

[eljose79]

Hello,

Thank you for sharing your question about the critical behavior of the two dimensional Ising model. I am happy to offer some insights and suggestions.

Firstly, let's briefly discuss the Ising model and its critical behavior. The Ising model is a mathematical model that describes the behavior of a collection of binary variables, such as the spin states of particles in a lattice. It is commonly used in statistical mechanics to study phase transitions, which are sudden changes in the properties of a system at a critical point. The critical behavior of the Ising model refers to the behavior of the system at the critical point, which is characterized by the emergence of long-range correlations among the particles.

Now, let's address your question about the phenomenological renormalization group (PRG) equation for the Ising model with different boundary conditions. The PRG equation is a recursion relation that describes the renormalization of the coupling constants in a system as its length scale is reduced. In the case of the Ising model, the PRG equation you mentioned is derived for the case of periodic boundary conditions. This means that the lattice is wrapped around in a periodic manner, with the last spin connected to the first spin.

If we impose different boundary conditions, such as the fixed or mixed boundary conditions you mentioned, the PRG equation may change. This is because the boundary conditions affect the interactions between the spins at the edges of the lattice, which in turn affects the overall behavior of the system. To derive the PRG equation for the Ising model with different boundary conditions, one can use the same approach as for the periodic boundary conditions, but with different boundary conditions taken into account.

In conclusion, the PRG equation for the Ising model with different boundary conditions can be derived by considering the specific boundary conditions and their effects on the interactions between the spins. I hope this helps to clarify your question. Best of luck in your studies!

 

[Entropia]

I would like to first commend your efforts in studying the critical behavior of the two-dimensional Ising model. The Ising model is a fundamental model in statistical mechanics and has been extensively studied for its critical behavior.

Regarding your question about the phenomenological renormalization group equation for the Ising model, it is important to note that this equation is a powerful tool used to study the behavior of systems at their critical points. It is derived from the concept of scale invariance and allows us to understand the behavior of a system at different length scales.

In the case of the Ising model, the PRG equation describes the behavior of the system with periodic boundary conditions. This is a common choice in theoretical studies as it simplifies the calculations and allows for analytical solutions. However, as you have rightly pointed out, the choice of boundary conditions can have an impact on the behavior of the system.

If we impose different boundary conditions, such as fixed or mixed boundary conditions, the PRG equation may change accordingly. This is because the boundary conditions affect the symmetry of the system and can lead to different critical behavior. Therefore, it would be necessary to derive a separate PRG equation for each type of boundary condition to fully understand the critical behavior of the Ising model.

In conclusion, the phenomenological renormalization group equation is a powerful tool in studying the critical behavior of the two-dimensional Ising model. However, the choice of boundary conditions can have an impact on the behavior of the system and may require separate PRG equations for each type of boundary condition. I hope this helps in your continued studies of the Ising model.