I 2D Ising Model Average Number of Domain Walls

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Thread starter thatboi
Start date Sep 20, 2024
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Classical Domain

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The discussion centers on the classical Ising model on a 2-D square lattice with no external field, focusing on the average internal energy per site at criticality, which is calculated to be u = -√2 when the coupling constant J is set to 1. The query involves determining the average number of domain walls per site, given that each domain wall contributes an energy of 2. The ground state energy is established as -2N, and the goal is to find the number of domain walls needed to raise the energy to -√2N. The derived equation, n_walls/N = (2 - √2)/2, is proposed as a solution to this energy transition. The logic and calculations presented appear to be sound and consistent with the principles of the Ising model.

Sep 20, 2024

thatboi

Hi all,
I'm trying to see if my question/logic makes sense. Suppose I have a classical Ising model on a 2-D Square lattice with ##N## sites and 0 external field. There is an exact formula for the average internal energy per site, and at criticality it turns out to be ##u = -\sqrt{2}## where I have set the coupling constant ##J=1##. From here, I'd like to know if it is possible to find the average number of domain walls per site. I know each domain wall contributes energy ##2##. Now suppose the system is in its ground state (say all spin-up). Then the ground state energy is ##-2N##. I want to put in domain walls until the system energy increases to ##-\sqrt{2}N##, so we just need to solve ##-2N + 2n_{\text{walls}} = -\sqrt{2}N## and I get ##\frac{n_{\text{walls}}}{N} = \frac{2-\sqrt{2}}{2}##. Does this make sense?

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