Decimation of a triangular lattice

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In summary, we discussed a network of spins and its Hamiltonian, as well as a method for finding a recursion relation for the renormalised coupling. The matrix size for the transfer matrix approach would be 8x8, and we would use the partition function and a derivative to calculate the average value of the bond action.
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CAF123

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Homework Statement


Consider the network of spins shown below. The Hamiltonian is given by $$H = - \sum_{\langle i j k \rangle} [J \sigma_i \sigma_j \sigma_k + J_0]$$ with ##J,J_o \geq 0## and ##\langle i j k \rangle## denoting spin in the same triangle (the triangles under consideration are each highlighted by a looping arrow). Decimate over the crosses and find a recursion relation for the renormalised coupling ##J'##.

FdvHZ.png

Homework Equations



Given below:

The Attempt at a Solution


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I was going to proceed by the method of transfer matrix but then I am not sure what size my matrix would be in this case. The bond action is $$e^{w(\sigma, \sigma' \sigma'')} = e^{J_o} e^{J \sigma \sigma' \sigma''}$$ and I want to find a new (renormalised) bond action $$e^{w'(\mu \mu' \mu'')} = e^{J_o'} e^{J' \mu \mu' \mu''}$$

I think $$e^{w'(\mu \mu' \mu'')} = \sum_{\sigma \sigma' \sigma''} = e^{\mu \sigma \sigma'} e^{\sigma' \mu'' \sigma''} e^{\sigma'' \sigma \mu'}$$ if I understood the set up correctly. I would proceed by writing down the matrix representation of these quantities but I am not sure what size to make them. There are ##2 \times 2 \times 2## possible combinations of spins.

Thanks for any tips!
 
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Hello,

Thank you for sharing your thoughts on this problem. I think your approach using transfer matrix is a good one. However, I believe the size of your matrix would be 8x8, as there are 8 possible combinations of spins (since each spin can take on two values: up or down).

To find the recursion relation for the renormalised coupling ##J'##, you could use the transfer matrix to calculate the partition function and then take the derivative with respect to ##J'##. This would give you the average value of the bond action, which can be related to the renormalised coupling.

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

1. What is a triangular lattice?

A triangular lattice is a two-dimensional geometric structure made up of equilateral triangles arranged in a repeating pattern. It is often used to model the arrangement of atoms in a crystal lattice or the structure of a honeycomb.

2. How is a triangular lattice decimated?

Decimation of a triangular lattice refers to the process of reducing the size of the lattice by removing a certain percentage of its vertices. This can be done in various ways, such as randomly removing vertices or removing them in a specific pattern.

3. What are the applications of decimating a triangular lattice?

Decimation of a triangular lattice has many applications in materials science, physics, and computer science. It can be used to study the behavior of materials under stress, simulate the spread of diseases, or create efficient data structures for computer algorithms.

4. What effect does decimation have on the properties of a triangular lattice?

The properties of a triangular lattice, such as its density, rigidity, and conductivity, are affected by decimation. Removing vertices can alter the lattice's overall structure and change how forces and energy are distributed within it.

5. Are there any limitations to decimating a triangular lattice?

Decimation can be a useful tool, but it also has its limitations. It may not accurately represent real-world systems, and the results obtained from decimated lattices may not always be applicable to the original lattice. Additionally, the choice of decimation method can greatly influence the outcomes, so careful consideration is necessary.

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