A typical problem in physics is that physical quantities can be calculated for a series of finite systems, but ultimately, we would like to know the results for the infinite system. Hulthen (1938) studied the one-dimensional Spin Heisenberg model with the Hamiltonian
where is the spin at the site and is the total number of sites in the system. From the eigen equation
Hulthen obtained the ground-state energy per site, for a series of finite systems with
Now assume that the ground-state energy per site is given by
Truncate the above series at and find by solving the linear equation set numerically.
To find the ground-state energy per site, , for a series of finite systems, we can start by truncating the given series at some finite value, . Then, we need to solve the linear equation set numerically.
Here are the steps to find :
1. First, truncate the series at some finite value, . This means we will consider only the terms up to .
2. Write the truncated series for the ground-state energy per site as:
\[E_0(N) = E_0(\infty) + \sum_{n=1}^N a_n \left(\frac{e^{-\alpha_n}}{(\alpha_n)^2}\right)\]
In this equation, is the ground-state energy per site for the infinite system, and are coefficients that need to be determined.
3. Define a linear equation set using the values of and the coefficients :
\[E_0(N) = E_0(\infty) + A \mathbf{a}\]
where is a vector containing the coefficients and is the matrix of terms
\[A_{ij} = \left(\frac{e^{-\alpha_j}}{(\alpha_j)^2}\right)\]
4. Now, we need to solve the linear equation set numerically to find the values of the coefficients .
5. To solve the linear equation set numerically, we can use numerical methods like the Gaussian elimination method, LU decomposition, or matrix inversion.
6. Using the appropriate numerical method and the given values of , solve the linear equation set to find the coefficients .
7. Once we have the coefficients , we can substitute them back into the truncated series to find the value of the ground-state energy per site, .
By following these steps, we can find the approximate value of the ground-state energy per site, , for a series of finite systems by truncating the series and solving the linear equation set numerically.