- #1
Thor90
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Homework Statement
I am trying to solve the model analitically just for 2 sites to have a comparison between computational results.
The problem is my professor keeps saying that the result should be a singlet ground state and a triplet of excited states, but when I compute it explicitally I find four different eigenvalues. Where am I wrong?
Homework Equations
My Hamiltonian (for 2 sites) is ## H = \sigma_0^x \sigma_1^x + \lambda (\sigma_0^z + \sigma_1^z) ## where the lower index stands for a site index.
The Attempt at a Solution
I have written all my independent site states on the computational basis, that is
|00> = (0,0,0,1) , |10> = (0,0,1,0) , |01> = (0,1,0,0) , |11> = (1,0,0,0)
(I have written states as row vectors because it was simpler while they are actually column vectors on which H acts, and as usual 0 stands as a down spin on the respective site, while 1 as a spin up) in a way to reduce the problem to a simple algebric task, and then I used the matrix tensor product to evaluate esplicitally the sigma matrices product (in which a single sigma stands for a product of a sigma and an identity that acts on the other remaining site) and at the end I found my Hamiltonian operator in the form
##
\begin{bmatrix}
2\lambda & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & -2\lambda
\end{bmatrix}
##
which has autovalues ##(-1,1,-\sqrt{4\lambda^2+1},\sqrt{4\lambda^2+1})## that are not degenerate.
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