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# Ferromagnet problem

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Ferromagnet

https://brainmass.com/physics/energy/ferromagnet-problem-123589

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I have not followed the Hint in question 3 as I think it is too cumbersome, and instead I offer a solution I like better for its simplicity.

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Here is the plain TEX source

centerline{bf Spin \$1over 2\$ Ferromagnet }

bf 1. rm

The dimension of the Hilbert space at each site is \$2cdot{1over 2} +1 = 2\$, therefore the dimension of the Hilbert space for the entire system is \$2^N\$.

bf 2. rm

The Hamiltonian contains only the pair interactions
\$\$
vecsigma_acdotvecsigma_b = 4 vec S_acdotvec S_b
= 2l[ (vec S_a + vec S_b)^2 - vec S_a^2 - vec S_b^2 r]
= 2l[ vec S_{ab}^2 - vec S_a^2 - vec S_b^2 r] =
\$\$
\$\$
= 2l[ vec S_{ab}^2 - S_a(S_a+1) - S_b(S_b+1) r]
= 2l[ vec S_{ab}^2 - {3over 4} - {3over 4} r]
= 2vec S_{ab}^2 - 3,
eqno(2.1)
\$\$
where \$vec S_{ab} = vec S_a + vec S_b\$.
For every such pair \$a,b\$, the state \$|+_a,+_bra\$ belongs to the \$S_{ab}=1\$ triplet, so that equation (2.1) gives
\$\$
vecsigma_acdotvecsigma_b = 2S_{ab}(S_{ab}+1) - 3
= 1.
eqno(2.2)
\$\$
Taking the full Hamiltonian and the full state \$|+_1,+_2,...,+_Nra\$, we see that the full state is an eigenstate of every pair term with eigenvalue \$-J\$, and therefore an eigenstate of the whole Hamiltonian with eigenvalue
\$\$
E_g = -Jsum_{j=1}^N 1 = ...

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