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"Total spin conservation" means global $SU(2)$ spin-rotation symmetry (a continuous symmetry) of the Heisenberg model, and "spin wave" indicates an ordered ground state that spontaneously breaks the spin-rotation symmetry. Thus, according to Goldstone theorem, there must be a gapless mode for spin wave.

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Suppose you have a signal that varies sinusoidally in time. We can write $f(t) = \sin(\omega\, t)$, where $\omega$ is the angular frequency of the signal. Similarly we can have a wave that is moving through space, $f(x) = \sin( k\,x)$. Here the value $k$ tells us how quickly the signal moves from one maximum to the next when $x$ changes -- the so-called ...

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The material may become opaque for radiation that has frequency close to resonance frequencies of the material. The electrons in matter are sensitive to radiation in certain ranges of frequencies (absorption peaks or bands) and can get excited. This is accompanied by stronger absorption of the radiation. The electrons are sensitive to certain resonance ...

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Let $I \sim \sum_{\vec R} e^{i\left(\vec{k'}-\vec{k}\right)\vec R} \int_{V_{UC}} d^3r \Psi^*_{n\vec{k}}\left(\vec r\right) \Psi_{n'\vec{k'}}\left(\vec r\right)$ The term $\sum_{\vec R} e^{i\left(\vec{k'}-\vec{k}\right)\vec R}$ gives you a $\sim \delta(\vec{k} - \vec{k'})$ term. Now, you have : $\Psi^*_{n\vec{k}}\left(\vec r\right) ... 1 In a real-life misshapen blob of metal, strictly speaking the cyclic boundary conditions cannot be applied, since the blob only has a trivial group of spatial symmetries. However, the blob is approximately invariant under lattice translations (with the only mismatch occurring with the extremely small number of atoms at the surface) so it is tempting to ... 1 In a plasma it is hard to specify the state as solid, liquid, or gas but a good way to think about it is to ask what is driving the interactions in the plasma. Literature often cites the plasma coupling parameter$\Gamma=Z^2 e^2/(k_b T a)$where a is proportional to$(number \ density)^{-1/3}\$, T is the temperature, and Z is the charge number. This can ...

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