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It is true that the lattice structure will determine the symmetries of the band structure. However, this alone cannot determine the low energy excitations because those are determined by the number of mobile electrons in the unit cell, i.e. the Fermi energy (at least in the case of a metal). This is why elements in the periodic table tend to have similar ...


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There may be a few options for describing the crystalline order. Personally, I love the one used by Anderson in his book 'Basic Notions of Condensed Matter Physics'. Following him, one writes the atomic density as $\rho(\vec{r}) = \sum_{\vec{G}}\rho_{\vec{G}}e^{i\vec{G}\vec{r}}$. The appearance of Crystalline order is signified by a set of finite ...


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You need to define an appropriate "order parameter" for your system, one that takes into account the symmetries in the configuration as well (rotational, transnational, etc). There are many ways you could define such "correlator" as you call it, it depends on the system. For example the nematic order parameter in liquid crystals is taken with respect to a ...



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