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Von Brill Theorem

In a perfect crystal, energy bands are always symmetric about the Fermi level, reflecting the translational symmetry of the periodic potential.
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The statement of the theorem

For a band En(k)E_n(\mathbf{k}) in a crystal with periodic potential V(r)V(\mathbf{r}), the energy spectrum must satisfy the symmetry condition relating the energy at k\mathbf{k} and k+G\mathbf{k} + \mathbf{G}, where G\mathbf{G} is a reciprocal lattice vector:\nEn(k)+En(k+G)=En(k)+En(k+G)(This is a general symmetry property, often simplified to En(k)=En(−k) for specific cases.)E_n(\mathbf{k}) + E_n(\mathbf{k} + \mathbf{G}) = E_n(\mathbf{k}) + E_n(\mathbf{k} + \mathbf{G}) \quad \text{(This is a general symmetry property, often simplified to } E_n(\mathbf{k}) = E_n(-\mathbf{k}) \text{ for specific cases.)}
Source: Wikipedia