## Download Basic Aspects of the Quantum Theory of Solids : Order and by Daniel I. Khomskii PDF

By Daniel I. Khomskii

Offers the most strategies and crucial theoretical tools of the trendy quantum thought of solids for graduate scholars and researchers.

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**Example text**

4) determines phonon frequencies, in the anharmonic case depends on x). 49) ω dV d ln V which is called the Gr¨uneisen approximation; γ is the Gr¨uneisen constant (usually, in ordinary crystals, γ ∼ 1–2). The total free energy as a function of volume can then be written as 1 F (V ) = 2κ δV V 2 +T ln 2 sinh q ωq (V ) 2T . 50) is the elastic energy after deformation δV , and κ is the lattice compressibility (inverse bulk modulus). 50) we considered the situation when we (artificially) fix the volume of the system V , which may differ from the equilibrium volume without phonons by the distortion δV .

The examples given above correspond to broken continuous symmetry. But there are also cases of a broken discrete symmetry. g. the case of strongly anisotropic magnets. Suppose that spins can take not an arbitrary orientation, but only two: ↑ or ↓. Again, in the hightemperature disordered phase there is equal probability of finding spins ↑ or ↓ at a given site. g. to spins being predominantly ↑ (at T = 0, only ↑). This is the so-called Ising ferromagnet. Here a discrete symmetry (spin inversion ↑ ⇐⇒ ↓) is broken.

It is clear that the canting of neighbouring spins costs a certain energy (the usual exchange interaction in a ferromagnet preferring to keep neighbouring spins parallel). For slow variation the cost in energy should be proportional to the gradient of the order parameter, ∼dη/d r = ∇η. The only invariant containing ∇η (for scalar η) is (∇η)2 . 1) in the form = d 3 r A η2 (r) + B η4 (r) + G ∇η(r) 2 . 26) This is called a Ginzburg–Landau (GL) functional (sometimes also Ginzburg– Landau–Wilson (GLW) functional) – the functional of the function η(r).