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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Additional info for Basic Aspects of the Quantum Theory of Solids : Order and Elementary Excitations
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).