By Daniel I. Khomskii

Aimed toward graduate scholars and researchers, this ebook covers the main points of the trendy quantum concept of solids, together with updated rules similar to quantum fluctuations and powerful electron correlations. It provides normally ideas of the trendy quantum concept of solids, in addition to a basic description of the fundamental theoretical equipment required whilst operating with those structures. varied issues comparable to common concept of part transitions, harmonic and anharmonic lattices, bose condensation and superfluidity, sleek features of magnetism together with resonating valence bonds, electrons in metals, and powerful electron correlations are handled utilizing unifying ideas of order and effortless excitations. the most theoretical instruments used to regard those difficulties are brought and defined in an easy means, and their purposes are verified via concrete examples.

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**Example text**

For noninteracting particles with the spectrum εk , the energy of the quantum state, in which there are nk particles, is Ek,nk = εk nk (we incorporated in the index k also spin indices, and other indices if they exist). 33), the thermodynamic potential for this quantum state is equal to k = −T ln e(μ−εk )/T nk . e. e. we necessarily have μ < εk (or μ < 0 if εk ∼ k 2 /2m), and after summation we obtain B k = T ln 1 − e(μ−εk )/T . 31) the average occupation of the state k is n¯ Bk = − ∂ Bk 1 . e.

Here I give only a few: in an isotropic ferromagnet these are spin waves, with the spectrum shown qualitatively in Fig. 16. In crystals these are the usual phonons, Fig. 17. We will meet many other such examples later in this book. 28 General theory of phase transitions w w q Fig. 16 q Fig. g. in a ferromagnet all directions of spontaneous magnetization are possible). All these states have the same energy and are equivalent. g. e. there exists gapless excitation. This is the content of the Goldstone theorem; the corresponding gapless modes are often called Goldstone modes.

In crystals these are the usual phonons, Fig. 17. We will meet many other such examples later in this book. 28 General theory of phase transitions w w q Fig. 16 q Fig. g. in a ferromagnet all directions of spontaneous magnetization are possible). All these states have the same energy and are equivalent. g. e. there exists gapless excitation. This is the content of the Goldstone theorem; the corresponding gapless modes are often called Goldstone modes. 5 Critical points In principle there may exist phase transitions without a change of symmetry, but only the first-order ones.