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Like those of sand, quartz, or salt, those grains are very likely to be themselves crystals which as said do not imply they are perfect: they may contain lots of impurities and defects. But there are two particular aspects of crystals we are concerned with here. The first is that unlike complex systems, which may display emergent structures at each scale think, e. The second is that since the building blocks obey quantum mechanics, crystals inherit the quantum character despite being themselves macroscopic. As recent experiments have shown, whereas most interactions but gravity are effectively short-ranged, there is no limit for quantum correlations; and this fact makes them the most important fact to account for in modeling.

Quantum correlations manifest themselves in many ways, but the by far dominant one comes from the indistinguishability of identical particles.


Unless the crystal is a monolayer, the state vector of a system of many indistinguishable particles must be either totally symmetric or totally antisymmetric a determinant under exchange. In the first case, the particles obey Bose-Einstein statistics and are called bosons. In the second, the particles obey Fermi-Dirac statistics and are called fermions. The requirement that the state vector of a system with many fermions be totally antisymmetric is the celebrated exclusion principle , postulated by Pauli. At present, there is no question that atoms are distinguishable. They can even be individually manipulated.

Section 2 keeps within the framework of first quantization. It is assumed that neither electrons we mean crystal electrons, with effective masses nor holes can be either created or destroyed. The Fermi level is the chemical potential of such a gas. The exclusion principle can make it so high that for white dwarfs and neutron stars, the pressure it generates prevents the system from becoming a black hole.

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The main assumption of the tight-binding approach to band spectra is that atoms in a crystal interact only very weakly. However, neglecting almost all interaction terms and overlap integrals atomic states at different lattice sites need not be orthogonal to each other may be too drastic an approximation.

Thus Section 2 is devoted to a thorough discussion of the issue. Instead, the framework of Section 3 is that of second quantization. Again, our view of the crystal is that of tight-binding atoms do not lose their identities. But we deal with it in the style of quantum field theory, by allowing at most one electron of each spin projection per atom. The purpose of this section is to illustrate an efficient Monte Carlo scheme that implements this strategy to find the ground state of many-electron systems.

Section 4 explores the boundaries of the concept of solid.

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Perhaps, it should be regarded as a metaphor of this concept. We illustrate a non-equilibrium spatiotemporal pattern formation process, akin to resonant crystal structures, in arrays of FitzHugh-Nagumo cells.

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For the benefit of those readers who are unfamiliar with the standard formalism of quantum mechanics, we review its main facts: Dynamical states are vectors : one can account for the wavelike behavior of quantum objects e. In few problems e. But most problems entail infinite sequences e. Dynamical magnitudes are linear operators L , which take a vector into another vector.

Theoretical Solid State Physics, Volume 1: Perfect Lattices in Equilibrium

Unitary evolution : in order to conserve the probabilistic interpretation, the dynamic evolution of the state is accomplished by a unitary operator. Eigenstates corresponding to different eigenvalues are automatically orthogonal. It thus makes sense to write up the lattice Hamiltonian in terms of projection operators as.

The minus sign in the second term ensures crystal stability energy is released by forming a crystal. Using Eqs. This allows to rearrange the sums their indices become dummy , and Eq. What has been left behind? This assumption is correct in the absence of interatomic interaction, but not necessarily when atoms interact. The contribution of the S ij known as overlap integrals to the band spectrum is our main concern in this section.

Variation of Eq.

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Note however that the number of multicenter integrals to be computed is immense! Because of that, most tight-binding calculations plainly ignore almost all the multicenter integrals keeping only those involving nearest neighbors and neglect orbital non-orthogonality. This way, the familiar cosine spectrum is obtained. Often, multicenter integrals are just regarded as parameters to fit the results of more sophisticated calculations made by other methods at the highest symmetry points of the Brillouin zone.

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In the following, we compute all the multicenter integrals exactly in the framework of a simple model for the atomic potential. The results help get an intuition on the effect on band spectrum of neglecting overlap integrals and distant-neighbor interactions. We restrict ourselves to a 1D monoatomic crystal and assume the interatomic distance a to be larger than the effective range of the screened Coulomb potential representing the atomic core.

The only two spatial scales involved in this problem are x 0 and the lattice spacing a. All the multicenter integrals can be computed analytically in terms of t. Explicit evaluation of Eq. Moreover, the multicenter integrals neglected in the cosine spectrum shift unevenly the top and bottom of the exact spectrum. Hence, the approximation performs worse for the top than for the bottom of the band. The state vectors dealt with in Section 1 represent pure states. They are the ones which display the spectacular effects seen in recent experiments. Since in this section, we will allow creation annihilation of electron states, we must work in the framework of the grand canonical ensemble.

Then we may numerically compute E from. Optical Properties.

Polarons and Excitons. Defects and Disordered Systems. Thermal conductivity of superconductors. Polaron energy in continuum polarization model. Small polaron motion in molecular crystal model.