Hamiltonian three-dimensional crystal

We now consider exchange interactions which extend throughout a three-dimensional crystal lattice. Obviously, the transition to this long-range order is a phase transition and as such it is characterized by a specific heat anomaly and peculiar behaviour of the magnetic susceptibility. As in section 1.1.3.7, the interaction may be correctly described by the Hamiltonian Eq. (49). However, there exists no exact solution to this Hamiltonian for a three-dimensional problem, although numerical calculations are available [163]. It is therefore customary to discuss the properties of three-dimensionally ordered substances on a classical basis. 

The medium considered by Holstein is a one-dimensional crystal that contains a single excess electron. The Hamiltonian of the system is composed of three... 

The electron Hamiltonian (15) describes the so-called orbital exchange coupling in a three-dimensional (3D) crystal lattice. The Pauli matrices, cr O ), have the same properties as the z-component spin operator with S = As a i) represents not a real spin but orbital motion of electrons, it is called pseudo spin. For the respective solid-state 3D-exchange problem, basic concepts and approximations were well developed in physics of magnetic phase transitions. The key approach is the mean-fleld approximation. Similar to (8), it is based on the assumption that fluctuations, s(i) = terms quadratic in s i) can be neglected. We do not go into details here because the respective solution is well-known and discussed in many basic texts of solid state physics (e.g., see [15]). 

When rotations of the coordinate frame in the full three-dimensional space are considered, other invariants appear. If denotes the collection of parameters AZf — k < q < k) the crystal field Hamiltonian for a single 4f electron can be written as F ). Although it may seem strange to construct a tensor out of nu-... 

We will discuss the case where the motion of heavy atoms is confined to two dimensions, while the motion of light atoms can be either two- or three-dimensional. It will be shown that the Hamiltonian 10.76 with Ues in 10.44 supports the first-order quantum gas-crystal transition at T = 0 [68], This phase transition resembles the one for the flux lattice melting in superconductors, where the flux lines are mapped onto a system of bosons interacting via a two-dimensional Yukawa potential [73]. In this case Monte Carlo studies [74,75] identified the first-order liquid-crystal transition at zero and finite temperatures. Aside from the difference in the interaction potentials, a distinguished feature of our system is related to its stability. The molecules can undergo collisional relaxation into deeply bound states, or form weakly bound trimers. Another subtle question is how dilute the system should be to enable the use of the binary approximation for the molecule-molecule interaction, leading to Equations 10.76 and 10.44. 

We start the discussion by formulating the Hamiltonian of the system and the equations of motion. The concept of force constants needs further examination before it can be applied in three dimensions. We shall discuss the restrictions on the atomic force constants which follow from infinitesimal translations of the whole crystal as well as from the translational symmetry of the crystal lattice. Next we introduce the dynamical matrix and the eigenvectors this will be a generalization of Sect.2.1.2. In Sect.3.3, we introduce the periodic boundary conditions and give examples of Brillouin zones for some important structures. In strict analogy to Sect.2.1.4, we then introduce normal coordinates which allow the transition to quantum mechanics. All the quantum mechanical results which have been discussed in Sect.2.2 also apply for the three-dimensional case and only a summary of the main results is therefore given. We then discuss the den-... 

In the Holstein model the molecular crystal is described as a regular one-dimensional (ID) array of diatomic molecules. The Hamiltonian of the system is a sum of three terms Hi, H, and H t. The lattice... 

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