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International Journal of Modern Physics B

QUASI-CLASSICAL SELF-CONSISTENT THEORY OF ANHARMONIC CRYSTALS

V.P. Zubov

Published Feb 1, 1992• 16 citations
Publisher

Abstract

Using the unsymmetrized self-consistent field method the quasi-classical expansion of the one-particle density matrices is considered. The integral equations for the first quantum corrections to the classical one-particle probability densities are derived. The exact solution for the perfect crystals with the strong anharmonicity of the fourth order is found and used to calculate the first quantum correction to the effective amplitudes of the strongly anharmonic vibrations of atoms. The solution is obtained also for the semi-infinite linear chain as the simplest model of the anharmonic crystal with the surface.

DOI
10.1142/s0217979292000189

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Using the unsymmetrized self-consistent field method the quasi-classical expansion of the one-particle density matrices is considered. The integral equations for the first quantum corrections to the classical one-particle probability densities are derived. The exact solution for the perfect crystals with the strong anharmonicity of the fourth order is found and used to calculate the first quantum correction to the effective amplitudes of the strongly anharmonic vibrations of atoms. The solution is obtained also for the semi-infinite linear chain as the simplest model of the anharmonic crystal with the surface.
Numerous examples in condensed matter physics possess an effective Hamiltonian structure involving two-body anharmonic interactions between elementary excitations. The effective second-quantised Hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting. The author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking. He introduces quantum field operators and calculates their equations of motion. The interaction coefficients are expanded in a Taylor series about the critical mode to second order to illustrate the method, but he gives the expansion to infinite order for completeness. It is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations (PDE) for the field operator. He shows that the form of the second-order equation is exact due to renormalisation theory, in the case where the Hamiltonian coefficients represent operators in real space which are isotropic, and incorporates symmetries and topologies which would appear in an expansion to infinite order. This special situation arises in a very wide range of physical examples. These equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non-linear analysis. In particular, the method of symmetry reduction for PDE provides a complete set of symmetry variables for the type of equations used in the study.
Irreversible phenomena involving phonons in a slightly anharmonic crystal are investigated using coherent states as a basis. This basis retains the advantages of the classical theory, in which phase relations are clearly exhibited, for a quantum-mechanical system. In the coherent-state basis the equation of motion for the density matrix has an obvious correspondence with the classical Liouville equation. In particular, the connection with the action-angle variables of Brout and Prigogine is elucidated. A new, rapid proof of the Brout-Prigogine equation is given using the method of semi-invariants. This technique exhibits higher-order corrections in a usable form. The quantum corrections to the classical equation of motion for the density function are shown to be due to extra terms involving second derivatives in the action-angle variables. The Peierls master equation is then derived from the Brout-Prigogine equation. The important problem of elastic scattering of phonons is treated by our method in an appendix.
The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same setting, in which thermodynamic phases of the model appear as probability measures on path spaces. Then the possibility of a phase transition to occur is related to the existence of multiple phases at the same values of the relevant parameters. Other definitions of phase transitions, based on the nondifferentiability of the free energy density and on the appearance of ordering, are also discussed.
A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.
The theory of solution for quantum field functional equations as developped in II and III for a suitable test problem of quantum mechanics is investigated in low approximations. In Sect. 1 the functional formulation of the anharmonic oscillator is once more given and in Sect. 2 general translational equivalent functional equations. The expansion of the physical state functional into series of unsymmetrical and symmetrical base functionals and the representation of the functional equations for such expansions are discussed in Sect. 3. In the next Sect. 4 the unsymmetrical DYSON representation is investigated and the explicit representation of the smeared out functional equation by an infinite system of equations is derived. Then in Sect. 5 and 6 the system of equations is truncated for N = 3 and the corresponding eigenvalue equation is considered. The same is done in Sect. 7 and 8 for the HERWITTE representation. In the following Sect. 9 the original functional equation in a not smeared out form is treated in the DYSON representation and the corresponding system of unsymmetrized equations is given. Furthermore in Sect. 10 the N = 3 approximation together with other possibilities is investigated again. Finally the numerical results of our calculations for eigenvalues are stated and discussed. In the appendices technical details are derived.