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Russian Physics Journal

Many-phonon processes and the interaction of spinless quasiparticles with lattice vibrations at finite temperature

N. V. Tkach, N. V. Lopushanskaya, A. D. Val'

Published Oct 1, 1986• 0 citations
Publisher
DOI
10.1007/bf00900755

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The Hamiltonian most often used to describe a system of electrons interacting with lattice vibrations has the asymmetry that the electrons are represented by the creation and annihilation operators which arise from quantum field theory whereas the lattice vibrations are represented by the one-particle ladder operators of the one-dimensional harmonic-oscillator problem. Although the equal spacing of the harmonic-oscillator energy levels and the commutation relations satisfied by the oscillator ladder operators have led to an interpretation of this formalism in terms of the "creation and annihilation" of phonons, the phonons thus defined are not the quasiparticles which are obtained by a field-theoretical formulation of the lattice Hamiltonian in the harmonic approximation. A Hamiltonian is derived in this paper in which both the electrons and the lattice (in the harmonic approximation) are represented in field-theoretical formalism. A lattice creation operator in this formulation actually creates a vibrational mode (in a given state) from the physical vacuum, in contrast to a ladder operator which merely generates transitions between two states of an already existing vibrational mode. As a consequence, the number operator for the lattice quasiparticles commutes with the Hamiltonian, a property which is useful in certain applications. It is shown that even though our Hamiltonian is formally different from the ordinary one, it yields equivalent results for the state vectors of physical interest.
The electron-lattice interaction term of the Bloch-FrÖhlich Hamiltonian, written in the occupation number representation, is bilinear in the electron creation and annihilation operators but linear in the phonon raising and lowering operators. This asymmetry leads to the non-commutability of the phonon number operator and the interaction term of the Hamiltonian. It is shown that by introducing a bilinear form for each phonon operator the commutability of these two terms can be achieved. Formally, the new lattice operators arise from the quantum field-theoretical treatment of the normal modes. Specifically, in the harmonic approximation and coordinate representation of the lattice Hamiltonian in phonon space, this is achieved by the replacement of wave functions by a coordinate-dependent operator field. One obtains then such a field for every normal mode of the lattice. The operator field expansion coefficients are the lattice vibrational mode creation and annihilation operators which appear only in a bilinear combination throughout the theory. To obtain an equivalence relation between the field-theoretical and wave-mechanical formulations, one requires that only one lattice quasiparticle per lattice vibrational mode be allowed. For this case, the relation between the lattice field operators and the conventional phonon operators assumes a simple form. Finally, the entire Bloch-FrÖhlich Hamiltonian is rewritten in terms of the new set of lattice operators and the conservation of the new lattice quasiparticles is demonstrated. Thus, both the number of electrons and the number of lattice quasiparticles are shown to be constants of motion of an interacting electron-lattice system.
We use determinant quantum Monte Carlo to study the single-particle properties of quasiparticles and phonons in a variant of the two-dimensional Holstein model that includes an additional nonlinear electron-phonon (e-ph) interaction. We find that a small positive nonlinear interaction reduces the effective coupling between the electrons and the lattice, suppresses charge-density-wave (CDW) correlations, and hardens the effective phonon frequency. Conversely, a small negative nonlinear interaction can enhance the e-ph coupling resulting in heavier quasiparticles, an increased tendency towards a CDW phase at all fillings, and a softened phonon frequency. An effective linear model with a renormalized interaction strength and phonon frequency can qualitatively capture this physics; however, the quantitative effects of the nonlinearity on both the electronic and phononic degrees of freedom cannot be captured by such a model. These results are significant for typical nonlinear coupling strengths found in real materials, indicating that nonlinearity can have an important influence on the physics of many e-ph coupled systems.