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Physical Review

Quasiparticle-Conserving Hamiltonian for Interacting Electrons and Phonons

C. Alton Coulter, D. W. Howgate, R. A. Shatas

Published Dec 13, 1965• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.140.a2000

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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.
Abstract. 1-dimensional hamiltonian with ionization energy (ξ) is shown to be exactly the same with the total energy from the standard harmonic oscillator hamiltonian, with the harmonic oscillator potential, mω 2 x 2 /2. Commutation relations for the ladder operators, energy at the n th state and the expectation value for the potential are analyzed and shown to be consistent with the principle of ξ. We also derive 1-dimensional wave functions for Dirac delta (−αδ(x)) potential and with V (x) = 0. Formal justifications are given as to why the total energy in ξ based Fermi-Dirac statistics is exactly the same with the standard Fermi-Dirac statistics. Energy levels and Bohr radius of a Hydrogen atom are also shown to be consistent with the ionization energy concept. This concept is also applied in a free Mn atom that represent a free many-electron atom. 1.
The phonon is discussed as an example of a collective mode which restores a symmetry property (in this case, translational invariance) to a system whose Hamiltonian is invariant under the symmetry operation, but in whose ground state the symmetry is broken. The crystal lattice is first studied within the time-independent Hartree approximation. It is then shown that allowing small time-dependent changes in the Hartree field generates an equation for the normal modes of vibration. The three k=0 modes with $\ensuremath{\omega}=0$ are shown to represent uniform translations of the solid, as expected. The $\mathbf{k}\ensuremath{\ne}0$ modes are analyzed to extract those three which may be identified as one-phonon modes, and the contribution of these modes to the free energy is computed. The recent theory of Brout is found to be equivalent to the Hartree approximation with the further assumption that the atoms are infinitely heavy. No restrictions are made here on the interatomic potential other than that a hard core is absent. The reduction of the present theory to well-known results in the case of harmonic forces is demonstrated. An extension to include hard cores, analogous to the Brueckner theory, is discussed.
The quasiparticle Fock-space coupled-cluster (QFSCC) theory, introduced by us in 1985, is described. This is a theory of many-electron systems which uses the second-quantisation formalism based on the algebraic approximation: one chooses a finite spin-orbital basis, and builds a fermionic Fock space to represent all possible antisymmetric electronic states of a given system. The algebraic machinery is provided by the algebra of linear operators acting in the Fock space, generated by the fermion (creation and annihilation) operators. The Fock-space Hamiltonian operator then determines the system's stationary states and their energies. Within the QFSCC theory, the Fock space and its operator algebra are subject to a unitary transformation which effectively changes electrons into some fermionic quasiparticles. A generalisation of the coupled-cluster method is achieved by enforcing the principle of quasiparticle-number conservation. The emerging quasiparticle model of many-electron systems offers useful physical insights and computational effectiveness. The QFSCC theory requires a substantial reformulation of the traditional second-quantisation language, by making full use of the algebraic properties of the Fock space and its operator algebra. In particular, the role of operators not conserving the number of electrons (or quasiparticles) is identified.
A field theoretical formulation of the rather easy problem of N identical, one-dimensional bosons, interacting pairwise via harmonic oscillator potentials of equal force constants is given, in a way similar to the way the nonlinear Schrodinger model provides a field theoretical formulation of a one-dimensional, delta -function Bose gas. The nonlinear integro-differential equation obtained for the complex scalar field of the model is treated classically, and solved in the semiclassical limit of a large number of particles. Working in the centre of mass frame of the field, a sequence of operators that would raise or lower the energy of a state by definite amounts without changing its particle number are constructed from certain bilinear expressions in the field. These operators have, in the limit of a large number of particles, commutation relations similar to those of the Fourier modes of a compact free field, except that the first mode is missing as a result of the fact that the field in the centre of mass frame is subject to the obvious constraint that its centre of mass always coincides with the coordinate origin. The quantum states can be labelled by the total centre of mass momentum, and the quasi-particle numbers in each of the independent oscillator modes. The energy spectrum of the system is calculated, giving the same results as the exact, quantum-mechanical N-body analysis.
We show the use of the theory of Lie algebras, especially their oscillator realizations, in the context of quantum mechanics. One can construct oscillator realizations from matrix realizations. In the case of symplectic and special orthogonal algebra, we demonstrate an alternative method of obtaining oscillator realizations from symmetric or exterior power of a vector space of annihilation and creation bosonic or fermionic operators. We find Lie algebra of polynomials of degree at most two in phase space of a mechanical system, which form the semi-direct product of the Heisenberg algebra and symplectic algebra. It is shown that a classical system with Hamiltonian function in this algebra can be quantized by two equivalent representations - Schrödinger or Bargmann-Fock representation. The second mentioned representation generates the same operators of symplectic algebra as we got from their previous formal construction from symmetric power of a vector space of bosonic operators. Quantization is demonstrated on the bosonic harmonic oscillator. We use the similarities between bosonic and fermionic oscillator realizations to define the fermionic harmonic oscillator. Some properties of spinor representations of special orthogonal algebra are illustrated on its state space. Powered by TCPDF (www.tcpdf.org)