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arXiv:quant-ph/0406095 · Physical Review B

Interacting fermions and bosons with definite total momentum

Ofir E. Alon, Alexej I. Streltsov, L. S. Cederbaum

Published Mar 17, 2005• 8 citations
arXivHTMLPDF

Abstract

Any exact eigenstate with a definite momentum of a many-body Hamiltonian can be written as an integral over a symmetry-broken function $\ensuremath{\Phi}$. For two particles, we exactly express $\ensuremath{\Phi}$ in terms of (single-particle) orbitals for all energy levels and any interparticle interaction. Especially for the ground state, $\ensuremath{\Phi}$ is given by the simple Hartree-Fock and Hartree Ans\"atze for fermions and bosons, respectively. Implications for several and many particles as well as a numerical example for interacting bosons are provided.

DOI
10.1103/physrevb.71.125113

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Any exact eigenstate with a definite momentum of a many-body Hamiltonian can be written as an integral over a symmetry-broken function \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ. For two particles, we exactly express \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ in terms of (single-particle) orbitals for all energy levels and any interparticle interaction. Especially for the ground state, \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ is given by the simple Hartree-Fock and Hartree Ans"atze for fermions and bosons, respectively. Implications for several and many particles as well as a numerical example for interacting bosons are provided.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
A system of NNN antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
In treating a system of NNN antisymmetric particles, it is shown that, if the total Hamiltonian Hop{\mathcal{H}}_{\mathrm{op}}Hop​ is degenerate, the eigenstates of the operator used for classifying the corresponding degenerate states may be selected by means of a "projection operator" O\mathcal{O}O. If the total wave function is approximated by such a projection of a single determinant, the description of the system may be reduced to the ordinary Hartree-Fock scheme treating this determinant, if the original Hamiltonian is replaced by a complete Hamiltonian ${\ensuremath{\Omega}}_{\mathrm{op}}={\mathcal{O}}^{\ifmmode\dagger\else\textdagger\fi{}}{\mathcal{H}}_{\mathrm{op}}\mathcal{O}$ containing also many-particle interactions. This approach corresponds to a "fixed" configurational interaction, but the scheme has preserved the physical simplicity and visuality of the Hartree-Fock approximation. The idea of "doubly filled" orbitals is abandoned, and the orbitals associated with different spins will automatically try to arrange themselves in such a way that particles having antiparallel spins will tend to avoid each other due to their mutual repulsion.
It is shown that the two component Fermi or Bose many-body Hamiltonian, such as the two-orbit fermion pairing and two-site Bose-Hubbard model with arbitrary finite higher order terms can always be solved exactly by using Bethe ansatz vector construction based on the permutation of two components of bosons or fermions involved. As examples of the solution, the extended one-dimensional dimer Bose -Hubbard model with multi-body interactions and the mean-fifield plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits are demonstrated with the eigenstates and the eigen-energy and the related Bethe ansatz equations. It is shown that the main feature of the solutions lies in the fact that the Bethe ansatz vectors can be expressed in terms of binomials of the boson or fermion operators times the related symmetric functions. As the consequence, two-component quantum many-body systems , such as the extended Lipkin-Meshkov-Glick model with higher-order interactions, can be solved in a similar way.
We describe briefly a new method of approach to the interacting fermion and boson systems. Namely, we determine the explicit form of the single-particle wave functions {wi_{i}i​(r)} appearing in the microscopic parameters of models in the second-quantization representation. The method is illustrated on the examples of H2_{2}2​ molecule and He atom, the Hubbard chain, and a ring of N ≤\leq≤ 10 atoms.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.