Quasiparticle Operators Research Articles

Phenomenological Hamiltonian in the Theory of SuperfluidHe4

We show that a many-body Hamiltonian, expressed in terms of renormalized quasiparticle operators, has interactions only on the energy shell. Our result justifies the method of Havlin and Luban, who resolve the discrepancy between the theoretical and experimental velocity of sound in $^{4}\mathrm{He}$. It also follows that the phonon spectrum must be concave at low momenta.

On heat diffusion mode of structural phase transition by two-fluid model

With the help of the two-fluid model developed by Gotze and Michel for phonons it is shown for a simple model Hamiltonian that in the low temperature phase the optical soft mode becomes isothermal, the heat diffusion mode is dominant near the transition temperatureTc and the quasiparticle interaction is of great importance in determining the thermodynamic quantities nearTc. Green function techniques are applied to describe the two-fluid model functions in a microscopic way. The simplest approximations are discussed for the model equations describing nonequilibrium phenomena of the soft optical phonon mode in the low temperature phase. The quasiparticle interaction operator can be related to the interaction operator between quasiparticles and the condensed mode. This relation enables one to understand the behaviour of the thermodynamic quantities near the transition temperature on a microscopic way. The first order displacive phase transition is also discussed.

The Bogoliubov-Valatin quasiparticles in a finite superconductor

The strong-coupling limit of the BCS model of a finite superconductor is analyzed mathematically and the exact Bogoliubov-Valatin quasiparticle operators are found. These quasiparticle operators are “number conserving.” The coefficientsu k ,v k and the quasiparticle energy are operators rather than constants, and they depend on the electron number density and the energy density. In the limit of large volume these densities becomec numbers in an irreducible representation, and the quasiparticles are described by the usual Bogoliubov-Valatin transformation.

The boson method and the antisymmetrization problem

Abstract The transcription of the quasiparticle in the ideal space is studied for systems containing up to five quasiparticles. A way is found to introduce properly the Pauli exclusion principle and the states are completely antisymmetrized. The quasiparticle operators and their products appearing in the Hamiltonian are also transcribed in that space. The matrix elements of the products of quasiparticle operators evaluated in the ideal space are exactly the same as those evaluated in the quasiparticle space. This enables one to determine many anharmonic corrections.

Quasi-particles in atomic shell theory

A new scheme is described for defining and classifying the states of the electronic configurations l N . The spaces for which the spin orientation is either up or down are both factored into two parts. Each of these parts (distinguished by a symbol Ɵ) corresponds to the irreducible representatio n (½ ½ ... ½ ) of the rotation group R Ɵ (2 l +1). The generators for this group are constructed from quasi-particle creation and annihilation operators. The angular momentum quantum numbers l Ɵ arising from the decomposition of (½ ½ ... ½) into representations of R Ɵ (3) can be used to couple the four parts together. No ambiguities arise when l < 9, thereby giving a very satisfactory coupling scheme. No coefficients of fractional parentage (c. f. p.) are required in the calculation of matrix elements. Simple explanations are given for some null c. f. p. and for some repeated eigenvalues of an operator that had previously been used to classify the state s of g N .

Principle of Compensation of Dangerous Diagrams for Boson Systems. II. Minimum Quasiparticle Number

The principle of compensation of dangerous diagrams (PCDD) postulated by Bogoliubov to determine the coefficients in his canonical transformation in boson systems is obtained from four criteria: (1) the number of quasiparticles in the true ground state is a minimum; (2) the ``best'' approximation to the true density matrix and pair amplitude is made; (3) the expectation value of an arbitrary operator is diagonalized up to terms cubic in the quasiparticle operators; (4) the most convenient starting point for dressing the quasiparticles is used. Quasiparticle Green's functions are introduced to obtain a diagrammatic expansion of the PCDD. The reducibility of the diagrams is also discussed.

Electron-Phonon Interaction in Ionized Impurity-Pairs in Silicon

A microscopic theory is developed to treat the adiabatic as well as the non-adiabatic interaction of a carrier trapped by a pair of impurities in Si with the lattice vibrations. Two complementary methods, which are called the electron (the adiabatic) and the polaron (the non-adiabatic) approaches, are proposed and formulated with the use of the canonical transformations known in the theories of small polarons. The Hamiltonian is rewritten in terms of the quasi-particle ( i. e. , “electron” and “polaron”) operators to obtain the zeroth-order energies. The residual interactions are treated as perturbations which shift and broaden the levels with the use of the thermal Green's functions. Approximate “single-phonon” Hamiltonians are derived from the exact “multi-phonon” Hamiltonians to clarify the physical natures of the problems involved since multi-phonon processes are not important in pairs in Si. The adiabatic criterion is discussed with the use of these single-phonon Hamiltonians. Completely homopolar pa...

Symmetry breaking quasi-particle method for baryons and for quarks I. Bound state problem

The quasi-particle method (essentially a non-perturbational approximation of quantum field theory) is applied to an SU 3 symmetric four-fermion interaction between octet-baryons N and antibaryons N (of between quarks and antiquarks). The method is developed in such a way, that the physical particles and resonances themselves appear as quasi-particles in a bound state approximation. The main weight is attached to the understanding of the principles of the method. The individual quasi-particle operators obtained are of the form Ω = u N + ν N ∗ . Their eigenvalue spectrum is compared with the observed baryon resonances. Spontaneous dynamical breaking of SU 3 symmetry appears and leads primarily to a GMO formula for the gap constants, irrespective of the coupling strength. (Similar results are obtained for quarks.) The GMO formula for the baryon masses is obtained in the strong coupling limit. (Furthermore mass formulae for quarks are derived.) Further it is shown in detail that the origin of the symmetry breaking is the fact that the quasi-particle method is an averaging approximation. Quasi-particles of higher order are considered in a preliminary way. Especially the meson spectra are treated in the lowest approximation. It is shown, that the first two of the three typical difficulties of the quasi-particle approximation [(i) inequivalent representations and degenerate vacua, (ii) parity mixed vacuum and (iii) spurious states ≡ Goldstone mesons) are avoided by our bound state approximation. The method is applicable to all compact symmetry groups.

Quasi-particle excitations and the many-boson problem

A many-boson theory is developed that is applicable to systems with intermediate as well as weak coupling strength. A transformation of basis from the single particle representation to that of an independent collection of quasi-particles, which represent the normal modes of the system, is accomplished in an approximate way. The results are in the form of coupled, nonlinear, integral equations for the ground state energy, the excitation energy, the quasi-particle excitation operator, and the momentum distribution function. No a priori assumptions concerning the population in any single low-lying momentum state is required nor is any artificial separation of the single particle zero-momentum state from states of higher momentum necessary. Contact with the results of weak-coupling theories is established by imposing additional restrictions on the interaction potential and on the momentum distribution of particles. It is shown that the weak-coupling equations are valid under more general circumstances than was previously apparent. A general theorem is provided for which the existence of a zero in the excitation spectrum at zero-momentum is required. It is also shown that the low-lying excitations follow a dispersion rule that is linear in momentum, independent of the form of the interaction. The theory is applied to the charged boson gas and, for the case where the zero-momentum state is appreciably depleted, the resulting excitation spectrum is quite different at low momentum from that derived in previous calculations.

Quasi-particle operators for the condensed Bose system

The concept of quasi-particle operators is used for the condensedBose system. The operators depend on the relative motion of the superfluid and the normal component.

On the perturbation-theoretic calculation of the one-particle excitation spectrum in a large Bose system

Perturbation theoretic calculation of the one-particle Green’s functions is studied for a large system of bosons at zero temperature. We start from a splitting of the Hamiltonian in which the unperturbed part is chosen to be diagonal in theBogolyubov quasi particle operators. The self-consistent form of the perturbation expansion required by the appearance of inequivalent representations of canonical commutation relations is formulated and discussed.

Pseudospin Model for Hard-Core Bosons with Attractive Interaction. Zero Temperature

We consider the many-body ground state and elementary excitations of the Matsubara-Matsuda cell model, which can be interpreted as a first approximation to Siegert's exact but untractable operator algebra for hard-core boson fields. The model has unphysical aspects, but allows treatment of some many-body effects of the interaction. It represents the hard core by restricting cell occupation numbers to \ensuremath{\le}1, which changes the algebra from Bose to Pauli type; kinetic energy and attraction appear as interactions between nearest-neighbor cell pseudospins, equivalent to a Heisenberg ferromagnet with anisotropy in pseudospin space. Our treatment, valid for scattering length ${f}_{0}\ensuremath{\ge}0$, splits the Hamiltonian into an isotropic part ${H}_{0}$, describing a system with hard core plus attraction of strength making ${f}_{0}=0$, and $A{H}_{1}$, consisting of the attraction's deviation from this strength, the parameter $A\ensuremath{\ge}0$ giving the magnitude of this (repulsive) deviation. The exact ground state of ${H}_{0}$, for any density, is the state symmetric in all pseudospins, with the appropriate eigenvalue of the total pseudospin component ${S}^{(3)}$ which measures $N$. These states exhibit what corresponds to incomplete Bose-Einstein condensation, the excluded volume effect of the hard-core constraint producing relative depletion ${\ensuremath{\xi}}_{0}$ of the condensate, proportional to $\ensuremath{\rho}$. Exact single excitations of ${H}_{0}$ are density fluctuations ${\ensuremath{\rho}}_{\mathrm{k}}$ with, however, free particle-like excitation spectrum, the ground-state energy being density-independent. Relaxing the restriction to eigenstates of $N$ permits definition, by rotation of the total pseudospin from the vacuum, of a quasiparticle vacuum and operators, for any mean density. These serve as starting points for treating the full Hamiltonian by the equations-of-motion method in the random-phase approximation. The excitation spectrum is now phonon-like for small $k$, with $s\ensuremath{\sim}{(\ensuremath{\rho}A)}^{\frac{1}{2}}$, ${f}_{0}$ being expressible exactly in terms of $A$, $\frac{{E}_{0}}{N}$ can be written in terms of ${f}_{0}$, $\ensuremath{\rho}$, and ${\ensuremath{\xi}}_{0}$, self-consistent in the RPA to order ${{f}_{0}}^{\frac{5}{2}}$. In the low-density limit, ${\ensuremath{\xi}}_{0}\ensuremath{\rightarrow}0$, there results the well-known expansion in ($\ensuremath{\rho}{{f}_{0}}^{3}$), but there are higher density corrections including a term $\ensuremath{\sim}\ensuremath{\rho}{f}_{0}{(\ensuremath{\rho}{{f}_{0}}^{3})}^{\frac{1}{3}}{{\ensuremath{\xi}}_{0}}^{\frac{2}{3}}$, due to the strong interaction included in the unperturbed many-body ground state.