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Progress of Theoretical Physics

Excitation-Modes and Their Interactions in a Many-Boson System. II

Shuichiro Yamasaki, T. Kebukawa, Sigenobu Sunakawa

Published Apr 1, 1979• 1 citation
Publisher

Abstract

By making use of the expressions for ρ-k and gk obtained in the preceding paper, the interaction Hamiltonian among the excitations described by the operators bl and bl* is derived up to the order of N-3/2. It is shown that all interaction terms have convergent forms and they always give finite corrections to all physical quantities.
DOI
10.1143/ptp.61.1012

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Abstract The dynamical evolution of open systems can be conveniently expressed in the operator sum representation in terms of Kraus operators. Here we introduce an expression for the canonical Kraus representation in terms of the eigenstates |γ〉 and eigenvalues Eγ of the total Hamiltonian, thus making a stronger connection to the tools in chemistry. The method is exact, and hence valid for non‐Markovian evolution of the system to all orders of the coupling to the bath. Significantly, it provides a starting point for developing approximate Kraus operators based upon perturbative approximations to |γ〉 and Eγ. As an example, the explicit form of these operators is derived for a d‐level system interacting with a single‐mode thermal bath.
The formalism of the preceding paper is applied to a system of uniform flow. Explicit expressions are obtained for the specific heat: C V , the superfluid-density tensor: ρ, and the spin susceptibility of the singlet pairing: χ. It is the quasiparticle excitation near the excitation threshold that is responsible for these quantities. The three expressions for v →0 reduce to the normal-state results of Luttinger; the Leggett expressions of ρ and χ are reproduced by neglecting the feedback effects.
1) Equation (2·8) should read к×vк=0. 2) The summand in the first term ori the right-hand side of Eq. (2·9) should be supplemented by a factor δ0,p+q. 3) Every summation symbol Σ in Eqs. (3·7) and (3·14) should be replaced by Σ′. 4) The inner product (p·s) in the expression for FII(3, 11) (p, q, r; s) in Eq. (3·25) should be followed by a factor √λqλrλs/λp in place of √λpλrλs/λp. 6) On the right-hand side of (4·22), the factor 2 should be dropped from the second term in the square bracket [⋯]. 7) The denominator (Zλ0)2 on the right-hand side of (4·23) should be replaced by (4λ0)2). 8) The first term in the square bracket [⋯] in Eq. (4·24) should read (1-λ02) in place of (1-λ0)2. 10) The last term on the right-hand side of Eq. (A·1) should be multiplied by 2.
The previous configuration-space cluster integral treatment of the properties of the ground state of a many-boson system is modified by including those higher order diagrams consistent with the pair excitation approximation, which were previously omitted. The cluster expansions involve a parameter, analogous to the fugacity in classical expansions, whose definition automatically accounts for the depletion of the free particle ground state. The resulting expectation value for the Hamiltonian is in agreement with that obtained by field-theoretic methods for states of pair-excitation type.