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Register a new userRelation of $a^dagger a$ terms to higher-order terms in the adiabatic expansion for large-amplitude collective motion
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Authors
Koichi Sato
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We investigate the relation of $a^dagger a$ terms in the collective operator to the higher-order terms in the adiabatic self-consistent collective coordinate (ASCC) method. In the ASCC method, a state vector is written as $e^{ihat G(q,p,n)}|phi(q)rangle$ with $hat G(q,p,n)$ which is a function of collective coordinate $q$, its conjugate momentum $p$ and the particle number $n$. According to the generalized Thouless theorem, $hat G$ can be written as a linear combination of two-quasiparticle creation and annihilation operators $a^dagger_mu a^dagger_ u$ and $a_ u a_mu$. We show that, if $a^dagger a$ terms are included in $hat G(q,p,n)$, it corresponds to the higher-order terms in the adiabatic expansion of $hat G$. This relation serves as a prescription to determine the higher-order collective operators from the $a^dagger a$ part of the collective operator, once it is given without solving the higher-order equations of motion.
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We propose a new set of equations to determine the collective Hamiltonian including the second-order collective-coordinate operator on the basis of the adiabatic self-consistent collective-coordinate (ASCC) theory. We illustrate, with the two-level Lipkin model, that the collective operators including the second-order one are self-consistently determined. We compare the results of the calculations with and without the second-order operator and show that, without the second-order operator, the agreement with the exact solution becomes worse as the excitation energy increases, but that, with the second-order operator included, the exact solution is well reproduced even for highly excited states. We also reconsider which equations one should adopt as the basic equations in the case where only the first-order operator is taken into account, and suggest an alternative set of fundamental equations instead of the conventional ASCC equations. Moreover, we briefly discuss the gauge symmetry of the new basic equations we propose in this paper.
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