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arXiv:nucl-th/0404042 · Physical Review C

Classification of states of single- j fermions with J -pairing interaction

Y. M. Zhao, A. Arima

Published Sep 21, 2004• 21 citations
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Abstract

In this paper we show that a system of three fermions is exactly solvable for the case of a single-$j$ in the presence of an angular momentum-$J$ pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-$j$ symbols. When the Hamiltonian contains only an interaction between pairs of fermions coupled to spin $J={J}_{\mathrm{max}}=2j\ensuremath{-}1$, the ``non-integer'' eigenvalues of three fermions with angular momentum $I$ around the maximum appear as ``non-integer'' eigenvalues of four fermions if $I$ is around (or larger than) ${J}_{\mathrm{max}}$. This pattern is also found in five and six fermion systems. A boson system with spin $l$ exhibits a similar pattern.

DOI
10.1103/physrevc.70.034306

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In this paper we show that a system of three fermions is exactly solvable for the case of a single-jjj in the presence of an angular momentum-JJJ pairing interaction. On the basis of the solutions for this system, we obtain new sum rules for six-jjj symbols. When the Hamiltonian contains only an interaction between pairs of fermions coupled to spin J=Jmax=2j\ensuremath−1J={J}_{\mathrm{max}}=2j\ensuremath{-}1J=Jmax​=2j\ensuremath−1, the non-integer'' eigenvalues of three fermions with angular momentum $I$ around the maximum appear as non-integer'' eigenvalues of four fermions if III is around (or larger than) Jmax{J}_{\mathrm{max}}Jmax​. This pattern is also found in five and six fermion systems. A boson system with spin lll exhibits a similar pattern.
We investigate Hamiltonians with attractive interactions between pairs of fermions coupled to angular momentum JJJ. We show that pairs with spin JJJ are reasonable building blocks for the low-lying states. For systems with only a J=JmaxJ={J}_{max}J=Jmax​ pairing interaction, eigenvalues are found to be approximately integers for a large array of states, in particular, for those with total angular momenta $I\ensuremath{\leqslant}2j$. For I=0I=0I=0 eigenstates of four fermions in a single-jjj shell we show that there is only one nonzero eigenvalue. We address these observations using the nucleon pair approximation of the shell model and relate our results with a number of currently interesting problems.
The eigenvalue problem in a one-dimensional system of many fermions interacting via a delta-function potential is solved exactly in a field theoretical framework. Simultaneous eigenstates of spin and energy are represented in a systematic and compact form, that is, in such a form that the orbital and spin eigenfunctions are unified in harmony without the aid of Young’s tableau. The eigenstate and energy eigenvalue are specified by two sets of quantum numbers, particle momenta and spin momenta, besides two eigenvalues of a total spin angular momentum and its z-component. It is shown that for given particle momenta and spin momenta the energy eigenvalue is given by the solutions of coupled equations.
In this paper we study the number of states with given spin for identical particles, denoted as DIn(l){D}_{I}^{n}(l)DIn​(l) for bosons and as DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j) for fermions, where III is the total angular momentum of the system, nnn is the particle number, lll is the spin of a single boson, and jjj is the spin of a single fermion. We derive relations of DIn(l){D}_{I}^{n}(l)DIn​(l) and DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j); in particular, we obtain recursive and explicit formulas of DIn(l){D}_{I}^{n}(l)DIn​(l) and DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j) in terms of those for $n\ensuremath{-}1$ particles and apply them to systems of two, three, and five identical particles.
We propose a simple approach to predict the angular momentum I ground state (I(I(I g.s.) probabilities of many-body systems that does not require the diagonalization of Hamiltonians with random interactions. This method is found to be applicable to all cases that have been discussed: even and odd fermion systems (both in single-jjj and many-jjj shells), and boson (both sd\mathrm{sd}sd and sdg)sdg)sdg) systems. A simple relation for the highest angular momentum g.s. probability is found. Furthermore, it is suggested for the first time that the 0 g.s. dominance in boson systems and in even-fermion systems is given by two-body interactions with specific features.
In the framework of exactly solvable quantum many-body systems we study models of interacting spin one-half Fermions in one dimension. The first part deals with systems of spin one-half Fermions which interact via repulsive contact interaction. A reformulation of the Bethe-Ansatz solvable many-body wave function is presented. This simplifies considerably the calculations for the highly imbalanced case, where very few particles of one species (minority Fermions) are present. For the other particle species (majority Fermions) the thermodynamic limit is taken. We assume the majority Fermions to be in the ground state such that their non-interacting momentum distribution is a Fermi-sea. Upon this we consider excitations where the particles of the minority species may occupy an arbitrary state within the Fermi-sea. In the case of only a single minority Fermion, the many-body wave function can be expressed as a determinant. This allows us to derive exact thermodynamic expressions for several expectation values as well as for the density-density correlation function. Moreover it is possible to find closed expressions for the single particle Green's function. All of the above mentioned quantities show a non-trivial dependence on the minority particle's momentum. In particular the Green's function in the Tonks-Girardeau regime of hardcore interaction is shown to undergo a transition from the one of impenetrable Bosons to that of free Fermions as the extra particle's momentum varies from the core to the edge of the Fermi-sea. This transition becomes manifest in an algebraic asymptotic decay of the Green's function. If two minority Fermions are present, the many-body wave function turns out to be more complicated. Nevertheless it is possible to derive exact expressions for the two and the three particle density-density correlation functions. Furthermore we calculate the system's total energy and based on that, identify terms which have a natural interpretation as effective interaction energy for the two minority Fermions in the presence of the Fermi-sea. The second part is devoted to the study of one-dimensional systems consisting of two fermionic particle species with different masses. We show that for specific kinds of interaction potentials and for certain relations between the masses and the coupling constants, the particle creation and annihilation operators of such a system can be constructed exactly.