Particle-hole parity

Particle-hole symmetry and particle-hole parity... 

Hence in his 1984 paper [7], when Ceulemans referred to the parity of a half-filled shell state with respect to a linear particle-hole conjugation operator he was actually referring to the quasispin character of the state, see equation (34). For a half-filled shell state Q is always integral, hence (— l)e = ttq = 1. This is just the result derived by Ceulemans in 1984 [7] with no knowledge whatsoever of the concept of quasispin. The governing equation from which Ceulemans 84 selection rules stem, expressed in terms of half-filled shell states, may be labelled with the quasispin scheme proposed by Ceulemans in 1994 [10] ... 

For the 7r-electron models this means that the average number of 7r-electrons per site is unity. This result is proved in Appendix B. A second property is that singlet particle-hole excitations that are negative under a particle-hole transformation have an even particle-hole spatial parity, while singlet particle-hole... 

There is an important connection between particle-hole symmetry and the relative parity of the particle-hole pair. Consider a basis state created by the removal of an electron from a valence band Wannier orbital on the repeat unit at i — r/2 and the creation of an electron on a conduction band Wannier orbital at R + r/2. This is illustrated in Fig. 6.1. This particle-hole pair has a centre-of-mass coordinate, R, and a relative coordinate, r ... 

There is an important observation to be made about this effective-particle model. This is that since the exchange interaction, X, is local (i.e. it is only nonzero when r = 0), we immediately see that this term vanishes for odd parity excitons (namely, (r) = —i/ ni—r)), as tl>n 0) = 0. Now, since the parity of the exciton is determined by the particle-hole symmetry, and odd singlet and... 

It is instructive to compare the approximate weak-coupling theory to essential exact, numerical (density matrix renormalization group) calculations on the same model (namely the Pariser-Parr-Pople model). The numerical calculations are performed on polymer chains with the polyacetylene geometry. Since these chains posses inversion symmetry the many-body eigenstates are either even (Ag) or odd By). As discussed previously, the singlet exciton wave function has either even or odd parity when the particle-hole eigenvalue is odd or even. Conversely, the triplet exciton wavefunction has either even or odd parity when the particle-hole eigenvalue is even or odd. As a consequence, we can express a B state as... 

The particle-hole separations are shown in Fig. 6.6 at 102 sites. The jumps in the separation occur at p = 9 and p = 8 for the even and odd parity excitons, respectively, corresponding to the j = 1 branches of the n = 3 and n = 4 families of excitons. Notice that, as predicted in Appendix E, the particle-hole separations decrease with increasing j for the same n. 

Fig. E.l. The dispersion curves of the four lowest bound states (solid and dashed) for a regularized Coulomb potential with a K = 0) = 1. Even (odd n) parity states (solid curves) and odd (even n) parity states (dashed curves). The particle-hole continuum is bounded by the dotted curves. The energies are in units of Ej.

The boson creation operators are Ypi where p = iv,v indicates proton or neutron and i = L,M the angular momentum of the basis bosons. We consider even L, even parity for the bosons, which are supposed to represent pairs of like particles (or holes). The numbers Np = Ejnpi are assumed to equal 1/2 the number of proton or neutron particles or holes from the nearest closed shell, and are thus well defined functions of N and Z, to the... 

In Eqs. (98)—(102), cyclic notation was used, with / denoting the identity permutation and x, representing either particles /3, or holes w,-. The trivial permutation operator, P = /, will not be explicitly used. It should be noted that the signs are in agreement with the parities of the permutation cycles, so that the criterion of correct phase for noncanonical diagrams is implemented on the single index set permutation operator level. 

Comparison of the experimental spins and parities of some odd-A nuclei with the predictions of the independent particle model. The central part of the table reproduces the level sequence, shown In O Fig. 2.9, for easier comparison of experimental data with theory. The exponents over the symbols of the orbits stand for multiple occupancy. Negative exponents mean occupancy with respect to full (sub)shells, "hole" in a (sub)shell... 

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