Particle-hole conjugation

Antilinear Particle-Hole Conjugation Operators in Jahn-Teller Theory... 

Four decades ago, Bell [3] introduced a particle-hole conjugation operator CB into nuclear shell theory. Its operator algebra is essentially isomorphic to that of Cq (for example, CB is unitary), the filled Dirac sea now corresponding to systems with half-filled shells. This was later extended to other areas of physics. For example,... 

Our procedure for securing basis independence follows a group-tensor algebraic approach to shell theory, and examines the algebraic interplay of particle-hole conjugation operators with quasispinors and quasispin tensors. The problem with Cs may be remedied while retaining an antilinear transformation only by replacing Z with another antilinear operation which is physical. Apart from an unimportant phase this can be identified as time reversal T, so that C = CT. Hence, the operators to be examined for physical interest are just two in number C and CT. In a later work we will explore the consequences of the work of Ceulemans [7,8,10] from this perspective. 

C(A) is unitary and linear, commuting with the metric (so that C(A) C (A) = —a A, etc.) and indeed with all annihilation or creation operators for states outside the subshell A. The combination operator C = n< A C(A) will jointly perform particle-hole conjugation separately within each subshell. 

In manifestly preserving transformation character, the above confirm that both quasispin and particle-hole conjugation are scalars in orbital and spin space. 

Ceulemans paper of 1984 generalizes the work of Griffith [25,26] on particle-hole conjugation for the specific case of d" electrons split by an octahedral symmetry field. He relies on the use of matrices and determinants, in particular Laplace s expansion of the determinant in terms of complementary minors, for the analysis. He bases his selection mle analysis on the properties of a novel particle-hole conjugation operator... 

Ceulemans considers a dn electron state, split by an octahedral field into the e and t2 levels, so that all the n electrons are in the t2 subshell. In the notation of Sugano et al. [27], rjj(t2SrMsMr) is the multi-electronic wavefunction, with SMs irrep labels for the total spin and rMr irrep labels in the octahedral group for the orbital state. We use a real orbital basis in which all njm factors take their simplest possible forms and suppress S, r and Mr below. It takes six electrons (three pairs each of opposed spin) to fill this t2 subshell. Ceulemans [7] particle-hole conjugation operator 0() has the effect of conjugating the occupancies within this subshell, and of... 

In the following CL denotes the generalized form of the particle-hole conjugation operator C(A) which runs over conjugate pairs and is defined as... 

The similarity between CL and T on half-filled shells should be noted (see the above equations they are identical to within a phase). This example strengthens the identification of CL as 0(4>). the particle-hole conjugation operator for half-filled shells defined by Ceulemans where (9(< ) is characterized by the properties listed in Ref. [9]. Ceulemans states that 0(< >) shares a number of properties with time reversal (and consequently with CL). 

The proof of these rules follows directly from the composite nature of the antilinear particle-hole conjugation operator <9(< ) of Ceulemans, which, it is emphasized here again, is equivalent to within a phase to CL = C T. It is an interesting point to note that this means that these rules stem from not just a linear choice of particle-hole conjugation operator, nor from just the antilinear choice, but from the effects of both separately and together. 

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] ... 

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