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65
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85038971360
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Note that the expression (20) has, just as (Formula presented) dimension of energy, since the (Formula presented) there are dimensionless. This is different from Eq. (11) where the quantities (Formula presented) have the dimension of the inverse temperature such that (Formula presented) is dimensionless
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Note that the expression (20) has, just as (Formula presented) dimension of energy, since the (Formula presented) there are dimensionless. This is different from Eq. (11) where the quantities (Formula presented) have the dimension of the inverse temperature such that (Formula presented) is dimensionless.
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66
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85038935153
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We note that the summation over (Formula presented) and (Formula presented) in Eq. (23) is unrestricted, i.e., is performed over all single-particle states. If one would restrict the summation to, say, states (Formula presented) that are occupied in the ground state and states (Formula presented) that are unoccupied, a factor of (Formula presented) would occur
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We note that the summation over (Formula presented) and (Formula presented) in Eq. (23) is unrestricted, i.e., is performed over all single-particle states. If one would restrict the summation to, say, states (Formula presented) that are occupied in the ground state and states (Formula presented) that are unoccupied, a factor of (Formula presented) would occur.
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68
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85038900761
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If the spin flip is performed with respect to the (Formula presented) plane, the corresponding operator acting on the carrier states is just the operator (Formula presented) of complex conjugation which changes the sign of (Formula presented) but leaves the other Pauli matrices invariant. In the case of the (Formula presented) plane or (Formula presented) plane the spin-flip operator is (Formula presented) and (Formula presented) respectively. For a general orientation of this plane one has to use the appropriate linear combination of these operators. If all Mn spins lie in such a plane, the parabolic-band Hamiltonian is invariant under this spin-flip operation
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If the spin flip is performed with respect to the (Formula presented) plane, the corresponding operator acting on the carrier states is just the operator (Formula presented) of complex conjugation which changes the sign of (Formula presented) but leaves the other Pauli matrices invariant. In the case of the (Formula presented) plane or (Formula presented) plane the spin-flip operator is (Formula presented) and (Formula presented) respectively. For a general orientation of this plane one has to use the appropriate linear combination of these operators. If all Mn spins lie in such a plane, the parabolic-band Hamiltonian is invariant under this spin-flip operation.
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69
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85038888143
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Note that the sum rules, Eqs. (33) and (34), are independent of our choice to take all wave functions to be real since possible common phases in the up or down spins cancel in each term in the summation over (Formula presented)
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Note that the sum rules, Eqs. (33) and (34), are independent of our choice to take all wave functions to be real since possible common phases in the up or down spins cancel in each term in the summation over (Formula presented)
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