SingleExcitationBra file

The operators \(B=c_{i\sigma_1}(c_{i\sigma_1}^{\dagger})\) to generate the bra single excited state \(B|\phi\rangle\), used to compute the off-diagonal dynamical Green’s function

(4.23)\[G_{BA}(z) = \langle \phi | B^{\dagger} \frac{1}{z-H} A | \phi \rangle ,\]

where the ket operator \(A\) is given in SingleExcitation. If this file is omitted, the ordinary diagonal Green’s function (\(B=A\)) is calculated.

The file format and parameters are the same as the SingleExcitation file; only the keyword for the number of operators is NSingleExcitationBra. An example:

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NSingleExcitationBra    1
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====== Single Excitation Bra ===
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    0     0     0    1.0    0.0

Use rules

  • Headers cannot be omitted; the format and parameters are identical to the SingleExcitation file.

  • The operator written here is \(B\), not \(B^{\dagger}\). The computed quantity is \(\langle\phi|B^{\dagger}(z-H)^{-1}A|\phi\rangle\).

  • The off-diagonal Green’s function is available only with the shifted BiCG method (method="CG") and CalcSpec="Normal" (restart/save spectrum modes are not supported together with a bra operator).

  • The ket (SingleExcitation) and bra operators must map \(|\phi\rangle\) into the same excited Hilbert sector, and all operators within one file must share the same sector change; otherwise the program is terminated. What “same sector” means depends on the model:

    • Hubbard (canonical): the same change of particle number and \(S_z\).

    • HubbardGC (grand canonical): a single excitation carries no sector shift, so the whole Fock space is one Hilbert space and the ket and bra (of either spin) always share it.

    • HubbardNConserved (\(S_z\) free): only the same change of particle number is required; the \(S_z\) change may differ.

  • Supported models: Hubbard, HubbardGC (grand canonical), and HubbardNConserved (canonical, fixed electron number with free \(S_z\)). (A single excitation is not defined for spin systems, and other models are not yet supported for the off-diagonal Green’s function; the program is terminated for them.)

  • HubbardNConserved is selected from the Hubbard model (CalcModel=0) by giving Ncond (the electron number) without 2Sz in the ModPara file. Because \(S_z\) is then free, this enables the cross-spin (2Sz-free) off-diagonal route (used, for example, for spin-orbit-coupled impurities): the excited Hilbert sector is fixed by the electron number alone, so the ket and bra need only share the same change of particle number, and a cross-spin \(G_{BA}\) is obtained from a single solve.

  • When SpectrumNumBra \(>1\) is set in the ModPara file, the additional bra operator sets 1 … \((B-1)\) are read from files named single_ex_bra_1.def … single_ex_bra_\((B-1)\).def in the run directory, using the same format as this file; bra set 0 is this SingleExcitationBra file given in the namelist.