Pairing Benchmark Observable Contract
This solver shares observable definitions with the number-conserving solver
at src/number_conserving/.
Definitions that exist in the no-pairing benchmark must keep the same
normalization, symbols, and \(K\)-point convention here.
Shared Triangular Geometry
Both solvers use the triangular primitive vectors
and reciprocal vectors
Thus \(\mathbf b_i\cdot\mathbf a_j=2\pi\delta_{ij}\). On a \(3\times3\) lattice the benchmark \(K\) point is
implemented as Latt%inv_cell_list(2*Nlx/3+1, Nly/3+1).
Observables Inherited From The No-Pairing Benchmark
| symbol | DQMC/ED key | definition |
|---|---|---|
| \(\rho\) | density_total |
\(\langle N_b+N_c\rangle/N_s\) |
| \(E\) | energy_density |
paper-facing total physical energy \(E=N_s\,\texttt{energy\_density}=\langle H_{\rm phys}\rangle\), excluding \(-\mu N\); plots use \(-E\) |
| \(D_{bc}\) | doubleOcc |
\(N_s^{-1}\sum_i\langle n_{b,i}n_{c,i}\rangle\) |
| \(M_b^{(2)}\) | onsite_n2_up |
\(\sum_i\langle n_{b,i}^2\rangle\) |
| \(M_c^{(2)}\) | onsite_n2_do |
\(\sum_i\langle n_{c,i}^2\rangle\) |
| \(N_b^2\) | numsquare_up |
\(\langle(\sum_i n_{b,i})^2\rangle\) |
| \(N_c^2\) | numsquare_do |
\(\langle(\sum_i n_{c,i})^2\rangle\) |
| \(S_{\rm SF}(K)\) | S_SF_K / sf_K |
\(N_s^{-2}\sum_{ij,\alpha}e^{iK\cdot(r_i-r_j)}\langle a^\dagger_{\alpha i}a_{\alpha j}\rangle\) |
| \(S_{\rm DW}(K)\) | S_DW_K / dw_K |
\(N_s^{-2}\langle n_K n_{-K}\rangle\), \(n_i=n_{b,i}+n_{c,i}\) |
| \(S_{\rm PSF}(\Gamma)\) | S_PSF_Gamma / psf_Gamma |
\(N_s^{-2}\sum_{ij}\langle b_i^\dagger c_i^\dagger c_j b_j\rangle\) |
| \(\mathrm{IPR}_\rho\) | IPR |
\(\sum_i\rho_i^2/(\sum_i\rho_i)^2\), \(\rho_i=\langle n_{b,i}+n_{c,i}\rangle\) |
The \(K\)-point structure factors are Hermitian positive structure-factor channels with exact real non-negative thermal averages. DQMC files remain two-column complex files because the Fourier sums are accumulated as complex numbers. The real part is the comparison channel; the imaginary part is checked with the analysis gate \(|{\rm Im}|\le \max(10^{-14},3\,{\rm SEM}_{\rm Im})\), where the absolute floor prevents roundoff-scale values from being amplified by a numerically tiny SEM. A violation of this imaginary-part gate, or a real part that is negative beyond the same roundoff/block-error scale, marks the observable unreliable.
Pairing-Specific Extensions
The pairing benchmark adds observables that do not exist in the no-pairing
model. The operators in this table follow the solver convention
b_code=b_paper, c_code=-c_paper, in which the real pair coefficient is
positive. Consequently the manuscript's anomalous amplitude per site is
\(P_{\rm paper}=N_s^{-1}\sum_i\langle b_i c_i+b_i^+c_i^+\rangle_{\rm paper}
=-\texttt{pair_equal}\), and its energy contribution per site is
\(e_\Delta=-\Delta P_{\rm paper}=\Delta\,\texttt{pair_equal}\).
The four plotted benchmark observables are invariant under this phase change.
See the Hamiltonian conventions.
| symbol | key | definition |
|---|---|---|
| \(P_\Delta\) | pair_equal |
\(N_s^{-1}\sum_i\langle b_i c_i+b_i^\dagger c_i^\dagger\rangle\) |
| \(e_\Delta\) | pairing_energy_density |
\(\Delta P_\Delta\), per site; total \(E_\Delta=N_s e_\Delta\) |
| \(e_U\) | interaction_energy_density |
\((U_1+U_2)(M_b^{(2)}+M_c^{(2)})/N_s+2(U_1-U_2)D_{bc}\), per site |
| \(e_\mu\) | chemical_energy_density |
\(-\mu\rho\), per-site diagnostic only |
| \(E_{\rm grand}\) | grand_energy_density |
\(N_s\,\texttt{grand\_energy\_density}=E+N_s e_\mu\), diagnostic only |
The paper-facing energy panel remains \(-E\), not
\(-e\) and not \(-E_{\rm grand}\). The internal DQMC/ED key
energy_density is kept only as a storage and comparison convention.
Extending observables
When changing a shared observable, compare the implementations in
src/pairing/src/obser_equal.f90 and
src/number_conserving/src/obser_equal.f90 as well as their ED operators.
For K-point quantities, check reciprocal coordinates, Hermiticity, and
normalization. Add the corresponding small ED identity or limiting-case test.