Pairing

The pairing matrix [17, 18, 19] gives access to the correlations of spin-singlet Cooper pairs.

We define the back-to-back two-baryon operators

\[\tilde{\Delta}^\dagger_{S,m,k} = \tilde{\psi}^\dagger_{\alpha,k} \left(P^\dagger_{S,m}\right)_{\alpha\beta} \tilde{\psi}^\dagger_{\beta,-k}\]

where \(S\) and \(m\) are spin quantum numbers.

Singlet Pairing

tdg.observable.pairing.P_singlet = tensor([[ 0.0000+0.j, -0.7071+0.j],         [ 0.7071+0.j,  0.0000+0.j]])

\(P_{0,0} = \frac{-i \sigma_2}{\sqrt{2}}\)

Triplet Pairing

There are three possible triplet operators. These correspond to two-fermion operators with definite \(S\) and \(S_z\).

tdg.observable.pairing.P_triplet_plus = tensor([[1.+0.j, 0.+0.j],         [0.+0.j, 0.+0.j]])

\(P_{1,+1} = \frac{1}{2}\left( \sigma_0 + \sigma_3 \right)\).

tdg.observable.pairing.P_triplet_zero = tensor([[0.0000+0.j, 0.7071+0.j],         [0.7071+0.j, 0.0000+0.j]])

\(P_{1,0} = \frac{\sigma_1}{\sqrt{2}}\)

tdg.observable.pairing.P_triplet_minus = tensor([[0.+0.j, 0.+0.j],         [0.+0.j, 1.+0.j]])

\(P_{1,-1} = \frac{1}{2}\left( \sigma_0 - \sigma_3 \right)\).

Pairing Matrices

We define pairing matrices of operators with good spin quantum numbers,

\[\begin{split}\begin{align} \tilde{M}^{S,m}_{k,q} =& \left\langle \tilde{\Delta}^\dagger_{S,m,k} \tilde{\Delta}_{S,m,q} \right\rangle \nonumber\\ & - \left(P^\dagger_{S,m}\right)_{\alpha\beta} \left(P^{\phantom{\dagger}}_{S,m}\right)_{\sigma\tau} \left\{ \left\langle \tilde{\psi}^\dagger_{\alpha, k} \tilde{\psi}_{\tau, q} \right\rangle \left\langle \tilde{\psi}^\dagger_{\beta, -k} \tilde{\psi}_{\sigma, -q} \right\rangle - \left\langle \tilde{\psi}^\dagger_{\alpha, k} \tilde{\psi}_{\sigma, -q} \right\rangle \left\langle \tilde{\psi}^\dagger_{\beta, -k} \tilde{\psi}_{\tau, q} \right\rangle \right\}. \end{align}\end{split}\]

Note

\(\tilde{M}\) is intentionally constructed to be zero in the free case, to ensure that it only encodes interactions.

We find that eigenvalues of \(\frac{\left\langle\Delta^\dagger_{S,m,k} \Delta_{S,m,q} \right\rangle}{L^4 (N/2)}\) are dimensionless, have good continuum limits, and are at largest intensive.

Each channel ∈ {singlet, triplet_plus, triplet_minus, triplet_minus} has the same observables. We present the documentation in a generic way; just replace channel appropriately.

tdg.observable.pairing.pair_pair_channel(ensemble)

\(\left\langle \tilde{\Delta}^\dagger_{S,m,k} \tilde{\Delta}_{S,m,q} \right\rangle\).

Configurations first, then momenta \(k\) and \(q\).

tdg.observable.pairing.pair_pair_eigenvalues_channel(ensemble)

Eigenvalues of \(\frac{\left\langle\Delta^\dagger_{S,m,k} \Delta_{S,m,q} \right\rangle}{L^4 (N/2)}\), which are dimensionless, have good continuum limits, and are at largest intensive.

Bootstraps first, then eigenvalues from least to greatest. The eigenvalues are positive definite.

tdg.observable.pairing.pairing_channel(ensemble)

\(\frac{M}{L^4 (N/2)}\).

Bootstraps first, then momenta \(k\) and \(q\).

tdg.observable.pairing.condensate_fraction_channel(ensemble)

Largest eigenvalue of \(\frac{M}{L^4 (N/2)} = \texttt{pairing_channel}\).

One number per bootstrap.

tdg.observable.pairing.pairing_wavefunction_channel(ensemble)

Eigenvector of \(\frac{M}{L^4 (N/2)} = \texttt{pairing_channel}\) that has eigenvalue condensate_fraction_channel().

Bootstrap first, then momentum.