    C = −ηx,2 − ηt,2 + (Ux† )21 ζ̄x,1 + (Ux† )22 ζ̄x,2 + (Ut† )21 ζ̄t,1 + (Ut† )22 ζ̄t,2 ,                           (A.4)
        1h                                                                                       i
    D=     − (Ux )12 η̄x,1 − (Ux )22 η̄x,2 − a+ (Ut )12 η̄t,1 − a+ (Ut )22 η̄t,2 + ζx,2 + a− ζt,2 ,                  (A.5)
        2
with a± = e±µ pt (n). After expanding out the second line in eq. (A.1), one can compare like
terms in eqs. (A.1) and (2.10),
                           h    then obtain the elementsi of coefficient tensor F . When a
                      P      T                     T
diquark source term n λ χ (n)σ2 χ(n) + χ̄(n)σ2 χ̄ (n) /2 is added to the action, then the
Grassmann tensor F in eq. (2.10) is given by ABCD +m(AB +CD)+iλ(AC +BD)+m2 +λ2 .

B    Construction of ρ




                                                                                                                             JHEP03(2025)027
In section 3.2, we introduced a Hermitian matrix as ρB = MB† MB . It is computationally
demanding to obtain ρB directly through this formula since it is a multiplication between
a 16K × 163 K 3 matrix and a 163 K 3 × 16K matrix.
    In this study, we obtain ρB (and ρA by similar steps) following another equivalent method.
First, we perform the following SVD on the coefficient tensor Tn as
                                                                        2K2
                                                                      16X
                   (−1)   fx (ft +fx′ )+fx′ ft
                                                    (Tn )xtx′ t′ =            (UB )x′ ty (sB )y (VB† )yxt′ .         (B.1)
                                                                        y=1

Since the direct calculation of eq. (B.1) is also demanding, we employ a randomized SVD [5]
with the oversampling parameter p ∼ 0.07(16K)2 and the iteration number of QR decom-
position r′ = 7 in this step.
    Then we substitute eq. (B.1) into the definition of ρB :

                                    (MB∗ )(tx′ t′ )x (MB )(tx′ t′ )x̃
                          X
            (ρB )xx̃ =
                         t,x′ ,t′

                                    (−1)fx (ft +fx′ )+fx′ ft (Tn∗ )xtx′ t′ (−1)fx̃ (ft +fx′ )+fx′ ft (Tn )x̃tx′ t′
                          X
                     =
                         t,x′ ,t′

                                          (UB )∗x′ ty (sB )y (VB† )∗yxt′ (UB )x′ tỹ (sB )ỹ (VB† )ỹx̃t′
                          X X
                     =                                                                                               (B.2)
                         t,x′ ,t′ y,ỹ

                                        δy,ỹ (sB )y (VB† )∗yxt′ (sB )ỹ (VB† )ỹx̃t′
                         XX
                     =
                          t′   y,ỹ

                                        (sB )2y (VB† )∗yxt′ (VB† )yx̃t′ .
                         XX
                     =
                          t′        y

From the third to the fourth line of eq. (B.2), we used the property UB† UB = I of an SVD.
eq. (B.2) shows that ρB can be made once sB and VB† in eq. (B.1) are obtained.

Data Availability Statement. This article has no associated data or the data will not
be deposited.

Code Availability Statement. This article has no associated code or the code will not
be deposited.

Open Access. This article is distributed under the terms of the Creative Commons Attri-
bution License (CC-BY4.0), which permits any use, distribution and reproduction in any
medium, provided the original author(s) and source are credited.




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