                                                                                                                                                                7



           TΨ1 ···Ψd Ψτ Ψ̄τ Ψ̄d ···Ψ̄1
                                                                                              
              Y  d          X                        X              d
                                                                    Y             X                       X
          =                                                                                   
                σ=1 iσ,↑ ,iσ,↓ ,jσ,↑ ,jσ,↓      iτ,↑ ,iτ,↓                       0
                                                              σ=1 i0σ,↑ ,i0σ,↓ ,jσ,↑   0
                                                                                     ,jσ,↓             i0τ,↑ ,i0τ,↓
                  P             √      P                    0     0
          × (−1)     σ,s iσ,s
                         ( t)            σ,s (iσ,s +jσ,s +iσ,s +jσ,s )
            h
          × δ1,iτ,↓ + σ (iσ,↓ +jσ,↓
                     P          0   ) δ1,i0τ,↓ + σ (i0σ,↓ +jσ,↓ ) δ1,iτ,↑ + σ (iσ,↑ +jσ,↑
                                                P                          P          0   ) δ1,i0τ,↑ + σ (i0σ,↑ +jσ,↑ )
                                                                                                      P


          − (µ + 1)δ0,iτ,↓ +Pσ (iσ,↓ +jσ,↓
                                        0   ) δ0,i0τ,↓ + σ (i0σ,↓ +jσ,↓ ) δ1,iτ,↑ + σ (iσ,↑ +jσ,↑
                                                        P                          P          0   ) δ1,i0τ,↑ + σ (i0σ,↑ +jσ,↑ )
                                                                                                              P

          − (µ + 1)δ1,iτ,↓ +Pσ (iσ,↓ +jσ,↓
                                          0 ) δ1,i0τ,↓ + σ (i0σ,↓ +jσ,↓ ) δ0,iτ,↑ + σ (iσ,↑ +jσ,↑
                                                         P                          P             0  ) δ0,i0τ,↑ + σ (i0σ,↑ +jσ,↑ )
                                                                                                                 P
                                                                                                                                                  i
          − U  − (µ + 1)2 δ0,iτ,↓ +Pσ (iσ,↓ +jσ,↓
             
                                                         0  ) δ0,i0τ,↓ + σ (i0σ,↓ +jσ,↓ ) δ0,iτ,↑ + σ (iσ,↑ +jσ,↑0  ) δ0,i0τ,↑ + σ (i0σ,↑ +jσ,↑ )
                                                                         P                          P                           P

                                  !                              !                             !                               !
                              0
             iτ,↑
                  Y i
                         σ,↑ jσ,↑     i0τ,↑   Y i0 j
                                                       σ,↑   σ,↑       iτ,↓
                                                                            Y i
                                                                                    σ,↓ jσ,↓
                                                                                            0
                                                                                                    i0τ,↓    Y i0 j
                                                                                                                    σ,↓    σ,↓
          × ητ,↑      ησ,↑ ζ̄σ,↑ η̄τ,↑              η̄σ,↑ ζσ,↑ ητ,↓               ησ,↓ ζ̄σ,↓ η̄τ,↓               η̄σ,↓ ζσ,↓ ,                                (A7)
                        σ                                 σ                                  σ                                        σ




where we have assigned the indices iσ,s (n), jσ,s (n), and                                  dences both from the auxiliary Grassmann fields and the
iτ,s (n) as the labels of the Taylor expansion for Eq. (A2),                                indices of the Taylor expansion, introducing the notation
Eq. (A3), and Eq. (A4), respectively. They take just 0 or                                   i0ν,s (n) = iν,s (n − ν̂). Then we sort the auxiliary Grass-
1 because of the nilpotency of the Grassmann numbers.                                       mann fields in Eq. (A7) as those in Eq. (A8) and the
For simplicity, we have omitted the lattice site depen-                                     Grassmann tensor T is finally written as




                     TΨ1 ···Ψd Ψτ Ψ̄τ Ψ̄d ···Ψ̄1
                                                                                                            
                        Y  d          X                          X              d
                                                                                Y                X                       X
                    =                                                                                       
                            σ=1 iσ,↑ ,iσ,↓ ,jσ,↑ ,jσ,↓        iτ,↑ ,iτ,↓                           0
                                                                                σ=1 i0σ,↑ ,i0σ,↓ ,jσ,↑   0
                                                                                                       ,jσ,↓           i0τ,↑ ,i0τ,↓

                    × T(i1,↑ ,i1,↓ ,j1,↑ ,j1,↓ )···(id,↑ ,id,↓ ,jd,↑ ,jd,↓ )(iτ,↑ ,iτ,↓ )(i01,↑ ,i01,↓ ,j1,↑
                                                                                                         0 ,j 0 )···(i0
                                                                                                               1,↓
                                                                                                                            0     0     0      0     0
                                                                                                                      d,↑ ,id,↓ ,jd,↑ ,jd,↓ )(iτ,↑ ,iτ,↓ )
                                                                                                            
                         i1,↑ i1,↓ j1,↑ j1,↓                     id,↑ id,↓ jd,↑ jd,↓                iτ,↑ iτ,↓
                    × η1,↑     η1,↓ ζ1,↑ ζ1,↓ · · · ηd,↑              ηd,↓ ζd,↑ ζd,↓              ητ,↑    ητ,↓
                       i0 i0   j 0 j 0 i0 i0                                 j 0 j 0 i0 i0 
                          τ,↓     τ,↑         d,↓     d,↑    d,↓    d,↑              1,↓     1,↑     1,↓     1,↑
                    × η̄τ,↓    η̄τ,↑       ζ̄d,↓   ζ̄d,↑  η̄d,↓  η̄d,↑     · · · ζ̄1,↓   ζ̄1,↑    η̄1,↓   η̄1,↑    .                                         (A8)



In the above expression, the coefficients of the auxiliary                                  Grassmann fields are identified as a multi-rank tensor T .
                                                                                            When d = 1 (σ = 1), the coefficient tensor T is given by




                       T(iσ,↑ ,iσ,↓ ,jσ,↑ ,jσ,↓ )(iτ,↑ ,iτ,↓ )(i0σ,↑ ,i0σ,↓ ,jσ,↑
                                                                              0     0
                                                                                  ,jσ,↓ )(i0τ,↑ ,i0τ,↓ )
                               P          √       P                      0      0
                      = (−1) s iσ,s ( t) s (iσ,s +jσ,s +iσ,s +jσ,s )
                        h
                      × δ1,iτ,↓ +iσ,↓ +jσ,↓  0   δ1,i0τ,↓ +i0σ,↓ +jσ,↓ δ1,iτ,↑ +iσ,↑ +jσ,↑     0    δ1,i0τ,↑ +i0σ,↑ +jσ,↑
                      − (µ + 1)δ0,iτ,↓ +iσ,↓ +jσ,↓
                                                0   δ0,i0τ,↓ +i0σ,↓ +jσ,↓ δ1,iτ,↑ +iσ,↑ +jσ,↑
                                                                                          0   δ1,i0τ,↑ +i0σ,↑ +jσ,↑
                      − (µ + 1)δ1,iτ,↓ +iσ,↓ +jσ,↓
                                                0   δ1,i0τ,↓ +i0σ,↓ +jσ,↓ δ0,iτ,↑ +iσ,↑ +jσ,↑
                                                                                            0   δ0,i0τ,↑ +i0σ,↑ +jσ,↑
                                                                                                                                      i
                      − U  − (µ + 1)2 δ0,iτ,↓ +iσ,↓ +jσ,↓
                        
                                                                0   δ0,i0τ,↓ +i0σ,↓ +jσ,↓ δ0,iτ,↑ +iσ,↑ +jσ,↑
                                                                                                            0   δ0,i0τ,↑ +i0σ,↑ +jσ,↑
                                  R(i                                       0     0     0     0      0     0
                                      σ,↑ ,iσ,↓ ,jσ,↑ ,jσ,↓ )(iτ,↑ ,iτ,↓ )(iσ,↑ ,iσ,↓ ,jσ,↑ ,jσ,↓ )(iτ,↑ ,iτ,↓ )
                      × (−1)                                                                                       ,                                         (A9)
                                                                                                                                                       8

                                                                                            with




               R(iσ,↑ ,iσ,↓ ,jσ,↑ ,jσ,↓ )(iτ,↑ ,iτ,↓ )(i0σ,↑ ,i0σ,↓ ,jσ,↑
                                                                      0     0
                                                                          ,jσ,↓ )(i0τ,↑ ,i0τ,↓ )
                                         0
             = iσ,↑ iτ,↑ + iσ,↓ (iτ,↑ + jσ,↑ + i0τ,↑ + i0σ,↑ + jσ,↑ + iτ,↓ )
                             0
             + jσ,↑ (iτ,↑ + jσ,↑                                   0
                                 + i0τ,↑ + i0σ,↑ ) + jσ,↓ (iτ,↑ + jσ,↑                           0
                                                                       + i0τ,↑ + i0σ,↑ + iτ,↓ + jσ,↓ + i0τ,↓ + i0σ,↓ )
                      0
             + iτ,↓ (jσ,↑                             0
                          + i0τ,↑ + i0σ,↑ ) + i0τ,↓ (jσ,↑                    0
                                                          + i0τ,↑ + i0σ,↑ + jσ,↓            0
                                                                                 ) + i0τ,↑ jσ,↑    0
                                                                                                + jσ,↓   0
                                                                                                       (jσ,↑ + i0σ,↑ ) + i0σ,↓ i0σ,↑ .            (A10)




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