On the off-diagonal Wick’s theorem and Onishi formula

Abstract

The projected generator coordinate method based on the configuration mixing of non-orthogonal Bogoliubov product states, along with more advanced methods based on it, require the computation of off-diagonal Hamiltonian and norm kernels. While the Hamiltonian kernel is efficiently computed via the off-diagonal Wick theorem of Balian and Brezin, the norm kernel relies on the Onishi formula (or equivalently the Pfaffian formula by Robledo or the integral formula by Bally and Duguet). Traditionally, the derivation of these two categories of formulae relies on different formal schemes. In the present work, the formulae for the operator and norm kernels are computed consistently from the same diagrammatic method. The approach further offers the possibility to address kernels involving more general states in the future.

Appendices

Appendix A: Normal-ordered operator

An arbitrary rank-N particle-number-conserving fermionic operator $O$ can be written as

$$\begin{aligned} O\equiv \sum _{n=0}^N {O^{nn}} , \end{aligned}$$

(A. 1)

where

$$\begin{aligned} O^{mn}\equiv \frac{1}{m!} \frac{1}{n!} \sum _{\begin{array}{c} a_1\cdots a_m\\ b_1\cdots b_n \end{array}} o^{a_1\cdots a_m}_{b_1\cdots b_n} \, c^\dag _{a_1}\cdots c^\dag _{a_m}c_{b_n}\cdots c_{b_1} , \end{aligned}$$

(A. 2)

contains m(n) particle creation (annihilation) operators. The zero-body part $O^{00}$ is the scalar obtained as the expectation value of O in the particle vacuum

$$\begin{aligned} O^{00}=\langle 0|O|0 \rangle . \end{aligned}$$

(A. 3)

In Eq. (A. 2), matrix elements are fully anti-symmetric under the exchange of any pair of upper or lower indices, i.e.

$$\begin{aligned} o^{a_1\cdots a_m}_{b_1\cdots b_n} = \epsilon (\sigma _u) \epsilon (\sigma _l) \, o^{\sigma _u(a_1\cdots a_m)}_{\sigma _l(b_1\cdots b_n)} , \end{aligned}$$

(A. 4)

where $\epsilon (\sigma _u)$ ($\epsilon (\sigma _l)$) refers to the signature of the permutation $\sigma _u(\cdots )$ ($\sigma _l(\cdots )$) of the m (n) upper (lower) indices. In case the particle-number conserving operator is hermitian, each term $O^{nn}$ is hermitian with its matrix elements fulfilling

$$\begin{aligned} o^{a_1\cdots a_n}_{b_1\cdots b_n} = \left( o_{a_1\cdots a_m}^{b_1\cdots b_n}\right) ^*. \end{aligned}$$

(A. 5)

By virtue of standard Wick’s theorem [34], the operator O can be normal ordered with respect to the Bogoliubov vacuum $| \varPhi \rangle$

$$\begin{aligned} O&\equiv \sum _{n=0}^N{{\textbf {O}}}^{[2n]} \equiv \sum _{n=0}^N\sum _{\begin{array}{c} i,j=0\\ i+j=2n \end{array}}^{2n}{{\textbf {O}}}^{ij} \end{aligned}$$

(A. 6)

where the component

$$\begin{aligned} {{\textbf {O}}}^{ij}\equiv \frac{1}{i!j!}\sum _{\begin{array}{c} k_1\cdots k_i\\ l_1\cdots l_j \end{array}}{{\textbf {o}}}^{k_1\cdots k_i}_{l_1\cdots l_j}\beta ^\dagger _{k_1}\cdots \beta ^\dagger _{k_i}\beta _{l_j}\cdots \beta _{l_1} \end{aligned}$$

(A. 7)

contains i(j) quasi-particle creation (annihilation) operators. The zero-body part ${{\textbf {O}}}^{[0]}$ is the scalar obtained as the expectation value of O in the Bogoliubov vacuum

$$\begin{aligned} {{\textbf {O}}}^{00}= \langle \varPhi |O|\varPhi \rangle \, . \end{aligned}$$

(A. 8)

In Eq. (A. 7), matrix elements are fully anti-symmetric under the exchange of any pair of upper or lower indices, i.e.

$$\begin{aligned} {{\mathbf {o}}}^{k_1\cdots k_i}_{l_1\cdots l_j} = \epsilon (\sigma _u) \epsilon (\sigma _l) \, {{\mathbf {o}}}^{\sigma _u(k_1\cdots k_i)}_{\sigma _l(l_1\cdots l_j)} \, . \end{aligned}$$

(A. 9)

These matrix elements are functionals of the Bogoliubov matrices $({\mathcal {U}},{\mathcal {V}})$ associated with $| \varPhi \rangle$ and of the matrix elements $\{o^{a_1\cdots a_n}_{b_1\cdots b_n}\}$ initially defining the operator O. As such, the content of each operator ${{\mathbf {O}}}^{ij}$ depends on the rank N of O. For more details about the normal ordering procedure and for explicit expressions of the matrix elements up to $N=3$, see Refs. [16, 35,36,37].

In case the operator is hermitian, each component ${{\textbf {O}}}^{[2n]}$ is itself hermitian with ${{\textbf {O}}}^{ij} = {{\textbf {O}}}^{ji\dagger }$ such that matrix elements satisfy

$$\begin{aligned} {{\mathbf {o}}}^{k_1\cdots k_i}_{l_1\cdots l_j} = \left( {\mathbf{o}}^{l_1\cdots l_j}_{k_1\cdots k_i}\right) ^*\, . \end{aligned}$$

(A. 10)

Appendix B:Diagrammatic rules

The present appendix is dedicated to setting up the diagrammatic rules allowing one to compute matrix elements of the form

$$\begin{aligned}&\langle _s|^i \, _j| ^t \rangle \equiv \frac{1}{s!} \frac{1}{t!} \langle \varPhi | {{\textbf {L}}}^s {{\textbf {O}}}^{ij} {{\textbf {R}}}^t |\varPhi \rangle \, , \end{aligned}$$

(B.11a)

$$\begin{aligned}&\langle _s| \,|^t \rangle \equiv \frac{1}{s!} \frac{1}{t!} \langle \varPhi | {{\textbf {L}}}^s {{\textbf {R}}}^t |\varPhi \rangle \, , \end{aligned}$$

(B.11b)

where ${{\textbf {O}}}^{ij}$ takes the form given in Eq. (A. 7) and where ${{\textbf {R}}}$ (${{\textbf {L}}}$) is a one-body²³ excitation (de-excitation) operator as defined in Eqs. (26a, 26b)).

The diagrammatic rules are also worked out to compute a second category of matrix elements of present interest

$$\begin{aligned} \langle _s | ^{k_2} | \, |_{k_1} |^t \rangle&\equiv \frac{1}{s!} \frac{1}{t!} \langle \varPhi | {{\textbf {L}}}^s \beta ^{\dagger }_{k_2} \beta _{k_1} {{\textbf {R}}}^t |\varPhi \rangle \, , \end{aligned}$$

(B.12a)

$$\begin{aligned} \langle _s | _{k_2} | \, |_{k_1} |^t \rangle&\equiv \frac{1}{s!} \frac{1}{t!} \langle \varPhi | {{\textbf {L}}}^s \beta _{k_2} \beta _{k_1} {{\textbf {R}}}^t |\varPhi \rangle \, , \end{aligned}$$

(B.12b)

$$\begin{aligned} \langle _s | ^{k_2} | \, |^{k_1} |^t \rangle&\equiv \frac{1}{s!} \frac{1}{t!} \langle \varPhi | {{\textbf {L}}}^s \beta ^{\dagger }_{k_2} \beta ^{\dagger }_{k_1} {{\textbf {R}}}^t |\varPhi \rangle \, , \end{aligned}$$

(B.12c)

$$\begin{aligned} \langle _s | _{k_2} | \, |^{k_1} |^t \rangle&\equiv \frac{1}{s!} \frac{1}{t!} \langle \varPhi | {{\textbf {L}}}^s \beta _{k_2} \beta ^{\dagger }_{k_1} {{\textbf {R}}}^t |\varPhi \rangle \, . \end{aligned}$$

(B.12d)

These matrix elements differ from the two introduced in Eq. (B.11) by the presence of two “external/fixed” quasi-particle operators, i.e. quasi-particle operators whose indices are not summed over.

The diagrammatic rules derive from the straight application of standard Wick’s theorem with respect to the Bogoliubov vacuum $|\varPhi \rangle$. The application of Wick’s theorem delivers the complete set of fully contracted terms associated with the operator product entering the matrix element of interest. Given that the operators at play are all conveniently expressed in the quasi-particle basis associated with the Bogoliubov vacuum $| \varPhi \rangle$, the four possible elementary contractions take the simplest possible form

$$\begin{aligned} {\mathbf {R}}_{k_1k_2}&= \begin{pmatrix} \frac{\langle \varPhi |\, \beta ^{\dagger }_{k_2} \beta _{k_1} | \varPhi \rangle }{\langle \varPhi | \varPhi \rangle } &{} \quad \frac{\langle \varPhi |\, \beta _{k_2} \beta _{k_1} | \varPhi \rangle }{\langle \varPhi | \varPhi \rangle } \\ \frac{\langle \varPhi |\, \beta ^{\dagger }_{k_2} \beta ^{\dagger }_{k_1} | \varPhi \rangle }{\langle \varPhi | \varPhi \rangle } &{}\quad \frac{\langle \varPhi |\, \beta _{k_2} \beta ^{\dagger }_{k_1} | \varPhi \rangle }{\langle \varPhi | \varPhi \rangle } \end{pmatrix} \nonumber \\&\equiv \begin{pmatrix} R^{+-}_{k_1k_2} &{} \quad R^{--}_{k_1k_2} \\ R^{++}_{k_1k_2} &{} \quad R^{-+}_{k_1k_2} \end{pmatrix} \nonumber \\&= \begin{pmatrix} 0 &{}\quad 0 \\ 0 &{}\quad \delta _{k_1k_2} \end{pmatrix} \ , \end{aligned}$$

(B.13)

such that the sole non-zero contraction $R^{-+}_{k_1k_2} = \delta _{k_1k_2}$ needs to be considered.

1.1 Appendix B.1: Diagrammatic representation

Fig. 6

Diagrammatic representation of the normal-ordered operator ${{\textbf {O}}}^{ij}$ as a fully anti-symmetric Hugenholtz vertex

Fig. 7

Diagrammatic representation of the two Thouless operators ${{\textbf {L}}}$ (left) and ${{\textbf {R}}}$ (right)

The diagrammatic representation of the various contributions to the matrix elements of interest relies on the definition of the following building blocks

• 1. As illustrated in Fig. 6, the normal-ordered operator ${{\textbf {O}}}^{ij}$ entering the matrix element $\langle _s|^i \, _j| ^t \rangle$ is represented by a Hugenholtz vertex with i (j) lines traveling out of (into) it and representing quasi-particle creation (annihilation) operators. The algebraic factor ${{\textbf {o}}}^{k_1 \cdots k_i}_{l_1 \cdots l_j}$ is associated to the vertex while assigning indices $k_1 \cdots k_i$ consecutively from the leftmost to the rightmost line above the vertex and indices $l_{1} \cdots l_{j}$ consecutively from the leftmost to the rightmost line below the vertex.

• 2. As illustrated in Fig. 7, the operator ${{\textbf {L}}}$ (${{\textbf {R}}}$) entering all matrix elements of present interest is represented by a vertex of the ${{\textbf {O}}}^{02}$ (${{\textbf {O}}}^{20}$) type and carry the associated algebraic factor ${{\textbf {z}}}^{*}_{l_1 l_2}(l)$ (${{\textbf {z}}}^{k_1 k_2}(r)$).

• 3. The only non-zero contraction $R^{-+}_{k_1k_2} = \delta _{k_1k_2}$ is represented in Fig. 8 and connects two up-going lines associated with one annihilation and one creation operator, both carrying the same quasi-particle index. For simplicity, one can eventually represent the contraction as a line carrying a single up-going arrow along with one quasi-particle index.

Fig. 8

Diagrammatic representation of the single non-zero elementary contraction. The convention is that the left-to-right reading of a matrix element corresponds to the up-down reading of the diagram

1.2 Appendix B.2: Diagrams generation

With these building blocks at hand, one can construct the diagrams gathering all contributions to the matrix elements introduced in Eq. (B.11) and (B.12). The basic rules to do so are as follows

• 1. Diagrams contain s square vertices (${{\textbf {L}}}$) and t triangle vertices (${{\textbf {R}}}$), the former being located above the latter. This is consistent with the convention that the left-to-right reading of a matrix element corresponds to the up-down reading of the diagram.

• 2. Diagrams making up the two matrix elements introduced in Eq. (B.11) are vacuum-to-vacuum diagrams with no line leaving the diagram. In $\langle _s|^i \, _j| ^t \rangle$, a dot vertex (${{\textbf {O}}}^{ij}$) is located in between the square and triangle vertices. This is consistent with the convention that the left-to-right reading of a matrix element corresponds to the up-down reading of the diagram.

• 3. Diagrams making up the four matrix elements introduced in Eq. (B.12) are linked with two external lines associated with the operators $\beta ^{(\dagger )}_{k_2}$ and $\beta ^{(\dagger )}_{k_1}$. The two lines leave the diagram on the same side to the, e.g., left such that (i) both lines are asymptotically in between the square (${{\textbf {L}}}$) and triangle (${{\textbf {R}}}$) vertices and such that (ii) the line carrying index $k_2$ is asymptotically located above the line carrying index $k_1$. This is consistent with the convention that the left-to-right reading of a matrix element corresponds to the up-down reading of the diagram. The arrow carried by each of the two lines points towards the interior (exterior) of the diagram if it is associated with a quasi-particle creation (annihilation) operator.

• 4. The fact that $R^{-+}_{k_1k_2}$ is the sole non-zero contraction implies that the number of quasi-particle creation operators involved in a given matrix element is equal to the number of quasi-particle annihilation operators. Given that each operator ${{\textbf {L}}}$ (${{\textbf {R}}}$) contains two quasi-particle annihilation (creation) operators, this property require the following conditions to be fulfilled

  • (a) $\langle _s|^i \, _j| ^t \rangle$ demands $t = s + (i-j)/2$  ,

  • (b) $\langle _s| \, | ^t \rangle$ demands $t = s$   ,

  • (c) $\langle _s | ^{k_2} | \, |_{k_1} |^t \rangle$ demands $t=s$   ,

  • (d) $\langle _s | _{k_2} | \, |_{k_1} |^t \rangle$ demands $t=s+1$   ,

  • (e) $\langle _s | ^{k_2} | \, |^{k_1} |^t \rangle$ demands $t=s-1$   ,

  • (f) $\langle _s | _{k_2} | \, |^{k_1} |^t \rangle$ demands $t=s$   .

  such that $t\ge (i-j)/2$, $t\ge 1$ and $s\ge 1$ in case (a), (d) and (e), respectively.

• 5. Given the above considerations, one must construct all possible topologically distinct unlabelled diagrams from the building blocks; i.e., contract together the lines belonging to the s square (${{\textbf {L}}}$) vertices and to the t triangle (${{\textbf {R}}}$) vertices, along with those belonging to the dot (${{\textbf {O}}}^{ij}$) vertex or to the two external ($\beta ^{(\dagger )}_{k_2}$ and $\beta ^{(\dagger )}_{k_1}$) lines whenever applicable, in all possible ways. Unlabelled diagrams correspond to diagrams in which ${{\textbf {L}}}$ (${{\textbf {R}}}$) vertices are not distinguished by a label. Topologically distinct unlabelled diagrams cannot be obtained from one another via a mere displacement, i.e. translation, of the vertices in the plane of the drawing.

• 6. The above process is constrained by the following properties

  • (a) Because the operators ${{\textbf {L}}}$, ${{\textbf {R}}}$, and ${{\textbf {O}}}^{ij}$ are in normal-ordered form with respect to $| \varPhi \rangle$, self-contractions must be ignored.

  • (b) Because $R^{-+}_{k_1k_2}$ is the sole non-zero contraction, ${{\textbf {L}}}$ (${{\textbf {R}}}$) operators cannot display contractions among themselves, i.e. they necessarily contract with ${{\textbf {R}}}$ (${{\textbf {L}}}$) operators, along with the i (j) quasi-particle creation (annihilation) operators inside ${{\textbf {O}}}^{ij}$ or with $\beta ^{\dagger }_{k_2}$ ($\beta _{k_2}$) and/or $\beta ^{\dagger }_{k_1}$ ($\beta _{k_1}$) whenever applicable.

• 7. The diagrams making up the various matrix elements of interest display different typical topologies. Indeed, each contribution generated via the application of Wick’s theorem can be expressed as a product of strings of contractions, each of which involves a subset of the ${{\textbf {L}}}$ and ${{\textbf {R}}}$ operators at play. As for $\langle _s|^i \, _j| ^t \rangle$ with $(i+j) \ge 4$, several such strings actually involve quasi-particle operators belonging to ${{\textbf {O}}}^{ij}$, thus forming an overall closed string that is said to be connected to ${{\textbf {O}}}^{ij}$. Translated into diagrammatic language, closed strings correspond to topologically disjoint closed sub-diagrams. In the case of $\langle _s|^i \, _j| ^t \rangle$, any given diagram is thus made out of disjoint closed sub-diagrams, one of which is connected. As for the matrix elements introduced in Eq. (B.12), one set of contractions must form a connected string involving the operators $\beta ^{(\dagger )}_{k_2}$ and $\beta ^{(\dagger )}_{k_1}$. These two operators cannot belong to two disjoint strings given that any string necessarily involves an even number of quasi-particle operators. Eventually, the diagrams making up the matrix elements introduced in Eq. (B.12) are made out of disjoint closed sub-diagrams, one of which is connected to the external lines. Last but not least, each diagram contributing to $\langle _s| \, | ^t \rangle$ is made out of disjoint closed sub-diagrams, none of which is connected.

1.3 Appendix B.3: Diagrams evaluation

Once all the diagrams are drawn, one must compute their expressions. The rules to do so are the following

• 1. Label all quasi-particle lines and associate the appropriate factor to each vertex, i.e. a factor ${{\textbf {z}}}^*_{l_1l_2}(l)$ (${{\textbf {z}}}^{k_1k_2}(r)$) to each vertex ${{\textbf {L}}}$ (${{\textbf {R}}}$) and a factor ${{\textbf {o}}}^{k_1 \cdots k_i}_{l_1 \cdots l_j}$ to the vertex ${{\textbf {O}}}^{ij}$, respectively.

• 2. Sum over all internal line labels.

• 3. Include a factor $(n_e!)^{-1}$ for each set of $n_e$ equivalent internal lines. Equivalent internal lines are those connecting identical vertices.

• 4. For any topologically distinct unlabelled diagram $\varGamma$, a symmetry factor $S_{\varGamma }^{-1}$ must be considered. Given a labelled version of $\varGamma$, i.e. a version in which each operator ${{\textbf {L}}}$ and ${{\textbf {R}}}$ carries a specific label, $S_{\varGamma }$ is equal to the number of permutations of the ${{\textbf {L}}}$ and ${{\textbf {R}}}$ operators delivering a topologically equivalent labelled diagram. The most obvious cases correspond to equivalent subgroups of ${{\textbf {L}}}$ and ${{\textbf {R}}}$ operators whose overall permutations lead to topologically equivalent labelled diagrams. The simplest example concerns two ${{\textbf {L}}}$ (${{\textbf {R}}}$) operators that are doubly connected to ${{\textbf {O}}}^{ij}$ or singly connected to ${{\textbf {O}}}^{ij}$ and to the same operator ${{\textbf {R}}}$ (${{\textbf {L}}}$). These $n_o\equiv 2$ operators ${{\textbf {L}}}$ (${{\textbf {R}}}$) are equivalent and contribute a factor 2! to $S_{\varGamma }$. The next simplest example corresponds to $n_o>2$ operators ${{\textbf {L}}}$ (${{\textbf {R}}}$) fully connected to ${{\textbf {O}}}^{ij}$. These $n_o>2$ operators are indeed equivalent²⁴ such that their permutations contribute a factor $n_o!$ to $S_{\varGamma }$. Beyond those two examples, a set of ${{\textbf {L}}}$ and ${{\textbf {R}}}$ operators can form a string of contractions that is equivalent to other identical strings. Such $n_s$ strings are said to be equivalent and contribute a factor $n_s!$ to $S_{\varGamma }$. Eventually, less obvious permutations can deliver topologically equivalent labelled diagrams and thus contribute to $S_{\varGamma }$²⁵.

• 5. Provide the diagram with a sign $(-1)^{\ell _c}$, where $\ell _c$ is the number of line crossings in the diagram. For diagrams containing external lines, their potential crossing must be counted.

Appendix C: Asymmetric approach

The asymmetric approach constitutes the standard path to the off-diagonal Wick theorem at play in the computation of the connected operator kernel [2, 18]. Employing the simplified notation

$$\begin{aligned} {{\textbf {R}}}&\equiv {{\textbf {Z}}}^{20}(l,r) =\frac{1}{2}\sum _{k_1k_2}{{\textbf {z}}}^{k_1k_2}(l,r)\beta ^\dagger _{k_1}(l)\beta ^\dagger _{k_2}(l)\,, \end{aligned}$$

(C.14)

for the Thouless operator introduced in Eq. (18b) and satisfying $\langle \varPhi (l) | {{\textbf {R}}} = 0$, the connected operator kernel reads as

$$\begin{aligned} \frac{\langle \varPhi (l) | O | \varPhi (r) \rangle }{\langle \varPhi (l) | \varPhi (r) \rangle }&\equiv \sum _{n=0}^N\sum _{\begin{array}{c} i,j=0\\ i+j=2n \end{array}}^{2n} \langle \varPhi (l) |{{\textbf {O}}}^{ij} e^{{{\textbf {R}}}} | \varPhi (l) \rangle \, , \nonumber \\&= \sum _{n=0}^N\sum _{\begin{array}{c} i,j=0\\ i+j=2n \end{array}}^{2n} \langle \varPhi (l) | ^{R}{{{\textbf {O}}}^{ij}} | \varPhi (l) \rangle \,, \end{aligned}$$

(C.15)

where the operator

$$\begin{aligned} ^{R}{{{\textbf {O}}}^{ij}}&\equiv e^{-{{\textbf {R}}}} {{\textbf {O}}}^{ij} e^{{{\textbf {R}}}} \nonumber \\&= \frac{1}{i!j!}\sum _{\begin{array}{c} k_1 \cdots k_i\\ l_1 \cdots l_j \end{array}} {{\textbf {o}}}^{k_1\cdots k_i}_{l_1\cdots l_j} \, ^R\beta ^{\dagger }_{k_1} \cdots ^R\beta ^{\dagger }_{k_i} \, ^R\beta _{l_j} \cdots ^R\beta _{l_1} \, , \end{aligned}$$

(C.16)

formally reads as ${{\textbf {O}}}^{ij}$ but with the quasi-particle operators replaced by their similarity-transformed partners

$$\begin{aligned} \begin{pmatrix} ^R\beta _k\\ ^R\beta ^{\dagger }_k \end{pmatrix}&\equiv e^{-{{\textbf {R}}}}\begin{pmatrix} \beta _k\\ \beta ^\dagger _k \end{pmatrix} e^{{{\textbf {R}}}} \, . \end{aligned}$$

(C.17)

Given that the similarity-transformed quasi-particle operators satisfy anticommutation relations

$$\begin{aligned} \{^R\beta ^{\dagger }_{k_1},^R\beta ^{\dagger }_{k_2}\}&= e^{-{{\textbf {R}}}}\{\beta ^{\dagger }_{k_1},\beta ^{\dagger }_{k_2}\}e^{{{\textbf {R}}}} = 0\, , \end{aligned}$$

(C.18a)

$$\begin{aligned} \{^R\beta _{k_1},^R\beta _{k_2}\}&= e^{-{{\textbf {R}}}}\{\beta _{k_1},\beta _{k_2}\}e^{{{\textbf {R}}}} = 0 \, , \end{aligned}$$

(C.18b)

$$\begin{aligned} \{^R\beta _{k_1},^R\beta ^{\dagger }_{k_2}\}&= e^{-{{\textbf {R}}}}\{\beta _{k_1},\beta ^{\dagger }_{k_2}\}e^{{{\textbf {R}}}} = \delta _{k_1k_2}\, , \end{aligned}$$

(C.18c)

standard Wick’s theorem with respect to $| \varPhi (l) \rangle$ applies and can be used to compute the matrix elements entering the right-hand side of Eq. (C.15). This results into the standard set of fully contracted terms, except that the elementary contractions at play do not involve the original quasi-particle operators but rather the similarity-transformed ones. The latter are related to the former via a non-unitary Bogoliubov transformation that is now detailed to compute the relevant elementary contractions.

Using Baker-Campbell-Hausdorff (BCH) identity, one first evaluates Eq. (C.17) according to

$$\begin{aligned} ^R\beta ^{(\dagger )}_{k_1}&= \beta ^{(\dagger )}_{k_1} - [{{\textbf {R}}},\beta ^{(\dagger )}_{k_1}] + \frac{1}{2!} [{{\textbf {R}}},[{{\textbf {R}}},\beta ^{(\dagger )}_{k_1}]] + \cdots \, . \end{aligned}$$

(C.19)

Given the two elementary commutators

$$\begin{aligned} \Big [ \beta ^{\dagger }_{k}(l) \beta ^{\dagger }_{k'}(l) , \beta _{k_1}(l) \Big ]&= \beta ^{\dagger }_{k}(l) \delta _{k' k_1} - \beta ^{\dagger }_{k'}(l) \delta _{k k_1} \, , \end{aligned}$$

(C.20a)

$$\begin{aligned} \Big [ \beta ^{\dagger }_{k}(l) \beta ^{\dagger }_{k'}(l) , \beta ^{\dagger }_{k_1} (l)\Big ]&= 0 \,, \end{aligned}$$

(C.20b)

it is straightforward to prove

$$\begin{aligned} \Big [ {{\textbf {R}}} , \beta _{k_1} \Big ]&= -\sum _{k_2} \Big [U(l) {{\textbf {z}}}(l,r)\Big ]_{k_1k_2} \beta ^{\dagger }_{k_2} (l)\, , \\ \Big [ {{\textbf {R}}}, \Big [ {{\textbf {R}}} , \beta _{k_1} \Big ] \Big ]&= 0 \, , \\ \Big [ {{\textbf {R}}}, \ldots \Big [ {{\textbf {R}}} , \beta _{k_1} \Big ] \ldots \Big ]&= 0 \, , \\ \Big [ {{\textbf {R}}} , \beta ^{\dagger }_{k_1} \Big ]&= -\sum _{k_2} \Big [V(l) {{\textbf {z}}}(l,r)\Big ]_{k_1k_2} \beta ^{\dagger }_{k_2} (l) \, , \\ \Big [ {{\textbf {R}}}, \Big [ {{\textbf {R}}} , \beta ^{\dagger }_{k_1} \Big ] \Big ]&= 0 \, , \\ \Big [ {{\textbf {R}}}, \ldots \Big [ {{\textbf {R}}} , \beta ^{\dagger }_{k_1} \Big ] \ldots \Big ]&= 0 \, , \end{aligned}$$

such that

$$\begin{aligned} ^R\beta _{k_1} =&\sum _{k_2} U_{k_1k_2}(l) \beta _{k_2} (l) \nonumber \\&+ \Big [V^{*}(l) + U(l) {{\textbf {z}}}(l,r)\Big ]_{k_1k_2} \beta ^{\dagger }_{k_2} (l) \, , \end{aligned}$$

(C.21a)

$$\begin{aligned} ^R\beta ^{\dagger }_{k_1} =&\sum _{k_2} V_{k_1k_2}(l) \beta _{k_2} (l) \nonumber \\&+ \Big [U^{*}(l) + V(l) {{\textbf {z}}}(l,r)\Big ]_{k_1k_2} \beta ^{\dagger }_{k_2} (l) \, , \end{aligned}$$

(C.21b)

which can be compacted in matrix form according to the non-unitary Bogoliubov transformation

$$\begin{aligned} \begin{pmatrix} ^R\beta \\ ^R\beta ^{\dagger } \end{pmatrix}&= \begin{pmatrix} U(l) &{} V^{*}(l) + U(l) {{\textbf {z}}}(l,r) \\ V(l) &{} U^{*}(l) + V(l) {{\textbf {z}}}(l,r) \end{pmatrix} \begin{pmatrix} \beta (l)\\ \beta ^\dagger (l) \end{pmatrix} \, . \end{aligned}$$

(C.22)

As Eqs. (C.21–C.22) testify, the infinite expansion in Eq. (C.19), originating from the presence of $e^{{{\textbf {R}}}}$ in Eq. (23), naturally terminates, i.e. it stops after two terms. Eventually, the four elementary contractions read as

$$\begin{aligned} {\rho }_{k_1k_2}(l,r)&= \langle \varPhi (l) | ^R{\beta ^{\dagger }_{k_2}} \, ^R{\beta _{k_1}} | \varPhi (l) \rangle \nonumber \\&= \left[ V^{*}(l)V^T(l) + U(l){{\textbf {z}}}(l,r)V^T(l)\right] _{k_1k_2} \nonumber \\&= +{\rho }_{k_1k_2}(l,l) + \left[ U(l){{\textbf {z}}}(l,r)V^T(l)\right] _{k_1k_2} \, , \end{aligned}$$

(C.23a)

$$\begin{aligned} {\kappa }_{k_1k_2}(l,r)&= \langle \varPhi (l) | ^R{\beta _{k_2}} \, ^R{\beta _{k_1}} | \varPhi (l) \rangle \nonumber \\&= \left[ V^{*}(l)U^T(l) + U(l){{\textbf {z}}}(l,r)U^T(l)\right] _{k_1k_2} \nonumber \\&= +{\kappa }_{k_1k_2}(l,l) + \left[ U(l){{\textbf {z}}}(l,r)U^T(l)\right] _{k_1k_2} \, \end{aligned}$$

(C.23b)

$$\begin{aligned} -\bar{{\kappa }}^{*}_{k_1k_2}(l,r)&= \langle \varPhi (l) | ^R{\beta ^{\dagger }_{k_2}} \, ^R{\beta ^{\dagger }_{k_1}} | \varPhi (l) \rangle \nonumber \\&= \left[ U^{*}(l)V^T(l) + V(l){{\textbf {z}}}(l,r)V^T(l)\right] _{k_1k_2} \nonumber \\&= -\bar{{\kappa }}^{*}_{k_1k_2}(l,l) + \left[ V(l){{\textbf {z}}}(l,r)V^T(l)\right] _{k_1k_2} \, , \end{aligned}$$

(C.23c)

$$\begin{aligned} -{\sigma }^{*}_{k_1k_2}(l,r)&= \langle \varPhi (l) | ^R{\beta _{k_2}} \, ^R{\beta ^{\dagger }_{k_1}} | \varPhi (l) \rangle \nonumber \\&= \left[ U^{*}(l)U^T(l) + V(l){{\textbf {z}}}(l,r)U^T(l)\right] _{k_1k_2} \nonumber \\&= -{\sigma }^{*}_{k_1k_2}(l,l) + \left[ V(l){{\textbf {z}}}(l,r)U^T(l)\right] _{k_1k_2}, \end{aligned}$$

(C.23d)

where Eqs. (C.21–C.22) have been used. This completes the derivation of the off-diagonal Wick theorem where the explicit form of the elementary off-diagonal contractions in Eq. (C.23) reflects the asymmetric character of the approach, i.e. the expressions are anchored on the bra state $\langle \varPhi (l) |$ and are a functional of the Thouless matrix ${{\textbf {z}}}(l,r)$ associated with the transition Bogoliubov transformation of Eqs. (15–17).

Starting from Eq. (C.23) and using repeatedly relations associated with the unitarity of W(l) (Eq. (10)), one can symmetrize the elementary contractions by expressing them in terms of the Thouless matrices ${{\textbf {z}}}(l)$ and ${{\textbf {z}}}(r)$ associated withright states, respectively. Doing so, one recovers exactly Eqs. (39) and (41) obtained directly via the symmetric approach.

Appendix D: Connected kernel of a rank-3 operator

According to Eq. (38), the connected kernel associated with a rank-3 operator $O\equiv {{\textbf {O}}}^{[0]}+{{\textbf {O}}}^{[2]}+{{\textbf {O}}}^{[4]}+{{\textbf {O}}}^{[6]}$ reads in terms of the off-diagonal elementary contractions as

$$\begin{aligned}&\frac{\langle \varPhi (l)| O |\varPhi (r) \rangle }{\langle \varPhi (l)|\varPhi (r) \rangle } = {{\textbf {O}}}^{[0]}\ + \frac{1}{2} \sum _{k_1k_2} {{\textbf {o}}}^{k_1k_2} \bar{{\kappa }}^{*}_{k_1k_2}(l,r) + \sum _{k_1l_1} {{\textbf {o}}}^{k_1}_{l_1} \rho _{l_1k_1}(l,r) + \frac{1}{2} \sum _{l_1l_2} {{\textbf {o}}}_{l_1l_2} {\kappa }_{l_1l_2}(l,r) \nonumber \\&\quad + \frac{1}{8} \sum _{k_1k_2k_3k_4} {{\textbf {o}}}^{k_1k_2k_3k_4} \bar{{\kappa }}^{*}_{k_1k_2}(l,r) \bar{{\kappa }}^{*}_{k_3k_4} (l,r)+ \frac{1}{2} \sum _{k_1k_2k_3l_1} {{\textbf {o}}}^{k_1k_2k_3}_{l_1} \rho _{l_1k_1}(l,r) \bar{{\kappa }}^{*}_{k_2k_3}(l,r) \nonumber \\&\quad + \frac{1}{2} \sum _{k_1k_2l_1l_2} {{\textbf {o}}}^{k_1k_2}_{l_1l_2} \rho _{l_1k_1}(l,r) \rho _{l_2k_2}(l,r) + \frac{1}{4} \sum _{k_1k_2l_1l_2} {{\textbf {o}}}^{k_1k_2}_{l_1l_2}\bar{{\kappa }}^{*}_{k_1k_2}(l,r){\kappa }_{l_1l_2} (l,r)\nonumber \\&\quad + \frac{1}{2} \sum _{k_1l_1l_2l_3} {{\textbf {o}}}^{k_1}_{l_1l_2l_3} \rho _{l_1k_1} (l,r) {\kappa }_{l_2l_3}(l,r) + \frac{1}{8} \sum _{l_1l_2l_3l_4} {{\textbf {o}}}_{l_1l_2l_3l_4} {\kappa }_{l_1l_2}(l,r) {\kappa }_{l_3l_4}(l,r)\nonumber \\&\quad + \frac{1}{48} \sum _{k_1k_2k_3k_4k_5k_6} {{\textbf {o}}}^{k_1k_2k_3k_4k_5k_6} \bar{{\kappa }}^{*}_{k_1k_2}(l,r) \bar{{\kappa }}^{*}_{k_3k_4} (l,r) \bar{{\kappa }}^{*}_{k_5k_6} (l,r) \nonumber \\&\quad + \frac{1}{8} \sum _{k_1k_2k_3k_4k_5l_1} {{\textbf {o}}}^{k_1k_2k_3k_4k_5}_{l_1} \rho _{l_1k_1}(l,r) \bar{{\kappa }}^{*}_{k_2k_3}(l,r) \bar{{\kappa }}^{*}_{k_4k_5}(l,r) \nonumber \\&\quad + \frac{1}{16} \sum _{k_1k_2k_3k_4l_1l_2} {{\textbf {o}}}^{k_1k_2k_3k_4}_{l_1l_2} \bar{{\kappa }}^{*}_{k_1k_2}(l,r) \bar{{\kappa }}^{*}_{k_3k_4}(l,r) \kappa _{l_1l_2} (l,r)\nonumber \\&\quad + \frac{1}{4} \sum _{k_1k_2k_3k_4l_1l_2} {{\textbf {o}}}^{k_1k_2k_3k_4}_{l_1l_2} \rho _{l_1k_1}(l,r) \rho _{l_2k_2}(l,r) \bar{{\kappa }}^{*}_{k_3k_4} (l,r) \nonumber \\&\quad + \frac{1}{6} \sum _{k_1k_2k_3l_1l_2l_3} {{\textbf {o}}}^{k_1k_2k_3}_{l_1l_2l_3} \rho _{l_1k_1}(l,r) \rho _{l_2k_2}(l,r) \rho _{l_3k_3}(l,r)\nonumber \\&\quad + \frac{1}{4} \sum _{k_1k_2k_3l_1l_2l_3} {{\textbf {o}}}^{k_1k_2k_3}_{l_1l_2l_3} \rho _{l_1k_1}(l,r) \bar{{\kappa }}^{*}_{k_2k_3}(l,r) \kappa _{l_2l_3}(l,r) \nonumber \\&\quad + \frac{1}{16} \sum _{k_1k_2l_1l_2l_3l_4} {{\textbf {o}}}^{k_1k_2}_{l_1l_2l_3l_4} \bar{{\kappa }}^{*}_{k_1k_2}(l,r) \kappa _{l_1l_2} (l,r) \kappa _{l_3l_4} (l,r)\nonumber \\&\quad + \frac{1}{4} \sum _{k_1k_2l_1l_2l_3l_4} {{\textbf {o}}}^{k_1k_2}_{l_1l_2l_3l_4} \rho _{l_1k_1}(l,r) \rho _{l_2k_2}(l,r) \kappa _{l_3l_4}(l,r) \nonumber \\&\quad + \frac{1}{8} \sum _{k_1l_1l_2l_3l_4l_5} {{\textbf {o}}}^{k_1}_{l_1l_2l_3l_4l_5} \rho _{l_1k_1}(l,r) \kappa _{l_2l_3}(l,r) \kappa _{l_4l_5}(l,r) \nonumber \\&\quad +\frac{1}{48} \sum _{l_1l_2l_3l_4l_5l_6} {{\textbf {o}}}_{l_1l_2l_3l_4l_5l_6} \kappa _{l_1l_2}(l,r) \kappa _{l_3l_4}(l,r) \kappa _{l_5l_6}(l,r). \end{aligned}$$

(D.24)