Importance truncation in non-perturbative many-body techniques

Abstract

Expansion many-body methods correspond to solving complex tensor networks. The (iterative) solving of the network and the (repeated) storage of the unknown tensors requires a computing power growing polynomially with the size of basis of the one-body Hilbert space one is working with. Thanks to current computer capabilities, ab initio calculations of nuclei up to mass $A\sim 100$ delivering a few percent accuracy are routinely feasible today. However, the runtime and memory costs become quickly prohibitive as one attempts (possibly at the same time) (i) to reach out to heavier nuclei, (ii) to employ symmetry-breaking reference states to access open-shell nuclei and (iii) to aim for yet a greater accuracy. The challenge is particularly exacerbated for non-perturbative methods involving the repeated storage of (high-rank) tensors obtained via iterative solutions of non-linear equations. The present work addresses the formal and numerical implementations of so-called importance truncation (IT) techniques within the frame of one particular non-perturbative expansion method, i.e., Gorkov Self-Consistent Green’s Function (GSCGF) theory, with the goal to eventually overcome above-mentioned limitations. By a priori truncating irrelevant tensor entries, IT techniques are shown to reduce the storage to less than 1% of its original cost in realistic GSCGF calculations performed at the ADC(2) level while maintaining a 1% accuracy on the correlation energy. The future steps will be to extend the present development to the next, e.g., ADC(3), truncation level and to SCGF calculations applicable to doubly open-shell nuclei.

Appendix A: Energy-driven measure

1.1 Appendix A.1: Interaction and correlation ground-state energies

The objective is to define an energy-driven measure based on perturbation theory and expressed in terms of the tensors at play in ADC(2) Gorkov SCGF calculations. Let us thus introduce

$$\begin{aligned} \varOmega (\lambda )\equiv \varOmega _0+\lambda \varOmega _1, \end{aligned}$$

(A.1)

where $\lambda$ is the perturbation parameter leading the system from the uncorrelated state ($\lambda =0$) to the fully correlated (interacting) one ($\lambda =1$). The unperturbed grand potential $\varOmega _0$ is taken to be of one-body character such that its ground-state is the Bogoliubov product state $| \varPhi _0 \rangle$, i.e.,

$$\begin{aligned} \varOmega _0 | \varPhi _0 \rangle = \varOmega ^{0}_0| \varPhi _0 \rangle . \end{aligned}$$

(A.2)

Correspondingly, the ground state of $\varOmega (\lambda )$ fulfils Schrödinger’s equation

$$\begin{aligned} \varOmega (\lambda )| \varPsi _0(\lambda ) \rangle =\varOmega _0(\lambda )| \varPsi _0(\lambda ) \rangle . \end{aligned}$$

(A.3)

From Eqs. (A.2) and (A.3), one obtains

$$\begin{aligned} \varOmega _0(\lambda )&=\langle \varPsi _0(\lambda ) |\varOmega (\lambda )| \varPsi _0(\lambda ) \rangle \nonumber \\&=\langle \varPsi _0(\lambda ) |\varOmega _0| \varPsi _0(\lambda ) \rangle +\lambda \langle \varPsi _0(\lambda ) |\varOmega _1| \varPsi _0(\lambda ) \rangle \nonumber \\&\equiv \varOmega _{\text {noint}} (\lambda )+ \varOmega _{\text {int}}(\lambda ) . \end{aligned}$$

(A.4)

as well as

$$\begin{aligned} \varOmega _0(\lambda )&= \frac{\langle \varPhi _0|\varOmega |\varPsi _0(\lambda )\rangle }{\langle \varPhi _0|\varPsi _0(\lambda )\rangle } \nonumber \\&=\frac{\langle \varPhi _0|\varOmega _0|\varPsi _0(\lambda )\rangle }{\langle \varPhi _0|\varPsi _0(\lambda )\rangle } + \lambda \frac{\langle \varPhi _0|\varOmega _1|\varPsi _0(\lambda )\rangle }{\langle \varPhi _0|\varPsi _0(\lambda )\rangle } \nonumber \\&\equiv \varOmega _0^0 + \varDelta \varOmega _0(\lambda ) . \end{aligned}$$

(A.5)

The two above equations define the so-called interaction and correlation energies²⁰, respectively, such that $\varOmega _{\text {int}}(1)\equiv \varOmega _{\text {int}}$ and $\varDelta \varOmega _0(1)\equiv \varDelta \varOmega _0$ for the fully interacting system.

1.2 Appendix A.2: Perturbative expansion of $\varDelta \varOmega _0$

The correlation energy is amenable to a perturbative expansion on the basis of Goldstone’s theorem [51, 52]

$$\begin{aligned} \varDelta \varOmega _0=\langle \varPhi _0 |\varOmega _1\sum _{n=0}^{\infty }\bigg (\frac{1}{\varOmega _0^0-\varOmega _0}H_1\bigg )^n| \varPhi _0 \rangle _{C}, \end{aligned}$$

(A.6)

where the index C denotes the fact that only so-called connected terms are retained. However, this expression does not explicitly relate the correlation energy to the propagator, the self energy and thus the many-body tensors at play in Gorkov SCGF calculations.

1.3 Appendix A.3: Linking $\varDelta \varOmega _0$ to the self energy

To achieve the needed connection, one must first exploit Hellmann-Feynman’s theorem [53] to relate the correlation energy to the interaction energy according to

$$\begin{aligned} \varDelta \varOmega _0(\lambda )=\int _0^{\lambda }\frac{d\lambda '}{\lambda '}\varOmega _{\text {int}}(\lambda '), \end{aligned}$$

(A.7)

and further exploit the link the interaction energy entertains with the one-body propagator.

Expressing $\varOmega _{\text {int}}(\lambda )$ in terms of the two-body density matrix and exploiting the definition of the one-body self-energy connecting the one-body Green’s function to the two-times reduction of the two-body Green’s function²¹, one easily obtains

$$\begin{aligned} \varOmega _{\text {int}}(\lambda )= -\frac{1}{2} \text {Tr}\,\bigl \{\varvec{\varSigma }(\lambda )\,\varvec{\mathcal {G}}(\lambda )\bigr \}, \end{aligned}$$

(A.8)

where the triple trace over the one-body Hilbert space, Nambu space and the frequency defined through²²

$$\begin{aligned} \text {Tr} \,\equiv -\text {Tr}_{\mathcal {H}_1}\frac{1}{2}\text {Tr}_{\mathcal {G}}\oint \frac{d\omega }{2\pi i}, \end{aligned}$$

(A.9)

has been used to shorten the expression. Substituting Eq. (A.8) into Eq. (A.7) one obtains

$$\begin{aligned} \varDelta \varOmega _0(\lambda )=-\frac{1}{2}\int _0^{\lambda }\frac{d\lambda '}{\lambda '}\text {Tr}\,\bigl \{\varvec{\varSigma }(\lambda ')\,\varvec{\mathcal {G}}(\lambda ')\bigr \}. \end{aligned}$$

(A.10)

Next, the self-energy is formally expanded in powers of the residual interaction according to²³

$$\begin{aligned} \varvec{\varSigma }(\lambda )=\sum _{n=1}^{\infty }\lambda ^n\varvec{\varSigma }_n(\lambda ), \end{aligned}$$

(A.11)

such that Eq. (A.10) becomes

$$\begin{aligned} \varDelta \varOmega _0(\lambda )=-\frac{1}{2}\sum _{n=1}^{\infty }\int _0^{\lambda }d\lambda '\,\lambda '^{n-1}\text {Tr}\, \varvec{\varSigma }_n(\lambda ')\varvec{\mathcal {G}}(\lambda '). \end{aligned}$$

(A.12)

Following standard steps [54], the integration over the coupling strength can be performed in Eq. (A.12) to eventually obtain²⁴

$$\begin{aligned} \varDelta \varOmega _0(\lambda )=\varPhi [\varvec{\mathcal {G}}(\lambda )]+ \text {Tr}\,\bigg \{\frac{\varvec{\mathcal {G}}(\lambda )}{\varvec{\mathcal {G}}_0}-1\bigg \}-\text {Tr}\,\bigg \{\ln \frac{\varvec{\mathcal {G}}(\lambda )}{\varvec{\mathcal {G}}_0}\bigg \}, \nonumber \\ \end{aligned}$$

(A.13)

where the Luttinger-Ward functional [55, 56] adapted to Gorkov’s formalism was introduced

$$\begin{aligned} \varPhi [\varvec{\mathcal {G}}(\lambda )]\equiv -\frac{1}{2}\sum _{n=1}^{\infty }\frac{\lambda ^n}{n}\text {Tr}\,\varvec{\varSigma }_n(\lambda )\varvec{\mathcal {G}}(\lambda ). \end{aligned}$$

(A.14)

Taking $\lambda = 1$ in Eq. (A.13), one obtains the fully interacting correlation energy $\varDelta \varOmega _0$ under the form of a functional of the fully interacting propagator $\varvec{\mathcal {G}}(\lambda )$. The $\varPhi$-functional $\varPhi [\varvec{\mathcal {G}}(\lambda )]$ invokes the convolution of $\varvec{\varSigma }_n(\lambda )$, itself a functional of $\varvec{\mathcal {G}}(\lambda )$, with $\varvec{\mathcal {G}}(\lambda )$. As a result, $\varPhi [\varvec{\mathcal {G}}(\lambda )]$ is made out of the complete set of two-particle irreducible skeleton vacuum-to-vacuum diagrams whose lines denote the fully interacting propagator $\varvec{\mathcal {G}}(\lambda )$.

1.4 Appendix A.4: Perturbative connection

In the present context, we are interested in a perturbative connection between the correlation energy and the one-body self energy. Starting from Eq. (A.13), it is easy to prove that $\varDelta \varOmega _0$ is stationary with respect to any variation of the propagator, i.e., $\varvec{\mathcal {G}}(\omega )$ solution of Gorkov’s equation delivers an extremum of $\varDelta \varOmega _0$.

Concretely, this variational property translates into the possibility to alter the employed propagator without significantly affecting the correlation energy. This property was numerically validated in Ref. [57]. Exploiting this feature, the correlation energy is presently approximated by evaluating its expression at the uncorrelated propagator rather at the interacting one, i.e., using

$$\begin{aligned} \varDelta \varOmega _0\simeq \varDelta \varOmega _0[\varvec{\mathcal {G}}_0]=\varPhi [\varvec{\mathcal {G}}_0]. \end{aligned}$$

(A.15)

Based on this approximation, a perturbation-theory-like series is obtained for the correlation energy such that the $n^\text {th}$ order reads as

$$\begin{aligned} \varDelta \varOmega _0^{(n)}[\varvec{\mathcal {G}}_0]=\varPhi ^{(n)}[\varvec{\mathcal {G}}_0]=-\frac{1}{2n}\, \text {Tr}\,\varvec{\varSigma }_n[\varvec{\mathcal {G}}_0]\,\varvec{\mathcal {G}}_0. \end{aligned}$$

(A.16)

One must note that Eqs. (A.6) and (A.15) do not provide, in general, the same expansion, i.e., while the first one is exact when summed to all orders, the second one is approximate. However, employing an uncorrelated propagator $\varvec{\mathcal {G}}_0$ associated with the solution of the HFB mean-field equations, one can prove that the two expansions match exactly up to second-order. This exact matching has been validated numerically and is used in the following to compute the second-order correlation energy within Gorkov SCGF formalism.

The last step of the derivation thus consists of exploiting Eq. (A.16) for $n=2$ and express the result in terms of the many-body tensors making up the ADC(2) self energy. For a spherically symmetric implementation, the following identities hold

$$\begin{aligned} \text {Tr}\{\varSigma ^{12}\mathcal {G}^{21}\}&=\text {Tr}\{\varSigma ^{21}\mathcal {G}^{12}\}, \end{aligned}$$

(A.17a)

$$\begin{aligned} \text {Tr}\{\varSigma ^{11}\mathcal {G}^{11}\}&=\text {Tr}\{\varSigma ^{22}\mathcal {G}^{22}\}, \end{aligned}$$

(A.17b)

eventually leading to the second-order correlation energy expression

$$\begin{aligned} \varDelta \varOmega _0^{(2)}[\varvec{\mathcal {G}}_0]=&\frac{1}{4}\sum _{ab}\oint \frac{d\omega }{2\pi 1}\bigl \{\varSigma ^{11(2)}_{ab}(\omega )\mathcal {G}_{0\,ba}^{11}(\omega ) \nonumber \\&+\varSigma ^{12(2)}_{ab}(\omega )\mathcal {G}_{0\,ba}^{21}(\omega )\bigr \} . \end{aligned}$$

(A.18)

Inserting Eqs. (9) and (13) into Eq. (A.18), the integration over the frequency can be performed to generate the needed form

$$\begin{aligned} \varDelta \varOmega _0^{(2)}[\varvec{\mathcal {G}}_0]&=\frac{1}{4}\sum _{\begin{array}{c} k_1k_2k_3k_4\\ ab \end{array}}\bigg \{\frac{{\mathcal {V}}_b^{k_4}{\mathcal {V}}_a^{k_4}\mathcal {C}_a^{k_1k_2k_3}\mathcal {C}_b^{k_1k_2k_3}}{E_{k_1k_2k_3}+\omega _{k_4}} \nonumber \\&\quad +\frac{\mathcal {U}_b^{k_4}\mathcal {U}_a^{k_4}{\mathcal {D}}_a^{k_1k_2k_3}{\mathcal {D}}_b^{k_1k_2k_3}}{E_{k_1k_2k_3}+\omega _{k_4}} \nonumber \\&\quad +\frac{{\mathcal {U}}_b^{k_4}{\mathcal {V}}_a^{k_4}\mathcal {C}_a^{k_1k_2k_3}\mathcal {D}_b^{k_1k_2k_3}}{E_{k_1k_2k_3}+\omega _{k_4}} \nonumber \\&\quad +\frac{\mathcal {V}_b^{k_4}\mathcal {U}_a^{k_4}{\mathcal {D}}_a^{k_1k_2k_3}{\mathcal {C}}_b^{k_1k_2k_3}}{E_{k_1k_2k_3}+\omega _{k_4}}\bigg \} \nonumber \\&=\frac{1}{4}\sum _{k_1k_2k_3k_4}\frac{\Big [\sum _a\big (\mathcal {V}_a^{k_4}\mathcal {C}_a^{k_1k_2k_3}+\mathcal {U}_a^{k_4}\mathcal {D}_a^{k_1k_2k_3}\big )\Big ]^2}{E_{k_1k_2k_3k_4}} \nonumber \\&=\sum _{k_1k_2k_3} e_{\,\,_{2}}^{\,\,_{\text {IT}}}(k_1k_2k_3) , \end{aligned}$$

(A.19)

where the last equality simply amounts to isolating three-pole contributions²⁵$(k_1k_2k_3)$ and proceeds to a trivial rewriting. The above formula thus makes explicit the relation between the correlation energy and the energy-driven IT measure introduced in Eq. (26).