In-Medium Similarity Renormalization Group Approach to the Nuclear Many-Body Problem

Abstract

We present a pedagogical discussion of Similarity Renormalization Group (SRG) methods, in particular the In-Medium SRG (IMSRG) approach for solving the nuclear many-body problem. These methods use continuous unitary transformations to evolve the nuclear Hamiltonian to a desired shape. The IMSRG, in particular, is used to decouple the ground state from all excitations and solve the many-body Schrödinger equation. We discuss the IMSRG formalism as well as its numerical implementation, and use the method to study the pairing model and infinite neutron matter. We compare our results with those of Coupled cluster theory (Chap. 8), Configuration-Interaction Monte Carlo (Chap. 9), and the Self-Consistent Green’s Function approach discussed in Chap. 11 The chapter concludes with an expanded overview of current research directions, and a look ahead at upcoming developments.

Appendix: Products and Commutators of Normal-Ordered Operators

In this appendix, we collect the basic expressions for products and commutators of normal-ordered one- and two-body operators. All single-particle indices refer to the natural orbital basis, where the one-body density matrix is diagonal

$$\displaystyle{ \rho _{kl} = \langle \varPhi \vert \,a_{l}^{\dag }a_{ k}\,\vert \varPhi \rangle = n_{k}\delta _{kl}\,,\quad n_{k} \in \{ 0,1\}\,, }$$

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(notice the convention for the indices of ρ, cf. [117]). We also define the hole density matrix

$$\displaystyle{ \bar{\rho }_{kl} \equiv \langle \varPhi \vert \,a_{k}a_{l}^{\dag }\,\vert \varPhi \rangle =\delta _{ kl} -\rho _{kl} \equiv \bar{n}_{k}\delta _{kl} }$$

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whose eigenvalues are

$$\displaystyle{ \bar{n}_{k} = 1 - n_{k}\,, }$$

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i.e., 0 for occupied and 1 for unoccupied single-particle states. Finally, we will again use the permutation symbol $\hat{P}_{ij}$ to interchange the indices in any expression,

$$\displaystyle{ \hat{P}_{ij}g(\ldots,i,\ldots,j) \equiv g(\ldots,j,\ldots,i)\, }$$

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(see Sect. 10.3 and Chap. 8).

10.1.1 Operator Products

$$\displaystyle{ \big\{a_{a}^{\dag }a_{ b}\big\}\big\{a_{k}^{\dag }a_{ l}\big\} =\big\{ a_{a}^{\dag }a_{ k}^{\dag }a_{ l}a_{b}\big\} - n_{a}\delta _{al}\big\{a_{k}^{\dag }a_{ b}\big\} + \bar{n}_{b}\delta _{bk}\big\{a_{a}^{\dag }a_{ l}\big\} + n_{a}\bar{n}_{b}\delta _{al}\delta _{bk} }$$

(10.245)

$$\displaystyle\begin{array}{rcl} \big\{a_{a}^{\dag }a_{ b}\big\}\big\{a_{k}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}\big\}& =& \big\{a_{a}^{\dag }a_{ k}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}a_{b}\big\} + \left (1 -\hat{P}_{mn}\right )n_{a}\delta _{an}\big\{a_{k}^{\dag }a_{ l}^{\dag }a_{ m}a_{b}\big\} \\ & & -\left (1 -\hat{P}_{kl}\right )\bar{n}_{b}\delta _{bl}\big\{a_{a}^{\dag }a_{ k}^{\dag }a_{ n}a_{m}\big\} \\ & & -\left (1 -\hat{P}_{kl}\right )\left (1 -\hat{P}_{mn}\right )n_{a}\bar{n}_{b}\delta _{am}\delta _{bl}\big\{a_{k}^{\dag }a_{ n}\big\} {}\end{array}$$

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$$\displaystyle\begin{array}{rcl} & & \big\{a_{a}^{\dag }a_{ b}^{\dag }a_{ d}a_{c}\big\}\big\{a_{k}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}\big\} \\ & & =\big\{ a_{a}^{\dag }a_{ b}^{\dag }a_{ k}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}a_{d}a_{c}\big\} \\ & & + (1 -\hat{P}_{ab})(1 -\hat{P}_{mn})n_{a}\delta _{am}\big\{a_{b}^{\dag }a_{ k}^{\dag }a_{ l}^{\dag }a_{ n}a_{d}a_{c}\big\} \\ & & -(1 -\hat{P}_{cd})(1 -\hat{P}_{kl})\bar{n}_{c}\delta _{ck}\big\{a_{a}^{\dag }a_{ b}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}a_{d}\big\} \\ & & + (1 -\hat{P}_{mn})n_{a}n_{b}\delta _{am}\delta _{bn}\big\{a_{k}^{\dag }a_{ l}^{\dag }a_{ d}a_{c}\big\} + (1 -\hat{P}_{cd})\bar{n}_{c}\bar{n}_{d}\delta _{ck}\delta _{dl}\big\{a_{a}^{\dag }a_{ b}^{\dag }a_{ n}a_{m}\big\} \\ & & + (1 -\hat{P}_{ab})(1 -\hat{P}_{cd})(1 -\hat{P}_{kl})(1 -\hat{P}_{mn})n_{a}\bar{n}_{c}\delta _{am}\delta _{ck}\big\{a_{b}^{\dag }a_{ l}^{\dag }a_{ n}a_{d}\big\} \\ & & + (1 -\hat{P}_{ab})(1 -\hat{P}_{kl})(1 -\hat{P}_{mn})n_{b}\bar{n}_{c}\bar{n}_{d}\delta _{bn}\delta _{ck}\delta _{dl}\big\{a_{a}^{\dag }a_{ m}\big\} \\ & & + (1 -\hat{P}_{cd})(1 -\hat{P}_{kl})(1 -\hat{P}_{mn})\bar{n}_{d}n_{a}n_{b}\delta _{dl}\delta _{an}\delta _{bm}\big\{a_{k}^{\dag }a_{ c}\big\} \\ & & + (1 -\hat{P}_{kl})(1 -\hat{P}_{mn})n_{a}n_{b}\bar{n}_{c}\bar{n}_{d}\delta _{am}\delta _{bn}\delta _{ck}\delta _{dl} {}\end{array}$$

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10.1.2 Commutators

$$\displaystyle{ [\big\{a_{a}^{\dag }a_{ b}\big\},\big\{a_{k}^{\dag }a_{ l}\big\}] =\delta _{bk}\big\{a_{a}^{\dag }a_{ l}\big\} -\delta _{al}\big\{a_{k}^{\dag }a_{ b}\big\} + (n_{a} - n_{b})\delta _{al}\delta _{bk} }$$

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$$\displaystyle\begin{array}{rcl} [\big\{a_{a}^{\dag }a_{ b}\big\},\big\{a_{k}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}\big\}]& =& \left (1 -\hat{P}_{kl}\right )\delta _{bk}\big\{a_{a}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}\big\} -\left (1 -\hat{P}_{mn}\right )\delta _{am}\big\{a_{k}^{\dag }a_{ l}^{\dag }a_{ n}a_{b}\big\} \\ & & +\left (1 -\hat{P}_{kl}\right )\left (1 -\hat{P}_{mn}\right )(n_{a} - n_{b})\delta _{an}\delta _{bl}\big\{a_{k}^{\dag }a_{ m}\big\} {}\end{array}$$

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$$\displaystyle\begin{array}{rcl} & & [\big\{a_{a}^{\dag }a_{ b}^{\dag }a_{ d}a_{c}\big\},\big\{a_{k}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}\big\}] \\ & & \quad = \left (1 -\hat{P}_{ab}\right )\left (1 -\hat{P}_{mn}\right )\delta _{am}\big\{a_{b}^{\dag }a_{ k}^{\dag }a_{ l}^{\dag }a_{ n}a_{d}a_{c}\big\} \\ & & \qquad -\left (1 -\hat{P}_{cd}\right )\left (1 -\hat{P}_{kl}\right )\delta _{kc}\big\{a_{a}^{\dag }a_{ b}^{\dag }a_{ l}^{\dag }a_{ n}a_{m}a_{d}\big\} \\ & & \qquad + (1 -\hat{P}_{cd})\left (\bar{n}_{c}\bar{n}_{d} - n_{c}n_{d}\right )\delta _{ck}\delta _{dl}\big\{a_{a}^{\dag }a_{ b}^{\dag }a_{ n}a_{m}\big\} \\ & & \qquad + (1 -\hat{P}_{ab})\left (n_{a}n_{b} -\bar{n}_{a}\bar{n}_{b}\right )\delta _{am}\delta _{bn}\big\{a_{k}^{\dag }a_{ l}^{\dag }a_{ d}a_{c}\big\} \\ & & \qquad + (1 -\hat{P}_{ab})(1 -\hat{P}_{cd})(1 -\hat{P}_{kl})(1 -\hat{P}_{mn})\left (n_{b} - n_{d}\right )\delta _{bn}\delta _{dl}\big\{a_{a}^{\dag }a_{ k}^{\dag }a_{ m}a_{c}\big\} \\ & & \qquad + (1 -\hat{P}_{ab})(1 -\hat{P}_{mn})\left (n_{b}\bar{n}_{c}\bar{n}_{d} -\bar{n}_{b}n_{c}n_{d}\right )\delta _{bn}\delta _{ck}\delta _{dl}\big\{a_{a}^{\dag }a_{ m}\big\} \\ & & \qquad - (1 -\hat{P}_{cd})(1 -\hat{P}_{kl})\left (n_{d}\bar{n}_{a}\bar{n}_{b} -\bar{n}_{d}n_{a}n_{b}\right )\delta _{dl}\delta _{am}\delta _{bn}\big\{a_{k}^{\dag }a_{ c}\big\} \\ & & \qquad + (1 -\hat{P}_{ab})(1 -\hat{P}_{cd})\left (n_{a}n_{b}\bar{n}_{c}\bar{n}_{d} -\bar{n}_{a}\bar{n}_{b}n_{c}n_{d}\right )\delta _{am}\delta _{bn}\delta _{ck}\delta _{dl} {}\end{array}$$

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