On nuclear short-range correlations and the zero-energy eigenstates of the Schrödinger equation

Nuclear short-range correlations (SRC) have been studied extensively over the last few decades (see Refs. [1, 2] for recent reviews). Large momentum-transfer quasi-elastic electron and proton scattering reactions are the main experimental tools facilitating these studies [3, 4]. In such reactions, interpreted in a high resolution picture, back-to-back SRC nucleon pairs were clearly identified [5, 6, 7, 8, 9, 10], with a significant dominance of neutron-proton pairs [11, 12, 9, 13, 14]. Inclusive reactions where used to study the abundance of such SRC pairs [15, 16, 17, 18, 19]. Currently, ab-initio approaches are unable to directly calculate the cross sections of these reactions, in all but the lightest nuclei. Nevertheless, qualitatively similar conclusions were obtained in structure studies, that focused mainly on the high momentum tail of the nuclear momentum distribution [20, 21, 22, 23, 24, 25, 22, 26, 27, 28]. The study of nuclear three-body SRCs, i.e. three nucleons at close proximity, is still very preliminary at this stage [29, 30] and their impact on nuclear quantities is still mostly unknown.

Following Tan’s work on ultra-cold atoms [31, 32, 33, 34], the Generalized Contact Formalism (GCF) was introduced and utilized to analyze SRC effects in nuclei [35, 36, 37, 38]. It is based on the scale separation ansatz, assuming a factorization of the nuclear wave-function when two nucleons are close to each other. The GCF provides a framework to study both nuclear structure and nuclear reactions, and was successfully tested against ab-initio studies, providing a good description of both two-body densities at short distance and high-momentum tails of different momentum distributions [36, 39, 40]. In addition, the GCF is found to be in good agreement with exclusive electron scattering experiments and other reactions sensitive to SRC pairs [37, 41, 42, 13, 43, 12, 13, 44, 14, 45]. As such, the GCF allows for a quantitative comparison between ab-initio calculations and experimental results, with direct connection to the underlying nuclear interaction. The GCF results lead to a comprehensive and consistent picture of nuclear SRCs, where the only tension is with respect to the analysis of inclusive reactions [46]. Recently, shell-model calculations were combined with the GCF to calculate nuclear matrix elements for neutrinoless double beta decay [47], taking into account both short-range and long-range contributions consistently.

As pointed out, the GCF is based on the asymptotic factorization ansatz for the many-body nuclear wave-function $\Psi$, when nucleon $i$ is close to nucleon $j$ [36]

$$\Psi\xrightarrow[r_{ij}\rightarrow 0]{}\sum_{\alpha}\varphi_{ij}^{\alpha}\left(\boldsymbol{r}_{ij}\right)A_{ij}^{\alpha}\left(\boldsymbol{R}_{ij},\{\boldsymbol{r}_{k}\}_{k\not=i,j}\right).$$

(1)

In this picture, particles $i$ and $j$ are strongly interacting, and, therefore, described by a two-body function $\varphi_{ij}^{\alpha}$, decoupled from the rest of the system, which is described by the function $A_{ij}^{\alpha}$. In the GCF, $\varphi_{ij}^{\alpha}$ is assumed to be universal, i.e. independent of the nucleus or its many-body state, and is defined to be the zero-energy solution of the two-body Schrödinger equation with quantum numbers $\alpha$, obtained with the same nucleon-nucleon interaction model used for the many-body wave-function. A similar factorization should hold in momentum space, for pairs with high relative momentum $\boldsymbol{k}_{ij}$

$\displaystyle\tilde{\Psi}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},...,\boldsymbol{k}_{A})\xrightarrow[k_{ij}\rightarrow\infty]{}\sum_{\alpha}\tilde{\varphi}_{ij}^{\alpha}\left(\boldsymbol{k}_{ij}\right)\tilde{A}_{ij}^{\alpha}\left(\boldsymbol{K}_{ij},\{\boldsymbol{k}_{n}\}_{n\not=i,j}\right),$

(2)

where $\tilde{\varphi}_{ij}^{\alpha}$ and $\tilde{A}_{ij}^{\alpha}$ are respectively the Fourier transforms of $\varphi_{ij}^{\alpha}$ and $A_{ij}^{\alpha}$. Based on these asymptotic factorizations, nuclear contact matrices are defined as

$$C_{ij}^{\alpha\beta}=N_{ij}\langle A_{ij}^{\alpha}|A_{ij}^{\beta}\rangle.$$

(3)

Here, $ij$ stands for one of the pairs: proton-proton, neutron-neutron or neutron-proton, and $N_{ij}$ is the total number of $ij$ pairs in the nucleus. The diagonal contact elements $C_{ij}^{\alpha\alpha}$ are proportional to the number of SRC pairs with qunatum number $\alpha$ in a given nuclear state.

The asymptotic factorization, including the definition of the universal two-body functions, is the underlying assumption for the GCF predictions, and was verified numerically using ab-initio calculations [36, 39, 40]. It is also supported by the work of Refs. [48, 49, 50], based on renormalization group arguments. In view of its success, the two-body GCF is expected to be the leading order term of a short-range (or a high-momentum) expansion of the nuclear wave-function. However, next order corrections are currently not well understood, especially the role of the elusive SRC triplets.

In this work we study the asymptotic form of the nuclear wave-function using the Coupled Cluster (CC) expansion [51, 52], aiming to put the GCF on a more solid theoretical grounds. In addition, the CC expansion provides a systematic way to include higher order corrections, e.g. 3-body SRCs, beyond the leading 2-body SRC term of the asymptotic expansion of the many-body wave-function. Here, we limit our attention to Hamiltonians containing only 2-body interaction, postponing the discussion of 3-body forces to future works.

The paper is organized as follows. In section II we provide a short introduction to the CC expansion method. Then, in section III we discuss the momentum basis and its merits. The derivation of the high-momentum asymptotic equations governing the behavior of two-body and three-body SRCs is presented in section IV. In section V we focus on two-body correlations and analyze their universal behavior. Three-body effects are then analyzed in section VI, where we derive the appropriate universal equation and show its relation to the solution of the zero-energy three-body problem. For the sake of brevity some more technical details are presented in the appendix.

The general form of a Hamiltonian describing a many-particle system interacting via two-body potential $\hat{V}$ is given by

	$\displaystyle\hat{H}\equiv\hat{H}_{0}+\hat{U}+\hat{V}$		(4)
	$\displaystyle=\sum_{r}\epsilon_{r}\boldsymbol{r}^{\dagger}\boldsymbol{r}+\sum_{rr^{\prime}}U^{r}_{r^{\prime}}{\boldsymbol{r}}^{\dagger}{\boldsymbol{r}}^{\prime}+\frac{1}{4}\sum_{rsr^{\prime}s^{\prime}}V^{rs}_{r^{\prime}s^{\prime}}\boldsymbol{r}^{\dagger}\boldsymbol{s}^{\dagger}{\boldsymbol{s}}^{\prime}{\boldsymbol{r}}^{\prime},$		(5)

where $\hat{H}_{0}$ is the “zero-order” or unperturbed Hamiltonian (not necessarily the free Hamiltonian), and $\hat{U}$ is the residual one-body interaction. The operators ${\boldsymbol{r}},{\boldsymbol{s}},\ldots$ are the usual fermionic ladder operators corresponding to the single particle eigenstates $\ket{r},\ket{s},\ldots$ of $\hat{H}_{0}$, i.e. $\hat{H}_{0}\ket{{r}}=\epsilon_{r}\ket{{r}}$, or equivalently

$$[\hat{H}_{0},{\boldsymbol{r}}^{\dagger}]=\epsilon_{r}{\boldsymbol{r}}^{\dagger}\qquad[\hat{H}_{0},{\boldsymbol{r}}]=-\epsilon_{r}{\boldsymbol{r}},$$

(6)

where $[\hat{A},\hat{B}]$ is the regular commutator. They obey the anti-commutation relations

$$\{{\boldsymbol{r}},{\boldsymbol{s}}\}=0,\hskip 20.00003pt\{{\boldsymbol{r}}^{\dagger},{\boldsymbol{s}}^{\dagger}\}=0,\hskip 20.00003pt\{{\boldsymbol{r}}^{\dagger},{\boldsymbol{s}}\}=\delta_{rs}.$$

(7)

In the following we will use the notation $\ket{r_{1}r_{2}\ldots r_{A}}$ to denote normalized antisymmetrized $A$-body states and $|r_{1}r_{2}\ldots r_{A})$ to denote the simple, non-symmetrized, many-body states, e.g. $\ket{rs}=\frac{1}{\sqrt{2}}[|rs)-|sr)]$. The matrix elements of the 2-body potential $\hat{V}$ are then given by

$$V^{rs}_{r^{\prime}s^{\prime}}=\bra{rs}\hat{V}\ket{r^{\prime}s^{\prime}}=(rs|\hat{V}|r^{\prime}s^{\prime})-(rs|\hat{V}|s^{\prime}r^{\prime}).$$

(8)

The starting point of the CC method is a reference Slater-determinant state $\ket{\Phi_{0}}$, composed of $A$ single particle states. In general, a wave function $|\Psi\rangle$ is a linear combination of all such Slater determinants. These determinants can be organized in a systematic way, by considering first the determinants obtained replacing a state occupied in $\ket{\Phi_{0}}$ with a state not occupied in $\ket{\Phi_{0}}$, than replacing two such states, and so on. Following the convention of Shavitt & Bartlett [53], we use the letters $i,j,\ldots,n$ to denote ”hole” states, i.e. single-particle states that are occupied in $|\Phi_{0}\rangle$, and the letters $a,b,\ldots,f$ to denote ”particle” states, i.e. single-particle states that are not occupied in $|\Phi_{0}\rangle$. $r,s,\ldots,w$ will be used to denote both states. Therefore,

$\displaystyle\boldsymbol{i}^{\dagger}\ket{\Phi_{0}}=0\;,\quad\text{and}\quad\boldsymbol{a}\ket{\Phi_{0}}=0.{}$

(9)

The interacting many-body state $|\Psi\rangle$, an eigenstate of $\hat{H}$, is written in the CC formulation as

$$\ket{\Psi}=e^{\hat{T}}\ket{\Phi_{0}},\quad\text{where}\quad\hat{T}=\sum_{n}\hat{T}_{{n}},$$

(10)

and

$$\hat{T}_{{n}}=\frac{1}{n!^{2}}\sum_{a_{1}\ldots a_{n},i_{1}\ldots i_{n}}t^{a_{1}a_{2}\ldots a_{n}}_{i_{1}i_{2}\ldots i_{n}}\boldsymbol{a}_{1}^{\dagger}\boldsymbol{a}_{2}^{\dagger}\cdots\boldsymbol{i}_{2}\boldsymbol{i}_{1}$$

(11)

is the $n$-particle, $n$-hole ($npnh$) cluster operator.

To determine the amplitudes $t^{a_{1}a_{2}\ldots a_{n}}_{i_{1}i_{2}\ldots i_{n}}$, a set of non-linear equations, the CC equations, can be obtained by projecting the Schrödinger equation on an $npnh$ state $\ket{\Phi^{ab\cdots}_{ij\cdots}}\equiv{\boldsymbol{a}}^{\dagger}{\boldsymbol{b}}^{\dagger}\cdots{\boldsymbol{j}}{\boldsymbol{i}}\ket{\Phi_{0}}$. The full derivation of the CC equations is given, e.g., in Ref. [53]. Omitting the 1-body potential term $\hat{U}$ and the $1p1h$ cluster operator $\hat{T}_{{1}}$, the two- and three-body CC equations are given by

	$\displaystyle 0=\bra{\Phi_{ij}^{ab}}\hat{V}$	$\displaystyle+[\hat{H}_{0},\hat{T}_{{2}}]+[\hat{V},\hat{T}_{{2}}]+\frac{1}{2}[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{2}}]$		(12)
		$\displaystyle+[\hat{V},\hat{T}_{{3}}]+[\hat{V},\hat{T}_{{4}}]\ket{\Phi_{0}},$		(13)
	$\displaystyle 0=\bra{\Phi_{ijk}^{abc}}$	$\displaystyle[\hat{H}_{0},\hat{T}_{{3}}]+[\hat{V},\hat{T}_{{2}}]+\frac{1}{2}[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{2}}]+[\hat{V},\hat{T}_{{3}}]$		(14)
		$\displaystyle+[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{3}}]+[\hat{V},\hat{T}_{{4}}]+[\hat{V},\hat{T}_{{5}}]\ket{\Phi_{0}}.$		(15)

To study SRCs it is most convenient to work with single-particle basis states, i.e. the eigenstate of $\hat{H}_{0}$, that have well defined momentum. This choice is natural for an infinite system, like nuclear matter - see e.g. [54, 55], but it might seem rather odd for describing a bound nucleus which is a compact object. However, large nuclei have relatively constant density and far from the surface behave like an infinite nuclear system. Thus, we set the problem in a box of size $L$ with periodic boundary conditions. For $L$ larger than the nucleus size, the wave-function and the binding energy approach very fast the free space ($L\to\infty$) values and we need not worry about the impact of the boundary conditions on the nuclear SRCs.

Assuming ${\boldsymbol{p}}=(p_{1},p_{2},p_{3})$ to be a triad of integers, the basis states $\{\ket{{\boldsymbol{p}}}\}$

$$\bra{{\boldsymbol{x}}}\ket{{\boldsymbol{p}}}=\frac{1}{\sqrt{\Omega}}e^{i\frac{2\pi}{L}{\boldsymbol{x}}\cdot{\boldsymbol{p}}},\quad\bra{{\boldsymbol{p}}}\ket{{\boldsymbol{p}}^{\prime}}=\delta_{{\boldsymbol{p}},{\boldsymbol{p}}^{\prime}},$$

(16)

with $\Omega=L^{3}$, is a complete set of orthonormal states, which combined with the spin and isospin degrees of freedom form our single-particle basis. A natural choice for $\ket{\Phi_{0}}$, the starting point of the CC expansion, is a Slater determinant composed of the $A$ lowest kinetic energy single-particle states.

If there is a well defined Fermi momentum $p_{F}$, such that all the hole states are momentum states with momentum smaller than $p_{F}$, while particle states have momentum larger than $p_{F}$, then the system is called a closed shell system. To simplify matters, in the following we shall restrict our the discussion to such closed shell systems only.

Working with this single-particle momentum basis, $\hat{H}_{0}$ coincides with the kinetic energy operator and therefore $\hat{U}=0$. The Slater-determinant $\ket{\Phi_{0}}$, as well as the $npnh$ states $\ket{\Phi^{ab\cdots}_{ij\cdots}}$, are a product of single-particle momentum states, hence they are eigenstates of the total center of mass (CM) momentum operator $\hat{{\boldsymbol{P}}}_{CM}$. The two-body potential is translational invariant, hence the CM momentum is a good quantum number, and the wave-function $\ket{\Psi}$ is also an eigenstate of $\hat{{\boldsymbol{P}}}_{CM}$,

$\displaystyle\hat{{\boldsymbol{P}}}_{CM}\ket{\Psi}=\hat{{\boldsymbol{P}}}_{CM}e^{\hat{T}}\ket{\Phi_{0}}={\boldsymbol{P}}_{CM}\ket{\Psi}.{}$

(17)

Closing the last equation with $\bra{\Phi_{0}}$ and acting with $\hat{{\boldsymbol{P}}}_{CM}$ once to the left and once to the right, and noting that $\bra{\Phi_{0}}\ket{\Psi}\neq 0$, we must conclude that $\ket{\Phi_{0}}$ and $\ket{\Psi}$ share the same eigenvalue of the total momentum ${\boldsymbol{P}}_{CM}$.

We may now repeat the same argument for the $1p1h$ states. Closing Eq. (17) with $\bra{\Phi_{i}^{a}}$ and using $\langle\Phi_{i}^{a}|\Psi\rangle=t_{i}^{a}$ we get

$$({\boldsymbol{p}}_{a}-{\boldsymbol{p}}_{i})t^{a}_{i}=\boldsymbol{0},$$

(18)

which for all closed shell systems implies [56, 54]

$$t^{a}_{i}=0,$$

(19)

because $({\boldsymbol{p}}_{a}-{\boldsymbol{p}}_{i})\neq\boldsymbol{0}$, as ${\boldsymbol{p}}_{a}$ corresponds to a particle state while ${\boldsymbol{p}}_{i}$ to a hole state. Thus, with this choice of basis states, $\hat{T}_{{1}}$ is eliminated from the CC expansion, as was assumed in Eqs. (12) and (14).

Considering now the $2p2h$ states, multiplying Eq. (17) by $\bra{\Phi^{ab}_{ij}}$ one gets [55]

$$({\boldsymbol{p}}_{a}+{\boldsymbol{p}}_{b}-{\boldsymbol{p}}_{i}-{\boldsymbol{p}}_{j})t^{ab}_{ij}=\boldsymbol{0}.$$

(20)

This implies that $\hat{T}_{{2}}$ conserves momentum, i.e. $t_{ij}^{ab}=0$ if ${\boldsymbol{p}}_{a}+{\boldsymbol{p}}_{b}-{\boldsymbol{p}}_{i}-{\boldsymbol{p}}_{j}\neq\boldsymbol{0}$. It can be similarly shown that for a closed shell system all amplitude operators $\hat{T}_{{n}}$ must conserve momentum.

SRCs are associated with high momentum particles. To understand their role in the many-body wave-function we need to study the high momentum behavior of the CC amplitudes $\hat{T}_{{n}}$ as dictated by Eqs. (12) and (14). In the following we will assume $a,b$ and $c$ to be highly excited states corresponding to momenta $p_{a},p_{b},p_{c}\gg p_{\text{F}}$. We note that in order for the wave-function to be properly normalized the CC amplitudes $\hat{T}_{{n}}$ must vanish in this limit, e.g. $t^{abc}_{ijk}\to 0$ when $a,b,c\to\infty$.

For a system of fermions, we expect the CC amplitudes to admit the natural hierarchy, where double excitations are much more significant than three-body excitations which on their part are more important than the four-body excitations, etc. It follows that the contributions of $[\hat{V},\hat{T}_{{3}}]$ and $[\hat{V},\hat{T}_{{4}}]$ to the 2-body equation can be neglected. Similarily, the terms $[\hat{V},\hat{T}_{{4}}]$ and $[\hat{V},\hat{T}_{{5}}]$ can be neglected in the 3-body CC equation.

In order to understand the behaviour of the CC amplitudes in the high momentum limit, let us inspect the $\hat{T}_{{2}}$ equation, Eq. (12), in the limit $p_{a},p_{b}\to\infty$. In this case, the leading terms are the source term $V^{ab}_{ij}$ and the kinetic energy term $[\hat{H}_{0},\hat{T}_{{2}}]$. Retaining only these terms leads to the well known asymptotic result

$$t^{ab}_{ij}\rightarrow-\frac{1}{E^{ab}_{ij}}V^{ab}_{ij},$$

(21)

were $E^{ab}_{ij}$ is the excitation energy given by the relation

$$E^{a_{1}a_{2}\ldots a_{n}}_{i_{1}i_{2}\ldots i_{n}}\equiv(\epsilon_{a_{1}}+\epsilon_{a_{2}}+\ldots\epsilon_{a_{n}})-(\epsilon_{i_{1}}+\ldots+\epsilon_{i_{n}}).$$

(22)

If, for $p_{a},p_{b}\to\infty$, the potential matrix elements $V^{ab}_{ij}$ are independent of the exact holes states, i.e. $V^{ab}_{ij}\approx V^{ab}_{0_{i}0_{j}}$, with $0_{i}$ being a zero-momentum state (used loosely to indicate the lowest momentum state with the same quantum numbers as the state $i$), the asymptotic two-body amplitude presented in Eq. (21) is universal in the limited sense. That is, $t^{ab}_{ij}\approx-\frac{1}{E^{ab}_{00}}V^{ab}_{00}$ is independent of the number of nucleons $A$ and the specifics of the nuclear state. On the other hand, it depends on the potential - therefore its universality is limited. This form of asymptotic behaviour was first suggested by Amado [57] exploring the asymptotic form of the nuclear momentum distribution. It turns out, however, that although Eq. (21) is asymptotically correct, it is valid only at extremely high momentum, larger than 10 fm^-1, making it impractical for actual calculations [58]. Consequently, in order to get a reasonable description of the asymptotic nuclear wave-function, we must retain more terms besides the source term and the $[\hat{H}_{0},\hat{T}_{{n}}]$ terms in the CC equations.

With the 3, and 4-body amplitudes neglected, the CC $\hat{T}_{{2}}$ equation, Eq. (12), takes the form

$$0=\bra{\Phi_{ij}^{ab}}\hat{V}+[\hat{H}_{0},\hat{T}_{{2}}]+[\hat{V},\hat{T}_{{2}}]+\frac{1}{2}[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{2}}]\ket{\Phi_{0}}.$$

(23)

Comparing now the terms $[\hat{V},\hat{T}_{{2}}]$, and $[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{2}}]$ we note that for the latter we get the following matrix elements, ignoring combinatorical factors,

$$V^{kl}_{de}t^{ab}_{ik}t^{de}_{jl},\quad V^{kl}_{de}t^{ab}_{kl}t^{de}_{ij},\quad V^{kl}_{de}t^{ad}_{ik}t^{be}_{jl},\quad V^{kl}_{de}t^{ad}_{ij}t^{be}_{kl}.$$

(24)

Here, for brevity, we use the Einstein convention assuming implicit summation on repeated lower and upper indices. In the high momentum $p_{a},p_{b}\to\infty$ and low density $i,j,k,l\to 0$ limit, these terms take the form

$$2V^{00}_{de}t^{de}_{00}t^{ab}_{00},\quad\text{and}\quad 2V^{00}_{de}t^{ad}_{00}t^{be}_{00}.$$

(25)

The first of these terms is nothing but an energy shift, a correction to the excitation energy $E^{ab}_{ij}$, which we can neglect in the high momentum limit. We note that the second term is zero unless ${\boldsymbol{p}}_{d}=-{\boldsymbol{p}}_{a}$, ${\boldsymbol{p}}_{e}=-{\boldsymbol{p}}_{b}$ and ${\boldsymbol{p}}_{b}=-{\boldsymbol{p}}_{a}$. This term is clearly suppressed by a factor of $t^{ab}_{00}$ with respect to $[\hat{V},\hat{T}_{{2}}]$, and thus can be neglected as well.

Considering now the $\hat{T}_{{3}}$ equation, Eq. (14). After neglecting the $\hat{T}_{{4}}$, $\hat{T}_{{5}}$ as well as the $[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{3}}]\ll[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{2}}]$ terms, we remain with

	$\displaystyle 0=\bra{\Phi_{ijk}^{abc}}$	$\displaystyle[\hat{H}_{0},\hat{T}_{{3}}]+[\hat{V},\hat{T}_{{2}}]$
		$\displaystyle+\frac{1}{2}[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{2}}]+[\hat{V},\hat{T}_{{3}}]\ket{\Phi_{0}}.{}$		(26)

Comparing again the $[\hat{V},\hat{T}_{{2}}]$, and $[[\hat{V},\hat{T}_{{2}}],\hat{T}_{{2}}]$, we see that the only terms that survive in the high momentum/low density limit are respectively $V^{ab}_{e0}t^{ce}_{00}$, and $V^{a0}_{ef}t^{be}_{00}t^{cf}_{00}$. Thus as before, the double commutator term is suppressed by a factor of $t^{ab}_{00}$ and can be neglected.

Summing up, in the limit of high momenta, we expect the two- and three-body CC amplitudes to obey the equations:

	$\displaystyle 0$	$\displaystyle=\bra{\Phi_{ij}^{ab}}[\hat{H}_{0},\hat{T}_{{2}}]+[\hat{V},\hat{T}_{{2}}]+\hat{V}\ket{\Phi_{0}},$		(27)
	$\displaystyle 0$	$\displaystyle=\bra{\Phi_{ijk}^{abc}}[\hat{H}_{0},\hat{T}_{{3}}]+[\hat{V},\hat{T}_{{3}}]+[\hat{V},\hat{T}_{{2}}]\ket{\Phi_{0}}.$		(28)

In the following sections we will analyze these equations.

In section IV we argued that asymptotically the 2-body CC equation takes the form of Eq. (27). In order to evaluate this equation, we note that there can be no contractions between the operators $\boldsymbol{a}^{\dagger}\boldsymbol{b}^{\dagger}\boldsymbol{j}\boldsymbol{i}$ that appear in the bra state, hence all the labels $a,b,i$ and $j$ has to appear on the amplitude and potential operators. Therefore $\bra{\Phi_{ij}^{ab}}\hat{V}\ket{\Phi_{0}}=V^{ab}_{ij}$, and $\bra{\Phi_{ij}^{ab}}\left[{\hat{H}_{0},\hat{T}_{{2}}}\right]\ket{\Phi_{0}}=E^{ab}_{ij}t^{ab}_{ij}$. To evaluate the commutator $\left[{\hat{V},\hat{T}_{{2}}}\right]=\hat{V}\hat{T}_{{2}}-\hat{T}_{{2}}\hat{V}$, we note that all the operators of $\hat{V}$ in $\hat{V}\hat{T}_{{2}}$ can be moved to the right of $\hat{T}_{{2}}$ and then the term will cancel with $\hat{T}_{{2}}\hat{V}$. In the process, all possible contractions between $\hat{V}$ and $\hat{T}_{{2}}$ will arise, i.e. at least one contraction should be made between them. This commutators yield 5 distinct terms, that combined with the potential and the $\hat{H}_{0}$ terms results in the linear Coupled-Cluster doublets (CCD) equation

	$\displaystyle 0$	$\displaystyle=V^{ab}_{ij}+E^{ab}_{ij}t^{ab}_{ij}+\frac{1}{2}V^{ab}_{de}t^{de}_{ij}$
		$\displaystyle+\frac{1}{2}V^{kl}_{ij}t^{ab}_{kl}+V^{ak}_{id}t^{bd}_{jk}+V^{ak}_{kd}t^{bd}_{ij}+V^{lk}_{ik}t^{ab}_{jl}$
		$\displaystyle+\text{permutations}\;.{}$		(29)

The title ‘permutations’ stands for anti-symmetrization with respect to the indices $ab$ or $ij$ when not placed on the same matrix elements. The summation $V^{kv}_{kw}$ is performed only on hole states as the string $\boldsymbol{r}^{\dagger}\boldsymbol{s}$ of $\hat{V}$, where both ${\boldsymbol{r}},{\boldsymbol{s}}$ are particle operators, is already normal ordered and therefore its contraction is zero.

In the limit of high momentum/low density the 2nd line of (V) take either the form $V^{00}_{00}t^{ab}_{00}$ or $V^{a0}_{a0}t^{ab}_{00}$. In both cases these terms enter as small corrections to the excitation energy. It follows that these terms can be neglected for large $p_{a},p_{b}$ with respect to the terms appearing on the first line.

Refining this argument, due to momentum conservation we expect that in the limit $p_{a},p_{b}\to\infty$ all the $\hat{T}_{{2}}$ terms on the 2nd and 3rd lines of Eq. (V), will be either exactly or at least approximately equal to $t^{ab}_{ij}$ since the hole states carry only low momentum of the order of $p_{\text{F}}$ and we assume weak momenta dependance on the hole states quantum numbers. For example, in $V^{ak}_{id}t^{bd}_{jk}$ the momentum ${\boldsymbol{p}}_{d}$ associated with the state $d$ must be of the order ${\boldsymbol{p}}_{d}={\boldsymbol{p}}_{a}+\order{p_{\text{F}}}$ implying that $t^{bd}_{jk}\approx t^{ba}_{ij}$. Comparing these terms to the term $E^{ab}_{ij}t^{ab}_{ij}$, we see that for large enough excitations

$$E^{ab}_{ij}\gg\sum_{kl}V^{kl}_{ij},\;\sum_{kd}V^{ak}_{id},\;\sum_{kd}V^{ak}_{kd},\;\sum_{kl}V^{lk}_{ik}$$

(30)

and these terms can be neglected. It is important to observe that the neglected terms are all intensive and do not scale with the size of the system.

The resulting 2-body amplitude equation is then

$\displaystyle 0$

$\displaystyle=t^{ab}_{ij}+\frac{1}{E^{ab}_{ij}}V^{ab}_{ij}+\frac{1}{2E^{ab}_{ij}}V^{ab}_{de}t^{de}_{ij}\;,{}$

(31)

which is nothing but particle-particle ladder approximation of the CCD equation, applied for example in Ref. [54] to estimate the nuclear matter equation of state. In the following we will use the notation $\hat{T}^{\infty}_{{2}}$ to denote the solution of Eq. (31) in the non-symmetrized basis with $E^{ij}\equiv\epsilon_{i}+\epsilon_{j}\to 0$. As it is a linear equation, $\hat{T}^{\infty}_{{2}}$ is unique. We will show in appendix A that indeed $\bra{\boldsymbol{ab}}\hat{T}_{{2}}\ket{\boldsymbol{ij}}\to\bra{\boldsymbol{ab}}\hat{T}^{\infty}_{{2}}\ket{\boldsymbol{ij}}$ as $p_{a},p_{b}\to\infty$.

We can now discuss the properties of $\hat{T}^{\infty}_{{2}}$. As stated above, the cluster operator $\hat{T}^{\infty}_{{2}}$ is defined as the solution of the equation

$\displaystyle 0$

$\displaystyle=\left({t^{\infty}}\right)^{ab}_{ij}+\frac{1}{E^{ab}}V^{ab}_{ij}+\frac{1}{2E^{ab}}V^{ab}_{de}\left({t^{\infty}}\right)^{de}_{ij},{}$

(32)

where $E^{ab}=\epsilon_{a}+\epsilon_{b}$. A similar result was derived by Zabolitzky in Ref. [58]. To analyze this equation, it is convenient to introduce particle-particle and the hole-hole projection operators,

$${Q}_{{2}}=\sum_{de}|{d}{e})({d}{e}|,\quad{P}_{{2}}=\sum_{lm}|{l}{m})({l}{m}|,$$

(33)

and the Green’s function

$$\hat{G}_{0}(E)=\frac{1}{E-\hat{H}_{0}+i\varepsilon}.$$

(34)

Eq. (32) can then be written as

$$\hat{T}^{\infty}_{{2}}={Q}_{{2}}\hat{G}_{0}(0)\hat{V}\hat{T}^{\infty}_{{2}}+{Q}_{{2}}\hat{G}_{0}(0)\hat{V}{P}_{{2}},$$

(35)

and formally solved to yield

$$\hat{T}^{\infty}_{{2}}=\frac{1}{1-{Q}_{{2}}\hat{G}_{0}(0)\hat{V}}{Q}_{{2}}\hat{G}_{0}(0)\hat{V}{P}_{{2}}\;.$$

(36)

Clearly, ${P}_{{2}}\hat{T}^{\infty}_{{2}}=\hat{T}^{\infty}_{{2}}{Q}_{{2}}=0$ as expected from a cluster operator. Using the relation ${Q}_{{2}}\hat{H}_{0}{P}_{{2}}=0$ the solution (36) can be rewritten as (see appendix B)

$\displaystyle\hat{T}^{\infty}_{{2}}$

$\displaystyle=\frac{1}{{Q}_{{2}}(0+i\varepsilon-\hat{H}){Q}_{{2}}}{Q}_{{2}}\hat{H}{P}_{{2}}\;.{}$

(37)

Before proceeding, we note that ${P}_{{2}}$ is not eqivalent to $\bar{Q}_{{2}}=1-{Q}_{{2}}$ the complement of ${Q}_{{2}}$, as $\bar{Q}_{{2}}$ must include not only hole-hole states but also particle-hole states. For infinite nuclear matter we expect however, that $\hat{T}^{\infty}_{{2}}$ is translational invariant and therefore we can consider only pairs with zero CM momentum. For such pairs, there are no particle-hole contributions and we can replace ${P}_{{2}}$ by $\bar{Q}_{{2}}$. In this subspace

$$\hat{T}^{\infty}_{{2}}=\frac{1}{{Q}_{{2}}(0+i\varepsilon-\hat{H}){Q}_{{2}}}{Q}_{{2}}\hat{H}\bar{Q}_{{2}}\;.$$

(38)

Comparing now Eq. (38) with the Bloch-Horowitz equations [59],

	$\displaystyle\bar{Q}_{{2}}|\Psi\rangle$	$\displaystyle=\frac{1}{\bar{Q}_{{2}}(E+i\varepsilon-\hat{H})\bar{Q}_{{2}}}\bar{Q}_{{2}}\hat{H}{Q}_{{2}}|\Psi\rangle$		(39)
	$\displaystyle{Q}_{{2}}|\Psi\rangle$	$\displaystyle=\frac{1}{{Q}_{{2}}(E+i\varepsilon-\hat{H}){Q}_{{2}}}{Q}_{{2}}\hat{H}\bar{Q}_{{2}}|\Psi\rangle\;,$		(40)

it is clear that $\hat{T}^{\infty}_{{2}}$ is nothing but the zero-energy 2-body Bloch-Horowitz operator

$$\hat{O}_{2}^{\text{B.H.}}=\frac{1}{{Q}_{{2}}(0+i\varepsilon-\hat{H}){Q}_{{2}}}{Q}_{{2}}\hat{H}\bar{Q}_{{2}}.$$

(41)

This operator fulfills the relation $\hat{O}_{2}^{\text{B.H.}}\ket{\Psi_{2}}={Q}_{{2}}\ket{\Psi_{2}}$ for any zero energy eigenstate $\ket{\Psi_{2}}$ that obeys $\hat{H}\ket{\Psi_{2}}=0$. It follows that if $\hat{H}\ket{\Psi_{2}}=0$ and $\hat{\boldsymbol{P}}_{CM}\ket{\Psi_{2}}=0$, then

$${Q}_{{2}}\ket{\Psi_{2}}=\hat{T}^{\infty}_{{2}}\ket{\Psi_{2}}.$$

(42)

Inspecting eqs. (37) and (42), we can conclude that (i) The asymptotic 2-body behavior of $\hat{T}_{{2}}$, and therefore of the many-body wave-function, is related to the zero-energy solutions of the 2-body problem. (ii) The relation to the zero-energy solutions show the universality of the asymptotic behavior in the limited sense, that it does not depend on the system, but depends on the potential.

As we have argued in Sec. IV, the behavior of the 3-body amplitude $\hat{T}_{{3}}$ at high momentum is dictated by Eq. (28). Explicitly, this equation takes the form

	$\displaystyle 0$	$\displaystyle=E^{abc}_{ijk}t^{abc}_{ijk}-V^{la}_{ij}t^{bc}_{kl}-V^{ab}_{id}t^{cd}_{jk}$
		$\displaystyle+\frac{1}{2}V^{ab}_{de}t^{cde}_{ijk}+\frac{1}{2}V^{lm}_{ij}t^{abc}_{klm}+V^{al}_{dl}t^{bcd}_{ijk}+V^{al}_{id}t^{bcd}_{jkl}+V^{lm}_{il}t^{abc}_{jkm}$
		$\displaystyle+\text{permutations}\;.{}$		(43)

Here, the first term on the rhs is due to $[\hat{H}_{0},\hat{T}_{{3}}]$, the next two terms come from the $[\hat{V},\hat{T}_{{2}}]$ commutator, and the next five are due to the $[\hat{V},\hat{T}_{{3}}]$ commutator. The title ‘permutations’ stands for anti-symmetrization with respect to the indices $abc$ or $ijk$ when not placed on the same matrix elements. Due to momentum conservation, for very large $p_{a}$ the potential matrix elements $V^{la}_{ij}$ must vanish, leaving $V^{ab}_{id}t^{cd}_{jk}$ as the only source term. In addition, all terms coming from the $[\hat{V},\hat{T}_{{3}}]$ commutator, except for the first term in the second line (and its corresponding permutations), are approximately proportional to $t^{abc}_{ijk}$. Therefore, for excitation energy $E^{abc}_{ijk}$ large enough

$$E^{abc}_{ijk}\gg\sum_{lm}V^{lm}_{ij},\;\sum_{ld}V^{al}_{dl}\;,\sum_{ld}V^{al}_{id}\;,\sum_{ml}V^{lm}_{il}\;,$$

(44)

and the corresponding terms can be neglected in comparison to the free term $E^{abc}_{ijk}t^{abc}_{ijk}$. As a consequence only the terms $\frac{1}{2}V^{ab}_{de}t^{cde}_{ijk}$ remain. Utilizing these observations, and defining the symmetrization operator ${\cal\hat{S}}_{123}\equiv 1+(123)+(132)$ where $(123)$ is the permutation operator, equation (VI) takes the form

$\displaystyle 0=t^{abc}_{ijk}$

$\displaystyle+\frac{{\cal\hat{S}}_{abc}({\cal\hat{S}}_{ijk}(V^{ab}_{id}t^{dc}_{jk}))}{E^{abc}_{ijk}}+\frac{{\cal\hat{S}}_{abc}(V^{ab}_{de}t^{cde}_{ijk})}{2E^{abc}_{ijk}}\;.{}$

(45)

As in the 2-body case we define $\hat{T}^{\infty}_{{3}}$ to be the solution of eq. (45) in the limit $E^{abc}_{ijk}\to E^{abc}$, and $t^{dc}_{jk}\to\left({t^{\infty}}\right)^{dc}_{jk}$. We show in appendix C that $\bra{\boldsymbol{abc}}\hat{T}_{{3}}\ket{\boldsymbol{ijk}}\to\bra{\boldsymbol{abc}}\hat{T}^{\infty}_{{3}}\ket{\boldsymbol{ijk}}$ as $p_{a},p_{b},p_{c}\to\infty$.

To analyze $\hat{T}^{\infty}_{{3}}$ we write eq. (45) in first quantization using the non-symmetrized basis defined above. In the 3-body case, the relation between the anti-symmetrized matrix elements and the non symmetrized ones is

	$\displaystyle\bra{rst}$	$\displaystyle\hat{O}\ket{uvw}$
		$\displaystyle={\cal\hat{S}}_{uvw}\left[(rst|\hat{O}|uvw)-(rst|\hat{O}|vuw)\right],{}$		(46)

and for a 2-body operator closed by 3-particle states

$\displaystyle(rst|\hat{O}_{2}$

$\displaystyle|uvw)\equiv\sum_{i=1}^{3}(rst|\hat{O}_{2}(i)|uvw){}$

(47)

where $\hat{O}_{2}(i)$ does not act on the $i$’th particles, e.g. $(rst|\hat{O}_{2}(3)|uvw)=(rs|{\hat{O}_{2}}|uv)\delta_{t,w}$. With the projection operators

$${Q}_{{3}}=\sum_{def}|def)(def|,\quad{P}_{{3}}=\sum_{lmn}|lmn)(lmn|,$$

(48)

the asymptotic equation for $\hat{T}^{\infty}_{{3}}$ can be written as

	$\displaystyle\hat{T}^{\infty}_{{3}}$	$\displaystyle={Q}_{{3}}\hat{G}_{0}\left({0}\right)\hat{V}\hat{T}^{\infty}_{{3}}+{Q}_{{3}}\hat{G}_{0}\left({0}\right)\hat{V}\hat{T}^{\infty}_{{2}}{P}_{{3}}$
		$\displaystyle=\frac{1}{1-{Q}_{{3}}\hat{G}_{0}\left({0}\right)\hat{V}}{Q}_{{3}}\hat{G}_{0}\left({0}\right)\hat{V}\hat{T}^{\infty}_{{2}}{P}_{{3}}$
		$\displaystyle=\frac{1}{{Q}_{{3}}(0+i\varepsilon-\hat{H}){Q}_{{3}}}{Q}_{{3}}\hat{H}\hat{T}^{\infty}_{{2}}{P}_{{3}}\;.{}$		(49)

Comparing eq. (VI) with the 3-body Bloch-Horowitz equations [59] and noting that for 2-body interactions ${Q}_{{3}}H\bar{Q}_{{3}}={Q}_{{3}}H\left({{Q}_{{1}}{P}_{{2}}+{Q}_{{2}}{P}_{{1}}}\right)$ with $\bar{Q}_{{3}}=1-{Q}_{{3}}$

	$\displaystyle\bar{Q}_{{3}}|\Psi\rangle$	$\displaystyle=\frac{1}{\bar{Q}_{{3}}(E+i\varepsilon-\hat{H})\bar{Q}_{{3}}}\bar{Q}_{{3}}\hat{H}{Q}_{{3}}|\Psi\rangle$		(50)
	$\displaystyle{Q}_{{3}}|\Psi\rangle$	$\displaystyle=\frac{1}{{Q}_{{3}}(E+i\varepsilon-\hat{H}){Q}_{{3}}}{Q}_{{3}}\hat{H}\bar{Q}_{{3}}|\Psi\rangle\;,$		(51)

we can connect $\hat{T}^{\infty}_{{3}}$ to the zero-energy Bloch-Horowitz operator. Specifically, if $|\Psi_{3}\rangle$ is a zero-energy 3-body eigenstate of $\hat{H}$, and if there is a 3-hole state $|\alpha_{3}\rangle$ such that

$\displaystyle\hat{T}^{\infty}_{{2}}|\alpha_{3}\rangle$

$\displaystyle\approx\left({{Q}_{{1}}{P}_{{2}}+{Q}_{{2}}{P}_{{1}}}\right)|\Psi_{3}\rangle{}$

(52)

then

$$\hat{T}^{\infty}_{{3}}\ket{\alpha_{3}}\approx{Q}_{{3}}\ket{\Psi_{3}}$$

(53)

and we can identify the matrix-elements of $\hat{T}^{\infty}_{{3}}$ with the ${Q}_{{3}}$ components of the zero-energy solutions of the Schrödinger equation. In the next section we will argue that eq. (53) approximately holds.