††thanks<EMAIL_ADDRESS>Present affiliation: XCP-2: Eulerian Codes
Group, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

# Short-range expansion for the quantum many-body problem

Ronen Weiss  Theoretical Division, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA  Diego Lonardoni  Theoretical Division, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  Stefano Gandolfi
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA

###### Abstract

In this work we derive a systematic short-range expansion of the many-body
wave function. At leading order, the wave function is factorized to a zero-
energy $s$-wave correlated pair and spectator particles, while terms that
include energy derivatives and larger orbital angular momentum two-body
functions appear at subleading orders. The validity of the expansion is tested
for the two-body case, as well as the many-body case, where infinite neutron
matter is considered. An accurate and consistent description of both
coordinate-space two-body densities and the one-body momentum distribution is
obtained. These results show the possibility to utilize such an expansion for
describing different observables in strongly-interacting many-body systems,
including nuclear, atomic and condensed-matter systems. This work also enables
a systematic description of large momentum transfer reactions in nuclear
systems sensitive to short-range correlations, provides a link between such
experiments and low-energy nuclear physics, and motivates measurement of new
observables in these experiments.

††preprint: LA-UR-23-27155

Non-relativistic quantum many-body systems are the focus of different research
fields, including nuclear, atomic, and condensed-matter physics, and quantum
chemistry. Studying such systems requires solving the many-body Schrödinger
equation with reliable and systematic methods. Mean-field models are often
used as a starting point for different perturbative approaches. However, in
cases where strong correlations between the particles exist, these methods are
usually ineffective. Significant contribution to such correlations usually
arises due to strong interaction between particles at short distances.

In nuclear physics, for example, large short-range correlations (SRCs) prevent
the use of many numerical methods. Methods based on renormalization-group (RG)
techniques led to significant progress in the description of nuclei [1], but
they are mostly adequate to deal with relatively soft interactions. Quantum
Monte Carlo methods are able to handle hard interactions, but are usually
limited to relatively light systems [2]. Developing a better theoretical
understanding of SRCs is, therefore, important in order to make progress in
the description of quantum many-body systems.

SRCs have been the focus of many experimental and theoretical studies in
different fields, including atomic [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and
nuclear systems [14, 15, 16, 17]. It was generally revealed that the
interaction of two particles at short distances inside a many-body system
results in correlated pairs with high relative momentum in a back-to-back
configuration, that behave as an isolated two-body system. Different
properties of SRC pairs in nuclei, including their abundance and momentum
dependence, were studied in detail. Nevertheless, there is still no systematic
framework for describing SRCs and utilizing our experimental and theoretical
knowledge of SRC properties for the description of more general observables
that are affected by both long-range and short-range physics. The purpose of
this work is to derive a short-range expansion of the quantum many-body wave
function that connects short-range and long-range physics together in an
effort to develop such a framework.

We start with the two-body case, considering a two-body eigenstate with energy
$E$ and additional quantum numbers $\alpha$, denoted by
$\varphi^{E}_{\alpha}(\bm{r})$. $\varphi^{E}_{\alpha}(\bm{r})$ obeys the
Schrödinger equation

$\left[-\frac{\hbar^{2}}{2\mu}\nabla^{2}+V(\bm{r})\right]\varphi^{E}_{\alpha}(\bm{r})=E\varphi^{E}_{\alpha}(\bm{r}),$
(1)

where $\mu$ is the reduced mass, $V$ is a two-body potential and $\bm{r}$ is
the relative coordinate of the pair. At short distances, the kinetic energy
term is dominant compared to $E$, and, therefore, the function does not depend
on the value of $E$. We can thus conclude that

$\varphi^{E}_{\alpha}(\bm{r})\xrightarrow[r\rightarrow
0]{}\varphi^{E=0}_{\alpha}(\bm{r}),$ (2)

i.e., $\varphi^{E}_{\alpha}(\bm{r})$ coincides with the zero-energy eigenstate
at short distances. The zero-energy wave function can be identified as the
leading term in a Taylor expansion around $E=0$

$\varphi^{E}_{\alpha}(\bm{r})=\varphi^{E=0}_{\alpha}(\bm{r})+\left(\frac{d}{dE}\varphi^{E=0}_{\alpha}(\bm{r})\right)E+\frac{1}{2!}\left(\frac{d^{2}}{dE^{2}}\varphi^{E=0}_{\alpha}(\bm{r})\right)E^{2}+...$
(3)

This Taylor expansion is a short-range expansion, because
$\varphi^{E}_{\alpha}(\bm{r})$ does not depend on E at short distances and,
therefore, terms involving energy derivatives vanish for $r\rightarrow 0$. As
more terms are included, it is expected to describe
$\varphi^{E}_{\alpha}(\bm{r})$ at larger and larger distances.

These claims can be tested against exact numerical calculations. We consider
in Fig. 1 the nuclear two-body bound deuteron. The AV4’ potential [18] is used
for simplicity, as it does not induce coupled channels. It is a central
potential in each of the 4 two-body spin-isospin channels. The leading order
(LO), next-to-leading order (NLO) and next-to-next-to-leading order (N2LO)
expressions of the expansion of Eq. (3) are shown in Fig. 1 (where, e.g.,
terms with up to two energy derivatives are included at N2LO). As more terms
in the expansion are included, the wave function approaches that of the
detueron at larger and larger distances, reaching an excellent agreement up to
$r\approx 7$ fm at N2LO. We stress that there are no fitting parameters here,
as we use the exact binding energy of the deuteron in Eq. (3). Fig. 1 also
includes a line showing the long-range exponential decay of the deuteron,
which agrees with the full wave function for $r\gtrsim 3$ fm. Combining the
short-range expansion with the known long-range behavior, we obtain a complete
description of the deuteron at all distances.

Figure 1:  The deuteron bound-state wave function using the AV4’ potential
compared to the short-range expansion of Eq. (3). The LO, NLO, and N2LO terms
of the expansion are shown. The dotted line shows the exponential long-range
decay of the deuteron, where $\kappa=\sqrt{m|E|/\hbar^{2}}$.

We now move to the many-body case, considering an antisymmetric eigenstate of
a given Hamiltonian with $A$ fermions
$\Psi(\bm{r}_{1},\bm{r}_{2},...,\bm{r}_{A})$ (we can deal with the bosonic
case similarly). First, we expand $\Psi$ using the complete set of
antisymmetric two-body eignestates
$\left\\{\varphi^{E}_{\alpha}(\bm{r})\right\\}$ of the same Hamiltonian

$\Psi(\bm{r}_{1},\bm{r}_{2},...,\bm{r}_{A})=\sum_{\alpha,E}\varphi^{E}_{\alpha}(\bm{r}_{12})A_{\alpha}^{E}(\bm{R}_{12},\bm{r}_{3},...,\bm{r}_{A}).$
(4)

Here, $\bm{r}_{i}$ is the single-nucleon coordinate of particle $i$, and
$\bm{r}_{12}$ and $\bm{R}_{12}$ are the relative and center-of-mass (CM)
coordinates of particles 1 and 2. The functions $A_{\alpha}^{E}$ serve as
coefficients in this expansion. We note that this is an exact expansion and,
therefore, $\Psi$ remains antisymmetric. Next, using the Taylor expansion of
Eq. (3), we obtain

$\displaystyle\Psi$
$\displaystyle(\bm{r}_{1},\bm{r}_{2},...,\bm{r}_{A})=\sum_{\alpha}\varphi^{E=0}_{\alpha}(\bm{r}_{12})A_{\alpha}^{(0)}(\bm{R}_{12},\bm{r}_{3},...,\bm{r}_{A})+$
$\displaystyle\sum_{\alpha}\left(\frac{d}{dE}\varphi^{E=0}_{\alpha}(\bm{r}_{12})\right)A_{\alpha}^{(1)}(\bm{R}_{12},\bm{r}_{3},...,\bm{r}_{A})+$
$\displaystyle\sum_{\alpha}\left(\frac{d^{2}}{dE^{2}}\varphi^{E=0}_{\alpha}(\bm{r}_{12})\right)A_{\alpha}^{(2)}(\bm{R}_{12},\bm{r}_{3},...,\bm{r}_{A})+...,$
(5)

where

$A_{\alpha}^{(n)}(\bm{R}_{12},\bm{r}_{3},...,\bm{r}_{A})\equiv\frac{1}{n!}\sum_{E}E^{n}A_{\alpha}^{E}(\bm{R}_{12},\bm{r}_{3},...,\bm{r}_{A}).$
(6)

This is our short-range expansion for the many-body case. Like the two-body
case, we expect that as we go to larger values of $r_{12}$, more terms should
be included in the expansion. However, the many-body case is more complicated
because when organizing the terms in order of importance, we need to consider
both the number of energy derivatives and the pair quantum numbers given by
$\alpha$. In the limit $r_{12}\rightarrow 0$, channels with $s$-wave component
are dominant, and terms with energy derivatives are suppressed. Therefore,
assuming a single $s$-wave channel, we obtain

$\Psi(\bm{r}_{1},\bm{r}_{2},...,\bm{r}_{A})\xrightarrow[r_{12}\rightarrow
0]{}\varphi^{E=0}_{s}(\bm{r}_{12})A_{s}^{(0)}(\bm{R}_{12},\bm{r}_{3},...,\bm{r}_{A}),$
(7)

where the subscript $s$ denotes the two-body $s$-wave channel. This is a
factorized form of the many-body wave function at short distances.

An identical short-range factorization ansatz is the basis of the Generalized
Contact Formalism (GCF) [19, 20, 21]. The GCF, developed as an extension of
Tan’s theory for the zero-range model [3, 4, 5, 6], is an effective model used
to describe nuclear SRCs and their impact on different nuclear structure
properties and reactions [20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 21, 19,
32], including two-body densities, momentum distributions, electron-scattering
cross sections, and neutrinoless double beta decay matrix elements. The same
approach is also useful for the description of other systems, like the case of
Helium atoms [13]. The subleading terms that appear in Eq. (Short-range
expansion for the quantum many-body problem) provide corrections to this
factorization ansatz.

Within the GCF, contact parameters, that measure the probability of finding
SRC pairs in a nucleus, are defined as [20]

$C_{\alpha}^{00}=\frac{A(A-1)}{2}\langle
A_{\alpha}^{(0)}|A_{\alpha}^{(0)}\rangle.$ (8)

We can now generalize this definition to account for subleading terms

$C_{\alpha}^{mn}=\frac{A(A-1)}{2}S_{mn}\left[\langle
A_{\alpha}^{(m)}|A_{\alpha}^{(n)}\rangle+\langle
A_{\alpha}^{(n)}|A_{\alpha}^{(m)}\rangle\right],$ (9)

where $S_{mn}=1/2$ if $m=n$ and $S_{mn}=1$ if $m\neq n$.

This expansion and definition of contacts can be used to describe different
quantities. We start with the two-body density $\rho_{2}(r)$, i.e., the
probability of finding two particles at relative distance $r$ in a given
system. Based on Eqs. (Short-range expansion for the quantum many-body
problem) and (9), we can write a short-range expansion for this density

$\rho_{2}(r)=\sum_{\alpha}\sum_{m\leq
n}\phi_{\alpha}^{(m)*}(r)\phi_{\alpha}^{(n)}(r)C_{\alpha}^{mn},$ (10)

where $\phi_{\alpha}^{(n)}\equiv\frac{d^{n}}{dE^{n}}\phi^{E=0}_{\alpha}$ and
$\phi_{\alpha}^{E}(r)$ is the radial part of the two-body functions (see more
details in the supplemental materials). The two-body density of a many-body
wave function is given here using only two-body functions and numerical
coefficients $C_{\alpha}^{mn}$. The contacts $C_{\alpha}^{mn}$ depend on the
many-body state and, therefore, are generally not simple to calculate
directly. One of the important features of this expansion is that the same
contact parameters appear in the description of different quantities, so they
can be extracted from one quantity and used to predict another. We will
demonstrate it here.

In our expansion, the LO term involves the $s$-wave channel without energy
derivatives and the contact $C_{s}^{00}$. Going to larger distances, two
possible terms might be involved in NLO corrections: (i) the same $s$-wave
channel with one energy derivative involving the $C_{s}^{01}$ contact, and
(ii) a $p$-wave channel without energy derivatives, involving a contact
parameter $C_{p}^{00}$. To understand whether these two contributions enter at
the same order, we can analyze the two-body Schrödinger equation. Considering
a central potential, an $s$-wave solution behaves as $\phi_{s}\sim 1$ at short
distances. The first energy derivative behaves as $\phi_{s}^{(1)}\sim r^{2}$,
indeed suppressed compared to $\phi_{s}$. A $p$-wave solution behaves as
$\phi_{p}\sim r$. Therefore, both $\phi_{s}^{*}\phi_{s}^{(1)}$ and
$|\phi_{p}|^{2}$ behave as $r^{2}$ at short distances. Hence, $C_{s}^{01}$ and
$C_{p}^{00}$ are expected to enter together at NLO. Similarly,
$\phi_{s}^{(2)}\sim r^{4}$, $\phi_{p}^{(1)}\sim r^{3}$, and $\phi_{d}\sim
r^{2}$ ($d$-wave solution), so $|\phi_{s}^{(1)}|^{2}$,
$\phi_{s}^{*}\phi_{s}^{(2)}$, $|\phi_{d}|^{2}$ and
$\phi_{p}^{*}\phi_{p}^{(1)}$ all behave as $r^{4}$. Therefore, $C_{s}^{11}$,
$C_{s}^{02}$, $C_{d}^{00}$ and $C_{p}^{01}$ are expected to contribute at N2LO
(see more details in the supplemental materials).

With this understanding of the power counting, we can now test Eq. (10)
against exact numerical calculations. For this purpose we will consider the
two-body density of infinite neutron matter. We use the AV4’ two-body
interaction, together with the central UIXc three-body force [33]. Auxiliary-
field diffusion Monte Carlo (AFDMC) calculations of $\rho_{2}(r)$ for infinite
neutron matter at density of $0.16$ fm-3 are compared to the short-range
expansion in Fig. 2. At very short distances ($r\lesssim 0.5$ fm), the leading
order $s$-wave contribution provides a good description. At larger distances,
both $C_{s}^{01}$ and $C_{p}^{00}$ terms should be included, resulting in a
good description for $r\lesssim 1.3$ fm. At N2LO, $C_{s}^{11}$ and
$C_{p}^{01}$ are included, extending the agreement to $r\lesssim 2$ fm. In
principle, at N2LO we should also have contributions involving $C_{s}^{02}$
and $C_{d}^{00}$, but their $r$-dependence is very similar to $C_{s}^{11}$ for
the AV4’ potential, and, therefore, cannot be separated. The contact values
are fitted to the AFDMC calculations (see more details in the supplementary).
At larger distances ($r\gtrsim 2$ fm), Fermi-gas (FG) calculations are in good
agreement with the exact AFDMC calculations. Combining the short-range
expansion at N2LO level, which includes 5 terms and only two-body
calculations, together with the long-range asymptotics based on the FG model,
we obtain a good description of the many-body AFDMC calculations of neutron
matter.

Figure 2:  AFDMC calculations of $\rho_{2}(r)$ for infinite neutron matter at
density 0.16 fm-3 (red points), compared to the short-range expansion of Eq.
(10). In the legend, the labels in parenthesis are of the form $\ell mn$,
where $\ell$ is the orbital angular momentum quantum number. The FG result is
also shown (black dotted line). See details in the text.

The above analysis of the two-body density shows the validity of our short-
range expansion for the many-body case. The same expansion can also be used to
describe other quantities. We can look at the one-body momentum distribution
$n(k)$. Similarly to the derivation presented in Ref. [20], we obtain

$n(k)=2\sum_{\alpha}\sum_{m\leq
n}\tilde{\phi}_{\alpha}^{(m)*}(k)\tilde{\phi}_{\alpha}^{(n)}(k)C_{\alpha}^{mn},$
(11)

where
$\tilde{\phi}_{\alpha}^{(n)}\equiv\frac{d^{n}}{dE^{n}}\tilde{\phi}^{E=0}_{\alpha}$
and $\tilde{\phi}_{\alpha}^{E}(k)$ is the radial part of the two-body
functions in momentum space. In this case, this expression should provide a
high-momentum expansion of the one-body momentum distribution. We note that CM
motion of the pair and three-body correlations are neglected here. They should
become important around the Fermi momentum.

Notice that the same contacts appear in both Eqs. (10) and (11), as generally
discussed above. Therefore, contact values fitted to the two-body density
(Fig. 2) can be used to test the expansion for the one-body momentum
distribution and verify the consistency of the relations. This analysis is
shown in Fig. 3. We see that Eq. (11) provides a good description of the high-
momentum part of the AFDMC one-body momentum distribution. The LO $s$-wave
term describes the very high momentum tail, while next-order corrections are
important at lower momenta, leading to a very good agreement with the exact
calculations above the Fermi momentum. Specifically, the $p$-wave channel has
significant contribution around $k=2$ fm-1, where the $s$-wave function is
zero. We can also describe the momentum distribution below the Fermi momentum
as a constant fixed by the global normalization of $n(k)$, leading to a good
description of the momentum distribution for all momentum values. This is
relevant also in connection to a recent experimental study of the transition
from mean-field to SRC domains [34].

Figure 3:  AFDMC calculations of $n(k)$ for infinite neutron matter at density
0.16 fm-3 (red points), compared to the short-range expansion of Eq. (11)
(above the Fermi momentum). The vertical dashed black line denotes the Fermi
momentum. Below the Fermi momentum, a constant is fixed by normalization.

On top of the description of structure quantities, our expansion can be useful
for analyzing reactions that are sensitive to SRCs, such as electron
scattering in nuclear systems, neutron diffraction measurement of the static
structure factor in liquid 4He [35, 13], and radio-frequency spectroscopy in
ultra-cold atomic systems [8]. For example, large momentum transfer electron
scattering cross sections dominated by SRCs are well described by spectral
function calculations which are based on the GCF LO short-range factorization
of Eq. (7) [24, 25, 26, 27, 29, 30]. No other methods are currently available
to describe these reactions, beyond the very light nuclei. The above expansion
provides next-order corrections for such spectral function calculations. They
are important especially for relatively low momentum, similarly to the case of
the momentum distribution (Fig. 3). Therefore, it will provide a systematic
description of such experiments and should allow us to extract contact values
from experiments, including subleading terms. This will be the focus of future
studies. We note that measurements of the spin and/or angular momentum of
knocked-out pairs in such experiments will allow to separate the different
contributions, e.g., the $s$-wave and $p$-wave contributions, and enable a
more accurate extraction of contact values. Such contact values can then be
used to predict different quantities for the same system.

The fact that our short-range expansion combined with long-range models (like
the FG model) allows us to obtain a good description at all values of relative
distances and momenta opens the possibility for describing different
quantities that are affected by both mean-field physics and short- and long-
range correlations. For example, we should be able to obtain a good
description of the kinetic energy and two-body potential energy. Notice that
at N2LO level, our expansion requires 5 contact parameters for neutron matter,
but this number can be reduced. Assuming that we know the value of the leading
contact parameter ($C_{s}^{00}$), we can fix the values of the remaining 4
contact parameters by requiring continuous and smooth matching with long-range
FG description of both spin-zero and spin-one two-body densities. This
approach leads to a good description of both the two-body density and one-body
momentum distribution, similar to the description shown in Figs. 2 and 3,
obtained by fitting all 5 contact parameters, see Fig. 4. Using this
description of the densities, we obtain a two-body per-particle potential
energy of $-29.3$ MeV (with the AV4’ potential) and kinetic energy of $43.2$
MeV for the above neutron matter system. This is very close to the values
obtained in AFDMC calculations, $-30.1$ MeV for the potential energy and
$43.3$ MeV for the kinetic energy. We can see that with only knowledge of a
single parameter (the leading order contact), the short-range expansion allows
us to accurately calculate the kinetic energy and potential energy of the
system.

Figure 4:  AFDMC calculations of $\rho_{2}(r)$ for infinite neutron matter at
density 0.16 fm-3 (orange points), and the expansion of Eq. (10). The leading
$s$-wave contact is fitted and the remaining 4 contact parameters are fixed by
matching to the FG expression. The matching point is shown by the dashed line.
For larger distances the FG density is used. Inset: AFDMC calculations of
$n(k)$ for the same system and the expansion of Eq. (11) with the same contact
values used in the main figure. Below the Fermi momentum, indicated by the
dashed line, a constant is fixed by normalization.

To summarize, we have presented a systematic short-range expansion of quantum
many-body wave functions. Description of observables is obtained using two-
body functions and corresponding contact parameters. The same contacts are
relevant for different quantities of a given system, allowing us to extract
their values using one quantity and then predict others. We have identified an
appropriate power counting by analyzing analytically the two-body problem.
This expansion is relevant for various strongly-interacting many-body systems.

We have tested our approach against many-body ab-initio numerical
calculations, considering nuclear systems as an example. Combined with
asymptotic long-range models, a good description of both two-body density and
one-body momentum distribution is obtained at all distances and momenta,
enabling calculations of quantities that involve both long-range and short-
range physics, like total potential energy and kinetic energy. We have also
demonstrated the consistency of the different relations as a good description
of the one-body momentum distribution with a clear order-by-order convergence
is obtained using contact values fitted to the two-body densities.

This work also provides a systematic framework for the analysis of large
momentum transfer electron scattering experiments, focused on SRC physics,
connecting them to low-energy nuclear physics studies. It provides next order
corrections to the description of such experiments, including an important
$p$-wave contribution, and motivates new experiments, such as measurements of
the spin or orbital angular momentum of the outgoing pair, to isolate the
contribution of different channels.

We would like to thank J. Carlson, J. Martin, S. Novario, R. Somasundaram, I.
Tews, N. Barnea, B. Bazak, S. Beck, O. Hen, and E. Piasetzky for helpful
discussions. The work of R.W. was supported by the Laboratory Directed
Research and Development program of Los Alamos National Laboratory under
project number 20210763PRD1. The work of S. G. was supported by U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, under
Contract No. DE-AC52-06NA25396, by the DOE NUCLEI SciDAC Program, and by the
DOE Early Career Research Program. Computer time was provided by the Los
Alamos National Laboratory Institutional Computing Program, which is supported
by the U.S. Department of Energy National Nuclear Security Administration
under Contract No. 89233218CNA000001.

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