# Tan’s two-body contact in a planar Bose gas: experiment vs theory

Adam Rançon  Univ. Lille, CNRS, UMR 8523 – PhLAM – Laboratoire de Physique des
Lasers Atomes et Molécules, F-59000 Lille, France  Institute of Physics,
Bijenička cesta 46, HR-10001 Zagreb, Croatia  Nicolas Dupuis  Sorbonne
Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée,
LPTMC, F-75005 Paris, France

(August 27, 2024)

###### Abstract

We determine the two-body contact in a planar Bose gas confined by a
transverse harmonic potential, using the nonperturbative functional
renormalization group. We use the three-dimensional thermodynamic definition
of the contact where the latter is related to the derivation of the pressure
of the quasi-two-dimensional system with respect to the three-dimensional
scattering length of the bosons. Without any free parameter, we find a
remarkable agreement with the experimental data of Zou et al. [Nat. Comm. 12,
760 (2021)] from low to high temperatures, including the vicinity of the
Berezinskii-Kosterlitz-Thouless transition. We also show that the short-
distance behavior of the pair distribution function and the high-momentum
behavior of the momentum distribution are determined by two contacts: the
three-dimensional contact for length scales smaller than the characteristic
length $\ell_{z}=\sqrt{\hbar/m\omega_{z}}$ of the harmonic potential and, for
length scales larger than $\ell_{z}$, an effective two-dimensional contact,
related to the three-dimensional one by a geometric factor depending on
$\ell_{z}$.

##### Introduction.

Relating the macroscopic properties of a physical system to microscopic
interactions and degrees of freedom is one of the main goals of many-body
quantum physics. In ultracold atomic gases not all details of the interaction
potential between particles are required since low-energy collisions are
generally fully described by the $s$-wave scattering length. As a result, the
equation of state of a dilute gas takes a simple, universal, expression where
the microscopic physics enters only through two parameters, the mass of the
particles and the scattering length. Considering the latter as an additional
thermodynamic variable, besides the usual variables (e.g. the chemical
potential $\mu$ and the temperature $T$ in the grand canonical ensemble), one
can define its thermodynamic conjugate, the so-called Tan two-body contact $C$
Tan (2008a, b, c). In a dilute gas, the contact relates the (universal) low-
temperature thermodynamics to the (universal) short-distance behavior which
shows up in the two-body correlations or the momentum distribution function
Tan (2008a, b, c); Braaten and Platter (2008); Zhang and Leggett (2009);
Combescot  _et al._ (2009); Valiente  _et al._ ; Werner and Castin (2012a, b).
This simple description fails in a strongly interacting Bose gas where other
parameters (e.g. associated with three-body effective interactions) are
required for a complete description of the universal thermodynamics and short-
distance physics [Intwodimensionsandbelow;
theequationofstatecanbeexpressedonlyintermsofthescatteringlengthandnoadditionalparameterssuchasthethree-
bodyparameterarenecessary; see]Adhikari95.

There have been few measurements of the two-body contact in Bose gases. Apart
from experiments in the thermal regime Wild  _et al._ (2012); Fletcher  _et
al._ (2017) or the quasi-pure BEC one Wild  _et al._ (2012); Lopes  _et al._
(2017), the two-body contact has been determined in a planar Bose gas in a
broad temperature range including the normal and superfluid phases as well as
the vicinity of the Berezinskii-Kosterlitz-Thouless (BKT) transition Zou  _et
al._ (2021). The experimental data are in good agreement with theoretical
predictions in the high-temperature limit (normal gas) and in the low-
temperature limit (strongly degenerate superfluid). On the other hand there is
no theoretical explanation for the value of the contact obtained near the BKT
transition. In particular, the experimental data seem at odds with the
predictions of a classical field theory Prokof’ev and Svistunov (2002) used
earlier successfully for the equation of state of a two-dimensional Bose gas
Yefsah  _et al._ (2011); Desbuquois  _et al._ (2014).

In this Letter we compute the two-body contact in a planar Bose gas confined
by a harmonic potential using the nonperturbative functional renormalization
group (FRG), a modern implementation of Wilson’s RG Berges  _et al._ (2002);
Delamotte (2012); Dupuis  _et al._ (2021). This approach has proven to be very
accurate for determining the equation of state of such a system Rançon and
Dupuis (2012a). We consider the weak-coupling limit where the three-
dimensional scattering length $a_{3}$ of the bosons is much smaller than the
characteristic length $\ell_{z}=\sqrt{\hbar/m\omega_{z}}$ of the harmonic
potential. We use the three-dimensional thermodynamic definition of the
contact where the latter is expressed as a derivative of the pressure with
respect to the three-dimensional scattering length $a_{3}$ and pay special
attention to the quasi-two-dimensional structure of the system. We show that
the short-distance behavior of the pair distribution function and the high-
momentum behavior of the momentum distribution are determined by two contacts:
the three-dimensional contact for length scales smaller than $\ell_{z}$ and,
for larger length scales, an effective two-dimensional contact (obtained from
the derivative of the pressure with respect to the effective two-dimensional
scattering length of the confined bosons), related to the three-dimensional
one by a geometric factor depending on $\ell_{z}$. We then compare our results
with the experimental data of Ref. Zou  _et al._ (2021). Without any free
parameters, we find a remarkable agreement between theory and experiment from
low to high temperatures, including the vicinity of the BKT phase transition.
Finally, we show how to reconcile these experimental data with the classical
field simulations of Ref. Prokof’ev and Svistunov (2002), although the fit is
not as accurate as with the FRG.

##### Contact of a planar Bose gas.

We consider a quasi-two-dimensional system of surface $L^{2}$ obtained by
subjecting a three-dimensional Bose gas to a confining harmonic potential of
frequency $\omega_{z}=\hbar/m\ell_{z}^{2}$ along the $z$ direction. In the
low-temperature regime $k_{B}T\ll\hbar\omega_{z}$, the physical properties of
the planar gas can be obtained from the effective two-dimensional Hamiltonian
(from now on we set $\hbar=k_{B}=1$)

$\hat{H}=\int
d^{2}r\biggl{\\{}\frac{\boldsymbol{\nabla}\hat{\psi}^{\dagger}\cdot\boldsymbol{\nabla}\hat{\psi}}{2m}+\frac{g}{2}\hat{\psi}^{\dagger}\hat{\psi}^{\dagger}\hat{\psi}\hat{\psi}\biggr{\\}},$
(1)

with an ultraviolet momentum cutoff $\Lambda\simeq 0.54\,\ell_{z}^{-1}$ Petrov
and Shlyapnikov (2001); Lim  _et al._ (2008). The effective interaction
constant $g$ is related to the three-dimensional scattering length $a_{3}$ by

$mg=\sqrt{8\pi}\frac{a_{3}}{\ell_{z}}.$ (2)

Computing the low-energy scattering amplitude from (1), one obtains the
effective two-dimensional scattering length $a_{2}$ as a function of the
microscopic parameters of the gas Petrov and Shlyapnikov (2001); Lim  _et al._
(2008); Pricoupenko and Olshanii (2007),

$\frac{a_{2}}{\ell_{z}}\simeq
3.71\,e^{-\sqrt{\frac{\pi}{2}}\frac{\ell_{z}}{a_{3}}-\gamma},$ (3)

where $\gamma\simeq 0.577$ is the Euler constant.

In the low-temperature regime $T\ll\omega_{z}$, the pressure can be written in
the scaling form characteristic of a two-dimensional system Rançon and Dupuis
(2012a),

$P(\mu,T)=-\frac{\Omega(\mu,T)}{L^{2}}=\frac{T}{\lambda^{2}}{\cal
F}\left(\frac{\mu}{T},\tilde{g}(T)\right),$ (4)

where $\Omega(\mu,T)$ is the grand potential, ${\cal F}$ a universal scaling
function, $\lambda=\sqrt{2\pi/mT}$ the thermal de Broglie wavelength and

$\tilde{g}(T)=-\dfrac{4\pi}{\ln\left(\frac{1}{2}\sqrt{2ma_{2}^{2}T}\right)+\gamma}$
(5)

a temperature-dependent dimensionless interaction constant. Equation (4) is
valid for $|\mu|,T\ll\Lambda^{2}/2m\sim\omega_{z}$. The corrections to the
scaling form (4) are negligible when $ma_{2}^{2}T,ma_{2}^{2}|\mu|\ll 1$. The
dependence of the scaling function ${\cal F}$ on $\mu/T$ and $ma_{2}^{2}T$ can
be simply obtained by dimensional analysis using the fact that the scattering
length is the only characteristic length scale at low energies.
Renormalization-group arguments show that the dependence on $ma^{2}_{2}T$
arises only through $\tilde{g}(T)$ Rançon and Dupuis (2012a).

The universal scaling function ${\cal F}$ entering the equation of state (4)
of the two-dimensional Bose gas has been computed using the FRG approach
Rançon and Dupuis (2012a), and very good agreement with experimental data in
planar gases Hung  _et al._ (2011); Yefsah  _et al._ (2011); Zhang  _et al._
(2012), with or without an optical lattice, has been obtained.

The pressure depending only on $a_{3}$ (through $a_{2}\equiv
a_{2}(a_{3},\ell_{z})$) and not on the details of the interaction potential
between particles, it is natural to consider $a_{3}$ as an additional
thermodynamic variable besides $\mu$ and $T$ Tan (2008a). The quasi-two-
dimensional two-body contact $C$ is then essentially defined as the conjugate
variable to $a_{3}$,

$\frac{C}{L^{2}}=8\pi m\frac{\partial P}{\partial(1/a_{3})}\biggl{|}_{\mu,T}.$
(6)

$C$ is an extensive quantity with dimension 1/length. It can equivalently be
defined in the canonical ensemble by replacing the pressure by the energy
density in (6) and taking the derivative at fixed particle number and entropy.
The motivation for defining the contact from a derivative with respect to
$1/a_{3}$ as in a three-dimensional system, rather than with respect to $\ln
a_{2}$ as in a two-dimensional system, is that in the limit $a_{3}\ll\ell_{z}$
the collisions keep their three-dimensional character at length scales smaller
than $\ell_{z}$.

Using the equation of state (4), we can write the contact as

$\frac{C}{L^{2}}=-4(2\pi)^{5/2}\frac{T\ell_{z}}{\lambda^{4}}{\cal
F}^{(0,1)}\left(\frac{\mu}{T},\tilde{g}(T)\right)\tilde{g}^{\prime}(T),$ (7)

where we use the notation ${\cal
F}^{(i,j)}(x,y)=\partial^{i}_{x}\partial^{j}_{y}{\cal F}(x,y)$. On the other
hand, the scaling function ${\cal F}$ determines the two-dimensional particle
and entropy densities, $\bar{n}=\partial P/\partial\mu$ and $s=\partial
P/\partial T$. This allows us to express ${\cal F}^{(0,1)}$ in terms of
$P,\bar{n},s$, which leads to a relation between the contact and the
thermodynamic potentials $P$ and $E$ (as in isotropic systems Tan (2008b))

$\frac{C}{L^{2}}=4(2\pi)^{5/2}\frac{\ell_{z}}{\lambda^{2}T}\left(P-\frac{E}{L^{2}}\right),$
(8)

where $E/L^{2}=-P+Ts+\mu\bar{n}$ is the two-dimensional energy density.

In the weak-coupling limit, the scattering length $a_{2}$ is exponentially
small with respect to $\ell_{z}$ and $\tilde{g}(T)\simeq\tilde{g}=2mg$ is
nearly temperature independent except for exponentially small temperatures
$T\sim\omega_{z}e^{-\sqrt{\pi/2}(\ell_{z}/a_{3})}$. For this reason it is
often concluded that the two-dimensional Bose gas exhibits an approximate
scale invariance Prokof’ev and Svistunov (2002); Yefsah  _et al._ (2011);
Desbuquois  _et al._ (2014) (with no characteristic energy scales other than
$\mu$ and $T$): ${\cal F}(\mu/T,\tilde{g}(T))\simeq{\cal F}(\mu/T,\tilde{g})$
Rançon and Dupuis (2012a), and the normalized contact
$C\lambda^{4}/L^{2}\ell_{z}$ is a function of $\mu/T$ and $\tilde{g}$. However
the assumption of approximate scale invariance cannot be used in the
calculation leading to (8) since this would imply $\tilde{g}^{\prime}(T)=0$
and in turn $E=L^{2}P$ and $C=0$. The contact defined in (8) can thus be seen
as a measure of the breakdown of scale invariance in the planar Bose gas not
(a). The situation is different in an isotropic three-dimensional system; in
the unitary limit $a_{3}\to\infty$ where scale invariance is satisfied,
$E_{\rm 3D}=(3/2)L^{3}P_{\rm 3D}$, but the contact remains finite since
$(3/2)P_{\rm 3D}-E_{\rm 3D}/L^{3}={\cal O}(1/a_{3})$ whereas $C_{\rm
3D}\propto a_{3}[(3/2)P_{\rm 3D}-E_{\rm 3D}/L^{3}]$.

A remarkable feature of the contact defined from the pressure is that it also
determines the short-distance behavior of the pair distribution function as
well as the high-momentum limit of the momentum distribution function Tan
(2008a, b, c); Braaten and Platter (2008); Zhang and Leggett (2009); Combescot
_et al._ (2009); Valiente  _et al._ ; Werner and Castin (2012a, b). This
property still holds in the planar gas but there are two different contacts,
depending on whether one considers length scales larger or smaller than
$\ell_{z}$. For the pair distribution function, averaged over the position
${\bf R}=({\bf r}_{1}+{\bf r}_{2})/2$ of the pair center of mass, one has not
(a)

$\displaystyle g({\bf r})$ $\displaystyle=\int
d^{3}R\,\langle\hat{\psi}^{\dagger}({\bf r}_{1})\hat{\psi}^{\dagger}({\bf
r}_{2})\hat{\psi}({\bf r}_{2})\hat{\psi}({\bf r}_{1})\rangle,$
$\displaystyle=\left\\{\begin{array}[]{lll}\frac{C}{(4\pi)^{2}}\left(\frac{1}{|{\bf
r}|}-\frac{1}{a_{3}}\right)^{2}&\mbox{if}&|{\bf r}|\ll\ell_{z},\lambda,d,\\\
\frac{C}{\sqrt{2\pi}\ell_{z}}|\phi_{0}^{-}(z)|^{2}\left(\frac{\ln|\boldsymbol{\rho}/a_{2}|}{2\pi}\right)^{2}&\mbox{if}&\ell_{z}\ll|\boldsymbol{\rho}|\ll\lambda,d,\end{array}\right.$
(11)

where ${\bf r}={\bf r}_{1}-{\bf r}_{2}=(\boldsymbol{\rho},z)$ is the
coordinate of the relative motion of the pair (with $\boldsymbol{\rho}$ a two-
dimensional coordinate) and $d$ the mean interparticle distance.
$\phi_{0}^{-}$ denotes the ground state of a particle of reduced mass $m/2$ in
an harmonic potential of characteristic frequency $\omega_{z}$. The fact that
the contact (6) determines the pair distribution function at short length
scales, $|{\bf r}|\ll\lambda,\ell_{z},d$, is not surprising since
$C\propto\partial P/\partial(1/a_{3})$ is defined as in the three dimensional
case. On the other hand, the contact that appears in the regime
$\ell_{z}\ll|\boldsymbol{\rho}|\ll\lambda,d$ can be written as

$\frac{C}{\sqrt{2\pi}\ell_{z}L^{2}}=-4\pi m\frac{\partial P}{\partial\ln
a_{2}},$ (12)

and thus coincides with the usual thermodynamic definition of a two-
dimensional contact. Similarly, for the momentum distribution function
$\bar{n}_{{\bf k},k_{z}}$, one finds

$\bar{n}_{{\bf k},k_{z}}=\frac{C}{({\bf
k}^{2}+k_{z}^{2})^{2}}\quad\mbox{if}\quad\frac{1}{a_{3}},\frac{1}{d},\frac{1}{\lambda}\ll\sqrt{{\bf
k}^{2}+k_{z}^{2}},$ (13)

and

$\displaystyle\bar{n}^{\rm(2D)}_{{\bf k}}$
$\displaystyle=\int\frac{dk_{z}}{2\pi}\bar{n}_{{\bf k},k_{z}},$
$\displaystyle=\frac{1}{\sqrt{2\pi}\ell_{z}}\frac{C}{|{\bf
k}|^{4}}\quad\mbox{if}\quad\frac{1}{d},\frac{1}{\lambda}\ll|{\bf
k}|\ll\frac{1}{\ell_{z}},$ (14)

where we assume the normalization $\frac{1}{L^{2}}\sum_{{\bf
k}}\int\frac{dk_{z}}{2\pi}\bar{n}_{{\bf k},k_{z}}=N$ (with
$N\equiv\langle\hat{N}\rangle$ the total number of particles). Equation (13)
holds at any temperature while Eq. (14) is valid only in the low-temperature
regime $T\ll\omega_{z}$. Note that a result similar to (13,14) has been
obtained for fermions in quasi-one- and quasi-two-dimensional traps He and
Zhou (2019) (see also Decamp  _et al._ (2018); Bougas  _et al._ (2020) for
bosons in a quasi-one-dimensional trap).

##### FRG calculation of the contact.

Following Ref. Rançon and Dupuis (2012a) we compute the pressure of the planar
Bose gas at low temperatures $T\ll\omega_{z}$ using the two-dimensional
Hamiltonian (1). The main quantity of interest in the FRG approach is the
effective action (or Gibbs free energy) $\Gamma$, defined as the Legendre
transform of the Helmholtz free energy, and which is directly related to the
pressure: $P=-(T/L^{2})\Gamma$. We refer to Refs. Rançon and Dupuis (2012a);
Dupuis  _et al._ (2021) for a detailed description of the method.

Figure 1: Two-body contact $C\lambda^{4}/L^{2}\ell_{z}$ vs $\mu/T$ for
$2mg=0.32$ and $T/\omega_{z}\simeq 0.00145$ as obtained from the FRG (solid
(blue) line). The dashed (red) line shows the result obtained from (8) while
the symbols correspond to the Bogoliubov result (15) and the virial expansion
(16). The BKT transition occurs at $\mu/T\simeq 0.21$, shown as a vertical
dashed line, as estimated from FRG.

In Fig. 1, we show the contact $C$ normalized by $L^{2}\ell_{z}/\lambda^{4}$
obtained for $2mg=2\sqrt{8\pi}a_{3}/\ell_{z}=0.32$ and $T/\omega_{z}\simeq
0.00145$ by computing the pressure $P$ for two nearby values of $a_{3}$ and
taking a numerical derivative. We find a very good agreement with two limiting
cases not (a), the zero-temperature limit in the superfluid phase where

$\frac{C_{\rm Bog}}{L^{2}}=\sqrt{2\pi}(m\mu)^{2}\ell_{z}$ (15)

can be obtained from the Bogoliubov theory (including the Lee-Huang-Yang
correction not (b)), and the dilute normal gas where

$\frac{C_{\rm
virial}}{L^{2}}=\frac{z^{2}\ell_{z}}{\lambda^{4}}\frac{2(2\pi)^{5/2}}{\left|\ln\left(\frac{a_{2}}{\lambda}\sqrt{\frac{\pi}{2}}e^{\gamma}\right)\right|^{2}},$
(16)

(with $z=e^{\beta\mu}$ the fugacity) can be obtained from the virial
expansion.

In the figure we also show the contact deduced from (8), the two-dimensional
energy density $E/L^{2}=-P+T\partial P/\partial T+\mu\partial P/\partial\mu$
being obtained from numerical derivatives. The apparent disagreement for
$\mu/T\lesssim 0.5$ is due to a lack of precision in the numerical
calculation, the values of $P$ and $E/L^{2}$ being extremely close and the
normalized contact very small.

##### Comparison with the ENS experiment.

The thermodynamic definition (6) of the contact was realized experimentally in
the ENS group by means of a Ramsey interferometric method Zou  _et al._
(2021). The measurements were performed on a Bose gas of 87Rb atoms confined
by a harmonic potential with $2mg=2\sqrt{8\pi}a_{3}/\ell_{z}=0.32$ in a broad
temperature range across the BKT transition. In Fig. 2 we compare the
experimental data with the FRG result obtained for $2mg=0.32$ and
$T/\omega_{z}=0.00145$. We show the contact normalized in terms of the
Bogoliubov contact, $C/C_{\rm Bog}$, as a function of the phase-space density
${\cal D}=\bar{n}\lambda^{2}$. To leading order, the Bogoliubov result is
deduced from (15) using the relation between $\mu$ and $\bar{n}$ not (a, b)

$\frac{C_{\rm
Bog}}{L^{2}}=\frac{(2\pi)^{5/2}\bar{n}^{2}\ell_{z}}{\left|\ln\left(a_{2}\sqrt{\pi\bar{n}}e^{\gamma+1/2}\right)\right|^{2}},$
(17)

although in practice we invert numerically $\bar{n}(\mu)$ to compute $C_{\rm
Bog}$.

The normalized contact varies between one for ${\cal D}\to\infty$ and

$\displaystyle\frac{C_{\rm virial}}{C_{\rm Bog}}$
$\displaystyle=2\left|\frac{\ln\left(a_{2}\sqrt{\pi\bar{n}}e^{\gamma+1/2}\right)}{\ln\left(\frac{a_{2}}{\lambda}\sqrt{\frac{\pi}{2}}e^{\gamma}\right)}\right|^{2}$
$\displaystyle=2-4\sqrt{\frac{2}{\pi}}\frac{a_{3}}{\ell_{z}}\left[\ln\sqrt{\frac{\pi^{2}{\cal
D}}{2}}+\frac{1}{2}\right]+{\cal
O}\left(\frac{a_{3}^{2}}{\ell_{z}^{2}}\right)$ (18)

for ${\cal D}\to 0$. The agreement between FRG and the experimental data is
very good in spite of the absence of any fit parameter. Note that there is no
sign of the BKT transition occurring for ${\cal D}\simeq 7.7$ (classical field
simulations Prokof’ev and Svistunov (2002)) and ${\cal D}\simeq 6.7$ (FRG),
marked as a vertical dashed line in Fig. 2. The bump near ${\cal D}=3$ is due
to the rapid change of $\bar{n}$, and therefore $C_{\rm Bog}$, for small and
positive values of $\mu/T$ Yefsah  _et al._ (2011); Desbuquois  _et al._
(2014).

Figure 2: Normalized two-body contact $C/C_{\rm Bog}$ as a function of the
phase-space density ${\cal D}=\bar{n}\lambda^{2}$ for $2mg=0.32$ and
$T/\omega_{z}=0.00145$. The square (green) symbols show the experimental data
from Ref. Zou  _et al._ (2021), the solid line the FRG result and the round
(orange) symbols the estimate from classical field simulations. The BKT
transition is at ${\cal D}\simeq 6.7$ (FRG), shown as a vertical dashed line
(classical field simulations etimate ${\cal D}\simeq 7.7$).

By showing $C/C_{\rm Bog}$ rather than $C/C_{0}$ as in Ref. Zou  _et al._
(2021) (with $C_{0}/L^{2}=4(2\pi)^{3/2}\bar{n}^{2}a_{3}^{2}/\ell_{z}$ the
mean-field contact) we ensure that the normalized contact tends to unity for
${\cal D}\gg 1$ even at temperatures where logarithmic corrections are not
fully negligible and $\tilde{g}(T)$ starts deviating from $2mg$. For $T\simeq
0.03\omega_{z}$, $C_{\rm Bog}\simeq C_{0}$ as a result of the approximate
scale invariance ($\tilde{g}(T)\simeq 2mg$) and whether one uses $C/C_{\rm
Bog}$ or $C/C_{0}$ does not matter. In the large ${\cal D}$ limit the ratio
$C/C_{0}$ becomes smaller than one when the temperature is decreased below
$0.03\omega_{z}$ since the logarithmic corrections make $C\simeq C_{\rm Bog}$
differ from $C_{0}$.

The accuracy of our calculation can in principle be estimated by using a more
refined approximation scheme for the exact FRG flow equation and checking
whether the results have converged. More simply a rough estimate can be
obtained from our previous work on the Bose-Hubbard model where the same
approximation scheme was used to obtain the transition line between the
superfluid phase and the Mott insulator with an accuracy of order 1-3% Rançon
and Dupuis (2011a, b). We expect a similar accuracy in the computation of the
equation of state and the contact of the planar Bose gas.

Finally we would like to discuss the apparent failure of the classical field
simulations Prokof’ev and Svistunov (2002) in explaining the experimental data
of the ENS group, as reported in Ref. Zou  _et al._ (2021). There the
theoretical value of the contact was obtained from the expression
$C/L^{2}=4(2\pi)^{3/2}(a_{3}^{2}/\ell_{z})\langle(\hat{\psi}^{\dagger})^{2}(\hat{\psi})^{2}\rangle$
(which follows from (6)) and using the classical limit of the quantum
expectation value $\langle(\hat{\psi}^{\dagger})^{2}(\hat{\psi})^{2}\rangle$
provided in Ref. Prokof’ev and Svistunov (2002). Away from the superfluid
region, the classical field simulations are however not expected to give a
reliable estimate of the expectation value
$\langle(\hat{\psi}^{\dagger})^{2}(\hat{\psi})^{2}\rangle$. Alternatively one
can start from the density $\bar{n}=\partial P/\partial\mu$ provided in Ref.
Prokof’ev and Svistunov (2002) to obtain the pressure and the contact
$C/L^{2}=-8(2\pi)^{3/2}(a_{3}^{2}/\ell_{z})\partial P/\partial g$. This
necessitates one fitting parameter, corresponding to the contact at the BKT
transition, see not (a) for details. The result, presented in Fig. 2, is in
good agreement with the experimental data although the fit is not as accurate
as with the FRG; this can be explained by the lower accuracy of the density
estimate and therefore of the normalized contact $C/C_{\rm Bog}$.

##### Conclusion.

The theoretical calculation of the two-body contact of a planar Bose gas in
the framework of the nonperturbative FRG is in remarkable agreement with the
experimental data of the ENS group Zou  _et al._ (2021). The FRG provides us
with an accurate value of the contact over a broad temperature range including
the vicinity of the BKT transition. We have also shown how the apparent
contradiction between the experimental data of Ref. Zou  _et al._ (2021) and
the classical field simulations Prokof’ev and Svistunov (2002) can be
overcome, although the latter do not provide as accurate a fit as the FRG.

An interesting aspect of the contact theory in a quasi-two-dimensional gas is
that the high-momentum tail, $\sqrt{{\bf k}^{2}+k_{z}^{2}}\gg
1/a_{3},1/\lambda,1/d$, of the three-dimensional momentum distribution is
controlled by the three-dimensional contact (as defined in (6)) while the
intermediate momentum range, $1/d,1/\lambda\ll|{\bf k}|\ll 1/\ell_{z}$, is
controlled by the two-dimensional contact $C/\sqrt{2\pi}\ell_{z}$ [Eqs.
(13,14) and Refs. He and Zhou (2019); Bougas  _et al._ (2020)]. A measure of
the momentum distribution, for instance by ballistic expansion of the atomic
cloud or rf spectroscopy Stewart  _et al._ , would allow us to confirm this
essential property of the contact theory.

##### Acknowledgment.

We thank J. Beugnon, J. Dalibard, S. Nascimbene for providing us with the
experimental data. We are grateful to them as well as F. Werner for
enlightening discussions and a critical reading of the manuscript. AR is
supported by the Research Grants QRITiC I-SITEULNE/ANR-16-IDEX-0004 ULNE.

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Tan’s two-body contact in a planar Bose gas
– Supplemental Material –

Adam Rançon1 and Nicolas Dupuis2

1Université de Lille, CNRS, Laboratoire de Physique des Lasers Atomes et
Molécules, PhLAM, F-59000 Lille, France

2Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière
Condensée, LPTMC, F-75005 Paris, France

In the Supplemental Material, we discuss the calculation of the two-body
contact of a quasi-two-dimensional (quasi-2D) dilute Bose gas in the zero-
temperature superfluid phase within the Bogoliubov theory and in the normal
(non-degenerate) phase within the virial expansion (Secs. I and II). In
particular the virial expansion shows that the short-distance behavior of the
pair distribution function and the high-momentum behavior of the momentum
distribution are determined by two contacts: the 3D contact for length scales
smaller than $\ell_{z}$ and, for larger length scales, an effective 2D contact
(obtained from the derivative of the pressure with respect to the effective 2D
scattering length of the confined bosons), related to the 3D one by a
geometric factor depending on $\ell_{z}$. We generalize these results beyond
the virial expansion in Sec. II.D. Finally we show how the theoretical value
of the contact can be deduced from the classical field simulations of Ref.
Prokof’ev and Svistunov (2002) (Sec. III).

We start from the (grand canonical) Hamiltonian

$\hat{H}=\int
d^{3}r\biggl{\\{}\frac{\boldsymbol{\nabla}\hat{\psi}^{\dagger}({\bf
r})\cdot\boldsymbol{\nabla}\hat{\psi}({\bf
r})}{2m}+[V(z)-\mu]\hat{\psi}^{\dagger}({\bf r})\hat{\psi}({\bf
r})+\frac{g_{3}}{2}\hat{\psi}^{\dagger}({\bf r})\hat{\psi}^{\dagger}({\bf
r})\hat{\psi}({\bf r})\hat{\psi}({\bf r})\biggr{\\}},$ (S1)

where $V(z)=\frac{1}{2}m\omega_{z}^{2}z^{2}$ is a harmonic trapping potential
along the $z$ direction and $\ell_{z}=1/\sqrt{m\omega_{z}}$ is the width of
the trap (we set $\hbar=k_{B}=1$). The surface $L^{2}$ of the quasi-2D system
is assumed much larger than all characteristic length scales of the system. We
denote by $g_{3}$ the interaction between bosons, assumed to be local, and
regularize the model by a UV momentum cutoff $\Lambda_{3}$. In the absence of
the trapping potential ($V(z)=0$), the $s$-wave scattering length is defined
by

$\frac{m}{4\pi a_{3}}=\frac{1}{g_{3}}+\int_{{\bf
k},k_{z}}\frac{1}{2\epsilon_{{\bf k},k_{z}}},$ (S2)

where $\epsilon_{{\bf k},k_{z}}=({\bf k}^{2}+k_{z}^{2})/2m$, $\int_{{\bf
k},k_{z}}=\int\frac{d^{2}k}{(2\pi)^{2}}\int\frac{dk_{z}}{2\pi}$ (with ${\bf
k}$ a 2D wavevector) and the momentum integral is restricted to $({\bf
k}^{2}+k_{z}^{2})^{1/2}\leq\Lambda_{3}$.

In presence of the trapping potential, scattering at low energy is effectively
2D (see Sec. II.A) with the 2D $s$-wave scattering length Petrov and
Shlyapnikov (2001); Lim  _et al._ (2008); Pricoupenko and Olshanii (2007)

$a_{2}\simeq
3.71\ell_{z}e^{-\sqrt{\frac{\pi}{2}}\frac{\ell_{z}}{a_{3}}-\gamma},$ (S3)

where $\gamma\simeq 0.577$ is the Euler constant. In the low-temperature
regime $T\ll 1/m\ell_{z}^{2}$, where $\ell_{z}$ is much smaller than the
thermal de Broglie wavelength $\lambda=\sqrt{2\pi/mT}$, only the lowest-energy
state of the trapping potential $V(z)$ is occupied and the low-energy
properties of the quasi-2D gas can be obtained from the effective 2D
Hamiltonian

$\hat{H}_{\rm eff}=\int
d^{2}\rho\biggl{\\{}\frac{\boldsymbol{\nabla}\hat{\psi}^{\dagger}(\boldsymbol{\rho})\cdot\boldsymbol{\nabla}\hat{\psi}(\boldsymbol{\rho})}{2m}-\mu\hat{\psi}^{\dagger}(\boldsymbol{\rho})\hat{\psi}(\boldsymbol{\rho})+\frac{g}{2}\hat{\psi}^{\dagger}(\boldsymbol{\rho})\hat{\psi}^{\dagger}(\boldsymbol{\rho})\hat{\psi}(\boldsymbol{\rho})\hat{\psi}(\boldsymbol{\rho})\biggr{\\}}$
(S4)

with an ultraviolet momentum cutoff $\Lambda$. Computing the low-energy
scattering amplitude from (S4), one obtains the effective 2D scattering length
$a_{2}$ as a function of the effective parameters $g$ and $\Lambda$

$a_{2}=\frac{2}{\Lambda}e^{-\frac{2\pi}{mg}-\gamma}.$ (S5)

This reproduces the scattering length (S3) setting $\Lambda\simeq
0.54\,\ell_{z}^{-1}$ and

$mg=\sqrt{8\pi}\frac{a_{3}}{\ell_{z}}.$ (S6)

In the planar Bose gas, we define the contact as in a 3D system,

$\frac{C}{L^{2}}=8\pi m\frac{\partial P}{\partial(1/a_{3})},$ (S7)

i.e. from the derivative of the pressure with respect to the scattering length
$a_{3}$.

##  I I. Zero temperature: Bogoliubov theory

In the zero-temperature degenerate limit ($\mu>0$) we expect most particles to
be in the condensate. In that case the contact should be well described by the
effective 2D Hamiltonian (S4) and the Bogoliubov theory.

Following Ref. Rançon and Dupuis (2012) we consider the effective potential
(or Gibbs free energy) $U(n)$ where $n$ is the 2D condensate density. At the
mean-field level, it is simply given by the Hamiltonian treating the operator
$\hat{\psi}\equiv\sqrt{n}e^{i\theta}$ (with $\theta$ an arbitrary phase) as a
classical field,

$U(n)=-\mu n+\frac{g}{2}n^{2},$ (S8)

assuming $n$ to be uniform. The equilibrium value of the condensate density,
$n_{0}=\mu/g$, is obtained by minimizing the effective potential:
$U^{\prime}(n_{0})=0$. This gives the pressure $P=-U(n_{0})=\mu^{2}/2g$ and
the particle density $\bar{n}=\partial P/\partial\mu=\mu/g$ at mean-field. We
thus obtain the contact

$\frac{C}{L^{2}}=\sqrt{2\pi}(m\mu)^{2}\ell_{z}.$ (S9)

The Lee-Huang-Yang correction to the mean-field result is given by the one-
loop expression Rançon and Dupuis (2012)

$U(n)=-\mu n+\frac{g}{2}n^{2}+\frac{1}{2\ell}\int_{\bf k}[-\xi_{\bf
k}-2gn+E_{\bf k}(n)],$ (S10)

where $\xi_{\bf k}=\epsilon_{\bf k}-\mu$ and

$E_{\bf k}(n)=\bigl{[}(\xi_{\bf k}+2gn)^{2}-(gn)^{2}\bigr{]}^{1/2}.$ (S11)

The parameter $\ell$ in (S10) is necessary to organize the loop expansion but
will eventually be set to unity. From (S10) we obtain

$\begin{split}U^{\prime}(n)&=-\mu+gn-\frac{g}{\ell}\int_{\bf
k}\left(1-\frac{E_{\bf k}^{\prime}(n)}{2g}\right),\\\
n_{0}&=\frac{\mu}{g}+\frac{1}{\ell}\int_{\bf k}\left(1-\frac{2\epsilon_{\bf
k}+\mu}{2E_{\bf k}}\right),\end{split}$ (S12)

where it is now understood that all calculations are performed to order
$1/\ell$ and $E_{\bf k}=E_{\bf k}(n=\mu/g)=[\epsilon_{\bf k}(\epsilon_{\bf
k}+2\mu)]^{1/2}$ is the value of $E_{\bf k}(n_{0})$ to leading order.
Performing the momentum integration with the UV cutoff $\Lambda$, we obtain

$\begin{split}n_{0}&=\frac{\mu}{g}+\frac{1}{\ell}\left[\frac{m\mu}{4\pi}\ln\left(\frac{\Lambda^{2}}{m\mu}\right)-\frac{m\mu}{2\pi}\right],\\\
P&=\frac{\mu^{2}}{2g}-\frac{1}{2\ell}\int_{\bf k}(E_{\bf k}-\epsilon_{\bf
k}-\mu)=\frac{\mu^{2}}{2g}-\frac{1}{\ell}\frac{m}{4\pi}\left[\frac{\mu^{2}}{4}-\frac{\mu^{2}}{2}\ln\left(\frac{\Lambda^{2}}{m\mu}\right)\right].\end{split}$
(S13)

Using the definition (S5) of the scattering length $a_{2}$ and setting
$\ell=1$, we deduce Popov (1983); Mora and Castin (2003); Pricoupenko

$\begin{split}P&=-\frac{m\mu^{2}}{8\pi}\ln\left(\frac{m\mu
a_{2}^{2}}{4}e^{2\gamma+\frac{1}{2}}\right),\\\
\bar{n}&=-\frac{m\mu}{4\pi}\ln\left(\frac{m\mu
a_{2}^{2}}{4}e^{2\gamma+1}\right).\end{split}$ (S14)

The contact as a function of $\mu$ retains the same expression as in the mean-
field approximation [Eq. (S9)], but the relation between $\mu$ and the density
$\bar{n}$ differs from the mean-field one. From (S14) and (S9) we obtain

$\mu\simeq\frac{2\pi\bar{n}/m}{\left|\ln\left(a_{2}\sqrt{\pi\bar{n}}e^{\gamma+\frac{1}{2}}\right)\right|},$
(S15)

and the zero-temperature value of the contact,

$\frac{C_{\rm
Bog}}{L^{2}}\simeq(2\pi)^{5/2}\frac{\bar{n}^{2}\ell_{z}}{\left|\ln\left(a_{2}\sqrt{\pi\bar{n}}e^{\gamma+\frac{1}{2}}\right)\right|^{2}},$
(S16)

as a function of the density.

From (S14) it is straightforward to recover the expression
$C/L^{2}=\sqrt{2\pi}(m\mu)^{2}\ell_{z}$ using Eq. (8) in the main text
relating the contact to the pressure and the energy density. This is not the
case in the mean-field theory; the latter gives a vanishing contact using (8)
but a nonzero one using (S7).

##  II II. Low-density limit: virial expansion

When the gas is in the normal (non-degenerate) phase, $\mu<0$ and $|\mu|\gg
T$, it is possible to compute the contact within an expansion in the fugacity
$z=e^{\beta\mu}$. This calculation not only gives an explicit expression of
the contact in that limit but also shows that the short-distance behavior of
the pair distribution function and the high-momentum behavior of the momentum
distribution are determined by two contacts: the 3D contact for length scales
smaller than $\ell_{z}$ and an effective 2D contact for length scales larger
than $\ell_{z}$. We show that the relation between these two contacts is valid
beyond the virial expansion.

###  II.1 A. Two-body problem and $T$ matrix

The Hamiltonian of a single particle in the harmonic potential reads

$\hat{H}=\frac{\hat{\bf
p}^{2}}{2m}+\frac{\hat{p}_{z}^{2}}{2m}+\frac{1}{2}m\omega_{z}^{2}\hat{z}^{2},$
(S17)

where ${\bf p}$ is a 2D momentum associated with the motion perpendicular to
the $z$ axis and we denote the 3D coordinate by ${\bf
r}=(\boldsymbol{\rho},z)$, while ${\bf k}$ is a 2D wavevector. The eigenstates
$|{\bf k},n\rangle$ and eigenenergies are given by

$\langle{\bf r}|{\bf k},n\rangle=\frac{e^{i{\bf
k}\cdot\boldsymbol{\rho}}}{L}\phi_{n}(z),\qquad\epsilon_{{\bf
k},n}=\epsilon_{\bf k}+\epsilon_{n}=\frac{{\bf
k}^{2}}{2m}+\left(n+\frac{1}{2}\right)\omega_{z}$ (S18)

with

$\phi_{n}(z)=\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega_{z}}{\pi}\right)^{1/4}H_{n}\left(\sqrt{m\omega_{z}}z\right)e^{-\frac{m\omega_{z}}{2}z^{2}}$
(S19)

where $H_{n}$ is a Hermite polynomial ($n\in\mathbb{N}$). For the two-body
problem it is convenient to introduce the center-of-mass variables and those
of the relative motion,

$\begin{split}&\hat{\bf P}=\hat{\bf p}_{1}+\hat{\bf
p}_{2},\quad\hat{P}_{z}=\hat{p}_{z1}+\hat{p}_{z2},\quad{\bf
R}_{\parallel}=\frac{\boldsymbol{\rho}_{1}+\boldsymbol{\rho}_{2}}{2},\quad\hat{Z}=\frac{\hat{z}_{1}+\hat{z}_{2}}{2},\\\
&\hat{\bf p}=\frac{\hat{\bf p}_{1}-\hat{\bf
p}_{2}}{2},\quad\hat{p}_{z}=\frac{\hat{p}_{z1}-\hat{p}_{z2}}{2},\quad\boldsymbol{\rho}=\boldsymbol{\rho}_{1}-\boldsymbol{\rho}_{2},\quad\hat{z}=\hat{z}_{1}-\hat{z}_{2},\end{split}$
(S20)

and write the two-body Hamiltonian as

$\displaystyle\hat{H}={}$ $\displaystyle\frac{\hat{\bf
P}^{2}}{4m}+\frac{\hat{P}_{z}^{2}}{4m}+\frac{1}{2}(2m)\omega_{z}^{2}\hat{Z}^{2}+\frac{\hat{\bf
p}^{2}}{m}+\frac{\hat{p}_{z}^{2}}{m}+\frac{1}{2}\left(\frac{m}{2}\right)\omega_{z}^{2}\hat{z}^{2}+g_{3}\,\delta(\hat{\boldsymbol{\rho}})\delta(\hat{z}).$
(S21)

The harmonic potential preserves the decoupling between the center-of-mass and
relative motions. The two-body eigenstates for $g_{3}=0$ can therefore be
written as $|{\bf k}_{1},n_{1};{\bf k}_{2},n_{2}\rangle=|{\bf
k}_{1},n_{1}\rangle\otimes|{\bf k}_{2},n_{2}\rangle$ with the corresponding
(non-symmetrized) wavefunctions

$\frac{e^{i{\bf k}_{1}\cdot\boldsymbol{\rho}_{1}+i{\bf
k}_{2}\cdot\boldsymbol{\rho}_{2}}}{L^{2}}\phi_{n_{1}}(z_{1})\phi_{n_{2}}(z_{2}),\qquad\epsilon_{{\bf
k}_{1},n_{1},{\bf k}_{2},n_{2}}=\epsilon_{{\bf k}_{1},n_{1}}+\epsilon_{{\bf
k}_{2},n_{2}}\qquad(n_{1},n_{2}\in\mathbb{N})$ (S22)

or, in the center-of-mass frame, $|{\bf K},N;{\bf k},n\rangle$ with the
corresponding wavefunctions

$\frac{e^{i{\bf K}\cdot{\bf R}_{\parallel}+i{\bf
k}\cdot\boldsymbol{\rho}}}{L^{2}}\phi^{+}_{N}(Z)\phi^{-}_{n}(z),\qquad\epsilon_{{\bf
K},N,{\bf k},n}=\epsilon^{+}_{{\bf K},N}+\epsilon^{-}_{{\bf
k},n}\qquad(N,n\in\mathbb{N}),$ (S23)

where $\phi_{n}^{\pm},\epsilon^{\pm}_{{\bf k},n}$ are deduced from
$\phi_{n},\epsilon_{{\bf k},n}$ by replacing the mass $m$ by $2m$ and $m/2$,
respectively.

In the center-of-mass frame, the $T$ matrix reads

$T_{nn^{\prime}}({\bf K},N,\tau-\tau^{\prime})=\langle{\bf K},N;{\bf
k},n|\hat{T}|{\bf K},N;{\bf
k}^{\prime},n^{\prime}\rangle=\Theta(\tau-\tau^{\prime})e^{-\epsilon^{+}_{{\bf
K},N}(\tau-\tau^{\prime})}t_{nn^{\prime}}(\tau-\tau^{\prime})$ (S24)

i.e.

$T_{nn^{\prime}}({\bf K},N,i\omega)=t_{nn^{\prime}}(i\omega-\epsilon^{+}_{{\bf
K},N}),$ (S25)

where $\omega$ is a Matsubara frequency (we suppress the index $n$ of
$\omega_{n}$ since we consider here the zero-temperature limit). The part of
the $T$ matrix that describes the relative motion can be easily obtained from
the Lippmann-Schwinger equation Lim  _et al._ (2008),

$t_{nn^{\prime}}(i\omega)=\phi_{n}^{-}(0)^{*}\phi_{n^{\prime}}^{-}{}(0)t(i\omega),\qquad\frac{1}{t(i\omega)}=\frac{1}{g_{3}}-\sum_{n}\int_{\bf
k}\frac{|\phi_{n}^{-}(0)|^{2}}{i\omega-\epsilon^{-}_{{\bf k},n}}.$ (S26)

Here and in the following the sum over $n$ extends over $\mathbb{N}$. We can
use the definition (S2) of the scattering length $a_{3}$ of the unconfined
bosons to rewrite (S26) as

$\frac{1}{t(i\omega)}=\frac{m}{4\pi a_{3}}-\sum_{n}\int_{\bf
k}\frac{|\phi_{n}^{-}(0)|^{2}}{i\omega-\epsilon^{-}_{{\bf k},n}}-\int_{{\bf
k},k_{z}}\frac{1}{2(\epsilon_{\bf k}+\epsilon_{k_{z}})},$ (S27)

where the UV cutoff $\Lambda_{3}$ can now be sent to infinity Lim  _et al._
(2008). At low frequencies, $|\omega|\ll\omega_{z}$, the $T$ matrix
$t_{00}(i\omega)$ takes the usual 2D form Petrov  _et al._ (2000); Lim  _et
al._ (2008)

$t_{00}(i\omega)=-\frac{2\pi/m}{\ln\frac{a_{2}}{2}\sqrt{-im\omega}+\gamma},$
(S28)

where the effective 2D scattering length $a_{2}$ is given by (S5). Finally we
note that Eq. (S27) implies that the retarded $T$ matrix
$t^{R}(\omega)=t(\omega+i0^{+})$ satisfies

$\displaystyle\frac{\partial}{\partial\omega}\frac{1}{t^{R}(\omega)}=\sum_{n}\int_{\bf
k}\frac{|\phi_{n}^{-}(0)|^{2}}{(\omega+i0^{+}-\epsilon^{-}_{{\bf k},n})^{2}},$
(S29)
$\displaystyle\frac{\partial}{\partial(1/a_{3})}\frac{1}{t^{R}(\omega)}=\frac{m}{4\pi}.$
(S30)

These two relations will be useful in the calculation of the contact.

###  II.2 B. Calculation of the contact from the thermodynamic definition

Let us consider the distribution of the in-plane momentum $\bar{n}_{\bf k}$.
To order $z^{2}$, and ignoring the part that comes from the non-interacting
propagator $G({\bf k},n,i\omega_{\nu})=(i\omega_{\nu}+\mu-\epsilon_{{\bf
k},n})^{-1}$ and does not depend on the interactions, it is sufficient to
consider Leyronas (2011)

$\delta\bar{n}_{\bf
k}=\frac{2}{\beta^{2}}\sum_{\omega_{n},\omega_{\nu}}\sum_{n_{1},n_{2}}\sum_{{\bf
K}}G({\bf k},n_{1},i\omega_{n})^{2}G({\bf K}-{\bf
k},n_{2},i\omega_{\nu}-i\omega_{n})T_{n_{1},n_{2};n_{1},n_{2}}({\bf
K},i\omega_{\nu}),$ (S31)

where

$\displaystyle T_{n_{1},n_{2};n_{1},n_{2}}({\bf K},i\omega_{\nu})$
$\displaystyle=\langle{\bf k}_{1},n_{1};{\bf K}-{\bf k}_{1},n_{2}|\hat{T}|{\bf
k}^{\prime}_{1},n_{1};{\bf K}-{\bf k}^{\prime}_{1},n_{2}\rangle$
$\displaystyle=\sum_{N,n}U_{n_{1},n_{2};N,n}T_{nn}({\bf
K},N,i\omega_{\nu})U^{\dagger}_{N,n;n_{1},n_{2}}$
$\displaystyle=\sum_{N,n}|U_{n_{1},n_{2};N,n}|^{2}|\phi_{n}^{-}(0)|^{2}t(i\omega_{\nu}-\epsilon^{+}_{{\bf
k},N}+2\mu)$ (S32)

is obtained using the eigenstates $|{\bf k}_{1},n_{1};{\bf
k}_{2},n_{2}\rangle=|{\bf k}_{1},n_{1}\rangle\otimes|{\bf k}_{2},n_{2}\rangle$
for the two-body transverse motion. We have used the fact that
$U_{n_{1},n_{2};N,n}=\langle n_{1},n_{2}|N,n\rangle$ is nonzero only if
$n_{1}+n_{2}=N+n$ since $|n_{1},n_{2}\rangle$ and $|N,n\rangle$ are orthogonal
when they do not belong to the same energy subspace. To order $z^{2}$, one can
use the zero-temperature limit of the $T$ matrix but one must shift its
argument by $2\mu$ to take into account the nonzero chemical potential. In the
following we include the zero-point energy $\omega_{z}/2$ in the chemical
potential so that the quantized levels of the harmonic oscillator with
frequency $\omega_{z}$ are given by $\epsilon_{n}=n\omega_{z}$
($n\in\mathbb{N}$). We thus obtain

$\displaystyle\delta\bar{n}_{\bf k}$ $\displaystyle=\sum_{n_{1},n_{2}\atop
N,n}|U_{n_{1},n_{2};N,n}|^{2}|\phi_{n}^{-}(0)|^{2}\frac{2}{\beta^{2}}\sum_{\omega_{n},\omega_{\nu}}\sum_{\bf
K}G({\bf k},n_{1},i\omega_{n})^{2}G({\bf K}-{\bf
k},n_{2},i\omega_{\nu}-i\omega_{n})t(i\omega_{\nu}-\epsilon^{+}_{{\bf
K},N}+2\mu)$ $\displaystyle=-\sum_{n_{1},n_{2}\atop
N,n}|U_{n_{1},n_{2};N,n}|^{2}|\phi_{n}^{-}(0)|^{2}\frac{2}{\beta}\sum_{\omega_{\nu}}\sum_{\bf
K}\frac{t(i\omega_{\nu}-\epsilon^{+}_{{\bf
K},N}+2\mu)}{(i\omega_{\nu}-\xi_{\bf k}-\xi_{{\bf K}-{\bf
k}}-(N+n)\omega_{z})^{2}}$
$\displaystyle=-\sum_{N,n}|\phi_{n}^{-}(0)|^{2}\frac{2}{\beta}\sum_{\omega_{\nu}}\sum_{\bf
K}\frac{t(i\omega_{\nu}-\epsilon^{+}_{{\bf
K},N}+2\mu)}{(i\omega_{\nu}-\xi_{\bf k}-\xi_{{\bf K}-{\bf
k}}-(N+n)\omega_{z})^{2}},$ (S33)

where we have kept only terms of order $z^{2}$ and used
$\sum_{n_{1},n_{2}}|U_{n_{1},n_{2};N,n}|^{2}=1$. Performing the sum over
$\omega_{\nu}$ by contour integral, we finally obtain

$\displaystyle\delta\bar{n}_{\bf k}$
$\displaystyle=-2\sum_{N,n}|\phi_{n}^{-}(0)|^{2}\sum_{\bf
K}\int_{0}^{\infty}\frac{d\omega}{\pi}n_{B}(\omega+\epsilon^{+}_{{\bf
K},N}-2\mu)\Im\left\\{\frac{t^{R}(\omega)}{(\omega+i0^{+}+\epsilon_{\bf
K}/2-\epsilon_{\bf k}-\epsilon_{{\bf K}-{\bf k}}-n\omega_{z})^{2}}\right\\}$
$\displaystyle=-2z^{2}\sum_{N,n}|\phi_{n}^{-}(0)|^{2}\sum_{{\bf
K}}e^{-\beta\epsilon^{+}_{{\bf
K},N}}\int_{0}^{\infty}\frac{d\omega}{\pi}e^{-\beta\omega}\Im\left\\{\frac{t^{R}(\omega)}{(\omega+i0^{+}+\epsilon_{\bf
K}/2-\epsilon_{\bf k}-\epsilon_{{\bf K}-{\bf k}}-n\omega_{z})^{2}}\right\\}$
(S34)

to order $z^{2}$. The contribution of the interactions to the pressure to the
same order is obtained by considering the 2D density
$\delta\bar{n}=\frac{1}{L^{2}}\int_{\bf k}\delta\bar{n}_{\bf k}=\partial\delta
P/\partial\mu$, i.e.

$\displaystyle\delta P$
$\displaystyle=-\frac{z^{2}}{\beta}\sum_{N,n}|\phi_{n}^{-}(0)|^{2}\int_{{\bf
K},{\bf k}}e^{-\beta\epsilon^{+}_{{\bf
K},N}}\int_{0}^{\infty}\frac{d\omega}{\pi}e^{-\beta\omega}\Im\left\\{\frac{t^{R}(\omega)}{(\omega+i0^{+}-2\epsilon_{\bf
k}-n\omega_{z})^{2}}\right\\}$ $\displaystyle=z^{2}\sum_{N}\int_{\bf
K}e^{-\beta\epsilon^{+}_{{\bf
K},N}}\int_{0}^{\infty}\frac{d\omega}{\pi}e^{-\beta\omega}\Im[\ln
t^{R}(\omega)],$ (S35)

where we have integrated by part in the integral over $\omega$ and used (S29).
This results is in agreement with a similar virial calculation done for a 2D
Fermi gas in Ngampruetikorn  _et al._ (2013). It is however different from
that of Ren (2004) for a two dimensional Bose gas, which we ascribe to the
fact that that calculation is not truly a virial expansion but a perturbative
expansion with ressumation of some of the $T$-matrix diagrams.

Equation (S35) yields the contact

$\displaystyle\frac{C}{L^{2}}$ $\displaystyle=-2m^{2}z^{2}\sum_{N}\int_{\bf
K}e^{-\beta\epsilon^{+}_{{\bf
K},N}}\int_{0}^{\infty}\frac{d\omega}{\pi}e^{-\beta\omega}\Im[t^{R}(\omega)]$
$\displaystyle=2m^{2}z^{2}\sum_{N}\int_{\bf K}e^{-\beta\epsilon^{+}_{{\bf
K},N}}\sum_{n}|\phi^{-}_{n}(0)|^{2}\int_{\bf k}e^{-\beta\epsilon^{-}_{{\bf
k},n}}|t^{R}(\epsilon^{-}_{{\bf k},n})|^{2},$ (S36)

where we have used (S30) and
$\Im[t^{R}(\omega)^{-1}]=\pi\sum_{n}|\phi_{n}^{-}(0)|^{2}\int_{\bf
k}\delta(\omega-\epsilon^{-}_{{\bf k},n})$.

In the low-temperature regime $T\ll\omega_{z}$, we can restrict ourselves to
$N=n=0$ in (S36). Performing the sum over ${\bf K}$ and using (S28), one finds

$\frac{C}{L^{2}}=4(2\pi)^{3/2}z^{2}\frac{\ell_{z}}{\lambda^{2}}\int_{0}^{\infty}dk\,k\frac{e^{-\beta
k^{2}/m}}{\left[\ln\left(\frac{ka_{2}}{2}e^{\gamma}\right)\right]^{2}+\frac{\pi^{2}}{4}}.$
(S37)

Performing the integral in the limit
$|\ln(\frac{1}{2}\sqrt{ma_{2}^{2}T}e^{\gamma})|\gg 1$ not , we finally obtain

$\frac{C_{\rm
virial}}{L^{2}}=2(2\pi)^{5/2}\frac{\bar{n}^{2}\ell_{z}}{\left|\ln\left(\frac{a_{2}}{\lambda}\sqrt{\frac{2}{\pi}}e^{\gamma}\right)\right|^{2}},$
(S38)

where $\bar{n}=z/\lambda^{2}+{\cal O}(z^{2})$. The result (S37) can also be
obtained from Eq. (8) in the main text, relating the contact to the pressure
and the energy density, using the expression (S35) of the pressure.

###  II.3 C. Calculation of the contact from the pair distribution function

####  II.3.1 1\. Scattering states

The scattering states associated with the relative motion can be deduced from
the Lippmann-Schwinger equation. They satisfy

$|\psi_{{\bf k},n}\rangle=|{\bf k},n\rangle+\frac{1}{L^{2}}\sum_{{\bf
k}^{\prime},n^{\prime}}\frac{t^{R}_{n^{\prime}n}(\epsilon^{-}_{{\bf
k},n})}{\epsilon^{-}_{{\bf k},n}+i0^{+}-\epsilon^{-}_{{\bf
k}^{\prime},n^{\prime}}}|{\bf k}^{\prime},n^{\prime}\rangle,$ (S39)

where $t^{R}(\omega)=t(\omega+i0^{+})$ is the retarded $T$ matrix defined by
(S27). Using

$\int_{{\bf k}^{\prime}}\frac{e^{i{\bf
k}^{\prime}\cdot\boldsymbol{\rho}}}{\epsilon^{-}_{{\bf
k},n}+i0^{+}-\epsilon^{-}_{{\bf
k}^{\prime},n^{\prime}}}=-i\frac{m}{4}H_{0}^{(1)}(k_{n^{\prime}}\rho),$ (S40)

where $k_{n^{\prime}}^{2}={\bf k}^{2}+(n-n^{\prime})m\omega_{z}$,
$\rho=|\boldsymbol{\rho}|$ and $H_{0}^{(1)}$ is a Hankel function of the first
kind, we obtain

$\psi_{{\bf k},n}({\bf r})=\frac{e^{i{\bf
k}\cdot\boldsymbol{\rho}}}{L}\phi^{-}_{n}(z)-i\frac{m}{4L}\sum_{n^{\prime}}t^{R}(\epsilon^{-}_{{\bf
k},n})\phi^{-}_{n^{\prime}}(0)^{*}\phi^{-}_{n}(0)\phi^{-}_{n^{\prime}}(z)H_{0}^{(1)}(k_{n^{\prime}}\rho).$
(S41)

When $k_{n^{\prime}}^{2}<0$, $k_{n^{\prime}}$ in (S40) and (S41) must be
understood as $i|k_{n^{\prime}}|$ and
$H_{0}^{(1)}(i|k_{n^{\prime}}|\rho)=-(2i/\pi)K_{0}(|k_{n^{\prime}}|\rho)$
decays exponentially for large $\rho$, which corresponds to a closed
scattering channel.

####  II.3.2 2\. Pair distribution function

We now consider the pair distribution function, averaged over the position
${\bf R}=({\bf r}_{1}+{\bf r}_{2})/2$ of the pair center of mass,

$g({\bf r}_{1}-{\bf r}_{2})=\int d^{3}R\,\langle\hat{\psi}^{\dagger}({\bf
r}_{1})\hat{\psi}^{\dagger}({\bf r}_{2})\hat{\psi}({\bf r}_{2})\hat{\psi}({\bf
r}_{1})\rangle.$ (S42)

To obtain the contact to order $z^{2}$, it is sufficient to consider states
with only two particles. The contribution to the pair distribution function of
the two-particle scattering state $|\psi^{(s)}_{{\bf K},N;{\bf k},n}\rangle$,
defined by its properly symmetrized wavefunction

$\psi^{(s)}_{{\bf K},N;{\bf k},n}({\bf R},{\bf
r})=\frac{1}{\sqrt{2}}\frac{e^{i{\bf K}\cdot{\bf
R}_{\parallel}}}{L}\phi^{+}_{N}(Z)[\psi_{{\bf k},n}({\bf r})+\psi_{-{\bf
k},n}({\bf r})],$ (S43)

reads $\int d^{3}R\,2|\psi^{(s)}_{{\bf K},N;{\bf k},n}({\bf R},{\bf r})|^{2}$.
To order $z^{2}$, the function $g({\bf r})$ is obtained by summing over all
states $\psi^{(s)}_{{\bf K},N;{\bf k},n}$ with the Boltzmann weight

$\frac{1}{Z}e^{-\beta(\epsilon^{+}_{{\bf K},N}+\epsilon^{-}_{{\bf
k},n}-2\mu)}=z^{2}e^{-\beta(\epsilon^{+}_{{\bf K},N}+\epsilon^{-}_{{\bf
k},n})}+{\cal O}(z^{3})$ (S44)

(using $Z=1+{\cal O}(z)$), i.e.

$g({\bf r})=\frac{1}{2}\sum_{{\bf K},{\bf
k}}\sum_{N,n}z^{2}e^{-\beta(\epsilon^{+}_{{\bf K},N}+\epsilon^{-}_{{\bf
k},n})}\int d^{3}R\,2|\psi^{(s)}_{{\bf K},N;{\bf k},n}({\bf R},{\bf r})|^{2},$
(S45)

where a factor $1/2$ has been introduced in the sum over ${\bf k}$ since
$\psi^{(s)}_{{\bf K},N;{\bf k},n}$ and $\psi^{(s)}_{{\bf K},N;-{\bf k},n}$ are
the same state.

Let us first consider the short distance limit $r=|{\bf
r}|=\sqrt{\rho^{2}+z^{2}}\ll\lambda,\ell_{z}$. In that case one has Petrov
_et al._ (2000)

$\frac{i}{4}\sum_{n^{\prime}}\phi^{-}_{n^{\prime}}(z)\phi^{-}_{n^{\prime}}(0)^{*}H_{0}^{(1)}(k_{n^{\prime}}\rho)\simeq\frac{1}{4\pi}\left(\frac{1}{r}-\frac{1}{a_{3}}\right)+\frac{1}{mt^{R}(\epsilon^{-}_{{\bf
k},n})}$ (S46)

and thus

$\psi_{{\bf k},n}({\bf r})\simeq-\frac{m}{4\pi
L}\phi_{n}^{-}(0)t^{R}(\epsilon^{-}_{{\bf
k},n})\left(\frac{1}{r}-\frac{1}{a_{3}}\right).$ (S47)

Equation (S45) then gives

$g({\bf r})=\frac{2m^{2}z^{2}}{(4\pi rL)^{2}}\sum_{{\bf
K},N}e^{-\beta\epsilon^{+}_{{\bf K},N}}\sum_{{\bf
k},n}e^{-\beta\epsilon^{-}_{{\bf
k},n}}|\phi_{n}^{-}(0)|^{2}|t^{R}(\epsilon^{-}_{{\bf
k},n})|^{2}=\frac{C}{(4\pi)^{2}}\left(\frac{1}{r}-\frac{1}{a_{3}}\right)^{2},$
(S48)

where $C$ is the contact (S36). At short length scales,
$r\ll\lambda,\ell_{z}$, the pair distribution function behaves as in an
isotropic 3D system with contact $C$.

Let us now consider the intermediate length scales,
$\ell_{z}\ll\rho\ll\lambda$, in the low-temperature regime $T\ll\omega_{z}$.
Since only the states with $\epsilon^{-}_{{\bf K},N},\epsilon^{-}_{{\bf
k},n}\lesssim T$ contribute significantly to the pair distribution function
(S45), we can restrict ourselves to $N=n=0$ and thus consider only the
scattering states

$\psi_{{\bf k},0}({\bf r})=\frac{e^{i{\bf
k}\cdot\boldsymbol{\rho}}}{L}\phi^{-}_{0}(z)-i\frac{m}{4L}\sum_{n^{\prime}}t^{R}(\epsilon^{-}_{{\bf
k}})\phi^{-}_{n^{\prime}}(0)^{*}\phi^{-}_{0}(0)\phi^{-}_{n^{\prime}}(z)H_{0}^{(1)}(k_{n^{\prime}}\rho)$
(S49)

with $|{\bf k}|\lesssim 1/\lambda$. If $n^{\prime}>0$ then
$k_{n^{\prime}}^{2}<0$ and
$H_{0}^{(1)}(k_{n^{\prime}}\rho)=H_{0}^{(1)}(i|k_{n^{\prime}}|\rho)\sim
e^{-\sqrt{n^{\prime}}\rho/\ell_{z}}$ decays exponentially for
$\rho\gg\ell_{z}$. On the other hand, if $n^{\prime}=0$,

$H_{0}^{(1)}(k_{n^{\prime}}\rho)=H_{0}^{(1)}(|{\bf
k}|\rho)\simeq\frac{2i}{\pi}\left(\ln\frac{|{\bf
k}|\rho}{2}+\gamma-i\frac{\pi}{2}\right)$ (S50)

for $\rho\ll\lambda$ (since $|{\bf k}|\lesssim 1/\lambda$). Thus for
$\ell_{z}\ll\rho\ll\lambda$, we can just consider the contribution of
$n^{\prime}=0$ in (S49) and using (S28) we obtain

$\psi_{{\bf k},0}({\bf r})\simeq\frac{m}{2\pi L}t^{R}(\epsilon^{-}_{{\bf
k}})|\phi^{-}_{0}(0)|^{2}\phi^{-}_{0}(z)\ln(\rho/a_{2}).$ (S51)

From (S45) we finally deduce

$\displaystyle g({\bf r})$
$\displaystyle=\frac{2m^{2}z^{2}}{L^{2}}\left(\frac{\ln(\rho/a_{2})}{2\pi}\right)^{2}\sum_{{\bf
K},{\bf k}}e^{-\beta(\epsilon^{+}_{{\bf K}}+\epsilon^{-}_{{\bf
k}})}|t^{R}(\epsilon^{-}_{{\bf
k}})|^{2}|\phi^{-}_{0}(0)|^{4}|\phi^{-}_{0}(z)|^{2}$
$\displaystyle=\left(\frac{\ln(\rho/a_{2})}{2\pi}\right)^{2}C|\phi^{-}_{0}(0)|^{2}|\phi^{-}_{0}(z)|^{2}$
$\displaystyle=\left(\frac{\ln(\rho/a_{2})}{2\pi}\right)^{2}\frac{C}{\sqrt{2\pi}\ell_{z}}|\phi^{-}_{0}(z)|^{2},$
(S52)

where $C$ is the low-temperature limit of the contact (S36). If we average
over $z$, we recover the standard result for a 2D system with a contact
$C/\sqrt{2\pi}\ell_{z}$. The latter is related to the pressure by

$\frac{C}{\sqrt{2\pi}\ell_{z}L^{2}}=4\sqrt{2\pi}\frac{m}{\ell_{z}}\frac{\partial
P}{\partial(1/a_{3})}=-4\pi m\frac{\partial P}{\partial\ln a_{2}}$ (S53)

and therefore coincides with the usual thermodynamic definition of a 2D
contact.

We shall see in Sec. D that the short-distance behavior of the pair
distribution function, Eqs. (S48) and (S52), is actually valid only when $r$
and $\rho$ are much smaller than the interparticle distance $d$.

####  II.3.3 3\. Momentum distribution

The knowledge of the short-distance behavior of the pair distribution function
allows one to obtain the high-momentum behavior of the momentum distribution
function $n_{{\bf k}}$. The calculation is standard and yields

$\bar{n}_{{\bf k},k_{z}}=\frac{C}{({\bf
k}^{2}+k_{z}^{2})^{2}}\quad\mbox{if}\quad\frac{1}{d},\frac{1}{\lambda},\frac{1}{a_{3}}\ll\sqrt{{\bf
k}^{2}+k_{z}^{2}},$ (S54)

and

$\bar{n}_{\bf k}=\int\frac{dk_{z}}{2\pi}\bar{n}_{{\bf
k},k_{z}}=\frac{1}{\sqrt{2\pi}\ell_{z}}\frac{C}{|{\bf
k}|^{4}}\quad\mbox{if}\quad\frac{1}{d},\frac{1}{\lambda}\ll|{\bf
k}|\ll\frac{1}{\ell_{z}}\quad\mbox{and}\quad T\ll\omega_{z},$ (S55)

where we assume the normalization $\frac{1}{L^{2}}\sum_{{\bf
k}}\int_{k_{z}}\bar{n}_{{\bf k},k_{z}}=N$ (with $N\equiv\langle\hat{N}\rangle$
the total number of particles). Equation (S55) can also be easily obtained
from (S34). Again we find that the 3D contact (S7) governs the high-momentum
behavior but the intermediate range $1/d,1/\lambda\ll|{\bf k}|\ll 1/\ell_{z}$
involves the 2D contact $C/\sqrt{2\pi}\ell_{z}$.

###  II.4 D. Beyond the virial expansion

Results similar to (S48,S52) and (S54,S55) have been obtained in Refs. He and
Zhou (2019); Bougas  _et al._ (2020); Decamp  _et al._ (2018) for fermions and
bosons in quasi-1D and quasi-2D traps. They are thus valid beyond the virial
expansion as can be easily shown following the method of Werner and Castin
Werner and Castin (2012a, b). We sketch the proof here.

Standard contact theory assumes that when two particles are close to each
other ($b\ll r_{ij}\ll d$ with $b$ the range of the interaction potential, $d$
the typical interparticle distance, and $r_{ij}=|{\bf r}_{ij}|=|{\bf
r}_{i}-{\bf r}_{j}|$), the many-body wavefunctions take the asymptotic form

$\Psi({\bf r}_{1},\ldots,{\bf r}_{N})\simeq\psi^{(s)}_{0}({\bf r}_{ij}){\cal
A}_{ij}({\bf R}_{ij},({\bf r}_{k})_{k\neq i,j}),$ (S56)

where $\psi^{(s)}_{0}({\bf r}_{ij})$ is the (properly symmetrized) zero-energy
scattering state. Since the range $b$ of the interaction potential and the 3D
$s$-wave scattering length $a_{3}$ are much smaller than the harmonic
oscillator length $\ell_{z}$, $\psi^{(s)}_{0}({\bf r})$ varies as
$1/r-1/a_{3}$ when $b\ll r\ll\ell_{z},d$, as for an isotropic 3D system.
Normalizing $\psi^{(s)}_{0}({\bf r})$ such that $\psi^{(s)}_{0}({\bf r})\simeq
1/r-1/a_{3}$ in that limit, the contact of the state $\Psi$ is then given by

$C_{\Psi}=32\pi^{2}\sum_{i<j}\int\prod_{k\neq i,j}d^{3}r_{k}\int d{\bf
R}_{ij}|{\cal A}_{ij}({\bf R}_{ij},({\bf r}_{k})_{k\neq i,j}|^{2},$ (S57)

which, thanks to the normalization of $\psi^{(s)}_{0}$, is related to the
energy $E_{\Psi}$ of the state $\Psi$ by $C_{\Psi}=8\pi m\frac{\partial
E_{\Psi}}{\partial(1/a_{3})}$, and the pair distribution function behaves as
$g_{\Psi}({\bf r})=[C_{\Psi}/(4\pi)^{2}](1/r-1/a_{3})^{2}$ for $b\ll
r\ll\ell_{z},d$. Averaging over the states with a Boltzmann weight, we recover
the thermodynamic contact discussed in the main text.

In the quasi-2D case, there are two points to consider. First, the zero-energy
limit must be taken carefully (since the $T$ matrix vanishes at zero energy).
Second, when $\ell_{z}\ll d$, the short distance behavior will cross-over from
a 3D to a 2D behavior depending on whether $r_{ij}\ll\ell_{z}$ or
$r_{ij}\gg\ell_{z}$ (in the opposite regime, $\ell_{z}\gg d$, only the 3D
behavior is visible at short distances $r_{ij}\ll d$).

Focusing on the quasi-2D regime $T\ll\omega_{z}$ in the harmonic trap, the
short-distance behavior of $\psi^{(s)}_{0}({\bf r})$ is obtained from that of
$\psi_{{\bf k},0}({\bf r})$ in the limit $|{\bf k}|\to 0$, see Eq. (S49). In
the short-distance limit, the symmetrization only changes the wavefunction
normalization by a factor $\sqrt{2}$. As this factor is unimportant here (it
will be absorbed in ${\cal A}$ and is the same in the 3D and 2D scattering
regimes), we discard it.

In the 3D scattering regime, we can use Eq. (S47) with $n=0$ and $|{\bf k}|\to
0$. Therefore, to obtain a correctly normalized $\psi^{(s)}_{0}({\bf r})$, we
need to normalize $\psi_{{\bf k},0}({\bf r})$ by a factor
$\mathcal{N}=-\frac{m}{4\pi L}\phi^{-}_{0}(0)t^{R}(\epsilon^{-}_{{\bf k},0})$.
We note that this normalization vanishes at low energies, as expected for a
system which is effectively 2D, see Appendix A in Werner and Castin (2012a).

In the 2D scattering regime, $\rho\gg\ell_{z}$, we can use Eq. (S51), and the
properly normalized scattering state (using the factor $\mathcal{N}$ computed
above) reads

$\psi^{(s)}_{0}({\bf
r})\simeq-2\phi^{-}_{0}(z)\phi^{-}_{0}(0)^{*}\left[\log(\rho/a_{2})+\ldots\right]$
(S58)

and is finite at zero energy. To summarize, we find that when two particles
are close to each other, the many-body wavefunction behaves as

$\Psi({\bf r}_{1},\ldots,{\bf r}_{N})={\cal A}_{ij}({\bf R}_{ij},({\bf
r}_{k})_{k\neq
i,j})\times\begin{cases}\left(\frac{1}{r_{ij}}-\frac{1}{a_{3}}\right)&\text{if
}b\ll r_{ij}\ll\ell_{z},d,\\\
-2\phi^{-}_{0}(0)^{*}\phi^{-}_{0}(z_{ij})\log(\rho_{ij}/a_{2})&\text{if
}\ell_{z}\ll\rho_{ij}\ll d,\end{cases}$ (S59)

with $z_{ij}=z_{i}-z_{j}$ and
$\rho_{ij}=|\boldsymbol{\rho}_{i}-\boldsymbol{\rho}_{j}|$. Since
$|\phi_{0}^{-}(0)|^{2}=1/\sqrt{2\pi}\ell_{z}$, this implies that the pair
distribution function of an arbitrary many-body state $\Psi$ behaves as

$g_{\Psi}({\bf
r})=\begin{cases}\frac{C_{\Psi}}{(4\pi)^{2}}\left(\frac{1}{r}-\frac{1}{a_{3}}\right)^{2}&\text{if
}b\ll r\ll\ell_{z},d,\\\
\frac{C_{\Psi}}{\sqrt{2\pi}\ell_{z}}|\phi^{-}_{0}(z)|^{2}\left(\frac{\ln(\rho/a_{2})}{2\pi}\right)^{2}&\text{if
}\ell_{z}\ll\rho\ll d.\end{cases}$ (S60)

By integrating over $z$ we recover the usual expression of the 2D case with a
contact $C_{\Psi}/\sqrt{2\pi}\ell_{z}$. This generalizes the conclusion drawn
from the virial expansion in Sec. II.C.

##  III III. Contact and classical field simulations

Figure S1: Normalized two-body contact $C\lambda^{4}/L^{2}\ell_{z}$ vs $\mu/T$
for $2mg=0.32$ and $T/\omega_{z}\simeq 0.00145$ as obtained from the FRG
(solid (blue) line) and classical field simulations (orange symbols). The
vertical dashed line shows the position of the BKT transition as estimated
from (S62).

The contact of the planar Bose gas can be obtained from the classical field
(Monte Carlo (MC)) simulations of Prokof’ev and Svistunov (2002) using the
thermodynamic definition (S7), following the method of Ozawa and Stringari
(2014); Ota and Stringari (2018).

Ref. Prokof’ev and Svistunov (2002) gives the density near the BKT transition
as

$\bar{n}_{\rm MC}=\bar{n}_{c}+F\left(X\right),$ (S61)

where $\lambda^{2}F$ is a universal function, $X=(\mu-\mu_{c})/mgT$ measures
the distance from the transition point at $\mu=\mu_{c}$ (neglecting the
renormalization of the interaction in the weak-interaction regime) and

$\begin{split}\bar{n}_{c}&=\frac{mT}{2\pi}\log\left(\frac{\xi}{mg}\right),\\\
\mu_{c}&=\frac{mgT}{\pi}\log\left(\frac{\xi_{\mu}}{mg}\right),\end{split}$
(S62)

with $\xi\simeq 380$ and $\xi_{\mu}\simeq 13.2$. Integrating the density with
respect to the chemical potential numerically, we obtain the pressure up to a
constant,

$\Delta P_{\rm MC}=P_{\rm
MC}-P_{c}=mgT\biggl{(}X\bar{n}_{c}+\int_{0}^{X}dx\,F\left(x\right)\biggr{)}.$
(S63)

Finally, using the thermodynamic definition (S7) of the contact, i.e.

$\frac{C}{L^{2}}=-2\sqrt{2\pi}\ell_{z}(mg)^{2}\frac{\partial P}{\partial g},$
(S64)

we obtain

$\begin{split}\frac{C_{\rm
MC}}{L^{2}}=\frac{C_{c}}{L^{2}}+2\sqrt{2\pi}\ell_{z}m^{2}g\left[(\mu-\mu_{c})(\bar{n}_{\rm
MC}+\lambda^{-2})+\left(\mu_{c}-\frac{mgT}{\pi}\right)(\bar{n}_{\rm
MC}-\bar{n}_{c})-\Delta P_{\rm MC}\right].\end{split}$ (S65)

The constant $C_{c}$, corresponding to the contact at the BKT transition, is
fixed by minimizing the relative difference between FRG and classical field
theory calculations. Figure S1 shows the FRG and MC results for the same
parameters as used in the main text, $2mg=0.32$ and $T/\omega_{z}\simeq
0.00145$. We find a very good agreement between the two methods once $C_{c}$
is fixed.

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