#  Universality of Efimov states in highly mass-imbalanced cold-atom mixtures
with van der Waals and dipole interactions

Kazuki Oi  Department of Physics, Tohoku University, Sendai 980-8578, Japan
Pascal Naidon  Few-Body Systems Physics Laboratory, RIKEN Nishina Centre,
RIKEN, Wako, 351-0198 Japan  Shimpei Endo<EMAIL_ADDRESS>Department
of Engineering Science, University of Electro-Communications, Chofu, Tokyo
182-8585, Japan

###### Abstract

We study three-body systems in a mass-imbalanced two-component cold-atom
mixture, and investigate the three-body parameter of their Efimov states for
both bosonic and fermionic systems, with a major focus on the Er-Er-Li Efimov
states. For a system interacting solely via van der Waals interactions, the
van der Waals universality of the three-body parameter is analytically derived
using the quantum defect theory. With the addition of a perturbative dipole
interaction between the heavy atoms, the three-body parameters of the bosonic
and fermionic Efimov states are found to behave differently. When the dipole
interaction is as strong as the van der Waals interaction, corresponding to
realistic Er-Er-Li Efimov states, we show that the van der Waals universality
persists once the effects of the non-perturbative dipole interaction are
renormalized into the $s$-wave and $p$-wave scattering parameters between the
heavy atoms. For a dipole interaction much stronger than the van der Waals
interaction, we find that the universality of the Efimov states can be
alternatively characterized by a quasi-one-dimensional scattering parameter
due to a strong anisotropic deformation of the Efimov wavefunctions. Our work
thus clarifies the interplay of isotropic and anisotropic forces in the
universality of the Efimov states. Based on the renormalized van der Waals
universality, the three-body parameter is estimated for specific isotopes of
Er-Li cold-atom mixtures.

##  I Introduction

Recent progress in the field of cold atoms has significantly advanced our
understanding of strongly correlated quantum systems and their universal
behaviors. A prime example of quantum phenomenon realized in cold atoms is the
Efimov effect [1, 2, 3, 4, 5, 6], which allows the existence of weakly bound
three-body states featuring a discrete scale invariance. While these Efimov
states have been theoretically suggested to appear universally for various
physical systems [3, 7, 8], they require the particles to interact with large
$s$-wave scattering lengths. This condition can be achieved in cold atoms by
using a Feshbach resonance [9, 10], which enables the precise control of the
$s$-wave scattering length between the atoms. Efimov states have been observed
in systems of identical bosons [11], mass-imbalanced bosonic mixtures [12, 13,
14], and three-component Fermi systems [15, 16, 17]. An Efimov state has also
been observed as the excited state of the 4He trimer [18], thanks to a
naturally large $s$-wave scattering length between 4He atoms. Importantly, the
Efimov states observed for various different atomic species [19] and internal
spin states [20] have been found to have almost the same physical property:
the three-body parameter, characterizing the binding energy and hence the size
of the Efimov states, has been found to scale universally with the van der
Waals length between the atoms. This universality of the three-body parameter
has been termed “van der Waals universality” and actively investigated both
theoretically and experimentally for identical bosons [19, 20, 21, 22, 23] as
well as mass-imbalanced bosonic mixtures [24, 25]; It has been demonstrated to
hold true except for systems close to a narrow Feshbach resonance [26, 27, 28,
29, 30, 31, 32, 33, 34].

Efimov physics has also been explored in fermionic systems. The repulsion
between identical fermions due to the Pauli exclusion competes with the Efimov
attraction, and leads to a richer few-body physics. Theoretically, Efimov
trimers have been predicted to appear in a mass-imbalanced two-component
fermionic mixture when the mass ratio is large: $M/m>13.6\ldots$ [2, 35]. With
such a large mass imbalance, the Efimov attraction surpasses the repulsion and
forms the Efimov trimers, in stark contrast to a moderate mass ratio
$8.1\ldots<M/m<13.6\ldots$ for which there exist “Kartavtsev-Malykh trimers”
[36, 37] and their crossover into the Efimov trimers [38], and a small mass
ratio $M/m<8.1\ldots$ for which no trimer appears. While these fermionic
trimers are formed in the $L^{\Pi}=1^{-}$ state, distinct from the $L=0^{+}$
state of identical bosons, they require a fermionic cold atom mixture with a
well-controlled Feshbach resonance. Until recently, the 40K-6Li mixture has
been the major mass-imbalanced Fermi system. The mass ratio, however, is
smaller than that required for the appearance of the trimer, and only a
precursor of the trimer formation has been observed [39]. Furthermore, the
Feshbach resonance of 40K-6Li is of a relatively narrow character [40, 10].
Recently, Feshbach resonances have been realized in highly mass-imbalanced
Fermi mixtures of Er-Li [41, 42] and Cr-Li [43], where multiple broad
resonances have been found. This opens the possibility of observing the trimer
physics of fermions in cold atoms. In particular, Er-Li is well inside the
Efimov regime, and it is expected to be the first system to exhibit a
“rotating” Efimov state, i.e. Efimov state with non-zero orbital angular
momentum $L=1$. One major challenge of the Er-Li mixture is that there is a
strong dipole-dipole interaction between the Er atoms [44, 45]. The dipole
interaction couples states with different angular momentum, so that the
orbital angular momentum $L$ is no longer a good quantum number [46, 47].
Since the dipole interaction is as strong as the van der Waals interaction, we
are faced with the question of how the interplay of dipole and van der Waals
interaction can modify the universal properties of the Efimov states: “Is the
three-body parameter in the Er-Li system universal?”

We address this question by investigating the Efimov states in a highly mass-
imbalanced three-body system. Using the Born-Oppenheimer approximation valid
for a highly mass-imbalanced system, we study the universality of the three-
body parameter, i.e. the binding energy of the Efimov states in the unitary
limit. To elucidate the role of the statistics of the heavy atoms, we study
both bosonic and fermionic systems with a variable strength of dipole
interaction. First, we study the three-body system interacting solely with a
van der Waals interaction. Using the quantum defect theory (QDT), the three-
body parameter is explicitly expressed as a universal function of the van der
Waals length and $s$-wave scattering length ($p$-wave scattering volume)
between the heavy atoms for bosons (fermions), respectively. Second, for a
system interacting with the van der Waals interaction and a weak dipole
interaction, our perturbative analysis, together with numerical calculations,
show that the bosonic and fermonic Efimov states are affected in a distinct
manner by the dipole interaction. Third, for moderate dipole interaction
strengths, including the realistic dipole strengths between Er atoms, our
coupled-channel numerical calculations vindicate the universality of the
three-body parameter. For the bosonic system, the van der Waals universality
formula is found to persist even though the dipole interaction is so large
that the $s$-wave scattering length and the binding energy are significantly
modified; once the effects of the dipole interaction are renormalized into the
$s$-wave scattering length, the QDT formula derived for a purely van der Waals
system holds true, so that the $s$-wave scattering length between the heavy
atoms universally characterize the three-body parameter. Based on this
renormalized van der Waals universality, we estimate the three-body parameters
of the Efimov states for some specific isotopes of Er-Li systems. For the
fermionic systems, the $p$-wave scattering volume is no longer a viable
physical parameter, and we find an alternative low-energy scattering parameter
which characterizes the universality of the three-body parameter. Finally, for
a dipole interaction much stronger than the van der Waals interaction, the
above universal description with the three-dimensional scattering parameters
should break down, but we find that quasi-one-dimensional scattering
parameters between the heavy atoms can be used to universally describe the
Efimov states in both bosonic and fermionic systems.

This paper is organized as follows: in Sec. II, after introducing the Born-
Oppenheimer description of the mass-imbalanced three-body system, our QDT
analysis of non-dipolar system is presented. In Sec. III, the results of our
numerical calculations are presented and compared with our analytical results
for non-dipolar, weak dipole, and moderate dipole interactions, followed by
the analysis in the limit of strong dipole interactions. In Sec. IV, we
present our estimates of the three-body parameters of some Er-Li isotopes. We
conclude and present future perspectives in Sec. V. Throughout the paper, the
natural unit $\hbar=1$ is used.

##  II Born-Oppenheimer description of highly mass-imbalanced 3-body system

###  II.1 General Formulation

We consider a three-body system of two heavy particles (mass $M$) which
interact resonantly with a light particle (mass $m$). The heavy particles are
assumed to be either identical spin-polarized bosons or fermions, to model
166,168,170Er and 167Er atoms, respectively. As the light particle and heavy
particles interact with a large $s$-wave scattering length $a_{\mathrm{HL}}$,
a condition for the Efimov states to appear, we model the light-heavy
interaction by the zero-range contact interaction [48]. On the other hand, the
interaction between the heavy particles is assumed to be the sum of van der
Waals and dipole interactions. This amounts to neglecting the van der Waals
interaction length scale between the light and heavy particles, which should
be a reasonable assumption because the van der Waals length between the heavy
atoms is larger than that between the light-heavy atoms. Throughout this
paper, we focus on the unitary limit $1/a_{\mathrm{HL}}=0$, and model the
light-heavy interaction via the unitary Bethe-Peierls boundary condition [48].
We note that this treatment of the inter-species interaction is valid in
describing broad Feshbach resonances, but less so for narrower ones [10]

In a highly mass-imbalanced system, the Born-Oppenheimer approximation
provides a good description of the Efimov physics. Indeed, it successfully
reproduces the Efimov scale factor, and the critical mass ratio for the Efimov
effect $M/m=13.99\ldots$ [2] in good agreement with the exact value
$M/m=13.60\ldots$ [35]. The Born-Oppenheimer approximation has also been
confirmed to reproduce well the van der Waals universality of the Efimov
states without the dipole interaction for mass ratios $14\lesssim M/m\lesssim
29$ [24]. It is therefore a good approximation for an Er-Li mixture whose mass
ratio is $M/m\approx 28$. The light particle’s Schrödinger equation is solved
analytically and leads to the induced attractive interaction between the heavy
particles $-\Omega^{2}/2mr^{2}$, where $\Omega=0.5671\ldots$ [3]. The
Schrödinger equation between the heavy particles then reads

$\left[-\frac{\nabla_{\bm{r}}^{2}}{M}-\frac{\Omega^{2}}{2mr^{2}}-\frac{C_{6}}{r^{6}}+\frac{C_{dd}(1-3\cos^{2}\theta)}{r^{3}}\right]\psi(\bm{r})=E\psi(\bm{r}),$
(1)

where $\theta$ is an angle measured from the $z$-axis taken to be parallel to
the orientation of the dipoles (parallel to the external magnetic field).
Here, $C_{6}$ and $C_{dd}$ are the van der Waals and dipole coefficients, from
which we can define the van der Waals length
$r_{\mathrm{vdw}}=\dfrac{1}{2}(MC_{6})^{1/4}$ and the dipole length
$a_{dd}=MC_{dd}/3$ between the heavy atoms.

The binding energy of the Efimov states (i.e. three-body parameter) can be
obtained by solving Eq. (1). It needs to be supplemented with a short-range
boundary condition. For this purpose, we use a hard wall condition at
$R_{\mathrm{min}}$: $\psi(r=R_{\mathrm{min}})=0$. We will see later that the
results are insensitive to the way the boundary condition is introduced as
long as $R_{\mathrm{min}}$ is small enough. As the dipole interaction couples
different partial waves, the angular momentum is no longer a good quantum
number, but the parity $\Pi$ and the $z$-component of the angular momentum
$M_{z}$ are good quantum numbers. While various values of $M_{z}$ are allowed
in general, we restrict ourselves to the $M_{z}=0$ channel in this paper. For
the bosonic heavy atoms, the $M_{z}^{\Pi}=0^{+}$ state is allowed by the
symmetry. This state involves the $L=0,2,4,...$ angular momentum channels,
among which $L^{\Pi}=0^{+}$ channel shows the Efimov effect and the others do
not for the Er-Li mass ratio [2, 36, 49, 50, 3]. For the fermionic heavy
atoms, $M_{z}^{\Pi}=0^{-}$ channel is allowed by the symmetry. This state
involves $L=1,3,5,...$ angular momentum channels, among which the
$L^{\Pi}=1^{-}$ channel shows the Efimov effect and the others do not for the
mass ratio in question [2, 36, 49, 50, 3]. Therefore, in the following, we
study the $M_{z}^{\Pi}=0^{+}$ and $M_{z}^{\Pi}=0^{-}$ states for the bosonic
and fermionic Er atoms, respectively.

###  II.2 Quantum Defect Theory for non-dipolar system

For typical cold atomic species, $r_{\mathrm{vdw}}\gg a_{dd}$ so that the
dipole interaction is negligible in Eq. (1). While this is not the case for Er
atoms, it still provides us with a basis for understanding the universal
behavior of the Efimov states with the dipole interactions. Therefore, in this
section we analyze

$\left[-\frac{\nabla_{\bm{r}}^{2}}{M}-\frac{\Omega^{2}}{2mr^{2}}-\frac{C_{6}}{r^{6}}\right]\psi(\bm{r})=E\psi(\bm{r})$
(2)

with a short-range boundary condition $\psi(|\bm{r}|=R_{\mathrm{min}})=0$.
While this equation was studied numerically in Ref. [25] and the van der Waals
universality of the Efimov states was clarified for the mass-imbalanced
system, we show below that Eq. (2) can be solved analytically and provide
explicit formulae for the binding energies and wavefunctions of the Efimov
states.

Without the dipole interaction, the angular momentum is a good quantum number,
and Eq. (2) is essentially a single-channel Schrödinger equation with
$1/r^{6}$ potential:

$\left[-\frac{1}{M}\frac{d^{2}}{dr^{2}}+\frac{s_{\ell}^{2}-\frac{1}{4}}{Mr^{2}}-\frac{C_{6}}{r^{6}}\right]u_{\ell}(r)=Eu_{\ell}(r)$
(3)

with $\psi(\bm{r})=\dfrac{u_{\ell}(r)}{r}Y_{\ell m}(\bm{r})$ and

$s_{\ell}^{2}=\left(\ell+\frac{1}{2}\right)^{2}-\frac{M}{2m}\Omega^{2}.$ (4)

Since we are interested in the Efimov states, we focus on the $\ell=0$
($\ell=1$) channel where $s_{\ell}^{2}<0$ for the mass ratio of interests
$M/m\approx 28$ for bosons (fermions), respectively [2, 36, 3, 49, 50].
Equation (3) can be solved analytically using Ref. [51], with a remark that
the centrifugal term is attractive due to the Efimov attraction. In other
words, the angular momentum in Ref. [51] should be generalized to an imaginary
number. We can then apply the quantum defect theory (QDT) [52, 53] to obtain
the analytical behavior of the Efimov states; using the QDT parameter $K^{c}$
introduced as

$u_{\ell}(r)=f_{\ell}^{c}(r)-K^{c}g^{c}_{\ell}(r),$ (5)

where $f_{\ell}^{c}$ and $g_{\ell}^{c}$ are two independent solutions of Eq.
(3). The binding energies of the Efimov states can be obtained by imposing
$u_{\ell}(r\rightarrow\infty)=0$, from which we find [52]

$K^{c}=\frac{W^{c}_{f-}}{W^{c}_{g-}}=\frac{(1+M_{\ell})\sin\dfrac{\pi\nu}{2}\cos\theta_{\ell}+(1-M_{\ell})\cos\dfrac{\pi\nu}{2}\sin\theta_{\ell}}{(1-M_{\ell})\cos\dfrac{\pi\nu}{2}\cos\theta_{\ell}+(1+M_{\ell})\sin\dfrac{\pi\nu}{2}\sin\theta_{\ell}}.$
(6)

The parameters $\theta_{\ell}$, $\nu$, and $M_{\ell}$ can be obtained from the
standard procedure of solving the van der Waals two-body problem (see
Appendix. A). At low energy $|E|\ll 1/Mr^{2}_{\mathrm{vdw}}$, by performing
the low-energy expansion in a similar manner to Refs. [53, 54], we find (see
Appendix. B )

$|E|=\frac{4}{Mr_{\rm
vdw}^{2}}\exp\left[\frac{2}{|s_{\ell}|}\left\\{\arctan\left(\frac{K^{c}}{\tanh\dfrac{\pi|s_{\ell}|}{4}}\right)+\xi_{\ell}\right\\}\right]e^{-\frac{2n\pi}{|s_{\ell}|}},$
(7)

where
$\xi_{\ell}=\arg\left[\Gamma\left(\frac{i|s_{\ell}|}{2}\right)\Gamma(1+i|s_{\ell}|)\right]$
and $n$ is an arbitrary integer satisfying $|E|\ll 1/Mr^{2}_{\mathrm{vdw}}$.

The QDT parameter $K^{c}$ is related to the short-range phase, hence the
short-range boundary condition. Indeed, at short distance, one can substitute
the low-energy form of the wavefunction Eq. (28) because the energy is much
smaller than the potential, and imposing $u_{\ell}(r=R_{\mathrm{min}})=0$, one
finds

$\displaystyle K^{c}$
$\displaystyle=-\tanh\left(\frac{\pi|s_{\ell}|}{4}\right)\frac{\mathrm{Re}\left[J_{\frac{s_{\ell}}{2}}\left(\Phi\right)\right]}{\mathrm{Im}\left[J_{\frac{s_{\ell}}{2}}\left(\Phi\right)\right]}$
(8) $\displaystyle\simeq-\frac{1}{\tan\left[\Phi-\frac{\pi}{4}\right]},$ (9)

where $\Phi=2r_{\rm vdw}^{2}/R^{2}_{\mathrm{min}}$. In the second line, we
have used the asymptotic form of the Bessel function (or equivalently Eq.
(25)) valid for $R_{\mathrm{min}}\ll r_{\rm vdw}$. This relation between
$R_{\mathrm{min}}$ and $K^{c}$, substituted into Eq. (7) gives an explicit
analytical formula of the Efimov binding energies as a function of
$R_{\mathrm{min}}$. The wavefunction of the Efimov states is also analytically
obtained as ($x\equiv 2\left(r_{\rm vdw}/r\right)^{2}$)

$\displaystyle u_{\ell}(r)$
$\displaystyle=\sqrt{\frac{r}{2}}\left(\frac{\mathrm{Re}\left[J_{\frac{s_{\ell}}{2}}\left(x\right)\right]}{\cosh\frac{\pi|s_{\ell}|}{4}}+K^{c}\frac{\mathrm{Im}\left[J_{\frac{s_{\ell}}{2}}\left(x\right)\right]}{\sinh\frac{\pi|s_{\ell}|}{4}}\right)$
(10)
$\displaystyle=\sqrt{\frac{r}{2}}\frac{\mathrm{Re}\left[J_{\frac{s_{\ell}}{2}}\left(\Phi\right)\right]}{\cosh\frac{\pi|s_{\ell}|}{4}}\left(\frac{\mathrm{Re}\left[J_{\frac{s_{\ell}}{2}}\left(x\right)\right]}{\mathrm{Re}\left[J_{\frac{s_{\ell}}{2}}\left(\Phi\right)\right]}-\frac{\mathrm{Im}\left[J_{\frac{s_{\ell}}{2}}\left(x\right)\right]}{\mathrm{Im}\left[J_{\frac{s_{\ell}}{2}}\left(\Phi\right)\right]}\right),$
(11)

where the log-periodic behavior characteristic of the Efimov states appears
from the oscillation of the imaginary-index Bessel function.

While the $R_{\mathrm{min}}$ dependence of the binding energy and wavefunction
may look like a non-universal short-range dependence of the Efimov states, we
can recast it into universal forms by using the relation between
$R_{\mathrm{min}}$ and the two-body universal scattering parameter between the
heavy atoms; if we consider the two-body problem of the two heavy atoms

$\left[-\frac{1}{M}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)}{Mr^{2}}-\frac{C_{6}}{r^{6}}\right]u_{\ell}(r)=Eu_{\ell}(r),$
(12)

the only difference with Eq. (2) is the presence/absence of the $1/r^{2}$
attraction. At short distance $r\sim R_{\mathrm{min}}\ll r_{\mathrm{vdw}}$,
this attraction is much smaller than the van der Waals force so that the two
equations are essentially the same. More physically, the short-range phase of
the three-body system is dominated by the potential between the heavy atoms.
This assumption should be particularly true for a highly mass-imbalanced
system where the atomic polarizability of the heavy atoms, hence the van der
Waals force and higher-order forces are much stronger than those for the light
atom. Therefore, the QDT parameter of the three-body system can be
approximated by that of two heavy atoms $K^{c}_{\mathrm{3body}}\simeq
K^{c}_{\mathrm{2body}}$, which amounts to adopting the same short-range cutoff
$R_{\mathrm{min}}$ value for the two- and three-body systems.

The van der Waals two-body problem Eq. (12) with the QDT condition Eq. (5) has
been solved in Refs. [53, 54], and the following relations between the the QDT
parameter and the two-body scattering parameters have been obtained:

$\displaystyle\frac{a^{(\mathrm{HH})}}{r_{\rm vdw}}$
$\displaystyle=\frac{4\pi}{\Gamma^{2}\left(\frac{1}{4}\right)}\left[1+\frac{1+K^{c}\tan\dfrac{\pi}{8}}{K^{c}-\tan\dfrac{\pi}{8}}\right]$
(13) $\displaystyle\frac{v_{p}^{(\mathrm{HH})}}{r^{3}_{\rm vdw}}$
$\displaystyle=-\frac{1}{3}\frac{\Gamma\left(\frac{1}{4}\right)}{\Gamma\left(\frac{7}{4}\right)}\frac{1+\tan\dfrac{3\pi}{8}-\left(1-\tan\dfrac{3\pi}{8}\right)K^{c}}{\sqrt{2}\left(\tan\dfrac{3\pi}{8}-K^{c}\right)},$
(14)

where $a^{(\mathrm{HH})}$ and $v_{p}^{(\mathrm{HH})}$ are $s$-wave scattering
length and $p$-wave scattering volume between the heavy atoms, respectively.
Noting that the bosonic (fermionic) heavy atoms interact dominantly with the
$s$-wave scattering length ($p$-wave scattering volume) at low energy, and
substituting Eqs. (13) and (14) into Eq. (7), the binding energies of the
Efimov states are represented as

$|E|=\frac{4}{Mr_{\rm
vdw}^{2}}\exp\left[\frac{2}{|s_{0}|}\left\\{\arctan\left(\frac{1}{\tanh\dfrac{\pi|s_{0}|}{4}}\frac{\dfrac{a^{(\mathrm{HH})}}{r_{\rm
vdw}}\tan\dfrac{\pi}{8}+\dfrac{4\pi}{\Gamma^{2}(\frac{1}{4})}\left(1-\tan\dfrac{\pi}{8}\right)}{\dfrac{a^{(\mathrm{HH})}}{r_{\rm
vdw}}-\dfrac{4\pi}{\Gamma^{2}(\frac{1}{4})}\left(1+\tan\dfrac{\pi}{8}\right)}\right)+\xi_{0}\right\\}\right]e^{-\frac{2n\pi}{|s_{0}|}}$
(15)

for the bosonic heavy atoms ($\ell=0$), and

$|E|=\frac{4}{Mr_{\rm
vdw}^{2}}\exp\left[\frac{2}{|s_{1}|}\left\\{\arctan\left(\frac{1}{\tanh\dfrac{\pi|s_{1}|}{4}}\frac{\dfrac{v_{p}^{(\mathrm{HH})}}{r_{\rm
vdw}^{3}}\tan\dfrac{3}{8}\pi+\dfrac{1}{3\sqrt{2}}\dfrac{\Gamma(\frac{1}{4})}{\Gamma(\frac{7}{4})}\left(1+\tan\dfrac{3}{8}\pi\right)}{\dfrac{v_{p}^{(\mathrm{HH})}}{r_{\rm
vdw}^{3}}+\dfrac{1}{3\sqrt{2}}\dfrac{\Gamma(\frac{1}{4})}{\Gamma(\frac{7}{4})}\left(1-\tan\dfrac{3}{8}\pi\right)}\right)+\xi_{1}\right\\}\right]e^{-\frac{2n\pi}{|s_{1}|}}$
(16)

for the fermionic heavy atoms ($\ell=1$), where $n$ is an arbitrary integer
such that $|E|\ll 1/Mr_{\rm vdw}^{2}$.

As we will see in Sec. III.1, these analytical formulae excellently reproduce
the numerical results for non-dipolar systems. These analytical formulae
suggest that the binding energy of the Efimov states in the unitary limit,
i.e. the three-body parameter, is insensitive to the short-range details of
the system but universally determined by the van der Waals length and the
$s$-wave scattering length ($p$-wave scattering volume) for the bosonic
(fermionic) Efimov states. This van der Waals universality has been
experimentally demonstrated for equal-mass [19, 20] and mass-imbalanced
bosonic systems [29] close to a broad Feshbach resonance. The van der Waals
universality has also been demonstrated theoretically with numerical
calculations for equal-mass [23, 21, 22] and mass-imbalanced bosonic systems
[24, 25]. In particular, the same Born-Oppenheimer equation (2) with the same
short-range boundary condition has been employed in Ref. [25] and solved
numerically to study the universality of the Efimov states. Our QDT analysis
provides an analytical demonstration of the van der Waals universality of
hetero-nuclear Efimov states; namely, the assumption of
$K^{c}_{\mathrm{3body}}\simeq K^{c}_{\mathrm{2body}}$ is pivotal in
universally characterizing the three-body parameter with the two-body
scattering parameters between the heavy atoms. As we use the single-channel
QDT, Eqs. (15) and (16) are invalid for describing the Efimov states near a
narrow Feshbach resonance, which have been demonstrated both experimentally
[26, 27, 28, 29] and theoretically [30, 31, 32, 33, 34] to deviate from the
van der Waals universality; the multi-channel QDT [55, 56] is necessary for
capturing their sensitivity to the channel coupling and nature of the internal
spin states.

Figure 1: Binding energy of the Efimov states with the unitary heavy-light
interaction for (a) bosonic and (b) fermionic heavy atoms interacting via the
van der Waals interaction. The mass ratio are (a) $M/m=27.5855...$ and (b)
$M/m=27.7520...$, corresponding to 166Er-6Li and 167Er-6Li, respectively. The
numerical results for the bosons (blue circles) and fermions (orange squares)
with $\ell_{\mathrm{max}}=20$ are shown, together with the analytical curves
of Eq. (7) with Eq. (8) (black solid) and Eq. (9) (green dashed). The vertical
thin dotted lines (black) denote the thresholds at which the higher angular
momentum bound states of $\ell=2$ ($\ell=3$) start to appear for the bosons
(fermions), obtained from Eqs. (35) and (8).

##  III Numerical Results

We show the numerical solutions of Eq. (1) in this section. As we are
particularly interested in Er-Er-Li three-body systems, we perform the
calculations with Er-Li mass ratio: $M/m=27.5855...$ for the bosonic system
corresponding to 166Er-6Li, and $M/m=27.7520...$ for the fermions
corresponding to 167Er-6Li, respectively. The differential equation (1) is
discretized in the log-scaled coordinate space betwen $R_{\mathrm{min}}$ and
$R_{\mathrm{max}}$, where $R_{\mathrm{max}}$ is typically taken as $\sim$
400-1000 $r_{\mathrm{vdw}}$. We find that 4000 grid points and
$\ell_{\mathrm{max}}=$10-20 are enough to attain convergence for most cases,
while we take up to 30000 grid points and $\ell_{\mathrm{max}}\approx 40$ for
some cases of stronger dipole interactions.

###  III.1 Non-dipolar case $a_{dd}=0$

Figure 2: Binding energy of the Efimov states plotted against (a)(b) the
$s$-wave scattering length $a^{(\mathrm{HH})}$ for the bosons, and (c)(d)
$p$-wave scattering volume $v^{(\mathrm{HH})}$ for the fermions, obtained by
converting $R_{\mathrm{min}}$ in Fig. 1 into $a^{(\mathrm{HH})}$ and
$v^{(\mathrm{HH})}$ using Eqs. (8) (13) (14). The left (right) pannels are
negative (positive) scattering length/volume sides, which should smoothly
connect with each other in the unitary limit $1/a^{(\mathrm{HH})}=0$,
$1/v_{p}^{(\mathrm{HH})}=0$ corresponding to the border between the left and
right panels. The solid curves (black) are the analytical results of Eqs. (15)
(16).

First, we show the results without the dipole interaction: in Fig. 1, we show
the binding energy of the Efimov states as the short-range boundary
$R_{\mathrm{min}}$ is varied. As $R_{\mathrm{min}}$ gets smaller, the short-
range van der Waals attraction increases, leading to the successive appearance
of bound states and increase of the binding energy. The numerical data points
almost perfectly agree with the analytical curves obtained with the QDT in Eq.
(7), particularly at small energy. There are a few points which significantly
deviate from the analytical curves. They are ascribed to higher angular
momentum states of $\ell=2,4,...$ for the bosons and $\ell=3,5,...$ for the
fermions, respectively. The thresholds at which the $\ell=2$ ($\ell=3$) bound
states start to appear, denoted by the vertical dotted lines in Fig. 1 (a)
(and (b)), agree with most of the stray data points, supporting this
interpretation. A few stray points which agree with neither are ascribed to
higher angular momentum channels $\ell\geq 4$ ($\ell\geq 5$). This
classification of the states has also been confirmed from the shape of the
wavefunctions. Because the higher angular momentum states are localized to
shorter distances than the $s$-wave ($p$-wave) state owing to the centrifugal
barrier, they are more sensitive to $R_{\mathrm{min}}$, resulting in the
steeper binding energy curves. The binding energy curves of different partial
waves do not render an avoided crossing but simply cross because the angular
momentum is a good quantum number in the absence of the dipole interaction.

In Fig. 2, we show the binding energies plotted against the $s$-wave
scattering length $a^{(\mathrm{HH})}$ and $p$-wave scattering volume
$v^{(\mathrm{HH})}$ between the heavy atoms, obtained from Fig. 1 and Eqs. (8)
(13) (14). The numerical data with different values of $R_{\mathrm{min}}$
collapse into universal curves. Therefore, the three-body parameter of the
Efimov states are universally characterized by ($a^{(\mathrm{HH})}$,
$r_{\mathrm{vdw}}$) for the bosons and ($v_{p}^{(\mathrm{HH})}$,
$r_{\mathrm{vdw}}$) for the fermions, respectively. In other words, they are
insensitive to the short-range details, but are universally characterized by
the two-body scattering parameters which describe the long-range asymptotic
behavior of the two heavy atoms. The numerical data in Fig. 2 excellently
agree with the curves of Eqs. (15) (16), vindicating our QDT analytical
formulae. Some points in Fig. 2 deviating from the curves are higher angular
momentum states, whose energies change rapidly as $R_{\mathrm{min}}$ hence
$a^{(\mathrm{HH})}$ and $v^{(\mathrm{HH})}$ are varied; although they
significantly deviate from the analytical curves of Eqs. (15) (16), the data
points with different $R_{\mathrm{min}}$ still tend to collapse into a single
curve (Fig. 2 (b)(c)) thanks to the angular-insensitive nature of the short-
range phase originating from $R_{\mathrm{min}}\ll r_{\mathrm{vdw}}$ [52]. The
marginal disagreement of the large binding energy region $|E|\simeq
1/Mr_{\mathrm{vdw}}^{2}$ in Fig. 2 (c) is ascribed to the breakdown of the
low-energy condition used to derive Eqs. (15) (16).

We note that the universal behaviors of the bosonic system in Fig. 2 (a)(b)
were found numerically in Ref. [25]; we have found here that a similar
universal behavior holds true for the fermions (Fig. 2 (c)(d)), together with
their analytical descriptions in Eqs. (15) (16).

###  III.2 Weak dipole interaction $a_{dd}\ll r_{\mathrm{vdw}}$

We show in Fig. 3 the binding energies of the Efimov states in the presence of
a weak dipole interaction. As the angular momentum is no longer a good quantum
number, different partial wave states mix with each other, resulting in
avoided crossings. We have confirmed that the shape of the wavefunction
changes across the avoided crossing via a superposition of different partial
waves. We have also confirmed that the width of the avoided crossings
increases as $a_{dd}$ gets larger.

Figure 3: Binding energy of the Efimov states for (a) bosonic and (b)
fermionic heavy atoms interacting via the van der Waals interaction and weak
dipole interaction $a_{dd}/r_{\mathrm{vdw}}=0.40$. The other parameters and
notations are the same as those in Fig. 1.

As shown in Fig. 3, the numerical results generally get shifted toward larger
binding energy. This can be understood by the perturbation theory: we have
analytically found that the energy shift induced by the dipole interaction is
$E-E^{(0)}\propto a_{dd}^{2}$ for the bosons predominantly in the $L=0$
channel, and $E-E^{(0)}\propto a_{dd}$ for the fermions predominantly in the
$L=1$ channel, with both negative shifts $E-E^{(0)}<0$ (see Appendix. C). The
difference in the scaling originates from the absence (presence) of the
diagonal matrix element of the dipole interaction for $L=0$ ($L=1$) states:
$\langle L=0|V_{dd}|L=0\rangle=0$, $\langle L=1|V_{dd}|L=1\rangle\neq 0$. This
is confirmed in Fig. 4: the change in binding energy, i.e. three-body
parameter of the Efimov states, scales quadratically for the bosons, and
linearly for the fermions. Notably, in most previous studies on the Efimov
effect, the role of the statistics of the particles has been rather marginal,
at best to modify the value of $s_{\ell}$, and thereby change the
absence/presence of the Efimov effect through the condition $s_{\ell}^{2}<0$
as the mass ratio is varied [2, 35, 49, 50, 3, 4, 5, 6]. In other words, the
statistics of the particles has been essentially irrelevant once its effect is
incorporated into the value of $s_{\ell}$. Figure 4 is one of the few examples
where the quantum statistics of the particles plays an explicit role in
qualitatively changing the universal behavior of the Efimov states.

###  III.3 Moderate dipole interaction $a_{dd}\sim r_{\mathrm{vdw}}$

Erbium atoms interact with a strong dipole interaction owing to their large
magnetic moments: the dipole length of 166Er-166Er is $a_{dd}=75.5a_{0}$,
which is comparable to their van der Waals length $r_{\mathrm{vdw}}=65.5a_{0}$
[57, 58] (see Table 1). Figure 4 shows that this is away from the region where
the perturbation theory works well $a_{dd}/r_{\mathrm{vdw}}\lesssim 0.5$.

Figure 4: Energy shift induced via the dipole interaction with a fixed value
of $R_{\mathrm{min}}/r_{\mathrm{vdw}}=0.40$ for (a) the bosons (blue circles)
and (b) fermions (orange squares). The dashed lines (black) are the
perturbation results in Eqs (51) (53), where the unperturbed energy $E^{(0)}$
and the matrix elements of Eq. (50) are evaluated with the energies and
wavefunctions numerically obtained for the purely van der Waals system in Sec.
III.1.

We have therefore performed the numerical calculations for moderate strengths
of the dipole interaction in Fig. 5(a)-(f); we first show the binding energy
of the bosonic Efimov states plotted against the heavy-heavy $s$-wave
scattering length estimated without the dipole interaction $a_{\mathrm{no-
dd}}^{(\mathrm{HH})}$ (Eq. (8)). As the dipole strength is increased from a
smaller ((a)(b)) to a larger ((e)(f)) value, the avoided crossing between the
$\ell=0$ dominant state and the $\ell=2$ dominant state gets broader. Due to
the admixture of the $\ell=0$ state, the slope of the $\ell=2$ dominant state
appearing around $a^{(\mathrm{HH})}_{\mathrm{no-dd}}/r_{\mathrm{vdw}}\simeq 1$
gets gentler. Furthermore, the binding energies increase as the dipole
interaction gets larger, which is consistent with the perturbative analysis in
Sec. III.2. The deviation from the analytical curves of the non-dipolar system
in Eq. (15) (black curves) is so significant that they cannot well explain the
numerical results of the realistic dipole interaction strength (middle row
(c)(d)), let alone stronger ones (bottom row (e)(f)). Still, the binding
energy data of different short-range parameters $R_{\mathrm{min}}$ collapse
onto a single curve, suggesting the universality of the three-body parameter
of the bosonic Efimov states even in the presence of non-perturbatively large
dipole interactions.

Figure 5: Binding energy of the bosonic Efimov states for the mass ratio
$M/m=27.5855...$ corresponding to 166Er-166Er-6Li system. Weaker dipole
strength $a_{dd}/r_{\mathrm{vdw}}=0.40$ (top row (a)(b)(g)(h)), realistic
dipole strength for 166Er-166Er of $a_{dd}/r_{\mathrm{vdw}}=0.86755....$
(middle row (c)(d)(i)(j)), and stronger dipole strength
$a_{dd}/r_{\mathrm{vdw}}=1.5$ (bottom row (e)(f)(k)(l)). (a)-(f) The
horizontal axis is the $s$-wave scattering length between the heavy atoms
estimated by Eq. (8), i.e. without the dipole interaction. (g)-(l) The
horizontal axis is the $s$-wave scattering length between the heavy atoms
numerically obtained by solving the coupled-channel two-body equation in Eq.
(17). The left (right) panels in each column show the negative (positive)
$s$-wave scattering length. The solid curve (black) is the analytical QDT
formula for non-dipolar system in Eq. (15).

To further elucidate the nature of this universality, we convert the
horizontal axis into an authentic $s$-wave scattering length between the heavy
atoms $a^{(\mathrm{HH})}_{\mathrm{with-dd}}$ in Fig. 5(g)-(l); in the presence
of the dipole interaction, the $s$-wave scattering length between the heavy
atoms $a^{(\mathrm{HH})}_{\mathrm{with-dd}}$ gets significantly modified from
a purely van der Waals value of $a^{(\mathrm{HH})}_{\mathrm{no-dd}}$ in Eq.
(13) [59, 60]. As the experimentally accessible scattering length between the
Er atoms is $a^{(\mathrm{HH})}_{\mathrm{with-dd}}$ rather than
$a^{(\mathrm{HH})}_{\mathrm{no-dd}}$, it should be more physical to use
$a^{(\mathrm{HH})}_{\mathrm{with-dd}}$ to eliminate $R_{\mathrm{min}}$. In
Fig. 6, we show $a^{(\mathrm{HH})}_{\mathrm{with-dd}}$ (circles) obtained by
numerically solving the coupled-channel Schrödinger equation between the two
heavy atoms

$\left[-\frac{\nabla_{\bm{r}}^{2}}{M}-\frac{C_{6}}{r^{6}}+\frac{C_{dd}(1-3\cos^{2}\theta)}{r^{3}}\right]\psi(\bm{r})=E\psi(\bm{r}),$
(17)

with the boundary condition $\psi(|\bm{r}|=R_{\mathrm{min}})=0$, and compare
it with $a^{(\mathrm{HH})}_{\mathrm{no-dd}}$. While they are almost
indistinguishable for a weak dipole interaction (Fig. 6(a)) except for the
proximity of higher-partial wave induced resonances (dotted vertical lines)
[59, 61], $a^{(\mathrm{HH})}_{\mathrm{with-dd}}$ is markedly different from
$a^{(\mathrm{HH})}_{\mathrm{no-dd}}$ for a realistic dipole strength (Fig.
6(b)), and stronger one (Fig. 6(c)). This is expected because the realistic
dipole strength of the Er atoms $a_{dd}/r_{\mathrm{vdw}}=0.86755....$ in Fig.
6(b) (and stronger one in Fig. 6(c)) is away from the perturbative regime
which was confirmed to be $a_{dd}/r_{\mathrm{vdw}}\lesssim 0.5$ in Sec. III.2.

In Fig. 5(g)-(l), we show the binding energies of the Efimov states plotted
against $a^{(\mathrm{HH})}_{\mathrm{with-dd}}$. Similar to Fig. 5(a)-(f), the
data with different values of $R_{\mathrm{min}}$ collapse into a single curve,
demonstrating the universality. One notable difference is that the agreement
between the numerical data and the purely van der Waals analytical curves
(black) is much better in (g)-(l) than those in (a)-(f): except for the broad
avoided crossing at $a_{\mathrm{with-
dd}}^{(\mathrm{HH})}/r_{\mathrm{vdw}}\simeq 1$, a significant fraction of the
numerical results agree well with the van der Waals curves not only for the
weak dipole interaction, but also for the realistic dipole strength (Fig.
5(i)(j)) and stronger dipole interaction ((k)(l)). This suggests that the van
der Waals quantum QDT of the bosonic Efimov states in Eq. (15) can capture the
universal behavior of the Efimov states in the presence of the dipole
interaction, once the effects of the dipole interaction are incorporated into
the $s$-wave scattering length between the heavy atoms:
$a^{(\mathrm{HH})}=a^{(\mathrm{HH})}_{\mathrm{with-dd}}$. In other words, the
major role of the dipole interaction is to change, hence renormalize the value
of the $s$-wave scattering length between the heavy atoms, so that the van der
Waals universality of the Efimov states in Eq. (15) persists with the
renormalized $s$-wave scattering length $a^{(\mathrm{HH})}_{\mathrm{with-
dd}}$. This renormalized van der Waals universality for the dipolar bosonic
Efimov states we have found here seems non-pertubative; the dipole interaction
is so large that it significantly changes both the binding energies (Fig.
5(c)-(f)) and the $s$-wave scattering length (Fig. 6(b)(c))) from the purely
van der Waals system. Some horizontal data points in Fig. 5(g)(l) which are
not present in (a)-(f) are artefacts of the higher-partial-wave induced
resonances: the two-body problem between the heavy atoms in Eq. (17) not only
shows the $s$-wave dominated resonances, but also shows the resonant
enhancement of $a^{(\mathrm{HH})}$ induced by higher-partial-wave channels
[59, 61], whose approximate positions are indicated by the vertical black
dotted lines in Fig. 6. The narrow nature of these higher-partial-wave induced
resonances leads to an abrupt change of $a^{(\mathrm{HH})}_{\mathrm{with-dd}}$
as $R_{\mathrm{min}}$ is changed, resulting in flat energy spectra in Fig.
5(g)-(l). Equation (15) cannot explain these flat data points. We note however
that these flat energy spectra are absent in Fig. 5(a)-(b), and they should
therefore be interpreted as artefacts originaing from the two-body effect of
$a^{(\mathrm{HH})}$ rather than a genuine three-body phenomenon. Physically,
when we attempt to evaluate the binding energies of the Efimov states from an
experimentally obtained value of $a^{(\mathrm{HH})}$ (see Sec. IV), we can
simply use the van der Waals curves of Eq. (15) by neglecting the stray flat
spectra, except for the case where the two Er atoms are coincidentally in the
proximity of the $s$-wave resonance induced by a $d$-wave or higher-angular-
momentum molecular state.

Figure 6: $S$-wave scattering length between the heavy atoms as
$R_{\mathrm{min}}$ is varied, obtained numerically by solving Eq. (17). The
solid curve (black) is the $s$-wave scattering length without the dipole
interaction $a^{(\mathrm{HH})}_{\mathrm{no-dd}}$, i.e., the analytical formula
of Eqs (13) (8). The dotted vertical lines (black) are the same as those in
Fig. 1(a).

While the physical origin of the renormalized van der Waals universality is
yet to be clarified, we suspect that the angular momentum insensitive nature
of the short-range phase [52] is playing a crucial role; we have confirmed in
our three-body calculations that a substantial fraction of the wavefunction is
occupying $\ell\geq 2$ states. If the short-range phase shifts in various
angular momentum channels are directly related with each other, as in the
angular-momentun-insensitive QDT [52], $a^{(\mathrm{HH})}$ characterizing the
$s$-wave short-range phase can universally capture the scatterings occuring in
all the angular momentum channels. We also suspect that a universal three-body
repulsion appearing at moderately large distance may be another possible
mechanism [23, 21, 22, 46, 47]; in the three-body system of identical bosons
interacting via the van der Waals interaction, the appearance of a non-
adiabatic three-body repulsion at a large distance $r\simeq 2r_{\mathrm{vdw}}$
is ascribed to be the physical origin of the universality of the three-body
parameter of the Efimov states [23, 21, 22]. Furthermore, for three identical
bosons interacting via purely dipolar interactions, a three-body repulsion has
also been found to appear universally at $r\simeq a_{dd}$ [46]. The multi-
channel effects with an interplay of the van der Waals and dipole interaction
in our sysetm may lead to a similar three-body repulsion scenario. We also
remark that our universality of the Efimov states involving higher partial-
wave channels is analogous to the partial-wave phase-locking mechanism found
in the resonant exchange collisions [62, 63], where the phase shifts of
different scattering channels between the atoms are related with each other.

Since the $s$-wave scattering length between the heavy atoms characterizes the
universal behavior of the bosonic Efimov states so precisely, it is tempting
to use the $p$-wave scattering volume $v_{p}^{(\mathrm{HH})}$ to analyze the
fermionic Efimov states. However, $v_{p}^{(\mathrm{HH})}$ cannot be defined in
the presence of the dipole interaction, because the dipole interaction is
present even at large distance in the $\ell=1$ channel and the asymptotic
wavefunction is no longer a free plane wave, in stark contrast to the $\ell=0$
channel [60]. We therefore consider below an alternative scattering parameter
$K_{c,F}^{(3D)}$ which characterizes the large-distance asymptotic
wavefunction of the two heavy fermionic dipoles: we consider the Schrödinger
equation between the two heavy atoms

$\left[-\frac{1}{M}\frac{\partial^{2}}{\partial
r^{2}}+\frac{2}{Mr^{2}}-\frac{C_{6}}{r^{6}}-\frac{4}{5}\frac{C_{dd}}{r^{3}}\right]u_{1}(r)=Eu_{1}(r).$
(18)

We have only kept the van der Waals interaction and the diagonal $\ell=1$
element of the dipole interaction, which are reasonably assumed to play major
roles in determining the short- and long-range behaviors. At large distance,
the van der Waals interaction is negligible in Eq. (18), so that the
wavefunction behaves at vanishing energy as [64]

$u_{1}(r\rightarrow\infty)=\sqrt{r}\left[J_{3}\left({\sqrt{\frac{48}{5}\frac{a_{dd}}{r}}}\right)-K_{c,F}^{(3D)}Y_{3}\left({\sqrt{\frac{48}{5}\frac{a_{dd}}{r}}}\right)\right],$
(19)

where $J_{3}$ and $Y_{3}$ are the Bessel and Neuman functions corresponding to
two independent solutions under the central $1/r^{3}$ potential.
$K_{c,F}^{(3D)}$ is introduced in an analogous manner as the QDT parameter
$K^{c}$ in the van der Waals case (see Eq. (5)), and hence is expected to be a
universal parameter characterizing the short-range two-body phase shift. In
Fig. 7, we show $K_{c,F}^{(3D)}$ numerically calculated from Eqs. (18) (19)
with the hard-wall boundary condition $u_{1}(r=R_{\mathrm{min}})=0$. We note
that the van der Waals term is essential in determining the short-range phase:
$K_{c,F}^{(3D)}$ is markedly different from that obtained by neglecting the
van der Waals term (black solid), suggesting both the van der Waals and dipole
interactions are relevant for characterizing the scattering between the two
heavy fermionic atoms.

Figure 7: $K_{c,F}^{(3D)}$ between two heavy fermionic atoms for
$a_{dd}/r_{\mathrm{vdw}}=0.86755....$ corresponding to the realistic dipole
strength between 167Er atoms ($a_{dd}=75.7a_{0}$ and
$r_{\mathrm{vdw}}=65.9a_{0}$ [57, 58]). The solid curve (black) is
$K_{c,F}^{(3D)}$ calculated with purely dipole interaction (i.e. without the
van der Waals interaction) in Eq. (18):
$K_{c,F}^{(3D)}=J_{3}\left(q_{\mathrm{min}}\right)/Y_{3}\left(q_{\mathrm{min}}\right)$
with $\displaystyle
q_{\mathrm{min}}=\sqrt{\frac{48a_{dd}}{5R_{\mathrm{min}}}}$

In Fig. 8, we show the binding energies of the fermionic Efimov states plotted
against $K_{c,F}^{(3D)}$ as obtained above. The numerical data of different
short-range $R_{\mathrm{min}}$ values collapse into a single curve,
demonstrating the universality of the fermionic Efimov states. The
universality holds particularly well for a realistic $a_{dd}$ value of
167Er-167Er (Fig. 8(a)(b)), where only a few points of higher angular momentum
character deviate from the universal curve. In contrast to the bosonic case,
we cannot find an analytical expression which could explain the universal
energy spectra owing to the impossibility of using $v_{p}^{(\mathrm{HH})}$
hence Eq. (16). The avoided crossing at $K_{c,F}^{(3D)}\simeq-1$ originates
from the $\ell=3$ channel which is present without the dipole interaction at
around $r_{\mathrm{vdw}}/v_{p}^{(\mathrm{HH})}\simeq-0.3$ (Fig. 2(c)). It
shifts toward a larger negative $K_{c,F}^{(3D)}$ side with a decreased slope
as the dipole strength is increased from a realistic value of 167Er-167Er
(Fig. 8(a)(b)) toward larger values (Fig. 8(c)-(f)). The foothill of the
avoided crossing shows a larger variance for stronger dipole interactions
(Fig. 8(c)-(f)). This is a precursor of the breakdown of our assumption that
the single-channel short-range phase in $\ell=1$ universally characterizes the
three-body system (see Sec. III.4), especially for a very strong dipole
interaction and large $K_{c,F}^{(3D)}$ (Fig. 8(c)(e)(f)).

Figure 8: Binding energy of the fermionic Efimov states for the mass ratio
$M/m=27.7520...$ corresponding to 167Er-167Er-6Li system. $K_{c,F}^{(3D)}$ in
the horizontal axis is obtained numerically from Eqs. (18) (19). Realistic
dipole strength for 167Er-167Er of $a_{dd}/r_{\mathrm{vdw}}=0.8705....$ (top
row (a)(b)), and stronger dipole strengths of $a_{dd}/r_{\mathrm{vdw}}=1.6$
(middle row (c)(d)), and $a_{dd}/r_{\mathrm{vdw}}=2.0$ (bottom row (e)(f)) are
presented.

Although the $K_{c,F}^{(3D)}$ parameter has the advantage of capturing the
universal behavior of the fermionic Efimov states, it is marginally useful in
predicting the three-body parameter of experimental systems of interest, e.g.
167Er-167Er-6Li. This is primarily because it is not easy, though not
impossible, to determine the value of $K_{c,F}^{(3D)}$; it sensitively depends
on the position (i.e. $R_{\mathrm{min}}$) and shape of the short-range
repulsive potential, and it is therefore challenging to predict its value by
theoretical calculations [57, 65, 66]. Furthermore, as the $K_{c,F}^{(3D)}$
dependence appears as the next-next-next leading order if we perform the
asymptotic expansion of the $J_{3}$ and $Y_{3}$ at large distance in Eq. (19),
it contributes to the low-energy scattering cross section as a small
correction term modifying the dominant quasi-universal contribution [60].

###  III.4 Strong dipole interaction $a_{dd}\gg r_{\mathrm{vdw}}$

While the universal descriptions via partial-wave scattering parameters of
$a^{(\mathrm{HH})}$ (boson) and $K_{c,F}^{(3D)}$ (fermion) are shown in Sec.
III.3 to be valid for moderate dipole strengths including the realistic values
for Er atoms, they should break down in the limit of strong dipole
interactions. This is because the single partial-wave description in defining
the short-range two-body phase between the two heavy atoms becomes invalid
since all the even (odd) angular momentum channels almost equally contribute
for the bosons (fermions).

We therefore need to introduce an alternative scattering parameter which
should characterize the short-range scattering phase of two strong dipoles. As
the two dipoles prefer to be aligned in a head-to-tail, hence quasi-one-
dimensional, configuration in the strong dipole limit, we consider a one-
dimensional equation

$\left[-\frac{1}{M}\frac{\partial^{2}}{\partial
z^{2}}-\frac{C_{6}}{z^{6}}-\frac{2C_{dd}}{z^{3}}\right]u(z)=Eu(z),$ (20)

which corresponds to Eq. (17) constrained to the one-dimensional configuration
on the $z$-axis. We introduce the scattering parameter $K_{c}^{(1D)}$ between
the two dipoles inspired from the asymptotic behavior of the zero-energy
wavefunction of Eq. (20) at $|z|\rightarrow\infty$ [67]

$\displaystyle
u(z)\rightarrow(\mathrm{sgn}z)^{P}\sqrt{|z|}\Biggl{[}J_{1}\Biggl{(}{\sqrt{\frac{24a_{dd}}{|z|}}}\Biggr{)}-K_{c}^{(1D)}Y_{1}\Biggl{(}{\sqrt{\frac{24a_{dd}}{|z|}}}\Biggr{)}\Biggr{]},$
(21)

where $J_{1}$ and $Y_{1}$ terms are two independent solutions of Eq. (20)
without the van der Waals interaction. We impose a hard-wall boundary
condition $u(z=\pm R_{\mathrm{min}})=0$ corresponding to the three-dimensional
counterpart. Since the positive and negative $z$ regions are separated, the
bosonic and fermionic systems become equivalent, with the same value of
$K_{c}^{(1D)}$; their only difference appears in the overall parity, i.e.
$P=0$ for bosons and $P=1$ for fermions in Eq. (21).

Figure 9: Binding energy of the bosonic (blue circles) and fermionic (orange
squares) Efimov states for strong dipole interactions of
$a_{dd}/r_{\mathrm{vdw}}=50$ (top row (a)(b)), and
$a_{dd}/r_{\mathrm{vdw}}=100$ (bottom row (c)(d)). $K_{c}^{(1D)}$ in the
horizontal axis is obtained numerically from Eqs. (20) (21). The mass ratio is
taken as $M/m=27.5855...$ which corresponds to 166Er-6Li for the boson, while
for the fermion it is hypothetically taken to be the same value as the boson
to demonstrate the difference via the statistics of the particles.

In Fig. 9, the binding energies of the Efimov states obtained from Eq. (17)
are plotted as a function of $K_{c}^{(1D)}$ in Eqs. (20) (21). The bosonic
(blue circles) and fermionic (orange squares) systems, hypothetically assumed
to be the same mass ratio, show rather similar behaviors, in stark contrast to
the weak and moderate dipole systems of Sec. III.1-III.3. This is because the
parity of the wavefunction plays a marginal role when the two heavy atoms are
mostly aligned in a quasi-one-dimensional configuration along the $z$-axis,
and the $z>0$ and $z<0$ regions are separated by the hard-wall boundary
condition. This is particularly true when the size of the trimer is much
smaller than $a_{dd}$, beyond which the centrifugal repulsion in Eq. (17)
representing the three-dimensional nature becomes more dominant than the
dipole term. For the tightly bound states in Fig. 9, the bosonic and fermionic
systems become almost indistinguishable, supporting this scenario. In other
words, the statistics of the particles is irrelevant for the strongly dipolar
Efimov states as long as their binding energy is larger than the dipole scale
$1/Ma_{dd}^{2}$.

Figure 10: Contour plot of $\displaystyle|r\psi(\bm{r})|$ of the relative
position of the two heavy particles in the Efimov states, plotted in the $y=0$
plane. (a) Shallow bound states in the universal regime for the bosons, and
(b) for the fermions. (c) Shallow bound states in the non-universal regime
(i.e, a steep slope region in-between the avoided crossings in Fig. 9(a)) for
the bosons, and (d) tightly bound states for the bosons (plotted in a smaller
$xz$ region than (a)-(c)). For all (a)-(d), the dipole strength and mass ratio
are the same as those in Fig. 9 (a)(b), with the following parameters: (a)
$Mr_{\mathrm{vdw}}^{2}|E|=1.24\times 10^{-4}$, $(K_{c}^{(1D)})^{-1}=0.71...$
($R_{\mathrm{min}}/r_{\mathrm{vdw}}=0.25225...$), (b)
$Mr_{\mathrm{vdw}}^{2}|E|=1.23\times 10^{-4}$, $(K_{c}^{(1D)})^{-1}=2.58...$
($R_{\mathrm{min}}/r_{\mathrm{vdw}}=0.2617...$), (c)
$Mr_{\mathrm{vdw}}^{2}|E|=1.10\times 10^{-4}$, $(K_{c}^{(1D)})^{-1}=-1.14...$
($R_{\mathrm{min}}/r_{\mathrm{vdw}}=0.25758...$), and (d)
$Mr_{\mathrm{vdw}}^{2}|E|=9.72\times 10^{-3}$, $(K_{c}^{(1D)})^{-1}=-3.42...$
($R_{\mathrm{min}}/r_{\mathrm{vdw}}=0.2592...$).

The data points with a small slope in Fig. 9, including the weakly bound
states, collapse into a single curve for both the bosonic and fermionic
systems, suggesting that $K_{c}^{(1D)}$ universally characterizes the short-
range phase, hence the Efimov states with the strong dipoles. On the other
hand, the data points with steeper slopes do not seem to show the universality
with respect to $K_{c}^{(1D)}$. Since those states have significant
contributions from large angular momentum states $\ell\gg 1$ and the angular
insensitive nature of their short-range phase can break down, it is natural
that their features cannot be universally captured by the one-dimensional
scattering model in Eqs. (20) (21) where the angular momentum is absent.

In Fig. 10, we show the amplitude of the wavefunction. For a bosonic Efimov
state with a small binding energy (Fig. 10 (a)), it essentially comprises of
two regions; the inner region $R\lesssim a_{dd}$ where the dipole interaction
is so strong that the two heavy particles are preferentially aligned on the
$z$-axis, and the outer region $R\gtrsim a_{dd}$ where the dipole interaction
is smaller than the kinetic term, thus forming a three-dimensional halo state
of a predominantly $s$-wave nature. They are neither a perfect one-dimensional
geometry nor a spherical $s$-wave halo owing to the strong admixture of
angular momentum states, but their universal nature is essentially determined
by the phase in the short-range region, where the one-dimensional
configuration is dominant and hence $K_{c}^{(1D)}$ is a universal parameter.
For fermionic Efimov states with a similarly small value of the binding energy
(Fig. 10 (b)), the short-range and long-range natures are almost the same,
with the only notable difference being the presence of a node in the
wavefunction at $z=0$, representing the anti-symmetrization. In contrast, the
wavefunction in the non-universal regime (Fig. 10 (c)) shows a distinct
feature: the wavefunction in the inner region $R\lesssim a_{dd}$ no longer
shows a simple quasi-one-dimensional character, but rather a strongly
anisotropic shape, which is likely induced by an admixture of various angular
momentum states. This is consistent with the breakdown of the universality
with respect to $K_{c}^{(1D)}$. When the binding energy is so large that the
size of the wavefunction is smaller than $a_{dd}$ (Fig. 10 (d)), the whole
wavefunction is localized in the inner region $R\lesssim a_{dd}$, forming an
almost perfect quasi-one-dimensional state. In this limit, the hard-wall
boundary condition $R=R_{\mathrm{min}}$ necessarily produces a node in the
wavefunction at $z\approx 0$, so that the bososnic and fermionic wavefunctions
become the same except for the sign change. In this limit, the bosonic and
fermionic systems show the same universal behaviors. While this limit of
boson-fermion duality, analogous to the Tonks-Girardeau limit [68, 69], is
fascinating, it is challenging to achieve because it requires a very large
dipole interaction strength, which is impossible for cold atoms with magnetic
dipole interactions, but may be possible with polar molecules [70, 71] or
Rydberg atoms [72, 73].

##  IV Three-body parameters of Er-Er-Li Efimov states

In Table 1, we list some combinations of isotopes of the Er-Li systems,
together with their physical parameters. The dipole lengths are comparable to
the Er-Er van der Waals length $a_{dd}/r_{\mathrm{vdw}}\simeq 0.87$, which lie
outside the perturbative regime $a_{dd}/r_{\mathrm{vdw}}\lesssim 0.5$
discussed in Sec. III.2. Nevertheless, they are not strong enough to be in the
strong-dipole limit discussed in Sec. III.4; they are in the parameter region
well described by the renormalized van der Waals universality shown in Sec.
III.3. We can therefore use Eq. (15) to evaluate the binding energy of the
Efimov states for the bosonic Er atoms interacting with a 6Li atom in unitary
limit. The $s$-wave scattering length between the Er atoms $a^{(\mathrm{HH})}$
generally depends on the applied magnetic field, owing to the existence of a
series of broad and narrow Feshbach resonances [57, 65], and it is yet to be
determined by experiments or by sophisticated theoretical calculations. Still,
by using the background $s$-wave scattering lengths reported in the literature
[57, 58] as representative values, we can estimate the binding energy of the
Efimov states $|E|=\kappa_{*}^{2}/M$. The results are tabulated in Table 1; we
find that the difference in the values of $a^{(\mathrm{HH})}$ significantly
affect the three-body parameter $\kappa_{*}$ across the isotopes while the
mass and $a_{dd}$ are almost the same and therefore play marginal roles. The
values of $\kappa_{*}$ in Table 1 can be readily improved with better
knowledge of the Er-Er scattering length $a^{(\mathrm{HH})}$. We also note
that the larger binding energy states in Table 1 may not be so accurate
because we use the low-energy analytical formula in Eq. (15); this can also be
straightforwardly improved using the numerical results of Sec. III.3.

Table 1: Three-body parameters of the Er-Er-Li Efimov states: binding energy at the unitary Er-Li scattering length $|E|=\kappa_{*}^{2}/M$, and the Er-Li scattering length $a_{-}^{(\mathrm{HL})}$ ($a_{*}^{(\mathrm{HL})}$) at which the Efimov states dissociate into three atoms (an atom and a dimer), respectively. The van der Waals and dipole lengths between the Er atoms are from Refs [57, 58] with the mass scaling. $a^{(\mathrm{HH})}$ between the heavy atoms are estimated by the background scattering lengths $a_{bg}$ tabulated in Refs [58]. $a_{0}$ is the Bohr radius. Species |  $r_{\mathrm{vdw}}$[$a_{0}$] |  $a_{dd}$[$a_{0}$] |  $a^{(\mathrm{HH})}$ [$a_{0}$] | $\kappa_{*}r_{\mathrm{vdw}}$ | $\dfrac{a_{-}^{(\mathrm{HL})}}{r_{\mathrm{vdw}}}$ | $\dfrac{a_{*}^{(\mathrm{HL})}}{r_{\mathrm{vdw}}}$
---|---|---|---|---|---|---
166Er-6Li | 75.5 | 65.5 | 68 | 0.495 | -2.66 | 0.0712
0.107 | -12.4 | 0.331
2.29$\times 10^{-2}$ | -57.3 | 1.54
4.94$\times 10^{-3}$ | -266 | 7.13
168Er-6Li | 75.8 | 66.3 | 137 | 0.352 | -3.74 | 0.100
7.65$\times 10^{-2}$ | -17.2 | 0.461
1.66$\times 10^{-2}$ | -79.0 | 2.12
3.62$\times 10^{-3}$ | -363 | 9.74
170Er-6Li | 76.0 | 67 | 221 | 0.298 | -4.40 | 0.118
6.55$\times 10^{-2}$ | -20.0 | 0.538
1.44$\times 10^{-2}$ | -91.3 | 2.45
3.16$\times 10^{-3}$ | -416 | 11.2

While there are spectroscopic methods to directly associate the Efimov trimers
and observe their binding energies [74, 75, 76, 77], it is challenging to
perform such experiments, especially in the unitary limit. It is more
straightforward to observe the three-body loss rate of the atoms for variable
$s$-wave scattering lengths $a^{(\mathrm{HL})}$, and search for the negative
scattering length $a=a_{-}^{(\mathrm{HL})}$ at which the loss rate peak
appears [11, 12, 13, 14, 16, 19]. Alternatively, one can prepare a mixture of
heavy-light Feshbach molecules and heavy atoms, and observe the dimer-atom
loss rate to find the positive scattering length $a=a_{*}^{(\mathrm{HL})}$ at
which its peak appears [17]. The values $a_{-}^{(\mathrm{HL})}$ and
$a_{*}^{(\mathrm{HL})}$ correspond to the points at which the Efimov states
dissociate into three atoms, or a dimer and an atom, respectively. Notably,
any of $a_{-}^{(\mathrm{HL})}$, $a_{*}^{(\mathrm{HL})}$, or $\kappa_{*}$ can
be regarded as the universal scale characterizing the Efimov states because
they can be related with each other by the universal Efimov theory [1, 2, 3,
4, 5, 6]. Using the zero-range Efimov theory, we have estimated
$a_{-}^{(\mathrm{HL})}$, $a_{*}^{(\mathrm{HL})}$ from the $\kappa_{*}$ values.
The results are presented in the last two columns of Table 1. The values of
$a_{-}^{(\mathrm{HL})}$ corresponding to the most tightly bound Efimov states
are likely to be either around
$a_{-}^{(\mathrm{HL})}/r_{\mathrm{vdw}}\approx-3$-$4$ or $-12$-$20$. With the
low-energy formula Eq. (15) and the zero-range theory, we cannot conclude
which of them are the ground Efimov state. We also note that they can be
significantly affected by finite-range effects owing to their relatively large
binding energy, while results for the more weakly bound states in Table 1
should be more quantitatively reliable. Similarly, we expect that the
dissociation points of the excited Efimov states around
$a_{*}^{(\mathrm{HL})}/r_{\mathrm{vdw}}=1$-$3$ or $7$-$12$ are more likely to
be experimentally observed than those corresponding to the tightly bound
states of Table 1. This is analogous to the measurement of $a_{*}$ in systems
of three identical bosons, for which the loss-rate peak corresponding to the
excited Efimov states appear clearly, whereas those for the ground and first
excited states may not appear, or only barely [78, 79].

For the fermionic Efimov states explored in recent cold-atom experiments of
167Er-6Li mixture [42], the shortage of scattering data between the fermionic
167Er atoms prevents us from making a similar estimate. Since the fermionic
Efimov states show the universal behavior characterized by $K_{c,F}^{(3D)}$
(see Sec. III.3), we can make a theoretical prediction once the value of
$K_{c,F}^{(3D)}$ is known. However, since the low-energy scattering between
the fermionic 167Er atoms is dominated by the universal dipolar scattering,
the determination of $K_{c,F}^{(3D)}$ should be challenging, requiring the
determination of the sub-leading contributions in the 167Er-167Er scattering.

##  V Conclusion & Outlook

We have studied the universality of the Efimov states in a mass-imbalanced
three-body system of heavy-heavy-light atoms with a unitary interaction
between the heavy-light atoms, with the major motivation of elucidating the
interplay between the van der Waals and magnetic dipole interactions in Er-Er-
Li Efimov states. By reducing the three-body problem to a coupled-channel two-
body equation between the heavy atoms using the Born-Oppenheimer
approximation, we have investigated analytically and numerically the three-
body parameters (i.e. binding energies in the unitary limit) of the Efimov
states for various strengths of dipole interactions between the bosonic and
fermionic heavy atoms.

In the absence of dipole interaction, using the quantum defect theory, we
derive analytical formulae of the three-body parameters of the Efimov states,
which are represented as explicit universal functions of the van der Waals
length and $s$-wave scattering length ($p$-wave scattering volume) between the
heavy bosonic (fermionic) atoms. The analytical formulae excellently reproduce
the numerical results reported in Ref. [25], demonstrating the van der Waals
universality in a analytical manner.

In the presence of dipole interaction, different orbital angular momentum
states get coupled and we have performed coupled-channel numerical
calculations. For a weak dipole interaction, we find that the three-body
parameters are affected by the dipole interaction in a different manner for
the bosons and fermions. Due to the presence (absence) of the dipole matrix
element for bosons in the $L=0$ (fermions in $L=1$) states, the three-body
parameter gets a quadratic (linear) shift with respect to a perturbative
dipole interaction for the bosons (fermions), respectively.

When the dipole interaction is as strong as the van der Waals interaction, we
find that the values of the three-body parameters get significantly altered
from those of a non-dipolar system. However, once the $s$-wave scattering
length between the heavy atoms is calculated in the presence of both the
dipole and van der Waals interactions, the QDT formulae derived for a pure van
der Waals system can well reproduce the three-body parameters of the bosonic
Efimov states. Using this renormalized van der Waals universality, we have
estimated the three-body parameters of the Er-Er-Li Efimov states for the
bososnic Er isotopes. For the fermionic heavy atoms, on the other hand, the
$p$-wave scattering volume can no longer be defined, owing to the long-range
nature of the dipole interaction, so that the renormalized van der Waals
universality does not hold. However, they can still be described universally
with a low-energy scattering parameter between the two fermionic dipoles which
characterizes the phase shift of an asymptotic $p$-wave state, so that the
universality of the three-body parameter holds true.

For an extremely large dipole interaction, neither the $s$-wave scattering
length nor the $p$-wave low-energy scattering parameter can universally
characterize the Efimov states, because the two dipoles get preferentially
aligned in a linear geometry at short distance and they are neither $s$-wave
nor $p$-wave dominant. In this limit, a quasi-one-dimensional scattering
parameter is capable of capturing the short-range scattering phase. We show
that except for some regions in-between avoided crossings, the Efimov states
can be universally described by the one-dimensional scattering parameter.

With the above studies, we have elucidated how the interplay of the van der
Waals and dipole interactions can affect the universality of the Efimov
states. In particular, we have clarified the roles of the statistics of the
particles and the angular momentum channels, in the presence of both isotropic
and anisotropic forces. Our work is not only relevant for the current
challenge of observing Efimov states in the highly mass-imbalanced cold atom
mixture of Er-Li atoms [41, 42], but also can shed new light on halo nuclear
phenomena. The Efimov states are believed to appear universally in various
strongly interacting quantum systems [18, 8, 3]. In particular, Efimov-like
weakly bound three-body states have been actively studied in nuclear systems,
such as neutron-rich nuclei [7, 80, 81, 82], Hoyle state of 12C [83, 84], and
in near-breakup excited states of nuclei [85]. In nuclear physics, the tensor
force has been pointed out to play a crucial role in determining the stability
and reactions of nuclei. Notably, the tensor force has an anisotropic nature
similar to the dipole-dipole interaction [86]. Investigating the interplay of
the van der Waals interaction and dipole interaction is thus analogous to the
nuclear problems where the interplay of the central part of the strong force
and the tensor force is relevant. Our study on the dipole and van der Waals
interactions in strongly interacting quantum few-body systems lays the basis
for the quantum simulation of nuclear phenomena using dipolar cold-atom
mixtures [41, 42, 43, 58, 87].

###### Acknowledgements.

We thank Yoshiro Takahashi, Yousuke Takasu, Lucas Happ, and Emiko Hiyama for
fruitful discussions. This work was supported by JSPS KAKENHI Grant Numbers
JP22K03492, JP23H01174, and JP23K03292. KO acknowledges support from Graduate
Program on Physics for the Universe (GP-PU) of Tohoku University. SE
acknowledges support from Institute for Advanced Science, University of
Electro-Communications.

##  Appendix A  Analytical Solution of Eq. (2)

Equation (2) can be analytically solved in a similar manner as Ref. [51]:
introducing $x\equiv 2\left(\frac{r_{\rm vdw}}{r}\right)^{2}$, $\Delta\equiv
Mr_{\rm vdw}^{2}E/4$ and $u_{\ell}(r)\equiv\sqrt{r}\mathcal{U}_{\ell}(x)$, Eq.
(2) is written in the dimensionless form as

$\left[x^{2}\frac{d^{2}}{dx^{2}}+x\frac{d}{dx}+x^{2}-\nu_{0}^{2}\right]\mathcal{U}_{\ell}(x)=-\frac{2\Delta}{x}\mathcal{U}_{\ell}(x),$
(22)

where $\nu_{0}=s_{\ell}/2$. In the absence of the induced attraction via the
light particle, Eq. (2) is the van der Waals potential problem; the attraction
modifies the last term in the left-hand side from the quarter integer
$\nu_{0}=(2\ell+1)/4$ into $\nu_{0}=s_{\ell}/2$. When the Efimov effect is
present (absent) as in the $\ell=0$ (large $\ell$) channel, $\nu_{0}$ is pure
imaginary (real).

In a similar manner as Ref. [51], the two linearly independent solutions of
Eq. (2) are repretented as

$\displaystyle\begin{split}\overline{f_{\ell}}(r)&=\sqrt{r}\sum_{n}b_{n}J_{\nu+n}\left(x\right),\\\
\overline{g_{\ell}}(r)&=\sqrt{r}\sum_{n}b_{n}Y_{\nu+n}\left(x\right).\end{split}$
(23)

$b_{n}$ and $\nu$ are determined from the same trandental equations in Ref.
[51], with the modification
$\nu_{0}=(2\ell+1)/4\rightarrow\nu_{0}=s_{\ell}/2$. In order to have favorable
short-range asymptotc features, it is conveniet to consider $f_{\ell}^{c}$ and
$g_{\ell}^{c}$ which are defined as linear transforms of $\overline{f_{\ell}}$
and $\overline{g_{\ell}}$ as

$\displaystyle\begin{split}f_{\ell}^{c}(r)&=\frac{1}{\sqrt{2(X_{\ell}^{2}+Y_{\ell}^{2})}}\left[\cos\left(\frac{\nu\pi}{2}+\theta_{\ell}\right)\overline{f_{\ell}}-\sin\left(\frac{\nu\pi}{2}+\theta_{\ell}\right)\overline{g_{\ell}}\
\right],\\\
g_{\ell}^{c}(r)&=\frac{-1}{\sqrt{2(X_{\ell}^{2}+Y_{\ell}^{2})}}\left[\sin\left(\frac{\nu\pi}{2}+\theta_{\ell}\right)\overline{f_{\ell}}+\cos\left(\frac{\nu\pi}{2}+\theta_{\ell}\right)\overline{g_{\ell}}\
\right],\end{split}$ (24)

where $\tan\theta_{\ell}=Y_{\ell}/X_{\ell}$ with
$X_{\ell}=\sum_{n}(-1)^{n}b_{2n}$ and $Y_{\ell}=\sum_{n}(-1)^{n}b_{2n+1}$.

The short-distance asymptotic forms of $f_{\ell}^{c}$ and $g_{\ell}^{c}$ for
$r\ll r_{\rm vdw}$ are

$\displaystyle\begin{split}&f^{c}_{\ell}(r)\rightarrow\frac{1}{\sqrt{2\pi}}\frac{r^{3/2}}{r_{\rm
vdw}}\cos\left[x-\frac{\pi}{4}\right]\\\
&g^{c}_{\ell}(r)\rightarrow-\frac{1}{\sqrt{2\pi}}\frac{r^{3/2}}{r_{\rm
vdw}}\sin\left[x-\frac{\pi}{4}\right].\end{split}$ (25)

Notably, they are independent of angular momentum nor energy, which are
essential features for the angular-insensitive QDT [52].

The large-distance asymptotic forms of $f_{\ell}^{c}$ and $g_{\ell}^{c}$ for
$E=k^{2}/M>0$ are

$\left[\begin{array}[]{c}f^{c}_{\ell}\\\
g^{c}_{\ell}\end{array}\right]\to\sqrt{\frac{2}{\pi
k}}\left[\begin{array}[]{cc}Z^{c}_{fs}&Z^{c}_{fc}\\\
Z^{c}_{gs}&Z^{c}_{gc}\end{array}\right]\left[\begin{array}[]{c}\sin\left(kr-\frac{l\pi}{2}\right)\\\
-\cos\left(kr-\frac{l\pi}{2}\right)\end{array}\right],$ (26)

and for $E=-\kappa^{2}/M<0$ are

$\left[\begin{array}[]{c}f^{c}_{\ell}\\\
g^{c}_{\ell}\end{array}\right]\to\sqrt{\frac{1}{2\pi\kappa}}\left[\begin{array}[]{cc}W^{c}_{f+}&W^{c}_{f-}\\\
W^{c}_{g+}&W^{c}_{g-}\end{array}\right]\left[\begin{array}[]{c}-2e^{-\kappa
r}\\\ e^{\kappa r}\end{array}\right].$ (27)

The elements of the matrices $Z^{c}$ and $W^{c}$ are the same as those in Ref.
[51], with the modification
$\nu_{0}=(2\ell+1)/4\rightarrow\nu_{0}=s_{\ell}/2$.

At vanishing energy $\Delta\rightarrow 0$, the wavefunction is dominantly
described by the single term of the Bessel series in Eq. (23) and
$\nu\simeq\nu_{0}$:

$\begin{split}f^{c}_{\ell}(r)&=\sqrt{\frac{r}{2}}\left[\cos\frac{\pi\nu_{0}}{2}J_{\nu_{0}}\left(x\right)-\sin\frac{\pi\nu_{0}}{2}Y_{\nu_{0}}\left(x\right)\right]\\\
g^{c}_{\ell}(r)&=\sqrt{\frac{r}{2}}\left[-\sin\frac{\pi\nu_{0}}{2}J_{\nu_{0}}\left(x\right)-\cos\frac{\pi\nu_{0}}{2}Y_{\nu_{0}}\left(x\right)\right].\end{split}$
(28)

In the absence of the light particle, $\nu_{0}=(2\ell+1)/4$ and the wave
function is essentially the Bessel function with a quarter integer index as
found in Ref. [88]. In the presence of the light particle and hence the Efimov
effect, the wave function is still the Bessel function, but now with a pure-
imaginary index $\nu_{0}=s_{\ell}/2$; this reflects the log-periodic behavior
of the Efimov states. Using $Y_{\alpha}(z)=(J_{\alpha}(z)\cos\alpha\pi-
J_{-\alpha}(z))/\sin\alpha\pi$ and $J_{-iq}(r)=\left(J_{iq}(r)\right)^{*}$ for
real $q$ and $r$, Eq. (28) is rewritten as

$\left[\begin{array}[]{c}f^{c}_{\ell}\\\
g^{c}_{\ell}\end{array}\right]=\sqrt{\frac{r}{2}}\left[\begin{array}[]{c}\mathrm{Re}\left[J_{\frac{s_{\ell}}{2}}\left(x\right)\right]/\cosh\frac{\pi|s_{\ell}|}{4}\\\
-\mathrm{Im}\left[J_{\frac{s_{\ell}}{2}}\left(x\right)\right]/\sinh\frac{\pi|s_{\ell}|}{4}\end{array}\right].$
(29)

##  Appendix B  Low-energy QDT expansion

Following Refs. [53, 54], we can perform an expansion (QDT expansion) at low
energy. Indeed, when $|\Delta|\ll 1$,

$\nu=\nu_{0}+\frac{3}{2\nu_{0}}\frac{\Delta^{2}}{(\nu_{0}^{2}-1)(4\nu_{0}^{2}-1)}+O(\Delta^{4}),$
(30)

from which we find $X_{\ell}\simeq 1$ and

$\tan\theta_{\ell}\simeq Y_{\ell}\simeq-\frac{4\Delta}{(4\nu_{0}^{2}-1)}.$
(31)

We also obtain

$\begin{split}M_{\ell}&\simeq|\Delta|^{2\nu_{0}}\frac{2\pi^{2}}{[\Gamma(\nu_{0})\Gamma(1+2\nu_{0})]^{2}}\frac{1}{\sin(\pi\nu_{0})\sin(2\pi\nu_{0})},\end{split}$
(32)

where $M_{\ell}\equiv G_{\ell}(-\nu)/G_{\ell}(\nu)$ with $G_{\ell}$ is defined
as

$G_{\ell}(\nu)\equiv\lim_{n\rightarrow\infty}\sum_{s=0}^{\infty}(-1)^{n+s}\frac{n!\Gamma(2\nu+n+1)}{s!\Gamma(-\nu-
n-s+1)}\Delta^{-\nu-n}b_{-n-2s}.$ (33)

Notably, all the above equations remain unchanged from Refs. [53, 54] even
when $\nu$ is real non-quater integer value ($s_{\ell}^{2}>0$), or a pure
imaginary value ($s_{\ell}^{2}<0$).

For the van der Waals QDT expansion, or more generally for real $\nu$,
$\nu_{0}$, and $s_{\ell}$ values, Eq. (32) suggests

$M_{\ell}\propto|\Delta|^{2\nu_{0}}\rightarrow 0.$ (34)

Therefore, $M_{\ell}$ becomes small at low energy, and thus can be negligible
in evaluating the right-hand side of Eq. (6) at the leading order [89]

$K^{c}\simeq\tan\dfrac{\pi\nu_{0}}{2}.$ (35)

On the other hand, in the presence of the Efimov effect ($\nu_{0}=s_{\ell}/2$
with $s_{\ell}^{2}<0$), $\nu$ takes a pure imaginary value. Using

$\left|\Gamma(1+i|s_{\ell}|)\right|^{2}=\frac{\pi|s_{\ell}|}{\sinh\pi|s_{\ell}|}$
(36)

and introducing

$\xi_{\ell}\equiv\arg\left[\Gamma\left(\frac{i|s_{\ell}|}{2}\right)\Gamma(1+i|s_{\ell}|)\right],$
(37)

we now find

$M_{\ell}\simeq-|\Delta|^{i|s_{\ell}|}e^{-2i\xi_{\ell}}=-\exp\left(i\left[|s_{\ell}|\ln|\Delta|-2\xi_{\ell}\right]\right).$
(38)

In stark contrast to Eq. (34), the magnitude of $M_{\ell}$ remains unity at
low energy. The log-periodic oscillation is a manifestation of the log-
periodicity of the Efimov effect; indeed, substituting Eqs. (38) (30) (31)
into Eq. (6), we find

$K^{c}\simeq\tan\left[\frac{|s_{\ell}|}{2}\ln|\Delta|-\xi_{\ell}\right]\tanh\frac{\pi|s_{\ell}|}{4}.$
(39)

From above, we obtain the binding energy of the Efimov states as a function of
$K^{c}$

$|\Delta|=\exp\left[\frac{2}{|s_{\ell}|}\left\\{\arctan\left(\frac{K^{c}}{\tanh\dfrac{\pi|s_{\ell}|}{4}}\right)+\xi_{\ell}\right\\}\right]e^{-\frac{2n\pi}{|s_{\ell}|}},$
(40)

which reproduces the discrete scale invariant energy spectrum. The above
equation is valid at low energy $|\Delta|\ll 1$, which implies that it holds
well when the integer $n$ is large $n\gg 1$.

##  Appendix C Perturbative calculation at weak dipole limit $|a_{dd}|\ll
r_{\mathrm{vdw}}$

When the dipole-dipole interaction is small, we can regard the dipole
interaction as a small perturbation for the van der Waals Efimov states in
Sec. II.2. We start from

$\left[\hat{H}_{0}+\hat{V}_{dd}\right]\ket{\psi_{IL}}=E_{IL}\ket{\psi_{IL}}.$
(41)

As the angular momentum is the good quantum number when $\hat{V}_{dd}=0$, we
label the states with $L$ and $I$ as
$\ket{\psi_{IL}}\simeq\ket{\psi^{(0)}_{IL}}$ : $L$ is the angular momentum of
the unperturbed state and $I$ is the label of the state. The second order
perturbation theory reads

$\displaystyle E_{IL}$ $\displaystyle=E^{(0)}_{IL}+E^{(1)}_{IL}+E^{(2)}_{IL}$
(42) $\displaystyle\ket{\psi_{IL}}$
$\displaystyle=\ket{\psi^{(0)}_{IL}}+\sum_{i\ell\neq
IL}a^{(1)}_{IL,i\ell}\ket{\psi^{(0)}_{i\ell}}+\sum_{i\ell\neq
IL}a^{(2)}_{IL,i\ell}\ket{\psi^{(0)}_{i\ell}},$ (43)

where

$\displaystyle E^{(1)}_{IL}$
$\displaystyle=\bra{\psi^{(0)}_{IL}}\hat{V}_{dd}\ket{\psi^{(0)}_{IL}}$ (44)
$\displaystyle E^{(2)}_{IL}$ $\displaystyle=-\sum_{i\ell\neq
IL}\frac{\left|\bra{\psi^{(0)}_{IL}}\hat{V}_{dd}\ket{\psi^{(0)}_{i\ell}}\right|^{2}}{E^{(0)}_{i\ell}-E^{(0)}_{IL}}$
(45) $\displaystyle a^{(1)}_{{IL},i\ell}$
$\displaystyle=-\frac{\bra{\psi^{(0)}_{i\ell}}\hat{V}_{dd}\ket{\psi^{(0)}_{IL}}}{E^{(0)}_{i\ell}-E^{(0)}_{IL}}$
(46)
$\displaystyle\begin{split}a^{(2)}_{{IL},i\ell}&=-\frac{\bra{\psi^{(0)}_{i\ell}}\hat{V}_{dd}\ket{\psi^{(0)}_{IL}}\bra{\psi^{(0)}_{IL}}\hat{V}_{dd}\ket{\psi^{(0)}_{IL}}}{\left(E^{(0)}_{i\ell}-E^{(0)}_{IL}\right)^{2}}\\\
&+\sum_{i^{\prime}\ell^{\prime}\neq
IL}\frac{\bra{\psi^{(0)}_{i\ell}}\hat{V}_{dd}\ket{\psi^{(0)}_{i^{\prime}\ell^{\prime}}}\bra{\psi^{(0)}_{i^{\prime}\ell^{\prime}}}\hat{V}_{dd}\ket{\psi^{(0)}_{IL}}}{\left(E^{(0)}_{i\ell}-E^{(0)}_{IL}\right)\left(E^{(0)}_{i^{\prime}\ell^{\prime}}-E^{(0)}_{IL}\right)}.\end{split}$
(47)

The unperturbed state is
$\ket{\psi^{(0)}_{i\ell}}=\dfrac{u_{i\ell}(r)}{r}Y_{\ell m}(\theta,\phi)$,
where $u_{i\ell}$ is the solution of the unperturbed Hamiltonian (see Eq. (3))

$\left[-\frac{1}{M}\frac{d^{2}}{dr^{2}}+\frac{s_{\ell}^{2}-\frac{1}{4}}{Mr^{2}}-\frac{C_{6}}{r^{6}}\right]u_{i\ell}(r)=E_{i\ell}^{(0)}u_{i\ell}(r)$
(48)

satisfying the ortho-normalization condition $\displaystyle\int
dru_{i\ell}^{*}(r)u_{i^{\prime}\ell}(r)=\delta_{ii^{\prime}}$. As explained in
Sec. II.2 and Appendix A, $u_{i\ell}$ can be obtained with the QDT.

The matrix element can be evaluated as

$\begin{split}&\bra{\psi^{(0)}_{i\ell}}\hat{V}_{dd}\ket{\psi^{(0)}_{IL}}\\\
&=\frac{3a_{dd}}{M}\int dr\frac{u^{*}_{i\ell}(r)u_{IL}(r)}{r^{3}}\int d\Omega
Y_{\ell}^{m*}(1-3\cos^{2}\theta)Y_{L}^{m}\\\
&=\frac{3a_{dd}}{M}\int_{R_{\mathrm{min}}}^{\infty}dr\frac{u^{*}_{i\ell}(r)u_{IL}(r)}{r^{3}}\\\
&\times 2(-1)^{m+1}\sqrt{(2L+1)(2\ell+1)}\left(\begin{array}[]{ccc}\ell&2&L\\\
-m&0&m\end{array}\right)\left(\begin{array}[]{ccc}\ell&2&L\\\
0&0&0\end{array}\right),\end{split}$ (49)

where the bracket in the last line is the Wigner’s $3j$ symbol. We note that
the azimuthal number $m$ around the $z$ axis is conserved even in the presence
of the dipole interaction and thus unchanged. As the $3j$ symbol only takes
non-zero values for $\ell=L,L\pm 2$, the perturbation only couples $+2,0,-2$
angular momentum states. Introducing the matrix element of the radial part

$\langle
i,\ell|v_{r}|I,L\rangle=\int_{R_{\mathrm{min}}}^{\infty}dr\frac{u^{*}_{i\ell}(r)u_{IL}(r)}{r^{3}}$
(50)

for the $L=0$ state corresponding to bosonic heavy atoms, we obtain

$\displaystyle\begin{split}E_{I,L=0}&=E^{(0)}_{I,L=0}-\frac{4}{5}\left(\frac{3a_{dd}}{M}\right)^{2}\sum_{i}\frac{\left|\langle
i,\ell=2|v_{r}|I,L=0\rangle\right|^{2}}{E_{i,\ell=2}^{(0)}-E_{I,L=0}^{(0)}}\end{split}$
(51) $\displaystyle\ket{\psi_{I,L=0}}$
$\displaystyle=\ket{\psi^{(0)}_{I,L=0}}+\dfrac{2}{\sqrt{5}}\frac{3a_{dd}}{M}\displaystyle\sum_{i}\dfrac{\langle
i,\ell=2|v_{r}|I,L=0\rangle}{E_{i,\ell=2}^{(0)}-E_{I,L=0}^{(0)}}\ket{\psi^{(0)}_{i,\ell=2}},$
(52)

while for the $L=1$ state corresponding to fermionic heavy atoms, we obtain

$\displaystyle E_{I,L=1}=$ $\displaystyle
E^{(0)}_{I,L=1}-\frac{4}{5}\frac{3a_{dd}}{M}\langle I,L=1|v_{r}|I,L=1\rangle$
(53)
$\displaystyle\begin{split}\ket{\psi_{I,L=1}}=&\ket{\psi^{(0)}_{I,L=1}}+\frac{3a_{dd}}{M}\left[\frac{4}{5}\sum_{i\neq
I}\frac{\langle
i,\ell=1|v_{r}|I,L=1\rangle}{E^{(0)}_{i,\ell=1}-E^{(0)}_{I,L=1}}\ket{\psi^{(0)}_{i,\ell=1}}\right.\\\
&\left.+\sqrt{\frac{108}{175}}\sum_{i}\frac{\langle
i,\ell=3|v_{r}|I,L=1\rangle}{E^{(0)}_{i,\ell=3}-E^{(0)}_{I,L=1}}\ket{\psi^{(0)}_{i,\ell=3}}\right].\end{split}$
(54)

Notably, the effect of the dipole interaction is qualitatively different for
the bosons and fermions: they appear in the second order in $a_{dd}$ for the
bosons, and the first order for the fermions. This difference originates from
the difference in the matrix element of the dipole interaction: $\langle
LM|\hat{V}_{dd}|LM\rangle$=0 for $L=0$ state, while it can take a non-zero
value for $L=1$.

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