# Universal properties of dipolar Bose polarons in two dimensions

J. Sánchez-Baena  Departament de Física, Campus Nord B4-B5, Universitat
Politècnica de Catalunya, E-08034 Barcelona, Spain  Center for Complex Quantum
Systems, Department of Physics and Astronomy, Aarhus University, DK-8000
Aarhus C, Denmark  L. A. Peña Ardila  Institut für Theoretische Physik,
Leibniz Universität Hannover, Germany  School of Science and Technology,
Physics Divisio, University of Camerino, Via Madonna delle Carceri, 9B - 62032
(MC), Italy  G. E. Astrakharchik  Departament de Física, Campus Nord B4-B5,
Universitat Politècnica de Catalunya, E-08034 Barcelona, Spain  F. Mazzanti
Departament de Física, Campus Nord B4-B5, Universitat Politècnica de
Catalunya, E-08034 Barcelona, Spain

###### Abstract

We study the quasiparticle properties of a dipolar impurity immersed in a two-
dimensional dipolar bath. We use the ab-initio Diffusion Monte Carlo technique
to determine the polaron energy, effective mass and quasiparticle residue. We
find that these quantities follow a universal behaviour when properly scaled
in terms of the polarization angle and scattering length. This trend is
maintained over a wide range of values of the gas parameter, even in the
highly correlated regime. Additionally, we show that the anisotropy of the
impurity-bath interaction leads to an anisotropic effective mass that is
unexpectedly larger in the direction of minimum repulsion of the impurity-bath
interaction. Finally, we use our Monte Carlo results to check the accuracy of
perturbative approaches and determine their range of validity in terms of the
gas parameter.

##  I Introduction

Impurities interacting with a complex quantum-many-body environment have been
the subject of intense research in recent years. In the solid-state realm,
impurities interacting with an ionic crystal disrupt the media and are
screened by lattice phonons, forming quasiparticles known as polarons [1].
Polarons have been found to play an important role in semiconductor transport
[2], colossal magnetoresistance [3], as well as non-equilibrium phenomena such
as quantum heat transport [4]. In particular, two-dimensional (2D) solid-state
materials, such as graphene [5] and transition metal dichalcogenides (TMDs)
[6], have garnered significant attention recently due to their unique
properties and potential applications in various fields. The formation of
polarons with dipole-dipole interactions may affect the optical properties of
the material by shifting the absorption and emission spectra. For example,
repulsive dipole-dipole interactions between electric field-tunable, localized
interlayer excitons in the MoSe2/WSe2 heterobilayer may provide valuable
insights into the creation of excitonic few- and many-body states, such as
dipolar crystals with spin-valley spinors in van der Waals heterostructures
[7, 8]. The emergence of strongly correlated phases of matter, such as the
crossover from the Tonks-Girardeau phase to the dipolar crystal, is originated
from dipolar interactions between spatially indirect excitons [9]. The ability
to stack and manipulate multiple layers of 2D materials also presents
opportunities for developing advanced devices and novel functionalities,
paving the way for quantum photonics technologies and solid-state-based
quantum simulators [10].

Figure 1: Sketch of the system. Blue arrows depict dipolar atoms from the bath
confined to the $x-y$ plane and red arrow shows the impurity. An external
magnetic field is used to polarize all atoms and the impurity in the same
direction, forming the tilting angle $\alpha$ with the respect to the
direction normal to the plane. The azimuthal angle $\theta$ encodes the
anisotropy of the system.

Recent studies on Bose polarons in ultracold dipolar gases [11, 12, 13, 14,
15], and the attainment of Bose-Einstein condensates (BEC) and degenerate
quantum gases of atoms with large magnetic dipole moments, such as
$\mathrm{Cr}$, $\mathrm{Er}$, and $\mathrm{Dy}$, have garnered significant
attention in the field of low-temperature physics due to their dominant
dipolar interactions. Furthermore, the recent experimental advances achieved
with mixtures of species with large magnetic moments [16, 17], and with
ultracold bosonic polar molecules [18, 19, 20, 21, 22] have attracted
additional interest in dipolar systems. The study of polarons in 2D ultracold
dipolar gases is an active area of research as the quasiparticles that arise
when an impurity particle interacts with a many-body quantum environment can
exhibit unique properties in the presence of anisotropic dipolar interactions
[23, 24, 25]. These features include an anisotropic effective mass, which may
lead to unusual transport properties in 2D dipolar systems.

The ability to easily control the geometry of ultracold gases using trapping
potentials, and the presence of a rich number of Feshbach resonances [26] in
dipolar atomic species have opened up new perspectives for the realization of
dipolar polarons in constrained geometries. The polaron energy can be measured
using standard ejection radio-frequency spectroscopy, similar to the method
used for neutral atoms [27]. Additionally, the quasiparticle residue of the
polaron wavefunction overlap can be extracted from interferometric
measurements. The ability to manipulate the polarization field provides an
additional degree of control. By applying a magnetic or electric field, the
direction of carrier transport can be controlled, which has the potential to
provide insights into the effects of anisotropy in the effective mass of the
impurity.

In this work we study the ground-state properties of a dipolar polaron
immersed in a two-dimensional dipolar medium, where all particles (the
impurity and the ones in the bath) are polarized along the same direction in
space. The dipolar interaction depends on the angle $\alpha$ formed by the
polarization field and the $z$-axis, and the dipolar strength $C_{dd}$.
Denoting by I and B the impurity and a background atom, respectively, the
dipolar interaction reads

$\displaystyle
V_{\sigma\sigma^{\prime}}(\mathbf{r}_{ij})=C_{dd}^{\sigma\sigma^{\prime}}\left({1-3\sin^{2}\alpha\cos^{2}\theta_{ij}\over
r_{ij}^{3}}\right)\ ,$ (1)

where $\sigma,\sigma^{\prime}\in\\{I,B\\}$. In this expression
$\mathbf{r}_{ij}=\mathbf{r}_{i}-\mathbf{r}_{j}$ is the in-plane relative
position vector between any pair of atoms, while $r_{ij}=|{\bf r}_{i}-{\bf
r}_{j}|$ and $\theta_{ij}$ are the corresponding distance and relative
orientation angle, respectively, as shown in Fig. 1. Furthermore,
$C_{dd}^{\sigma\sigma^{\prime}}=6\pi\hbar^{2}\sqrt{d_{\sigma}d_{\sigma^{\prime}}}/\mu^{\sigma\sigma^{\prime}}$,
with $d_{\sigma}=m_{\sigma}C_{dd}^{\sigma\sigma}/(12\pi\hbar^{2})$ the
corresponding dipolar length, and
$\mu^{\sigma\sigma^{\prime}}=m_{\mathrm{\sigma}}m_{\mathrm{\sigma^{\prime}}}/(m_{\mathrm{\sigma}}+m_{\mathrm{\sigma^{\prime}}})$
the reduced mass between two atoms. The expression in Eq. (1) shows that the
dipolar interaction is anisotropic and depends on both the polarisation angle
$\alpha$ and the interaction strength $C_{dd}$, both of which determine the
scattering length. The opposite is also true, namely, that a given a set of
particles feel a different dipolar interaction when either $\alpha$ or the
scattering length is changed.

It is well known that all short-range interactions become universal in the low
gas parameter limit once properly scaled with the scattering length [28, 29,
30, 31]. In a previous work [32] we showed that beyond this universal limit, a
bulk dipolar system of bosons in two dimensions follows additional scaling
laws in terms of the scattering length and polarisation angles up to unusually
large values of the gas parameter and $\alpha$, even when the system is
strongly correlated. In this work we extend that study to the problem of one
dipolar impurity immersed in a background of bosonic dipolar atoms in two
dimensions. In order to do that we use the Diffusion Monte Carlo (DMC) method,
which is known to provide statistically exact observables for the ground state
of a many-body system of bosons. In particular, we calculate the polaron
energy, pair correlation functions, effective mass and quasiparticle residue,
and analyze how these quantities change with the polarization angle and
overall strength of the interaction. With this, we aim to gain a deeper
understanding of the behavior of dipolar polarons in dipolar baths and the
underlying physical mechanisms at play.

##  II System

The system consists of a single impurity interacting with a background of
bosonic dipoles in two dimensions at fixed density $n$ and at zero
temperature, described by the Hamiltonian

$\mathcal{H}=-\frac{\hbar^{2}}{2m_{B}}\sum_{i=1}^{N}\nabla_{i}^{2}-\frac{\hbar^{2}}{2m_{I}}\nabla_{I}^{2}+\sum_{i<j}V_{\mathrm{BB}}\left({\bf
r}_{ij}\right)+\sum_{i=1}^{N}V_{\mathrm{IB}}\left({\bf r}_{iI}\right)\ .$ (2)

The first two terms in this expression represent the kinetic energy of the
host bath and the impurity, while the last ones correspond to the dipolar
interactions between the background atoms and with the impurity, respectively.

In the following we consider the impurity and background bosons to have the
same mass, so we set $m_{\mathrm{I}}=m_{\mathrm{B}}=m$. This assumption is
well suited, for example, when we consider the atoms in the bath and the
impurity to be different isotopes of the same highly dipolar, heavy atom
isotopes as could be 162Dy and 164Dy. This is also a realistic assumption when
the impurity and the background particles correspond to the same isotopes, but
in different hyperfine states. We also restrict the analysis to tilting angles
$\alpha\in[0,0.615]$, as for larger values the system collapses in the absence
of additional hard core repulsive forces.

Within this model, the $s$-wave scattering length for both the {I,B} and {B,B}
interaction pairs become [33]

$a_{\sigma\sigma^{\prime}}(\alpha,C_{dd}^{\sigma\sigma^{\prime}})\simeq\frac{mC_{dd}^{\sigma\sigma^{\prime}}}{4\pi\hbar^{2}}\exp(2\gamma)\left(1-\frac{3\sin^{2}\alpha}{2}\right)$
(3)

where $\gamma$ is the Euler-Mascheroni constant $\gamma=0.577\cdots$. These
values fix another relevant parameter of the system,
$\beta=C_{dd}^{\mathrm{IB}}/C_{dd}^{\mathrm{BB}}=a_{\mathrm{IB}}/a_{\mathrm{BB}}$,
which sets the relative strength between the impurity-background and the
background-background interactions. In this way, the system properties are
governed by $\alpha$, $\beta$ and the background gas parameter
$x=na_{\mathrm{BB}}^{2}$.

In the subsequent DMC simulations we use a trial wave function of the Jastrow
form

$\psi_{T}(\mathbf{R})=\prod_{i=1}^{N}f_{\rm
IB}\left(\mathbf{r}_{iI}\right)\prod_{i<j}f_{{\rm
BB}}\left(\mathbf{r}_{ij}\right)\ ,$ (4)

where the two-body correlation factors $f_{\mathrm{IB}}$ (impurity-background)
and $f_{\mathrm{BB}}$ (background-background) are obtained from the solution
of the zero-energy two-body problem, as done in previous works [32, 33]. These
functions have been matched with suitable large-distance phononic tails in
order to recover the proper behavior of the many-body wave function.

Figure 2: (a) Rescaled polaron energy $\varepsilon(\alpha)/\varepsilon(0)$,
with $\varepsilon(\alpha)=E_{p}(\alpha)/(\hbar^{2}/ma_{{\rm BB}}^{2}(\alpha))$
and $a_{\mathrm{{\rm BB}}}$ the boson-boson scattering length, as a function
of the tilting angle $\alpha$ for different values of the gas parameter and
$\beta=10$. (b) Ratio between the DMC ($E_{p}$) and the first order
perturbation theory ($E_{p}^{(0)}$) polaron energies for $\beta=1.42$ and
$\beta=10$. (c) Polaron energy as a function of the impurity-boson coupling
$\beta$ for different values of the gas parameter. Energies have been rescaled
with respect to its corresponding value at $\beta=1$.

##  III Results

###  III.1 Polaron energy

The driving quantity in any DMC calculation is the ground state energy, which
therefore becomes in a natural way the first property to analyze. For a dilute
system, the mean field prediction for the polaron energy [34]

$E_{p}^{(0)}=-\frac{4\pi n\hbar^{2}}{m\ln(na_{\mathrm{IB}}^{2})}\ ,$ (5)

is expected to hold, irrespective of the details of the interaction. In the
present case, $a_{{\rm IB}}$ and $a_{{\rm BB}}$ present the same dependence on
the polarization angle, according to Eq. (3). Consequently, for fixed impurity
and bath, the ratio $a_{{\rm IB}}/a_{\rm BB}=\beta$ remains constant when
$\alpha$ changes, and the product $E_{p}^{(0)}(\alpha)a^{2}_{\rm BB}(\alpha)$
becomes a function of the gas parameter alone. This same behaviour is
displayed by the ratio of energies
$\varepsilon^{(0)}(\alpha)=E_{p}^{(0)}(\alpha)/(\hbar^{2}/ma_{\rm
BB}^{2}(\alpha))$. Being a function of the gas parameter alone, for a fixed
$x=na_{\mathrm{BB}}^{2}$, $\varepsilon^{(0)}(\alpha)$ is independent of the
polarization angle and thus $\varepsilon^{(0)}(\alpha)/\varepsilon^{(0)}(0)$
equals one for all values of $\alpha$.

Considering this property emerges from Eq. (5), it is in principle expected to
hold only at low $x$. Figure 2(a) displays the ratio
$\varepsilon(\alpha)=E_{p}(\alpha)/(\hbar^{2}/ma_{\rm BB}^{2}(\alpha))$ for
$\beta=10$ and different values of $\alpha$ and $x$, computed from the DMC
polaron energies $E_{p}(\alpha)$. This ratio has been rescaled with respect to
its result for $\alpha=0$ for visualization purposes, such that the curves for
all gas parameters start from unity. As it can be seen, the plot suggest an
almost perfect universal behavior for titling angles $\alpha\lesssim 0.4$.
Surprisingly, this trend holds even for values of the gas parameter as large
as $x=100$, for which the short-range details of the interaction potential
would be expected to have a huge impact. Moreover, for higher tilting angles,
the deviation from a perfect universal behavior is of the order of $5\%$,
meaning that even in this regime, universality is mostly preserved. These
results align with the previous findings for the bulk system, where a similar
universal behavior is observed [32].

Given this trend displayed by the polaron energy and the analytic expression
of the scattering length in Eq. (3), the dipolar polaron energy can be
considered to be a function of $\beta$ that can be obtained from a fit to the
corresponding Monte Carlo data. We have obtained these fits for $\alpha=0$ and
two relevant values of the coupling strength: $\beta=1.42$, corresponding to
the case of a Dy impurity immersed in an Er bath, and the extreme case of a
strongly interacting impurity, $\beta=10$. We have checked that the polaron
energy follows a law of the form

$E_{p}(\alpha=0)=\exp(a(\log(x)+c)^{d}+b)\epsilon_{d}\ ,$ (6)

with $a=0.94(8)$, $b=-16.30(1)$ for $\beta=1.42$ and $a=0.97(4)$,
$b=-15.79(4)$ for $\beta=10$. In both cases, $c=11.99(3)$, and $d=1.09(3)$.
Also, $\epsilon_{d}=\hbar^{2}/md_{B}^{2}$. In all cases the error of the fit
is fairly small, being at most slightly less than $10\%$.

A relevant question related to the previous results is the extent to which a
perturbative approximation accurately describes the ground state energy of the
system for the dipolar polaron. In a perturbative scheme, the bath is usually
described by a Bogoliubov Hamiltonian in the absence of the impurity, while
the impurity-bath interaction is considered to be the (weak) perturbation. For
the two-dimensional dipolar system considered in this work, the boson-boson
and impurity-boson interactions in momentum space are taken to be described by
the pseudopotentials [35]

$\displaystyle V^{(p)}_{\sigma\sigma^{\prime}}({\bf k})$
$\displaystyle=-\frac{4\pi\hbar^{2}}{m\log(na_{\sigma\sigma^{\prime}}^{2})}+\frac{C_{dd}^{\sigma\sigma^{\prime}}k\sin^{2}\alpha\cos
2\theta_{k}}{2}$ (7)

with $\theta_{k}$ the polar angle of the momentum vector. This pseudopotential
is built so as to guarantee that both the $s$ and $d$-wave scattering
properties are properly accounted for. Note also that this pseudopotential
incorporates finite-range effects via the anisotropic contribution in the
second term. Within perturbation theory and using the Fröhlich Hamiltonian,
one considers only processes where the impurity couples to a single excitation
of the medium at once. In order to quantify the accuracy of the perturbative
approach, we show in Fig. 2(b) the ratio of the DMC energies to the lowest
order perturbation prediction $E_{p}^{(0)}$ of Eq. (5).

Notice that, at this order, the polaron energy obtained with the
pseudopotential in Eq. (7) is equal to that obtained for contact interactions.
As it can be seen from the figure and as expected, the perturbative
approximation holds only in the dilute limit, corresponding to gas parameter
values $x\lesssim 0.01$ for $\beta=1.42$ and $x\lesssim 0.001$ for $\beta=10$.
For larger values of $x$, higher-order effects, neglected in the lowest-order
perturbative scheme, start to become important.

Finally, Fig. 2(c) shows the dependence of the polaron energy on the coupling
ratio $\beta$ for $\alpha=0$ and several values of the gas parameter. Energies
have been rescaled with respect to their values at $\beta=1$ for the sake of
comparison, which correspond to $E_{p}(x=10^{-5})=1.25\times
10^{-7}\epsilon_{d}$, $E_{p}(x=10^{-2})=3.92\times 10^{-4}\epsilon_{d}$ and
$E_{p}(x=10^{2})=58.18\epsilon_{d}$. We find that the relative variation of
the polaron energy grows with increasing gas parameter. This is a consequence
of the fully repulsive character of the dipole-dipole interaction and the fact
that, for a fixed polarization angle, increasing the value of the gas
parameter is equivalent to increasing the density of atoms of the bath.

Figure 3:  $s$\- and $d$\- partial wave modes of the impurity-boson pair-
distribution function for three characteristic values of the gas parameter:
$x=0.001$ (top row), $x=1$ (middle row), $x=100$ (bottom row). The impurity
strength ratio is fixed to $\beta=10$.

Similarly to what is done in the analysis of the dipolar bulk case, additional
insight into the dipolar polaron universality can be drawn from the relation
between the polaron energy and the boson-boson and impurity-boson pair
distribution functions. These quantities are defined as

$\displaystyle g_{\mathrm{BB}}({\bf r_{1}}-{\bf r_{2}})$ $\displaystyle=$
$\displaystyle\frac{N(N-1)}{n^{2}}\frac{\int d{\bf r_{3}}\cdots d{\bf
r_{N}}d{\bf r_{I}}\absolutevalue{\Psi({\bf R})}^{2}}{\int d{\bf
R}\absolutevalue{\Psi({\bf R})}^{2}}$ (8) $\displaystyle g_{\mathrm{IB}}({\bf
r_{I}}-{\bf r_{1}})$ $\displaystyle=$ $\displaystyle\frac{N}{n_{I}n}\frac{\int
d{\bf r_{2}}\cdots d{\bf r_{N}}\absolutevalue{\Psi({\bf R})}^{2}}{\int d{\bf
R}\absolutevalue{\Psi({\bf R})}^{2}}\ ,$ (9)

with ${\bf R}$ representing the set of all particle coordinates, $n=N/V$ and
$n_{I}=1/V$ being the average bath and impurity density, respectively.
Actually, these two functions can be expanded in partial waves and, due to the
anisotropy of the dipolar interaction, they present non-zero contributions
beyond the s-wave. We show in Figure 3 the first two modes of

$g_{\mathrm{IB}}({\bf r_{I}}-{\bf
r_{1}})=\sum_{l=0}^{\infty}g^{(2l)}_{\mathrm{IB}}(r)\cos 2l\theta$ (10)

for $\beta=10$ and different gas parameters and tilting angles. Results for
$g_{\mathrm{BB}}({\bf r_{1}}-{\bf r_{2}})$ are very similar to those obtained
for the dipolar bulk case in Ref. [32]. As it can be seen from the figure, the
isotropic mode is universal up to $\alpha\simeq 0.4$ for abnormally large
values of the gas parameter, while the first anisotropic mode does not show
any universality at all. However, we also see that, up to $\alpha\simeq 0.4$,
the isotropic mode clearly dominates over the anisotropic contribution unless
both $\alpha$ and $x$ are large, leading to an essentially universal
behaviour, similarly to what it was reported in [32] for the bulk. The pair
distribution functions is related to the potential energy of the system by the
relation

Figure 4:  Second order perturbation theory results for the quasiparticle
residue $Z$ for a bath with (purple dots) and without (green solid line) the
finite range, anisotropic contribution of the boson-boson interactions (see
Eq. 7). In both cases, $\alpha=0.6$ for the impurity-boson interaction.

$\displaystyle\langle V\rangle=n\int d{\bf r}V_{\mathrm{IB}}({\bf
r})g_{\mathrm{IB}}({\bf r})+\frac{nN}{2}\int d{\bf r}V_{\mathrm{BB}}({\bf
r})g_{\mathrm{BB}}({\bf r})\ ,$ (11)

where the first term comes from the interaction between the impurity and the
rest of the particles in the medium, while the second term accounts for the
contribution of the bath. From this expression, one can recover the total
energy of the system through the Hellmann-Feynman theorem [32, 36]

$\displaystyle E$ $\displaystyle=\int_{0}^{1}du\left\\{n\int d{\bf
r}V_{\mathrm{IB}}({\bf r})g_{\mathrm{IB}}({\bf r},u)\right.$
$\displaystyle\left.+\frac{nN}{2}\int d{\bf r}V_{\mathrm{BB}}({\bf
r})g_{\mathrm{BB}}({\bf r},u)\right\\}$ (12)

where $g_{\mathrm{IB}}({\bf r},u)$ and $g_{\mathrm{BB}}({\bf r},u)$ stand the
pair distribution functions corresponding to the Hamiltonian
$\hat{H}=\hat{H}_{kin}+\hat{H}_{pot}u$, with $\hat{H}_{kin}$ and
$\hat{H}_{pot}$ the kinetic and potential terms of the Hamiltonian in Eq. (2),
respectively. The polaron energy can then be recovered from the energy
difference

$\displaystyle E_{p}=E(N,1)-E(N,0)=\int_{0}^{1}du\left\\{n\int d{\bf
r}V_{\mathrm{IB}}({\bf r})g_{\mathrm{IB}}({\bf r},u)\right.$
$\displaystyle\left.+\frac{nN}{2}\int d{\bf r}V_{\mathrm{BB}}({\bf
r})g_{\mathrm{BB}}({\bf r},u)\right\\}-\frac{nN}{2}\int d{\bf
r}V_{\mathrm{BB}}({\bf r})\tilde{g}_{\mathrm{BB}}({\bf r},u)$ (13)

where $E(N_{B},N_{I})$ denotes the ground-state energy of a system with
$N_{B}$ bosons and $N_{I}$ impurities, and $\tilde{g}_{\mathrm{BB}}({\bf
r},u)$ is the boson-boson pair distribution function of the bulk system (i.e.
for absent impurity). In this way, the universality in the polaron energy can
be understood as being inherited from the corresponding behavior of the pair
distribution functions.

Figure 5:  (a) DMC results for the quasiparticle residue as a function of the
tilting angle for several values of the gas parameter for $\beta=10$. (b) DMC
results (dots) and perturbative results (solid lines) for $Z$ obtained for
$\alpha=0$ as a function of the gas parameter.

###  III.2 Quasiparticle residue

Another experimentally relevant quantity in the study of the polaron physics
is the quasiparticle residue $Z$, which quantifies the overlap between the
full wave function of the system and a state conformed by a non-interacting
impurity and a vacuum of excitations. A mixed estimator for this quantity can
be obtained in DMC from the long-range asymptotic behaviour of the one-body
density matrix associated to the impurity [34]

$Z=\lim_{r\rightarrow\infty}\rho(\mathbf{r})=\lim_{r\rightarrow\infty}\left\langle\frac{\psi_{T}(\mathbf{r}_{\mathrm{I}}+\mathbf{r},\mathbf{r}_{1},\cdots,\mathbf{r}_{N})}{\psi_{T}(\mathbf{r}_{\mathrm{I}},\mathbf{r}_{1},\cdots,\mathbf{r}_{N})}\right\rangle$
(14)

where $\psi_{T}$ is the many-body trial wave function guiding the simulation.
We report in Fig. 5(a) the dependence of $Z$ on the polarization angle
$\alpha$ for different values of the gas parameter and $\beta=10$. As can be
seen, $Z$ is also independent of $\alpha$ and seems to depend on the gas
parameter exclusively, even for the largest values of $x$ where inter-atomic
correlations play an important role. This surprising property can be hinted
already at the perturbative level using the simple model described above. To
second order and using the interaction in Eq. (7) one finds

$\displaystyle Z^{(2)}=\left(1+\frac{n}{(2\pi)^{2}}\int d{\bf
k}(V^{(p)}_{\mathrm{IB}}({\bf k}))^{2}\frac{\epsilon_{{\bf k}}}{E({\bf
k})}\frac{1}{\left(\epsilon_{{\bf k}}-E({\bf k})\right)^{2}}\right)^{-1}$ (15)

with $\epsilon_{{\bf k}}=\frac{\hbar^{2}k^{2}}{2m}$ and $E({\bf
k})=\sqrt{\epsilon_{{\bf k}}\left(\epsilon_{{\bf
k}}+2nV^{(p)}_{\mathrm{BB}}({\bf k})\right)}$ the excitation spectrum of the
bulk. Interestingly, getting rid of the anisotropic contribution to the bath-
bath pseudopotential (second term on the rhs of Eq. (7)) that enters $Z$
through the excitation spectrum of the medium leaves the quasiparticle residue
essentially unchanged. This is shown in Figure 4, where we compare the values
of $Z$ obtained with and without this contribution for different values of the
gas parameter and $\alpha=0.6$. In this way, the only relevant anisotropic
contribution to $Z$ comes from the impurity-bath interaction. However, the
lowest order anisotropic contribution is proportional to $\sin^{4}\alpha\ll
1$, since the term proportional to $\sin^{2}\alpha\cos 2\theta_{k}$ coming
from $(V^{(p)}_{\mathrm{IB}}({\bf k}))^{2}$ yields zero contribution when the
angular integration is evaluated. This means that, at the perturbative level,
the dependence of $Z$ on the density and the impurity-bath scattering length
is the same as that of an isotropic system with zero range interactions, and
thus $Z$ is a function of the gas parameter alone [34].

At this point, one can also compare the DMC prediction of the quasiparticle
residue to the results obtained with perturbation theory. We show in Fig. 5
(b) the DMC estimation of $Z$ together with the perturbative result obtained
from Eq. (14) for the two cases $\beta=1.42$ and $\beta=10$ and $\alpha=0$.
Because of universality, the same results hold for any other value of
$\alpha$. As it can be seen, the predictive power of the perturbative approach
worsens with increasing $x$ and/or $\beta$, as happens with the polaron
energy. In this case, though, the situation is worse as the perturbative
prediction ceases to reproduce the DMC data at lower gas parameter values, at
least for $\beta=10$. The results in Fig. 5 (b) are also useful to delimit the
regime of validity of the quasiparticle picture, which requires $Z$ to be
close to unity. Remarkably, for the lowest coupling case, this regime extends
up to $x\lesssim 10^{-2}$.

Figure 6: DMC (dots) and second order perturbation theory (solid line) results
for the inverse effective mass as a function of the gas parameter for
$\alpha=0.6$, $\beta=1.42$. DMC results correspond to $m/m_{x}^{*}\simeq
m/m_{y}^{*}$ since, in this regime, the DMC estimations of $m_{x}^{*}$ and
$m_{y}^{*}$ are indistinguishable within statistical noise.

###  III.3 Effective mass

The last quantity we address in this work is the polaron’s effective mass. In
order to obtain the effective mass in DMC, one can track the diffusion
movement of the polaron in imaginary time. This is done calculating its mean-
square displacement according to the expression
$\frac{m}{m^{\ast}}=\lim_{\tau\rightarrow\infty}\frac{\left\langle\left|\Delta\mathbf{r}_{\mathrm{I}}(\tau)\right|^{2}\right\rangle}{4D\tau}$
[34, 37]; with $D=\hbar^{2}/(2m)$ the diffusion constant of a free particle,
and $\langle\left|\Delta{\bf
r}_{\mathrm{I}}(\tau)\right|^{2}\rangle=\langle\left|{\bf
r}_{\mathrm{I}}(\tau)-{\bf r}_{\mathrm{I}}(0)\right|^{2}\rangle$,
$\tau=it/\hbar$ being the imaginary time of the simulation. Due to the
anisotropic character of the dipolar interaction, the effective mass turns out
to depend on the impurity’s momentum direction for $\alpha\neq 0$. In order to
quantify this effect, one can define an anisotropic effective mass by tracking
the position of the impurity in each direction separately, according to
$\frac{m}{m_{\chi}^{\ast}}=\lim_{\tau\rightarrow\infty}\frac{\left\langle\left|\Delta\chi_{\mathrm{I}}(\tau)\right|^{2}\right\rangle}{2D\tau}$
with $\chi=x\text{ or }y$. Notice that the previous expression contain a
factor of $2$ instead of $4$ in the denominator, as in this case the diffusion
is treated as one-dimensional.

Figure 7: DMC Results for the effective mass for $x=1$ (a) and $x=100$ (b) as
a function of the tilting angle. In both cases, $\beta=10$.

In any case and as done before, it is interesting to discuss first the
predictions of perturbation theory. To second order, the effective mass is
obtained from the second-order polaron energy

$\displaystyle E^{(2)}_{p}=\frac{\bra{{\bf
P},0}\hat{H}_{\mathrm{IB}}\ket{\Psi_{1}}}{\bra{\Psi}\ket{\Psi}}$ (16)

where $\bra{{\bf P},0}$ denotes a state with a non-interacting impurity with
momentum ${\bf P}$ and a zero-momentum medium. In much the same way,
$\ket{\Psi}=\ket{{\bf P},0}+\lambda\ket{\Psi_{1}}$ with $\lambda$ a
perturbative parameter proportional to the strength of the impurity-medium
interaction and $\ket{\Psi_{1}}$ the first order contribution to the total
wave function accounting for the perturbation. Evaluating Eq. (16) and
performing a Taylor expansion in terms of the impurity momentum, the $P^{2}$
contribution results in the following momentum-dependent correction to the
polaron energy

$\displaystyle\Delta E=-\frac{P^{2}}{2m}\frac{1}{(2\pi)^{2}}\int d{\bf
k}\text{ }n\left(V^{(p)}_{\mathrm{IB}}({\bf k})\right)^{2}\frac{\epsilon_{\bf
k}}{E_{\bf k}}\frac{2\hbar^{2}k^{2}}{\left(\epsilon_{\bf k}+E_{\bf
k}\right)^{3}}\cos^{2}\theta$ (17)

where $\theta$ is the angle between the impurity momentum ${\bf P}$ and the
integration momentum $\hbar{\bf k}$, which forms an angle $\phi$ with the $x$
axis. The effective mass is then obtained from the momentum-dependent part of
the polaron energy, which is given by the sum of impurity kinetic energy and
the correction in Eq. (17), i.e.

$\displaystyle\frac{P^{2}}{2m^{*}}$
$\displaystyle=\frac{P^{2}}{2m}\left(1-\frac{1}{(2\pi)^{2}}\int d{\bf k}\text{
}n\left(V^{(p)}_{\mathrm{IB}}({\bf k})\right)^{2}\right.$
$\displaystyle\left.\times\frac{\epsilon_{\bf k}}{E_{\bf
k}}\frac{2\hbar^{2}k^{2}}{\left(\epsilon_{\bf k}+E_{\bf
k}\right)^{3}}\cos^{2}\theta\right)\ .$ (18)

This result indicates that the inclusion of the second term in the
pseudopotential of Eq. (7) leads to an anisotropic effective mass, induced by
the angular dependence on the impurity’s momentum vector. In particular, for
an impurity moving along the $x$ axis, $m^{*}$ is replaced by $m^{*}_{x}$ in
Eq. (18) and this results into the substitution
$\cos^{2}\theta\rightarrow\cos^{2}\phi$ while for a impurity moving along the
$y$ axis, computing $m^{*}_{y}$ implies setting
$\cos^{2}\theta\rightarrow\sin^{2}\phi$. Furthermore and as happens with the
quasiparticle residue, the anisotropy of the boson-boson interactions that
enters Eq. (18) through the excitation spectrum of the background has little
to no impact on $m^{*}$. Since, as usual, the second order correction to the
polaron energy is negative, $m^{*}>m$ and the impurity acts as a heavier
quasiparticle in the medium.

Figure 6 shows the ratios $m/m_{x}^{*}$ and $m/m_{y}^{*}$ obtained with second
order perturbation theory for $\beta=1.42$ and $\alpha=0.6$ as a function of
the gas parameter. As one can see, the effective mass is larger when the
polaron moves along the $x$ axis. This is because the anisotropy in the
effective mass is determined by the anisotropy of the impurity-bath
interaction in momentum space, which is maximally repulsive along this
direction. The corresponding DMC predictions for the effective mass in the $x$
(or $y$) axis are also shown in the plot. Because the noise in the estimation
in the effective mass is large, it prevents a clear observation of its
anisotropic character but at large enough gas parameters and impurity-bath
coupling strengths. This implies that, in this regime, the DMC results for the
effective mass along the $x$ and $y$ axes are indistinguishable up to
statistical noise, which is why only a set of points is shown. This is an
issue even when long simulations, which accumulate a large quantity of
statistical data, are performed. Regardless, we see agreement between the
perturbative and DMC results in the regime where $m_{x}^{*}\simeq m_{y}^{*}$.

The anisotropic character of the effective mass is more clearly seen when
correlations are strong. In order to showcase that, we show in Fig. 7 the DMC
results for $m/m_{x}^{*}$ and $m/m_{y}^{*}$ as a function of the polarization
angle for $\beta=10$ and two values of the gas parameter, $x=1$ and $x=100$.
We can see that, even away from the regime of validity of perturbation theory
($\beta\gg 1$), anisotropic effects in the effective mass follow the
qualitative trends predicted by the perturbative calculation, showing indeed
that $m_{x}^{*}>m_{y}^{*}$.

##  IV Conclusions

To summarize, we have studied the quasiparticle properties of a dipolar
impurity immersed in a dipolar bath in two dimensions, both being subject to
an external polarization field that makes all dipole moments point in the same
direction. In order to do that, we have used the Diffusion Monte Carlo (DMC)
method, comparing the results to second order perturbation theory. We have
shown that, to a large extent, the polaron energy is a universal function of
the gas parameter alone, independent of the precise values of the density and
polarization angle. This is directly induced by the universality of the pair-
distribution function, where the isotropic mode dominates. We have also shown
that the quasiparticle residue is also a universal function of the gas
parameter, a result that is recovered through perturbation theory even when
anisotropic finite range effects are considered. Finally, we have shown that
the anisotropy of the dipole-dipole interaction leads to an anisotropic
effective mass which, surprisingly, is larger in the direction of minimum
repulsion of the dipole-dipole interaction in position space, which is a
consequence of its angular dependence in momentum space. For all the
aforementioned properties, we have established the regime of validity of
perturbation theory in terms of the gas parameter by direct comparison to the
DMC results.

###### Acknowledgements.

The work has been supported by grant PID2020-113565GB-C21 from
MCIN/AEI/10.13039/ 501100011033, and by the Danish National Research
Foundation through the Center of Excellence “CCQ” (Grant agreement no.:
DNRF156). J. Sánchez-Baena acknowledges funding by the European Union, the
Spanish Ministry of Universities and the Recovery, Transformation and
Resilience Plan through a grant from Universitat Politecnica de Catalunya.
L.A.P.A acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy– EXC-2123
QuantumFrontiers– 390837967, and FOR 2247. L.A.P.A also acknowledges by the
PNRR MUR project PE0000023 - NQSTI and the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy \- EXC -
2123 Quantum Frontiers-390837967 and FOR2247

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