# Exploring phonon-like interactions in one-dimensional Bose-Fermi mixtures

Axel Gagge  Department of Physics, Stockholm University, SE-106 91 Stockholm,
Sweden  Th. K. Mavrogordatos  Department of Physics, Stockholm University,
SE-106 91 Stockholm, Sweden  ICFO – Institut de Ciències Fotòniques, The
Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain  Jonas Larson<EMAIL_ADDRESS>Department of
Physics, Stockholm University, SE-106 91 Stockholm, Sweden

###### Abstract

With the objective of simulating the physical behavior of electrons in a
dynamic background, we investigate a cold atomic Bose-Fermi mixture confined
in an optical lattice potential solely affecting the bosons. The bosons,
residing in the deep superfluid regime, inherit the periodicity of the optical
lattice, subsequently serving as a dynamic potential for the polarized
fermions. Owing to the atom-phonon interaction between the fermions and the
condensate, the coupled system exhibits a Berezinskii-Kosterlitz-Thouless
transition from a Luttinger liquid to a Peierls phase. However, under
sufficiently strong Bose-Fermi interaction, the Peierls phase loses stability,
leading to either a collapsed or a separated phase. We find that the primary
function of the optical lattice is to stabilize the Peierls phase.
Furthermore, the presence of a confining harmonic trap induces a diverse
physical behavior, surpassing what is observed for either bosons or fermions
individually trapped. Notably, under attractive Bose-Fermi interaction, the
insulating phase may adopt a fermionic wedding-cake-like configuration,
reflecting the dynamic nature of the underlying lattice potential. Conversely,
for repulsive interaction, the trap destabilizes the Peierls phase, causing
the two species to separate.

Bose-Fermi mixture, Peierls instability, harmonic trap, lattice potential

###### pacs:

05.30.Rt, 42.50.Ct, 75.10.Kt

##  I Introduction

Over the past decades, we have witnessed a rampant growth of experimental
methods devised to cool and control dilute gases. The attainment of Bose-
Einstein condensates (BECs) Anderson et al. (1995) was soon to be followed by
in-depth explorations of BEC dynamics in light-induced periodic potentials
Morsch et al. (2001); Denschlag et al. (2002), paving the way for the
groundbreaking demonstration of the Mott-superfluid phase transition. This
exemplary quantum phase transition was predicted by the Bose-Hubbard model
Fisher et al. (1989); Jaksch et al. (1998) for an atomic gas loaded into an
optical lattice Greiner et al. (2002). Since then, trapped and cooled dilute
atomic gases have developed into a versatile laboratory where quantum matter
can be studied in a controlled and detailed fashion, constituting one of the
most promising platforms for the realization of analog quantum simulators
Lewenstein et al. (2007); Bloch et al. (2008); Daley et al. (2022).

Many of the open questions in the field of quantum phase transitions,
especially those revolving around the formation and characterization of exotic
quantum phases of matter, cannot, however, be directly addressed by the Bose-
Hubbard model per se. Although the classical laser field forming the optical
lattice is, in principle, dynamic, the back-action between the trapped
particles (modeled as beam splitters) and the lattice is typically very weak
Asbóth et al. (2007, 2008). One may then, to a very good approximation, treat
the lattice as the outcome of applying a periodic static potential. Certainly,
this approximation would be sufficient to emulate many paradigmatic lattice
models. However, such classical potentials do not take into account any back-
action between the lattice and the conducting matter, which we know is
essential to explain several phenomena like the Peierls distortion Peierls and
Peierls (1955) [see Figs. 1 (a) and (b)] and superconductivity Mahan (2013).
In solids, effects of that kind result from electron-phonon interactions
emerging from the very nature of a dynamical lattice.

A central theme in our analysis is that assigning tractable degrees of freedom
to the lattice renders its description dynamic and enables the simulation of
some analog of phonon-like interactions. One possibility is to couple the
atoms to the light field of an optical resonator Deng et al. (2014); Pan et
al. (2015), where substantial Stark shifts are observed even in the presence
of a few photons. However, in current experiments, only a few modes of the
resonator actively participate in the light-matter interaction, and it is,
therefore, not possible to locally modify the dynamical lattice. Instead, one
must turn to multi-mode cavities Ballantine et al. (2017), which still pose
difficulties in reaching configurations similar to those encountered in real
solids Lewenstein et al. (2006). A viable alternative arises when considering
atoms directly interacting with an ionic crystal Bissbort et al. (2013). Here,
harmonically trapped ions form a Wigner crystal, while additional neutral
atoms move within this lattice. This ion-atom system bears obvious
similarities to a real solid, albeit being experimentally challenging. In
fact, the crystalline structure is not necessary for exploring phonon-like
interactions. In mixtures of different atomic species — either different atoms
or different internal states of the same atomic species — the interplay
between subsystems can lead to intriguing effects.

Coming now to single atomic species, the use of Feshbach resonances allows
experimenters to control the strength, and even the sign, of all involved
interactions Stan et al. (2004); Best et al. (2009); Kawaguchi and Ueda
(2012); Park et al. (2012); Ferrier-Barbut et al. (2014); DeSalvo et al.
(2019), aiming to probe the resulting phase diagram. In mixtures of bosons and
spin-polarized fermions (hereinafter referred to as BF mixtures), it is well
known that an attractive BF interaction leads to a so-called pairing phase in
the strongly correlated regime. This phase has been studied for weak BF
interactions, $g_{bf}$, in one dimension (1D) Cazalilla and Ho (2003);
Miyakawa et al. (2004); Rizzi and Imambekov (2008), as well as in twoBüchler
and Blatter (2003); Klironomos and Tsai (2007) and three dimensions Titvinidze
et al. (2008). The phase in question collapses if the interaction becomes too
strong, resulting in clumping of the atoms and breaking of translational
invariance. The effect of optical lattices on BF mixtures has also been
investigated in Refs. Albus et al. (2003); Mathey et al. (2004); Roth and
Burnett (2004); Salerno (2005); Pazy and Vardi (2005); Bruderer et al. (2007);
Lan and Lobo (2014). For deep lattices and/or very strong interaction, such
systems can be described by a BF Hubbard model as they enter an insulating
phase of composite fermions Lewenstein et al. (2004); Pollet et al. (2006).
The physical behavior is typically described within a Wannier-basis expansion
for both species, where the bosons can be construed as agents of effective
onsite energy shifts for the lighter fermions. The imposed approximations in
such a scheme omit certain back-action between the two subsystems in
comparison to the self-consistent analysis we employ here. Nevertheless, it is
still possible to encounter Peierls phases, supersolids, and charge density
waves Büchler and Blatter (2003); Lewenstein et al. (2004). Furthermore, if
the repulsive boson-boson interaction is weak, the system can enter a regime
of phase separation where the bosons and fermions completely avoid each other
Büchler and Blatter (2004).

Reporting from the experimental front, an early study focused on how the
coherence of an atomic condensate –- held in place by a cubic optical lattice
–- is affected by the presence of fermionic atoms Günter et al. (2006).
Quantum degeneracy for both bosons and fermions was attained in Ref. Ferrier-
Barbut et al. (2014), where the two species were treated on equal footing, a
trend followed in a series of subsequent papers Delehaye et al. (2015); Roy et
al. (2017); DeSalvo et al. (2017); Yao et al. (2016). Further experimental
investigations have also considered a particular regime emerging for a
condensate in weak contact with much lighter fermionic atoms DeSalvo et al.
(2019); Edri et al. (2020); DeSalvo et al. (2017). Here, the fermions
effectively induce a so-called Ruderman–Kittel–Kasuya–Yosida (RKKY) long-range
boson-boson interaction Ruderman and Kittel (1954). For attractive Bose-Fermi
interaction, it was demonstrated that a self-sustained trap may emerge for
those fermions located inside the bosonic condensate DeSalvo et al. (2017), an
effect which may as well lead to the formation of soliton trains DeSalvo et
al. (2019). The fermion-mediated spin-spin interaction in a spinor condensate
has also been recently observed by means of microwave spectroscopy Edri et al.
(2020).

Figure 1: (color online) Schematic description of the Peierls distortion. At a
fermion filling corresponding to a wavenumber $k_{F}$, opening up a gap of the
dispersion at $k_{F}$ will lower the kinetic energy of the fermions, as
depicted in (a). The gap opening results from a lattice distortion of the
dynamical lattice, as shown in (b). Here, the period has been doubled through
shifting every second atom by a distance $\delta$, and consequently, the size
of the Brillouin zone has been halved. Thus, for half filling, we encounter a
realization of the Su-Schrieffer–Heeger (SSH) model, while for other fillings,
the lattice distortion will generate another periodicity. In the normal phase,
the condensate will share the same periodicity as the lattice and
predominantly populate the lattice minima (c). In the Peierls phase, for half
filling, the bosonic density will alternate between every second site (d).
Upon comparison of (b) and (d), we note that the periodicity has been broken
in two different ways: in (b), the densities are held fixed, but the locations
of the sites have been shifted, while in (d), the locations are fixed and the
densities have been altered. The former can be seen as a ‘phase modulation,’
and the latter as an ‘amplitude modulation’ of the densities. Finally, in (e),
we depict the collapsed phase where all atoms, bosons, and fermions have been
compressed to a small region of the lattice.

Carrying on with the thread of recent investigations on collective phenomena
brought about by a coherent interaction between Bose-Fermi atomic clouds, in
this report, we study a 1D atomic BF mixture where only the bosons are subject
to an optical lattice. The motivation for considering such a configuration is
drawn from the resemblance to an actual solid – the optical lattice orders the
bosons in a crystalline structure. Our spin-polarized fermions experience only
the periodic potential arising from the boson atomic density but no externally
imposed static potential, with the occasional exception of a harmonic trap. A
similar idea, although experimentally more challenging, was proposed in Ref.
Johnson et al. (2016), where the crystalline order for the bosons was
established from a rapidly rotating condensate that created a triangular
Abrikosov vortex lattice. We focus on the limit of a weak boson-boson (BB)
interaction and an optical lattice of amplitude $\leq 20E_{R}$ ($E_{R}$ is the
recoil energy), where the bosonic gas is expected to form a superfluid, see
Fig. 1(c). In this regime, we can work within a mean-field approximation for
the condensed bosons, which act as a classical dynamical lattice felt by the
fermions. The coupled system is solved self-consistently such that every back-
action, at the mean-field level, is taken into account (without imposing, for
instance, any single-band nor tight-binding approximations). At this hybrid
mean-field level, the mixture displays a rich phase diagram. For strong BF
interaction, with a coupling rate $g_{bf}$, the system either experiences a
collapse ($g_{bf}\gg 0$), in which the two species overlap and populate only a
small fraction of the lattice, or a separation ($g_{bf}\gg 0$) where instead
the two species avoid each other and populate different parts of the lattice.
These are first-order transitions, even though translational invariance is
spontaneously broken in both cases. For a non-zero interaction beyond a
critical coupling $g_{bf}=g_{bf}^{c}$, an instability occurs such that the
period of the state is different from that of the underlying lattice; for
example, for half-filling, a periodic doubling is found. The system then
transitions from a Luttinger liquid (LL) phase into a Peierls phase via a
Berezinskii-Kosterlitz-Thouless (BKT) transition. The Peierls phase manifests
itself through a non-zero gap $\Delta_{P}$ at the Fermi wavenumber
$k_{\mathrm{F}}$, i.e. $\Delta_{P}\neq 0$ for $g_{bf}$ beyond the critical
interaction strength $g_{bf}^{c}$.

We further present an account of the effects arising due to the presence of a
confining harmonic trap. For a sufficiently strong repulsive boson-fermion
interaction, it has already been demonstrated that, at the mean-field level,
the trap induces a separation of the two species Nygaard and Mølmer (1999).
This instance suggests ruling out a Peierls-like phase when a trap breaks the
translational invariance. Inside a trap, the atomic densities vary in space,
which, in a local density approximation, translates to a local Fermi
wavenumber. A spatially varying wavenumber has a direct impact on the phases,
such as by smoothing the discontinuous transition between the (possible)
Peierls and collapsed/separated phases. Naturally, the presence of a trap
entails a larger number of involved wavenumbers and thereby a broadening of
the Bragg peaks in a time-of-flight detection. Additionally, we find that the
interplay between the trap, the dynamical boson field, and the fermions
results in an insulator bearing a clear fermionic “wedding cake structure”,
which differs from the density profile obtained for a static periodic
potential.

Our narrative is structured as follows. Section II introduces the Bose-Fermi
system under study and the model we employ for its description. There we also
discuss some basic concepts and known results pertaining to the properties of
the involved phases. The numerical methods are developed in Sec. III, while
the associated results are presented in Sec. IV, first for the translationally
invariant case IV.1, and subsequently in the presence of a harmonic trap IV.2,
where we focus on the quantum coherence registered in the one-particle density
matrix and on the decay of spatial correlations. Concluding remarks in Sec. V
close our paper out, while the two appendixes present two possible paths to
the derivation of effective model Hamiltonians taking quantum fluctuations
into account for both species, alongside some details underlying the employed
hybrid mean-field method.

##  II Background: model Hamiltonian and phases

###  II.1 The Hamiltonian

The physical system we study is a mixture of two atomic gases, one bosonic and
one fermionic. Owing to tight transverse confinement (in the sense of
$\omega_{\perp}\gg\omega_{\parallel}$), excited states of the transverse modes
can be neglected and the low-energy physical description is quasi-one
dimensional Das (2003); Miyakawa et al. (2004). To begin with, we will take
the gas to be of infinite extent in the longitudinal direction, and later
consider the effect of a trap. We work with dimensionless quantities, where
energies are scaled by the boson recoil energy, $E_{R}=\hbar^{2}k^{2}/2m_{b}$
with $k$ the lattice wavenumber and $m_{b}$ the atomic boson mass, such that
half the optical lattice wavelength sets the characteristic length scale of
the problem. The behavior of such a system is modeled by the Hamiltonian

$\begin{array}[]{lll}\hat{\mathcal{H}}&=&\displaystyle{\int
dx\;\hat{\psi}_{b}^{\dagger}\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{b}(x)+\frac{g_{b}}{2}\hat{\psi}_{b}^{\dagger}\hat{\psi}_{b}\right)\hat{\psi}_{b}}\\\
\\\ &&\displaystyle{+\int
dx\;\hat{\psi}_{f}^{\dagger}\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+V_{f}(x)+g_{bf}\hat{\psi}_{b}^{\dagger}\hat{\psi}_{b}\right)\hat{\psi}_{f}},\end{array}$
(1)

where $r=m_{f}/m_{b}$ is the mass ratio between the two species, $g_{b}$ is
the effective 1D boson-boson (BB) interaction strength, and $g_{bf}$ is the
effective 1D BF interaction strength. Later on, we will assume the bosons to
be harmonically confined and subjected to an optical lattice, while the
fermions will be harmonically confined but assumed to have negligible
interaction with the optical lattice,

$\displaystyle V_{b}(x)$
$\displaystyle=\tfrac{1}{2}\omega_{b}^{2}x^{2}+V_{0}\sin^{2}(x),$ (2)
$\displaystyle V_{f}(x)$ $\displaystyle=r\tfrac{1}{2}\omega_{f}^{2}x^{2}.$ (3)

The last term on the RHS of Eq. (2) is the optical-lattice potential, also
denoted as $V_{OL}=V_{0}\sin^{2}(x)$, where the distance between potential
minima is $a=\pi$ in dimensionless units. At first, we remove the harmonic
confinement and keep only the periodic optical lattice. To ensure that the
lattice interacts only with the bosonic atoms, the light frequency should be
such that it is not quasi-resonant with any of the dipole transitions –
dictated by selection rules – of the fermionic atoms. The atomic field
operators $\hat{\psi}_{b,f}(x)$ and $\hat{\psi}_{b,f}^{\dagger}(x)$ obey the
standard commutation relations for bosonic and fermionic creation and
annihilation operators. Note that the Hamiltonian commutes with both particle
number operators ($\alpha=b,f$),

$\hat{N}_{\alpha}=\int
dx\;\hat{\psi}_{\alpha}^{\dagger}(x)\hat{\psi}_{\alpha}(x).$ (4)

For further use, we define the average particle number of bosons/fermions as

$\bar{n}_{\alpha}=\lim_{L\to\infty}\frac{1}{L}\int_{-L/2}^{L/2}dx\;\langle\hat{\psi}_{\alpha}^{\dagger}(x)\hat{\psi}_{\alpha}(x)\rangle,$
(5)

and the density $\nu_{\alpha}=a\bar{n}_{\alpha}=\pi\bar{n}_{\alpha}$ as the
filling of bosons/fermions per lattice site.

Hereinafter, we assume that the bosonic gas is dense enough to form a (quasi)
condensed state, but not so dense that the interaction energy becomes
dominant. In terms of the 1D and 3D scattering lengths $a_{s,1D}$ and
$a_{s,3D}$ this implies

$a_{s,3D}\ll\xi\ll a_{s,1D},$ (6)

where the healing length of the condensate is defined as $\xi=(8\pi
a_{s,3D}\rho_{b})^{-1/2}$, with $\rho_{b}$ the condensate density.
Furthermore, our approximations are consistent with limiting our attention to
optical lattice strengths satisfying $V_{0}\lesssim 20E_{R}$. We take the
fermions to be lighter than the bosons, i.e. $r=m_{f}/m_{b}<1$, which is
typically the case for the relevant experiments, e.g. for a Li6-Li7 mixture
Ferrier-Barbut et al. (2014) one has $r=0.86$ and for a Li6-Cs133 mixture
DeSalvo et al. (2017); Repp et al. (2013) one instead gets $r\approx 0.04$. We
will assume $g_{b}>0$ but allow any sign and magnitude of $g_{bf}$. Both the
BB and BF interactions should be possible to vary in the experiment by
deploying Feshbach resonances Stan et al. (2004); Best et al. (2009);
Kawaguchi and Ueda (2012); Park et al. (2012); Ferrier-Barbut et al. (2014).

###  II.2 Pairing phase and Peierls instability

For low bosonic densities, the occurrence of a so-called pairing phase for BF
mixtures with attractive BF interaction is well known. For weak interaction
strength and light fermions, the development of modulated densities can be
understood from the fact that the fermion-mediated long-range interaction
between two bosons is an ultracold atomic analog to the RKKY interaction
Ruderman and Kittel (1954); Mering and Fleischhauer (2010); De and Spielman
(2014). These spatial modulations of the boson density alter the potential
landscape for the fermions, and subsequently instigate back-action between the
two species. The spatial modulations will occur at the wavenumber $2k_{F}$ if
we assume an RKKY interaction from a “flat” background of fermions with a
Fermi wavevector $k_{F}$. In Cazalilla and Ho (2003), a 1D BF mixture without
optical lattice was investigated using bosonization and a transition was
identified between a two-component gapless LL phase and a gapped pairing phase
with periodic density modulations. The transition was found to “flow” towards
the BKT fixed point with the gap opening as

$\Delta\sim\exp(-1/\sqrt{|g_{bf}-g_{bf}^{c}|})\Theta(g_{bf}-g_{bf}^{c}),$ (7)

where $\Theta(x)$ is the Heaviside step function, and $g_{bf}^{c}$ is the
critical Bose-Fermi coupling strength. A study using discretization of the
spatial coordinate and the numerical DMRG method also predicted a stable LL
phase for small negative $g_{bf}$ Rizzi and Imambekov (2008). For repulsive
interaction ($g_{bf}>0$), the aforementioned works found no pairing phase.
However, in Ref. Miyakawa et al. (2004), a pairing phase was indeed reported
upon employing the method of a random phase approximation.

A gap opening at $2k_{F}$ and the corresponding periodic modulation in the
density are reminiscent of the Peierls distortion. As first demonstrated by
Peierls Peierls and Peierls (1955), a metal is unstable towards lattice
distortions, i.e. a small displacement of the atoms. This phenomenon can be
understood by treating the atomic displacements on the mean-field level in the
well-known SSH model. Opening up a gap $\Delta_{P}$ in the fermion dispersion
at the Fermi wavevector $k_{F}$, as depicted in Fig. 1 (a), lowers the total
energy and corresponds to a lattice modulation of wavenumber $k_{P}=2k_{F}$,
as sketched in Fig. 1 (b). There are however also some discrepancies. The gap
$\Delta_{P}$ was derived from a mean-field treatment of the ions, while the
pairing gap $\Delta$ is the gap of excitations in the full quantum system.
Still, it is interesting to observe that the gap in the SSH model also has an
exponential form similar to (7). Moreover, in the original work of Peierls,
the instability was demonstrated for any non-zero interaction Peierls and
Peierls (1955), while the pairing effect arises as a consequence of a phase
transition occurring at finite interaction strength.

An optical lattice is, on intuitive grounds, expected to stabilize the pairing
phase against a collapse, since it renders the bosons less mobile by
generating a larger effective mass. On the other hand, using the same
argument, the lattice may also extend a possible metallic phase to larger
regions in the phase diagram. We may note that for BF mixtures where both
species are subject to an optical lattice, it has been reported that the
pairing phase appears for both repulsive as well as attractive BF interactions
Pazy and Vardi (2005). Furthermore, one could in principle expect a beating
between the involved length scales, the Fermi wavenumber $k_{F}$ and the
optical lattice wavenumber $k$, that could, in principle, give rise to so-
called Devil’s staircase structures Chanda et al. (2022).

A last remark on the periodically modulated Peierls phase is in order. We have
envisioned the boson superfluid as the agent creating a dynamical lattice for
the fermions. We may consider the opposite viewpoint of a condensate living in
a partially dynamic background. Within this perspective, the Peierls phase is
reminiscent of a supersolid Büchler and Blatter (2003); Titvinidze et al.
(2008), i.e., a superfluid state that has spontaneously broken the periodicity
of the Hamiltonian. Somewhat similar scenarios have been studied in atomic
condensates confined within optical resonators Baumann et al. (2010); Léonard
et al. (2017).

###  II.3 Stability of the Bose-Fermi mixture

For sufficiently strong BF interaction, the system is unstable towards long-
wavelength fluctuations. The stability of BF mixtures without an optical
lattice has already been investigated in, e.g., Refs. Das (2003); Miyakawa et
al. (2004). Applying a hydrodynamic (mean-field) description of both fermions
and bosons, a linear stability analysis yields the condition

$\bar{n}_{f}\geq\frac{g_{bf}^{2}}{2g_{f}g_{b}},$ (8)

where the constant $g_{f}=\pi^{2}/r$ appears as an effective fermion
interaction strength. Note that the different definition compared to Ref. Das
(2003) is due to our use of rescaled dimensionless units. The fact that a high
fermion density stabilizes the mixture is particular to the 1D case.
Attractive interaction leads to a collapse of both species, while a repulsive
interaction induces a phase separation where the bosons and fermions either
avoid each other or lump together to form BF soliton mixtures Adhikari (2005);
Tylutki et al. (2016); Salerno (2005). More precisely, for repulsive boson-
fermion interaction, the bosons can form a dark soliton (density dip), filled
by a bright fermionic soliton. For attractive interaction, both species form a
bright soliton. This is reminiscent of the Townes solitons predicted for
electromagnetic waves Chiao et al. (1964), and recently demonstrated in
ultracold atomic Bose-Bose mixtures Chen and Hung (2021); Bakkali-Hassani et
al. (2021). Soliton solutions are known to be unstable beyond the mean-field
approach Krutitsky et al. (2010); Rubbo et al. (2012) but BF mixtures can form
stable self-bound systems (e.g., in Ref. Salasnich et al. (2007),) which in 1D
are stable within the mean-field approximation as well as to higher order
Rakshit et al. (2019). In the course of our analysis, we will find out that an
optical lattice increases the range of $g_{bf}$ values for which the mixture
is stable.

##  III Methods

###  III.1 A hybrid mean-field approximation

To investigate the pairing phase, we employ a “hybrid” approach that
intertwines quantum and classical dynamics, a method which has already been
followed in Refs.Wang et al. (2012); Karpiuk et al. (2004). We note that this
path bears similarities to the mean-field approximation employed to derive the
SSH model. It is also akin to the DFT method Wang et al. (2012). Due to its
affinity with the SSH model, we will refer to the gapped phase in the hybrid
approximation as the Peierls phase. For the ground state, we adopt the ansatz

$|\Psi\rangle=|\psi_{b}\rangle\otimes|\Psi_{f}\rangle,$ (9)

where $|\Psi_{f}\rangle$ is a general state of $N_{f}$ fermions, and
$|\psi_{b}\rangle$ is a (generalized) coherent state of the bosons satisfying
Solomon et al. (1999)

$\hat{\psi}_{b}(x)|\psi_{b}\rangle=\psi_{b}(x)|\psi_{b}\rangle,$ (10)

with the complex variable $\psi_{b}(x)$ called the condensate wavefunction or
order parameter. We impose the normalization
$\int_{0}^{L}dx\,|\psi_{b}(x)|^{2}=1$ for a finite system and thereby factor
out the total boson number $N_{b}$. If $m_{f}/m_{b}=r\ll 1$, it is justified
to assume an adiabatic evolution, similar to the Born-Oppenheimer
approximation in molecular physics Worth and Cederbaum (2004), where the
lighter fermions adjust approximately instantaneously according to the heavier
bosons. Under such conditions, any gauge potential Larson et al. (2020),
characterizing non-adiabatic corrections and arising due to back-action
between the condensate and the fermions, can be neglected. The fermionic part
of the Hamiltonian can then be diagonalized while keeping the bosonic degrees
of freedom fixed, thus acting as an “adiabatic potential”. The approximation
is equivalent to replacing the bosonic field operators with the condensate
wavefunction in the Hamiltonian, yielding the hybrid operator

$\hat{\mathcal{H}}[\psi_{b}]=\mathcal{E}_{b}[\psi_{b}]+\sum_{n}\epsilon_{f,n}\hat{\psi}_{f,n}^{\dagger}\hat{\psi}_{f,n},$
(11)

where the mean-field energy functional of the bosonic part is

$\mathcal{E}_{b}[\psi_{b}]=N_{b}\int_{0}^{L}dx\;\psi_{b}^{*}\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{b}+\frac{g_{b}N_{b}}{2}|\psi_{b}|^{2}\right)\psi_{b},$ (12)

and the second part is just the fermionic part of the Hamiltonian written in
diagonal form, $\epsilon_{f,n}$ being the eigenvalues of the Hartree equation
for the fermion orbitals,

$\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+g_{bf}N_{b}|\psi_{b}|^{2}\right)\phi_{f,n}=\epsilon_{f,n}\phi_{f,n}.$
(13)

Note that the solution of this Hartree equation exhibits a functional
dependence on the boson density, hence from now on we will be writing
$\epsilon_{f,n}=\epsilon_{f,n}[\psi_{b}]$. From Eq. (11), a nonlinear
Schrödinger equation for $\psi_{b}$ can be derived,

$i\dot{\psi}_{b}=\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{b}+g_{b}N_{b}|\psi_{b}|^{2}+g_{bf}n_{f}\right)\psi_{b},$ (14)

where the expectation value of the fermion density can be calculated as

$n_{f}(x)=\langle\hat{\psi}_{f}^{\dagger}(x)\hat{\psi}_{f}(x)\rangle=\sum_{n=1}^{N_{f}}|\phi_{f,n}(x)|^{2}.$
(15)

To determine the ground state, we resort to Eq. (14) and numerically propagate
an initial condensate wavefunction in imaginary time, employing the split-
operator method Feit et al. (1982). This procedure amounts to solving the
eigenvalue problem (13) at each time step.

To assess the Peierls phase for a system of infinite extent, we will use the
following prescription: Assume a filling $\nu_{f}=1/2$, so that we expect
density modulations with a period of two sites. Since the modulations are
commensurate with the optical lattice, the entire problem is periodic and the
solutions of the Hartree equation are Bloch waves, with a reduced unit cell in
reciprocal space due to the doubled periodicity in the Peierls phase. Again,
for the system evolution, we propagate in imaginary time, solving the Hartree
equation at each time instant. The case of incommensurate fermion filling is
interesting in its own right, as it may lead to coexisting spatially separated
regions of commensurable and incommensurate phases Molina et al. (2007).
However, our method does not allow for an incommensurate filling at present,
whence we leave the problem aside for later investigation.

For the Peierls phase, we appeal to the same order parameter as in the SSH
model, namely the energy gap of fermionic excitations above the Fermi surface,

$\Delta_{P}:=\lim_{\epsilon\to
0^{+}}\Big{(}E_{f}(k_{F}+\epsilon)-E_{f}(k_{F}-\epsilon)\Big{)},$ (16)

which in turn is identical to the gap of the hybrid Hamiltonian (11). As we
discuss in Sec. II.2, this quantity should approximate the excitation gap of
the entire system.

Figure 2: (color online) Phase properties in the absence of a trap. (a) Phase
diagram in the $(g_{bf},g_{b})$ plane, marking the different phases: LL,
Peierls, collapsed, and separated. The solid black lines mark the boundaries
of first-order transitions, while the solid red lines represent the BKT
transitions. The mean-field prediction (8) is plotted with dotted black lines.
The critical coupling $g_{bf}^{c}$ for the LL-to-Peierls transition is
calculated from a least square fit of the Peierls gap to the expression (7),
using the hybrid mean-field method. (b) Same as for (a) but in the
$(g_{bf},V_{0})$ plane. Here it becomes clear that the presence of the optical
lattice alters the phase diagram quantitatively, in particular through
stabilizing the Peierls phase. (c) The Peierls gap as a function of $g_{bf}$,
once more calculated using the hybrid code, for $V_{0}=0$ (yellow), $V_{0}=4$
(green), and $V_{0}=16$ (blue), fitted to the BKT formula (7) (dotted lines).
(d) Boson and fermion densities in the Peierls phase for the parameters
$V_{0}=1$, $g_{b}=0.4$, and $g_{bf}=-1$. In all plots, we consider half
filling, $\nu_{f}=1/2$. Figure 3: (Color online) Density profiles in the
presence of a harmonic trap. The top row shows the boson density (magenta) and
the fermion density (green), while the bottom row shows the fermion density
per site. Only $x>0$ is shown since the densities are symmetric around $x=0$.
Both plots pertain to the ground state in a BF mixture in a trap, obtained
using the hybrid method. (a): “Wedding-cake” structured fermionic insulator,
with parameters $V_{0}=4$, $g_{bf}=-48$, $\omega_{b}=0.03$ and
$\omega_{f}=0.3$. (b): Peierls phase, with parameters $V_{0}=1$, $g_{bf}=-4$,
$\omega_{b}=0.072$ and $\omega_{f}=0.0072$. The inset in the upper plot zooms
on the density profile in the region $0\leq x/a\leq 8$, evincing density
modulations in the bulk, with their origin in the Peierls instability. (c):
Separated phase with parameters $V_{0}=4$, $g_{bf}=4$, $\omega_{b}=0.08$ and
$\omega_{f}=0.8$. In all simulations, we have set the remaining parameters to
$g_{b}=0.4$, $r=0.04$, $N_{f}=32$, and $N_{b}=400$.

###  III.2 Thomas-Fermi model for the fermions

To ascertain the transition to the collapsed or separated phases, we resort to
a Thomas-Fermi model for the fermions (not to be confused with the Thomas-
Fermi approximation); for an introduction, see Refs. Parr and Yang (1989);
Spruch (1991) and for applications, see Refs. March et al. (2000); Salasnich
et al. (2002, 2007). We define a classical field
$\psi_{f}(x)=\sqrt{n_{f}(x)/N_{f}}$, called the Thomas-Fermi wavefunction,
where $n_{f}(x)$ has been defined in Eq. (15) under the normalization $\int
dx\,|\psi_{f}|^{2}=1$. The kinetic energy is approximated as a functional

$\mathcal{T}[\psi_{f}]=\int dx\;\frac{\pi^{2}N_{f}^{3}}{3r}|\psi_{f}|^{6}.$
(17)

This form can be motivated by means of a dimensional analysis, and is derived
in detail in Parr and Yang (1989); Spruch (1991). We obtain the Thomas-Fermi
energy functional as

$\mathcal{E}[\psi_{b},\psi_{f}]=\mathcal{E}_{b}[\psi_{b}]+\mathcal{E}_{f}[\psi_{b},\psi_{f}],$
(18)

where the first term on the right is given by (12) and the second by

$\begin{array}[]{lll}\mathcal{E}_{f}[\psi_{b},\psi_{f}]&=&\displaystyle{N_{f}\int
dx\Bigg{(}\frac{\pi^{2}N_{f}^{2}}{3r}|\psi_{f}|^{6}+\frac{1}{r}\left|\frac{\partial\psi_{f}}{\partial
x}\right|^{2}}\\\ \\\
&&+g_{bf}N_{b}|\psi_{b}|^{2}|\psi_{f}|^{2}\Bigg{)}.\end{array}$ (19)

Taking the functional derivative, we find an equation of a two-component
Gross-Pitaevskii type,

$\begin{array}[]{l}\displaystyle{i\partial_{t}\psi_{b}=\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{\text{OL}}+g_{b}N_{b}|\psi_{b}|^{2}+g_{bf}N_{f}|\psi_{f}|^{2}\right)\psi_{b}},\\\
\\\
\displaystyle{i\partial_{t}\psi_{f}=\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+\frac{\pi^{2}N_{f}^{2}}{r}|\psi_{f}|^{4}+g_{bf}N_{b}|\psi_{b}|^{2}\right)\psi_{f}.}\end{array}$
(20)

Employing once more the split-operator method Feit et al. (1982), we then
evolve an initial state in imaginary time to determine the ground state. Due
to the mass separation between the two species, one may argue that an
adiabatic elimination of the fermion field should be justified. This would
result in an effective bosonic model where the fermion-mediated RKKY-like
boson-boson interactions would appear. Such an approximation, however, has a
negligible benefit to the numerical solution, whence we solve the full mean-
field model formulated by Eqs. (20).

##  IV Results

###  IV.1 Translation-invariant BF mixture

We first focus on the periodic case in the absence of a trap, where we set
$V_{\rm{trap}}=0$. This idealization targets the thermodynamic limit in the
system response, which is meaningful when assessing the universal properties
of the phase transitions reported herein. A Bose-Fermi mixture with no lattice
was studied in Ref. Cazalilla and Ho (2003), where it was found that, for
negative $g_{bf}$, a BKT transition separates a LL from a Peierls phase; that
gap had the form dictated by Eq. (7).

In the upper two frames of Fig. 2, we depict the numerically extracted phase
diagrams; frame (a) shows a region in the $(g_{bf},g_{b})$ plane when keeping
$V_{0}=0$ and $r=0.04$ constant, while frame (b) displays a region in the
$(g_{bf},V_{0})$ plane for $g_{b}=0.4$ and $r=0.04$ constant. Both (a) and (b)
show the region of stability of the Peierls phase towards collapse or
separation (black solid lines), located via a calculation using the Thomas-
Fermi approximation and compared to the mean-field prediction (8) (black
dotted lines). The transition is identified by detecting the discontinuity in
the overlap between boson and fermion densities. The LL-to-Peierls transition
(red solid lines) is calculated via the hybrid method, and the critical
coupling $g_{bf}^{c}$ is obtained from a least square fit of the expression
(7) to the numerical data. From frame (a) we conclude that the region of
stability has an extent which abides by the inequality (8), but more
importantly that our method predicts a Peierls (i.e. pairing) phase for
$g_{bf}>0$, in disagreement with Ref. Cazalilla and Ho (2003). The reason for
such a disparity could be simply attributed to the fact that the hybrid method
entails the unphysical assumption that the solution is periodic, which hinders
the detection of long-wavelength fluctuations. From the literature on Bose-
Fermi-Hubbard systems, where both gases are subject to an optical lattice, we
expect however an optical lattice to stabilize the Peierls phase for repulsive
interaction Mathey (2007). From frame (b) we infer that this is indeed the
case: the optical lattice stabilizes the Peierls phase beyond the boundary set
by condition (8) for a homogeneous configuration. More precisely, the extent
of this phase is growing with the optical lattice potential while the LL,
collapsed, and separated phases are shrinking for the shown parameter ranges.
Furthermore, the Peierls gap also grows with increasing lattice amplitude,
meaning that it should be easier to observe in an experiment with a stronger
optical lattice, provided the full system does not enter an insulating phase
due to the inclusion of an optical lattice.

To illustrate the type of transition occurring between the LL and Peierls
phases, Fig. 2(c) shows three examples of the Peierls gaps as a function of
$g_{bf}$, for $V_{0}=0$, $4$, and $8$, keeping $g_{b}=0.4$ constant. As
demonstrated by the dotted curves, close to the critical value, the
numerically extracted gaps fit very well the analytical expression (7) for the
gap of a BKT transition. Away from the critical point, there is a notable
“knee” feature, which occurs when the bosons only populate every other site,
and solely for $V_{0}\neq 0$.

Figure 2(d) displays the density of bosons and fermions within the unit cell,
as a typical example for a particular point picked in the phase diagram:
$V_{0}=1$, $g_{b}=0.4$, and $g_{bf}=-1$. The doubling of the periodicity is in
evidence. Contrary to most earlier studies, our method is capable of capturing
both amplitude and phase modulations, however, we only find modulations in
amplitude. For the repulsive case, $g_{bf}>0$, the situation is similar to
that shown in (d), but with the two densities being “out-of-phase” instead.

In order to experimentally probe the phase diagram, one must detect the order
parameter or any other quantity capable of telling the different phases apart.
The gap could, in principle, be extracted via pump-probe experiments Ernst et
al. (2010); Heinze et al. (2011), while one could also imagine transport
experiments to differentiate insulating and conduction phases, i.e. Peierls
vs. LL. As will be evident in the next section, the trap induces a varying
Fermi wave number $k_{F}$ which makes the Peierls gap opening not that
pronounced. In a time-of-flight detection, for instance, the Bragg peaks are
smeared out, and the particular one corresponding to the gap is almost
invisible. Hence, in a realistic setup, including the trap, time-of-flight
measurements might not turn out to be the most optimal scheme. Instead, the
onsite densities, which display alternating variations, could be accessed via
single-site resolution detection Bakr et al. (2009). This technique has been
successfully implemented for cold fermionic gases, including even detection of
higher order correlators Haller et al. (2015); Boll et al. (2016); Scherg et
al. (2021).

###  IV.2 BF mixture in a harmonic trap

Current experiments rely on the application of harmonic potentials for the
attainment of sufficiently long trapping times. However, the trap may have a
relatively small frequency yet still confine the atoms, such that the system
locally experiences a periodic potential. To study these more realistic
situations, we include a harmonic trap in our analysis but still consider a
very tight transverse confinement and consequently a quasi-1D configuration.

Figure 4: From a Mott to a Peierls insulator with extended eigenstates in a
trap. One-particle density matrix,
$|\langle\hat{\psi}_{f}(x)\hat{\psi}_{f}^{\dagger}(x^{\prime})\rangle|^{2}$,
where only the top-right quadrant is plotted owing to the symmetry about
$x=0$. Two cases are shown: (a) a trapped mixture in the wedding cake phase,
for the same state as the one presented in the left column of Fig. 3; (b)
Peierls phase for the same state as in the central column of Fig. 3. The non-
diagonal part of the one-particle density matrix is significantly suppressed
in the wedding cake phase, where the decay is Gaussian and on a length scale
similar to the characteristic length of the optical lattice. In the Peierls
phase, there are instead periodic oscillations signifying coherence. Figure 5:
(Color online) Scaling of the two-point density matrix with the width of the
harmonic trap. The one-point density matrix $n(x,x^{\prime})$ is calculated
for $x^{\prime}=-x$ and plotted with $x$ scaled with $a\sqrt{k}$. The
oscillations of the one-point density matrix are fitted to the expression
$n(x,-x)=C\text{sinc}(bx)$ in line with Eq. (46). Left column: normal phase
($g_{bf}=0,\omega_{0,b}=0.005,\omega_{0,f}=0.07$). Right column: Peierls phase
($g_{bf}=-1.5,\omega_{0,b}=\omega_{0,f}=0.0036$). The harmonic confinements
have been set as $\omega_{b/f}=k\omega_{0,b/f}$, with $k=1,2,4$ for the top,
middle and bottom rows. In all simulations, we have fixed the remaining
parameters to $g_{b}=0.1$ and $V_{0}=0.25$.

We have used the hybrid method to investigate the different phases in the
aforementioned setup. Figure 3 presents the numerical results obtained for the
densities, along the main diagonal of the one-particle density matrix. In the
upper row, we present the real-space densities, and in the lower row, the
fermion density per site is depicted, which could, in principle, be accessed
in an experiment, as mentioned in the last section. Illustrated in the left
column is a novel insulating state not present in the translationally
invariant system. We refer to this profile as a “wedding cake”, drawing an
analogy to the wedding cake-like structure found in the harmonically confined
Bose-Hubbard model Jaksch et al. (1998) and also in fermionic systems Jördens
et al. (2008); Greif et al. (2016). In this state, the fermions form plateaus
(or “layers of the cake”) with an integer number of fermions per site. In the
figure, a plateau of three fermions per site is observed in the center of the
trap, transitioning in a step-like fashion to two and finally one fermion per
site as we move further out from the center. The wedding cake pattern results
from the interplay between kinetic, potential, and interaction energies and
only appears when an optical lattice is present. It cannot be accurately
modeled using a local density approximation due to the abrupt changes in
density.

In the central column of Fig. 3, we elaborate on the correspondence to the
Peierls phase occurring in the absence of a trap. Generally, the boson and
fermion clouds form a region of a size determined by the trapping strength
alongside the BB and BF interactions. The optical lattice induces a spatial
variation in the bosonic profile, imparted to the fermions via the BF
interaction. The Friedel oscillations of the fermions in this potential are
then imprinted on the bosons via the RKKY interaction. Because the Peierls
phase is a compressible supersolid, the wavelength varies continuously over
the width of the trap. Therefore, in the trapped system, the Peierls
modulations are generally not commensurate with the lattice. These modulations
may be detected using single-site addressing, leading to a spatial variation
in density that cannot be attributed to the profile of the trap or that of the
optical lattice. We note here that the beating between the fermion and the
optical lattice wavenumbers may give rise to local commensurate-incommensurate
transitions in which the fermionic wavenumber of an atomic density wave
spatially adjusts to the optical lattice period Molina et al. (2007).

For any positive value of $g_{bf}$ and any strength of the optical lattice, we
have observed that the system phases separate. The rightmost column of Fig. 3
displays the separated state, where the heavier atoms occupy the center of the
trap, surrounded by regions of the lighter species. Perhaps the most
surprising feature of Fig. 3 is the emergence of the wedding cake phase where
we would expect a collapsed phase. The integer filling per site clearly
indicates a fermionic Mott insulator state Rigol et al. (2003); Rigol and
Muramatsu (2004); Schneider et al. (2008). One way to ascertain whether this
instance is something more than a coincidence is to examine the off-diagonal
elements of the one-particle density matrix, given by

$\langle\hat{\psi}_{f}^{\dagger}(x)\hat{\psi}_{f}(x^{\prime})\rangle=\sum_{n=0}^{N_{f}-1}\phi_{f,n}^{*}(x)\phi_{f,n}(x^{\prime}).$
(21)

To derive the above expression, we have written

$\hat{\psi}_{f}(x)=\sum_{n}\phi_{f,n}(x)\hat{\psi}_{f,n},$ (22)

where $\hat{\psi}_{f,n}$ are fermionic operators and the orbitals
$\phi_{f,n}(x)$ are solutions of the Hartree equation (see Appendix B)

$\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+g_{bf}N_{b}|\psi_{b}|^{2}\right)\phi_{f,n}=\epsilon_{f,n}\phi_{f,n}.$
(23)

An insulating state is expected to have an exponentially decaying one-particle
density matrix Cloizeaux (1964); He and Vanderbilt (2001). However, it turns
out that this is not the case for the BF mixture in a harmonic trap.

In Fig. 4, we focus into a more detailed comparison between the wedding cake
[frame (a)] and Peierls [frame (b)] phases, showcasing the corresponding one-
body density matrix. As seen in Fig. 3(a), the wedding cake is distinguished
by having an integer number of fermions per site. The one-particle density
matrix provides an additional way of distinguishing the phases. In the
wedding-cake phase, it displays a Gaussian decay on a length scale similar to
the characteristic length of the optical lattice. This feature, in turn,
signals the opening of a more pronounced gap in the spectrum and the presence
of an insulating phase. On the other hand, the spatial decay of the matrix
elements in the Peierls phase reveals a sustained coherence originating from
the individual trapped-particle wavefunctions, which we will attempt to
elucidate.

A characteristic example of the spatially decaying one-point density matrix
$n(x,x^{\prime})$ is plotted in Fig. 5 for $x^{\prime}=-x$. It is evident that
in the normal phase, the “wavelength” of the fitted sinc function scales with
$\sqrt{k}$, while the relation is not so direct in the Peierls phase. To
motivate the mechanism underlying the periodically decaying oscillations, we
start by setting $g_{bf}=0$, in which case the fermions are free from their
interaction with the bosons in the same trap. In this case, we find (see
Appendix C for further details)

$n(x,-x)=\frac{1}{\pi}\frac{\sin(\sqrt{2N}\,2x)}{2x}.$ (24)

Equation (24) explicitly reveals an algebraically decaying envelope and a
$\sqrt{N_{f}}$-dependence of the spatial frequency, with $N_{f}$ being the
number of fermions confined in the trap, both attributes evidenced in the left
plots of Fig. 5. While gapped phases might suggest an exponential decay of the
single-particle density Sachdev (1999), the spatial confinement imposed by the
trap implies that the extent of excitations is typically of the same order as
the characteristic trap length (unless very weak trap frequencies are
considered). Thus, we observe an algebraic decay instead of an exponential
one. Turning on an attractive Bose-Fermi interaction $g_{bf}$ to access the
regime of Peierls instability in the right frames of Fig. 5 marks the
departure from the sinc profile and the $\sqrt{N_{f}}$ scaling, yet the
periodicity is still evident, and the frequency carries on increasing with the
extent of the trap. The counterdiagonal now defines a particular cut through
ripples visible in Fig. 4(d) – remnants of quantum interference developing
symmetrically to the main diagonal and signifying a Peierls phase with a
larger spatial extent than the normal. The overall amplitude of the density
matrix along the counterdiagonal, depicted in the right frames of the figure,
also increases with $N_{f}$, in line with the prediction of Eq. (46). Taking
$N_{f}\to\infty$, we obtain the delta function as the familiar limit of the
sinc function, writing $n(x,-x)\propto\delta(x)$.

##  V Concluding remarks

In this work, we have studied the response of a dilute mixture of two atomic
clouds of disparate masses, one fermionic and one bosonic, to an external
lattice potential alongside an atomic trap. The gases are confined in a
cylindrical trap so that the low-energy physical behavior of the system is
effectively one-dimensional, while the optical lattice imprints a period
structure onto the bosonic gas. We have assumed a weak optical lattice and a
high density of bosons, such that the bosons form a condensate, which amounts
to an effective interaction of the fermions with a coherent field.

We confirmed that a Peierls instability, manifested via a BKT-type phase
transition to a Peierls phase, persists when the optical lattice potential is
applied solely to the bosons. Moreover, the presence of this phase was found
to stabilize the system against collapse and separation, as well as to enhance
the significance of the Peierls effect, a property that is highly desirable
for experimental explorations of the associated phonon-like physics. The
presence of the Peierls phase was confirmed upon developing and applying a
hybrid mean-field method, which retained an amount of coherence able to
sustain sinc-type oscillations along the counterdiagonal of the one-particle
density matrix. It is interesting to note that such a method can reveal the
BKT nature of the phase transition, as well as the similarity to the SSH model
of the Peierls instability. The crucial scaling is captured by the method we
have developed owing to the fact that we take the deformation of the Fermi
surface into account, while we also retain the quantum degrees of freedom for
the fermions albeit within an adiabatic approximation. An algebraic decay of
correlations is found for all phases occurring in the trapped BF mixture, in
accordance with a BKT phase transition.

The natural step forwards would be to take quantum fluctuations of the boson
degrees of freedom into account. This entails the development of a formulation
and related numerical methods manifestly beyond any mean-field reduction, and
to that aim we outline possible directions for deriving the corresponding
many-body Hamiltonian in Appendix A below. Another promising direction for
future research concerns the investigation of fermion fillings incommensurate
with the optical lattice as well as studying the quenched time-evolution
problem of a single fermion in a dynamical bosonic potential. Furthermore, an
interesting extension replaces the optical lattice with a BB mixture, using
either trapped ions or dipole bosons. Finally, systems of the kind can be
analyzed in greater detail using matrix product states or bosonization
techniques.

##  Appendix A Routes towards known quantum many-body models

In Sec. III.1, we reported on the presence of a Peierls phase at a hybrid
mean-field level. We argue that this description accounts for the pairing
effect qualitatively. However, there are clear limitations to such an
approach. Effectively, the model is a free fermion theory, and as long as
there is a finite Peierls gap, $\Delta\neq 0$, however we find a nontrivial
insulator with algebraically decaying order for the trapped BF mixture. This
is in line with what one expects from a BKT transition, where one phase should
exhibit algebraic decay. Furthermore, the classical field stemming from the
boson condensation cannot build up quantum correlations with the fermions,
while a quantum field can. To quantitatively analyze such situations one must
go beyond mean-field. In this appendix we outline, without going into details,
two possible approaches in order to derive effective Hamiltonians for which
both species are treated quantum mechanically.

In Ref. Bruderer et al. (2007), the condensate wavefunction was calculated for
zero coupling between the two species, and subsequently, Bogoliubov
excitations around the mean-field response were taken into account, arising
from a weak interaction. Due to the presence of a deep external lattice
potential in their model, a Hubbard-Holstein model is obtained which can be
understood in terms of polarons. In our case, we instead find the fermion-
phonon Hamiltonian

$\displaystyle\hat{\mathcal{H}}=$
$\displaystyle\hat{\mathcal{H}}_{f}+\sum_{\mu}\omega_{\mu}\int
dx\;\left(M_{\mu}\hat{b}_{\mu}+M_{\mu}^{*}\hat{b}_{\mu}^{\dagger}\right)\hat{\rho}_{f}$
$\displaystyle+g_{bf}\int
dx|\phi_{b}|^{2}\hat{\rho}_{f}+\sum_{\mu}\omega_{\mu}\hat{b}_{\mu}^{\dagger}\hat{b}_{\mu},$
(25)

where

$\displaystyle M_{\mu}(x)=u_{\mu}(x)-v_{\mu}(x)$ (26)

are given in terms of the $u_{\mu}(x),v_{\mu}(x)$ of the Bogoliubov
transformation and $\omega_{\mu}$ are the energies of the Bogoliubov modes.
The third term will confine the fermions to their lowest Bloch band if the
interaction is strong enough, and in this limit, the model approaches the
Hubbard-Holstein model of Ref. Bruderer et al. (2007). However, the approach
above would have to be modified since it rests on the assumption that $g_{bf}$
is weak.

For a deep optical lattice one could come up with another method: expand the
boson field in the corresponding Wannier functions, $w_{j}(x)$ localized at
site $j$, as Lewenstein et al. (2007); Bloch et al. (2008)

$\hat{\psi}_{b}(x)=\sum_{j}\hat{a}_{j}w_{j}(x).$ (27)

If we expand the fermion field as
$\hat{\psi}_{f}(x)=\sum_{k}\hat{c}_{k}e^{ikx}$ and impose the single-band and
tight-binding approximations, we arrive at the many-body Hamiltonian

$\displaystyle\hat{\mathcal{H}}_{mb}$
$\displaystyle=\hat{\mathcal{H}}_{BH}+\sum_{k}\\!\frac{k^{2}}{r}\hat{c}_{k}^{\dagger}\hat{c}_{k}$
$\displaystyle+\hat{N}_{b}\sum_{k,k^{\prime}}\\!\left[D(k-k^{\prime})\hat{c}_{k}^{\dagger}\hat{c}_{k^{\prime}}+\text{H.c.}\right]\\!,$
(28)

where

$\displaystyle\hat{\mathcal{H}}_{BH}$
$\displaystyle=-J\sum_{j}\left(\hat{a}_{j}^{\dagger}\hat{a}_{j+1}+\text{H.c.}\right)$
$\displaystyle+\frac{U}{2}\sum_{j}\hat{n}_{j}\left(\hat{n}_{j}-1\right)$ (29)

is the Bose-Hubbard Hamiltonian, with $J$ and $U$ the tunneling rate and
onsite interaction strengths, respectively,
$\hat{n}_{j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}$, and $D(k-k^{\prime})$ is the
overlap integral (which is Gaussian in the harmonic approximation). Since
$\hat{N}_{b}$ is preserved, the quadratic fermionic Hamiltonian can be readily
diagonalized, and we find no back-action on the bosons due to the fermions. To
incorporate such effects one would need to go beyond the single-band or tight-
binding approximations. Alternatively, the approximations may be kept but
other additional degrees of freedom should be introduced corresponding to the
phonon modes. For heavy bosons, one can follow the idea of Ref. Maluckov et
al. (2013) to allow a variation in the position of the Wannier-function
centers $j\pi$, but keep the shape of the functions intact. Thus, we associate
a quantized shift $\hat{\delta}_{j}$ with each Wannier function,
$w_{j}(x)\rightarrow w_{j}(x-\hat{\delta}_{j})$. Assuming small shifts
$\hat{\delta x}_{j}\ll 1$ one may then expand to linear order in them and
derive an effective Fröhlich-like Hamiltonian Mahan (2013)

$\hat{\mathcal{H}}_{Fr}=\hat{\mathcal{H}}_{mb}+\hat{\mathcal{H}}_{BP}+\hat{\mathcal{H}}_{FBP},$
(30)

where the boson-phonon interaction is given by

$\begin{array}[]{lll}\hat{\mathcal{H}}_{BP}&=&\displaystyle{-J_{1}\sum_{j}\left(\hat{\delta}_{j+1}-\hat{\delta}_{j}\right)\left(\hat{a}_{j}^{\dagger}\hat{a}_{j+1}+\text{H.c.}\right)}\\\
\\\
&&\displaystyle{+\frac{U_{1}}{2}\sum_{j}\hat{\delta}_{j}\hat{n}_{j}\left(\hat{n}_{j}-1\right)}.\end{array}$
(31)

If we introduce the local phonon annihilation/creation operators
$\hat{d}_{j}/\hat{d}_{j}^{\dagger}$, the phonon displacement is expressed as

$\hat{\delta}_{j}=\left(\hat{d}_{j}+\hat{d}_{j}^{\dagger}\right)/2,$ (32)

while the fermion-boson-phonon interaction term takes the form

$\hat{\mathcal{H}}_{FBP}=\mu\hat{N}_{d}+\sum_{j}\sum_{k,k^{\prime}}\hat{n}_{j}D_{1}(k-k^{\prime})\hat{\delta}_{j}\hat{c}_{k}^{\dagger}\hat{c}_{k^{\prime}},$
(33)

where the first term is the bare phonon energy ($\mu$ is the characteristic
frequency and $\hat{N}_{d}=\sum_{j}\hat{d}_{h}^{\dagger}\hat{d}_{j}$). In
principle, within the harmonic approximation, the coefficients $J_{1}$,
$U_{1}$ and $D_{1}$ can be analytically determined. Both
$\hat{\mathcal{H}}_{BP}$ and $\hat{\mathcal{H}}_{FBP}$ describe phonon-
assisted tunneling, either between neighboring lattice sites (bosons) or
between different momentum modes (fermions). It may be noted that if
$g_{bf}=0$, then $D(k-k^{\prime})=D_{1}(k-k^{\prime})=0$ and $\hat{N}_{d}=0$
such that $\hat{\mathcal{H}}_{BP}=0$.

##  Appendix B The hybrid mean-field–quantum method

In this appendix, we provide some further details about the employed hybrid
mean-field ansatz. As already mentioned in Sec. III.1, a gap of the form (7)
opens up in the pairing phase. A superfluid of bosons is gapless and, to the
lowest order, is described by the condensate wavefunction $\psi_{b}(x)$ – the
mean-field order parameter. We define the generalized coherent state

$\ket{\psi_{b}}=\exp\left(\int
dx\;\psi_{b}(x)\hat{\psi_{b}}^{\dagger}(x)-\text{H.c.}\right)\ket{0},$ (34)

where $\ket{0}$ is the bosonic vacuum. Taking the expectation value of the
time-dependent Schrödinger equation, we obtain an effective hybrid Hamiltonian

$\displaystyle\hat{\mathcal{H}}_{\text{eff}}=\mathcal{E}_{b}[\psi_{b}]+\hat{\mathcal{H}}_{f}[\psi_{b}]-i\matrixelement{\psi_{b}}{\partial_{t}}{\psi_{b}},$
(35)

where the first term is given by (12), the second term is

$\displaystyle\hat{\mathcal{H}}_{f}[\psi_{b}]=\int
dx\;\hat{\psi}_{f}^{\dagger}\Big{(}-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+V_{f}+g_{bf}|\psi_{b}|^{2}\Big{)}\hat{\psi}_{f},$ (36)

and the third term is the Berry connection Larson et al. (2020). We will
assume that the state of the condensate is slowly modified, so that this term
can be neglected.

The above is equivalent to a product-state ansatz for the ground state

$\ket{\Psi_{0}}=\ket{\psi_{b}}\otimes\ket{\Psi_{f}},$ (37)

where $\ket{\Psi_{f}}$ is a general fermion state of $N_{f}$ fermions. Such an
ansatz neglects any entanglement built between the bosons and fermion
subsystems; correlations may actually arise within the non-condensed fraction
of bosons and, more importantly, different configurations of the condensate
may get entangled with the fermions. In general, such mixed quantum-classical
dynamics is interesting in a much wider context in condensed matter physics,
high-energy physics, and quantum gravity, and has been investigated in detail
in Refs. Prezhdo and Kisil (1997); Caro and Salcedo (1999); Oppenheim et al.
(2020). In Ref. Prezhdo and Kisil (1997), a multi-configurational mean-field
approximation based on hybrid quantum-classical theory was developed, with the
central object being the quantum-classical distribution function – a map from
the classical phase space to the set of quantum density operators, defined as

$\hat{\rho}(\mathbf{q},\mathbf{p})=\sum_{ij}\varrho_{ij}|\Psi_{i}\rangle\langle\Psi_{j}|\delta(\mathbf{q}-\mathbf{q}_{ij})\delta(\mathbf{p}-\mathbf{p}_{ij}),$
(38)

where $\mathbf{q},\mathbf{p}$ are the classical generalized coordinates and
momenta, and $|\Psi_{i}\rangle$ is a basis state of the quantum subsystem. In
this formalism, “non-diagonal” contributions correspond to the quantum
subsystem generating a coupling between different trajectories of the
classical subsystems. When considering only a single trajectory, the quantum-
classical distribution function can be written as a single delta-function
term,

$\hat{\rho}[\psi_{b}^{\prime}]=|\Psi_{f}\rangle\langle\Psi_{f}|\delta[\psi_{b}-\psi_{b}^{\prime}],$
(39)

where the brackets indicate a functional dependence. In other words, the
system is represented by a single point in classical phase space, evolving
with a quantum density matrix of a pure state.

We find the ground state of $\hat{H}_{\text{eff}}$ self-consistently first
observing that it can be readily diagonalized by selecting

$\hat{\psi}_{f}(x)=\sum_{n}\phi_{f,n}(x)\hat{\psi}_{f,n},$ (40)

where the orbitals are solutions of the Hartree (single-particle) equation

$\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+g_{bf}N_{b}|\psi_{b}|^{2}\right)\phi_{f,n}=\epsilon_{f,n}\phi_{f,n}.$
(41)

Second, we define a non-linear Schrödinger equation for the condensate,

$\begin{array}[]{lll}i\dot{\psi}_{b}&=&\displaystyle{\frac{\delta}{\delta\psi_{b}^{*}}\matrixelement{\Psi_{f}}{\hat{H}_{\text{eff}}}{\Psi_{f}}}\\\
\\\ &=&\displaystyle{\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{b}+g_{b}N_{b}|\psi_{b}|^{2}+g_{bf}n_{f}\right)\psi_{b}},\end{array}$
(42)

where

$n_{f}(x)=\matrixelement{\Psi_{f}}{\hat{\psi}_{f}^{\dagger}(x)\hat{\psi}_{f}(x)}{\Psi_{f}}=\sum_{n}|\phi_{f,n}(x)|^{2}$
(43)

depends explicitly on the bosons through (41). The ground state of (42) is
found from imaginary time propagation utilizing the split-operator method Feit
et al. (1982).

As mentioned in Sec. III.1, the above method can also be easily adapted to
study a translationally invariant system of infinite extent for fermion
filling $\nu_{f}=1/2$. In this case, we assume a periodic solution
$\psi_{b}(x+4\pi)=\psi_{b}(x)$. The solutions of the Hartree equations are
then Bloch waves $e^{iqx/2}u_{f,n}(q,x)$. Due to the assumed double
periodicity of $4\pi$, the first Brillouin zone is halved and the fermionic
filling is in direct correspondence to the edges of the Brillouin zone. It is
straightforward to find the fermion density as the integral

$n_{f}(x)=\int_{1BZ}dq\;|u_{f,0}(q,x)|^{2}.$ (44)

Another important difference here is that we have to diagonalize the
Bloch/Hartree equation for each quasi-momentum – in practice for a large
number of sampled values. Apart from these discrepancies between the two
methods, the solution to Eq. (42) is also found using imaginary time
propagation.

##  Appendix C Decay of correlations in the absence of Bose-Fermi interaction

For $g_{bf}=0$ (normal phase), the one-point density matrix can be calculated
analytically via the Cristoffel-Darboux formula, involving summed products of
individual orthogonal fermionic wavefunctions in terms of the Hermite
polynomials $H_{n}(x)$ (with $n=0,1,\ldots N_{f}$) derived from the quantum
harmonic oscillator eigenstates. Letting $N_{f}=N$ for convenience, the result
reads

$\displaystyle n(x,x^{\prime})$
$\displaystyle=\pi^{-1/2}\exp\left(-\frac{x^{2}+(x^{\prime})^{2}}{2}\right)$
$\displaystyle\times\frac{1}{N!2^{N+1}}\frac{H_{N}(x^{\prime})H_{N+1}(x)-H_{N}(x)H_{N+1}(x^{\prime})}{x-x^{\prime}}.$
(45)

For $x^{\prime}=-x$, along the counterdiagonal of the density matrix that we
select for Fig. 5, the above expression can be recast into the following form:

$\displaystyle n(x,-x)=\pi^{-1/2}\exp(-x^{2})\frac{1}{N!2^{N+1}}$ (46)
$\displaystyle\times\frac{2(-1)^{N}H_{N}(x)H_{N+1}(x)}{2x}\sim\frac{2^{2N}\Gamma((N+1)/2)\Gamma(N/2+1)}{\pi^{3/2}\,\,2^{N}\,N!}$
$\displaystyle\times\frac{\sin(\sqrt{2N}\,2x)}{2x}=\frac{1}{\pi}\frac{\sin(\sqrt{2N}\,2x)}{2x},$

which follows from the familiar asymptotic $N\gg 1$ expansion of quantum
harmonic oscillator eigenstates M. and I. (1983)

$e^{-x^{2}/2}H_{N}(x)\sim\frac{2^{N}}{\sqrt{\pi}}\Gamma\left(\frac{N+1}{2}\right)\cos(x\sqrt{2N}-N\pi/2)$
(47)

and the Legendre duplication formula for the gamma function. In fact, the
number of fermions confined in the trap need not be large. Even for
$N_{f}\gtrsim 10$ we note a reasonable agreement between the exact formula of
Eq. (C), and the asymptotic sinc profile of (46).

###### Acknowledgements.

We thank Maciej Lewenstein, Hannes Conners, and Christophe Salomon for helpful
discussions and comments. We acknowledge financial support from VR-
Vetenskapsrådet (The Swedish Research Council), and KAW (The Knut and Alice
Wallenberg foundation). Th. K. M. acknowledges support from: ERC AdG NOQIA;
Ministerio de Ciencia y Innovation Agencia Estatal de Investigaciones
(PGC2018-097027-B-I00/10.13039/501100011033,
EX2019-000910-S/10.13039/501100011033, QUANTERA DYNAMITE PCI2022-132919,
Proyectos de I+D+I “Retos Colaboración” QUSPIN RTC2019-007196-7); MICIIN with
funding from European Union NextGenerationEU(PRTR-C17.I1) and by Generalitat
de Catalunya; Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya
(European Social Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452,
QuantumCAT U16-011424, co-funded by ERDF Operational Program of Catalonia
2014-2020); Barcelona Supercomputing Center MareNostrum (FI-2022-1-0042); EU
Horizon 2020 FET-OPEN OPTOlogic (Grant No 899794); EU Horizon Europe Program
(Grant Agreement 101080086 — NeQST); ICFO Internal “QuantumGaudi” project;
European Union’s Horizon 2020 research and innovation program under the Marie-
Skłodowska-Curie grant agreement No 101029393 (STREDCH) and No 847648 (“La
Caixa” Junior Leaders fellowships ID100010434: LCF/BQ/PI19/11690013,
LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013). Views and
opinions expressed in this work are, however, those of the author(s) only and
do not necessarily reflect those of the European Union, European Climate,
Infrastructure and Environment Executive Agency (CINEA), nor any other
granting authority. Neither the European Union nor any granting authority can
be held responsible for them.

## References

  * Anderson et al. (1995) M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell.  Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor.  _Science, New Series_ , 269(5221):198–201, 1995\.
  * Morsch et al. (2001) OMJH Morsch, J. H. Müller, M. Cristiani, Donatella Ciampini, and Ennio Arimondo.  Bloch oscillations and mean-field effects of Bose-Einstein condensates in 1D optical lattices.  _Physical Review Letters_ , 87(14):140402, 2001\.
  * Denschlag et al. (2002) J. Hecker Denschlag, J. E. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, Donghyun Cho, Kristian Helmerson, S. L. Rolston, and William D. Phillips.  A Bose-Einstein condensate in an optical lattice.  _Journal of Physics B: Atomic, Molecular and Optical Physics_ , 35(14):3095, 2002.
  * Fisher et al. (1989) Matthew PA Fisher, Peter B Weichman, Geoffrey Grinstein, and Daniel S Fisher.  Boson localization and the superfluid-insulator transition.  _Physical Review B_ , 40(1):546, 1989.
  * Jaksch et al. (1998) Dieter Jaksch, Christoph Bruder, Juan Ignacio Cirac, Crispin W Gardiner, and Peter Zoller.  Cold bosonic atoms in optical lattices.  _Physical Review Letters_ , 81(15):3108, 1998\.
  * Greiner et al. (2002) Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W Hänsch, and Immanuel Bloch.  Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms.  _nature_ , 415(6867):39–44, 2002.
  * Lewenstein et al. (2007) Maciej Lewenstein, Anna Sanpera, Veronica Ahufinger, Bogdan Damski, Aditi Sen, and Ujjwal Sen.  Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond.  _Advances in Physics_ , 56(2):243–379, 2007.
  * Bloch et al. (2008) Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger.  Many-body physics with ultracold gases.  _Reviews of Modern Physics_ , 80(3):885–964, 2008\.  ISSN 00346861.  doi: 10.1103/RevModPhys.80.885.
  * Daley et al. (2022) Andrew J Daley, Immanuel Bloch, Christian Kokail, Stuart Flannigan, Natalie Pearson, Matthias Troyer, and Peter Zoller.  Practical quantum advantage in quantum simulation.  _Nature_ , 607(7920):667–676, 2022.
  * Asbóth et al. (2007) JK Asbóth, H Ritsch, and P Domokos.  Collective excitations and instability of an optical lattice due to unbalanced pumping.  _Physical review letters_ , 98(20):203008, 2007\.
  * Asbóth et al. (2008) JK Asbóth, H Ritsch, and P Domokos.  Optomechanical coupling in a one-dimensional optical lattice.  _Physical Review A_ , 77(6):063424, 2008.
  * Peierls and Peierls (1955) Rudolf Peierls and Rudolf Ernst Peierls.  _Quantum Theory of Solids_.  Oxford University Press, 1955.  ISBN 0-19-850781-X.
  * Mahan (2013) Gerald D Mahan.  _Many-Particle Physics_.  Springer Science & Business Media, 2013.
  * Deng et al. (2014) Y Deng, J Cheng, H Jing, and S Yi.  Bose-Einstein condensates with cavity-mediated spin-orbit coupling.  _Physical Review Letters_ , 112(14):143007, 2014\.
  * Pan et al. (2015) Jian-Song Pan, Xiong-Jun Liu, Wei Zhang, Wei Yi, and Guang-Can Guo.  Topological superradiant states in a degenerate Fermi gas.  _Physical Review Letters_ , 115(4):045303, 2015\.
  * Ballantine et al. (2017) Kyle E Ballantine, Benjamin L Lev, and Jonathan Keeling.  Meissner-like effect for a synthetic gauge field in multimode cavity qed.  _Physical Review Letters_ , 118(4):045302, 2017\.
  * Lewenstein et al. (2006) M Lewenstein, A Kubasiak, J Larson, C Menotti, G Morigi, K Osterloh, and A Sanpera.  Travelling to exotic places with ultracold atoms.  In _AIP Conference Proceedings_ , volume 869, pages 201–211. American Institute of Physics, 2006.
  * Bissbort et al. (2013) U. Bissbort, D. Cocks, A. Negretti, Z. Idziaszek, T. Calarco, F. Schmidt-Kaler, W. Hofstetter, and R. Gerritsma.  Emulating Solid-State Physics with a Hybrid System of Ultracold Ions and Atoms.  _Physical Review Letters_ , 111(8):080501, August 2013.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.111.080501.
  * Stan et al. (2004) CA Stan, MW Zwierlein, CH Schunck, SMF Raupach, and W Ketterle.  Observation of Feshbach resonances between two different atomic species.  _Physical review letters_ , 93(14):143001, 2004\.
  * Best et al. (2009) Th Best, S Will, U Schneider, L Hackermüller, D Van Oosten, I Bloch, and D-S Lühmann.  Role of interactions in rb 87- K 40 bose-fermi mixtures in a 3D optical lattice.  _Physical Review Letters_ , 102(3):030408, 2009\.
  * Kawaguchi and Ueda (2012) Yuki Kawaguchi and Masahito Ueda.  Spinor bose–einstein condensates.  _Physics Reports_ , 520(5):253–381, 2012.
  * Park et al. (2012) Jee Woo Park, Cheng-Hsun Wu, Ibon Santiago, Tobias G Tiecke, Sebastian Will, Peyman Ahmadi, and Martin W Zwierlein.  Quantum degenerate Bose-Fermi mixture of chemically different atomic species with widely tunable interactions.  _Physical Review A_ , 85(5):051602, 2012.
  * Ferrier-Barbut et al. (2014) Igor Ferrier-Barbut, Marion Delehaye, Sebastien Laurent, Andrew T. Grier, Matthieu Pierce, Benno S. Rem, Frédéric Chevy, and Christophe Salomon.  A Mixture of Bose and Fermi Superfluids.  _Science_ , 345(6200):1035–1038, August 2014\.  ISSN 0036-8075, 1095-9203.  doi: 10.1126/science.1255380.
  * DeSalvo et al. (2019) B. J. DeSalvo, Krutik Patel, Geyue Cai, and Cheng Chin.  Observation of fermion-mediated interactions between bosonic atoms.  _Nature_ , 568(7750):61–64, April 2019.  ISSN 0028-0836, 1476-4687.  doi: 10.1038/s41586-019-1055-0.
  * Cazalilla and Ho (2003) M. A. Cazalilla and A. F. Ho.  Instabilities in binary mixtures of one-dimensional quantum degenerate gases.  _arXiv:cond-mat/0303550_ , October 2003.  doi: 10.1103/PhysRevLett.91.150403.
  * Miyakawa et al. (2004) Takahiko Miyakawa, Hiroyuki Yabu, and Toru Suzuki.  Peierls instability, periodic Bose-Einstein condensates and density waves in quasi-one-dimensional boson-fermion mixtures of atomic gases.  _Physical Review A_ , 70(1):013612, July 2004\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.70.013612.
  * Rizzi and Imambekov (2008) Matteo Rizzi and Adilet Imambekov.  Pairing of one-dimensional Bose-Fermi mixtures with unequal masses.  _Physical Review A_ , 77(2):023621, February 2008\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.77.023621.
  * Büchler and Blatter (2003) H. P. Büchler and G. Blatter.  Supersolid versus Phase Separation in Atomic Bose-Fermi Mixture.  _Physical Review Letters_ , 91(13):130404, September 2003.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.91.130404.
  * Klironomos and Tsai (2007) F. D. Klironomos and S.-W. Tsai.  Pairing and density-wave phases in Boson-Fermion mixtures at fixed filling.  _Physical Review Letters_ , 99(10):100401, September 2007.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.99.100401.
  * Titvinidze et al. (2008) I. Titvinidze, M. Snoek, and W. Hofstetter.  Supersolid Bose-Fermi Mixtures in Optical Lattices.  _Physical Review Letters_ , 100(10):100401, March 2008.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.100.100401.
  * Albus et al. (2003) Alexander Albus, Fabrizio Illuminati, and Jens Eisert.  Mixtures of Bosonic and Fermionic Atoms in Optical Lattices.  _Physical Review A_ , 68(2):023606, August 2003\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.68.023606.
  * Mathey et al. (2004) L. Mathey, D.-W. Wang, W. Hofstetter, M. D. Lukin, and Eugene Demler.  Luttinger liquid of polarons in one-dimensional boson-fermion mixtures.  _arXiv:quant-ph/0401151_ , January 2004.  doi: 10.1103/PhysRevLett.93.120404.
  * Roth and Burnett (2004) Robert Roth and Keith Burnett.  Quantum phases of atomic boson-fermion mixtures in optical lattices.  _Physical Review A_ , 69(2):021601, 2004.
  * Salerno (2005) Mario Salerno.  Matter-wave quantum dots and antidots in ultracold atomic Bose-Fermi mixtures.  _Physical Review A_ , 72(6):063602, 2005.
  * Pazy and Vardi (2005) E. Pazy and A. Vardi.  Holstein model and Peierls instability in 1D boson-fermion lattice gases.  _Physical Review A_ , 72(3):033609, September 2005\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.72.033609.
  * Bruderer et al. (2007) Martin Bruderer, Alexander Klein, Stephen R. Clark, and Dieter Jaksch.  Polaron Physics in Optical Lattices.  _Physical Review A_ , 76(1):011605, July 2007\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.76.011605.
  * Lan and Lobo (2014) Zhihao Lan and Carlos Lobo.  Optical lattices with large scattering length: Using few-body physics to simulate an electron-phonon system.  _Physical Review A_ , 90(3):033627, September 2014\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.90.033627.
  * Lewenstein et al. (2004) M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann.  Atomic Bose-Fermi mixtures in an optical lattice.  _Physical Review Letters_ , 92(5):050401, February 2004.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.92.050401.
  * Pollet et al. (2006) Lode Pollet, Matthias Troyer, Kris Van Houcke, and Stefan M. A. Rombouts.  Phase diagram of Bose-Fermi mixtures in one-dimensional optical lattices.  _Physical Review Letters_ , 96(19):190402, May 2006.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.96.190402.
  * Büchler and Blatter (2004) HP Büchler and G Blatter.  Phase separation of atomic Bose-Fermi mixtures in an optical lattice.  _Physical Review A_ , 69(6):063603, 2004.
  * Günter et al. (2006) Kenneth Günter, Thilo Stöferle, Henning Moritz, Michael Köhl, and Tilman Esslinger.  Bose-Fermi mixtures in a three-dimensional optical lattice.  _Physical Review Letters_ , 96(18):180402, 2006\.
  * Delehaye et al. (2015) Marion Delehaye, Sébastien Laurent, Igor Ferrier-Barbut, Shuwei Jin, Frédéric Chevy, and Christophe Salomon.  Critical velocity and dissipation of an ultracold Bose-Fermi counterflow.  _Physical review letters_ , 115(26):265303, 2015\.
  * Roy et al. (2017) Richard Roy, Alaina Green, Ryan Bowler, and Subhadeep Gupta.  Two-element mixture of Bose and Fermi superfluids.  _Physical review letters_ , 118(5):055301, 2017\.
  * DeSalvo et al. (2017) BJ DeSalvo, Krutik Patel, Jacob Johansen, and Cheng Chin.  Observation of a degenerate fermi gas trapped by a bose-einstein condensate.  _Physical Review Letters_ , 119(23):233401, 2017\.  doi: 10.1103/PhysRevLett.119.233401.
  * Yao et al. (2016) Xing-Can Yao, Hao-Ze Chen, Yu-Ping Wu, Xiang-Pei Liu, Xiao-Qiong Wang, Xiao Jiang, Youjin Deng, Yu-Ao Chen, and Jian-Wei Pan.  Observation of coupled vortex lattices in a mass-imbalance Bose and Fermi superfluid mixture.  _Physical Review Letters_ , 117(14):145301, 2016\.
  * Edri et al. (2020) Hagai Edri, Boaz Raz, Noam Matzliah, Nir Davidson, and Roee Ozeri.  Observation of Spin-Spin Fermion-Mediated Interactions between Ultracold Bosons.  _Physical Review Letters_ , 124(16):163401, April 2020.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.124.163401.
  * Ruderman and Kittel (1954) Melvin A Ruderman and Charles Kittel.  Indirect exchange coupling of nuclear magnetic moments by conduction electrons.  _Physical Review_ , 96(1):99, 1954.
  * Johnson et al. (2016) TH Johnson, Y Yuan, W Bao, SR Clark, C Foot, and D Jaksch.  Hubbard model for atomic impurities bound by the vortex lattice of a rotating bose-einstein condensate.  _Physical Review Letters_ , 116(24):240402, 2016\.
  * Nygaard and Mølmer (1999) Nicolai Nygaard and Klaus Mølmer.  Component separation in harmonically trapped boson-fermion mixtures.  _Physical Review A_ , 59(4):2974, 1999.
  * Das (2003) Kunal K. Das.  Bose-Fermi Mixtures in One Dimension.  _Physical Review Letters_ , 90(17):170403, May 2003.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.90.170403.
  * Repp et al. (2013) M Repp, R Pires, J Ulmanis, R Heck, ED Kuhnle, M Weidemüller, and E Tiemann.  Observation of interspecies 6 li-133 cs feshbach resonances.  _Physical Review A_ , 87(1):010701, 2013.
  * Mering and Fleischhauer (2010) Alexander Mering and Michael Fleischhauer.  Fermion-mediated long-range interactions of bosons in the one-dimensional Bose-Fermi-Hubbard model.  _Physical Review A_ , 81(1):011603, January 2010\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.81.011603.
  * De and Spielman (2014) S De and IB Spielman.  Fermion-mediated long-range interactions between bosons stored in an optical lattice.  _Applied Physics B_ , 114(4):527–536, 2014.
  * Chanda et al. (2022) Titas Chanda, Daniel González-Cuadra, Maciej Lewenstein, Luca Tagliacozzo, and Jakub Zakrzewski.  Devil’s staircase of topological Peierls insulators and Peierls supersolids.  _SciPost Physics_ , 12(2):076, 2022.
  * Baumann et al. (2010) Kristian Baumann, Christine Guerlin, Ferdinand Brennecke, and Tilman Esslinger.  Dicke quantum phase transition with a superfluid gas in an optical cavity.  _nature_ , 464(7293):1301–1306, 2010.
  * Léonard et al. (2017) Julian Léonard, Andrea Morales, Philip Zupancic, Tilman Esslinger, and Tobias Donner.  Supersolid formation in a quantum gas breaking a continuous translational symmetry.  _Nature_ , 543(7643):87–90, 2017.
  * Adhikari (2005) Sadhan K. Adhikari.  Fermionic bright soliton in a boson-fermion mixture.  _Physical Review A_ , 72(5):053608, November 2005\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.72.053608.
  * Tylutki et al. (2016) Marek Tylutki, Alessio Recati, Franco Dalfovo, and Sandro Stringari.  Dark–bright solitons in a superfluid Bose–Fermi mixture.  _New Journal of Physics_ , 18(5):053014, 2016\.
  * Chiao et al. (1964) Raymond Y Chiao, E Garmire, and Charles H Townes.  Self-trapping of optical beams.  _Physical review letters_ , 13(15):479, 1964.
  * Chen and Hung (2021) Cheng-An Chen and Chen-Lung Hung.  Observation of scale invariance in two-dimensional matter-wave Townes solitons.  _Physical Review Letters_ , 127(2):023604, 2021\.
  * Bakkali-Hassani et al. (2021) B Bakkali-Hassani, C Maury, Y-Q Zou, É Le Cerf, R Saint-Jalm, Patricia Christina Marques Castilho, Sylvain Nascimbene, Jean Dalibard, and Jérôme Beugnon.  Realization of a Townes soliton in a two-component planar Bose gas.  _Physical Review Letters_ , 127(2):023603, 2021\.
  * Krutitsky et al. (2010) Konstantin V Krutitsky, Jonas Larson, and Maciej Lewenstein.  Dark solitons near the Mott-insulator–superfluid phase transition.  _Physical Review A_ , 82(3):033618, 2010.
  * Rubbo et al. (2012) Chester P Rubbo, Indubala I Satija, William P Reinhardt, Radha Balakrishnan, Ana Maria Rey, and Salvatore R Manmana.  Quantum dynamics of solitons in strongly interacting systems on optical lattices.  _Physical Review A_ , 85(5):053617, 2012.
  * Salasnich et al. (2007) Luca Salasnich, Sadhan K. Adhikari, and Flavio Toigo.  Self-bound droplet of Bose and Fermi atoms in one dimension: Collective properties in mean-field and Tonks-Girardeau regimes.  _arXiv:cond-mat/0701752_ , January 2007.
  * Rakshit et al. (2019) Debraj Rakshit, Tomasz Karpiuk, Paweł Zin, Mirosław Brewczyk, Maciej Lewenstein, and Mariusz Gajda.  Self-bound Bose–Fermi liquids in lower dimensions.  _New Journal of Physics_ , 21(7):073027, July 2019\.  ISSN 1367-2630.  doi: 10.1088/1367-2630/ab2ce3.
  * Wang et al. (2012) Hongmei Wang, Yajiang Hao, and Yunbo Zhang.  Density-functional theory for 1D harmonically trapped Bose-Fermi mixture.  _Physical Review A_ , 85(5):053630, May 2012.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.85.053630.
  * Karpiuk et al. (2004) T. Karpiuk, M. Brewczyk, S. Ospelkaus-Schwarzer, K. Bongs, M. Gajda, and K. Rzazewski.  Soliton trains in Bose-Fermi mixtures.  _Physical Review Letters_ , 93(10):100401, September 2004.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.93.100401.
  * Solomon et al. (1999) A. I. Solomon, Y. Feng, and V. Penna.  On the Structure of the Bose-Einstein Condensate Ground State.  _Physical Review B_ , 60(5):3044–3047, August 1999.  ISSN 0163-1829, 1095-3795.  doi: 10.1103/PhysRevB.60.3044.
  * Worth and Cederbaum (2004) Graham A Worth and Lorenz S Cederbaum.  Beyond born-oppenheimer: Molecular dynamics through a conical intersection.  _Annu. Rev. Phys. Chem._ , 55:127–158, 2004.
  * Larson et al. (2020) Jonas Larson, Erik Sjöqvist, and Patrik Öhberg.  _Conical Intersections in Physics_.  Springer, 2020.
  * Feit et al. (1982) M.D Feit, J.A Fleck, and A Steiger.  Solution of the Schrödinger equation by a spectral method.  _Journal of Computational Physics_ , 47(3):412–433, September 1982.  ISSN 00219991.  doi: 10.1016/0021-9991(82)90091-2.
  * Molina et al. (2007) Rafael A Molina, Jorge Dukelsky, and Peter Schmitteckert.  Commensurability effects for fermionic atoms trapped in 1D optical lattices.  _Physical review letters_ , 99(8):080404, 2007\.
  * Parr and Yang (1989) Robert G. Parr and Weitao Yang.  _Density-Functional Theory of Atoms and Molecules_.  Number 16 in International Series of Monographs on Chemistry. Oxford University Press ; Clarendon Press, New York : Oxford [England], 1989.  ISBN 978-0-19-504279-5.
  * Spruch (1991) Larry Spruch.  Pedagogic notes on Thomas-Fermi theory (and on some improvements): Atoms, stars, and the stability of bulk matter.  _Reviews of Modern Physics_ , 63(1):151, 1991\.
  * March et al. (2000) N.H. March, P. Senet, and V.E. Van Doren.  Non-local kinetic energy functional for an arbitrary number of Fermions moving independently in one-dimensional harmonic oscillator potential.  _Physics Letters A_ , 270(1-2):88–92, May 2000\.  ISSN 03759601.  doi: 10.1016/S0375-9601(00)00288-7.
  * Salasnich et al. (2002) L. Salasnich, A. Parola, and L. Reatto.  Effective wave-equations for the dynamics of cigar-shaped and disc-shaped Bose condensates.  _Physical Review A_ , 65(4):043614, April 2002\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.65.043614.
  * Mathey (2007) L. Mathey.  Commensurate mixtures of ultra-cold atoms in one dimension.  _Physical Review B_ , 75(14):144510, April 2007\.  ISSN 1098-0121, 1550-235X.  doi: 10.1103/PhysRevB.75.144510.
  * Ernst et al. (2010) Philipp T Ernst, Sören Götze, Jasper S Krauser, Karsten Pyka, Dirk-Sören Lühmann, Daniela Pfannkuche, and Klaus Sengstock.  Probing superfluids in optical lattices by momentum-resolved bragg spectroscopy.  _Nature Physics_ , 6(1):56–61, 2010.
  * Heinze et al. (2011) J Heinze, S Götze, JS Krauser, B Hundt, N Fläschner, D-S Lühmann, C Becker, and K Sengstock.  Multiband spectroscopy of ultracold fermions: observation of reduced tunneling in attractive bose-fermi mixtures.  _Physical Review Letters_ , 107(13):135303, 2011\.
  * Bakr et al. (2009) Waseem S Bakr, Jonathon I Gillen, Amy Peng, Simon Fölling, and Markus Greiner.  A quantum gas microscope for detecting single atoms in a hubbard-regime optical lattice.  _Nature_ , 462(7269):74–77, 2009.
  * Haller et al. (2015) Elmar Haller, James Hudson, Andrew Kelly, Dylan A Cotta, Bruno Peaudecerf, Graham D Bruce, and Stefan Kuhr.  Single-atom imaging of fermions in a quantum-gas microscope.  _Nature Physics_ , 11(9):738–742, 2015.
  * Boll et al. (2016) Martin Boll, Timon A Hilker, Guillaume Salomon, Ahmed Omran, Jacopo Nespolo, Lode Pollet, Immanuel Bloch, and Christian Gross.  Spin-and density-resolved microscopy of antiferromagnetic correlations in fermi-hubbard chains.  _Science_ , 353(6305):1257–1260, 2016.
  * Scherg et al. (2021) Sebastian Scherg, Thomas Kohlert, Pablo Sala, Frank Pollmann, Bharath Hebbe Madhusudhana, Immanuel Bloch, and Monika Aidelsburger.  Observing non-ergodicity due to kinetic constraints in tilted fermi-hubbard chains.  _Nature Communications_ , 12(1):4490, 2021.
  * Jördens et al. (2008) Robert Jördens, Niels Strohmaier, Kenneth Günter, Henning Moritz, and Tilman Esslinger.  A Mott insulator of fermionic atoms in an optical lattice.  _Nature_ , 455(7210):204–207, September 2008\.  ISSN 0028-0836, 1476-4687.  doi: 10.1038/nature07244.
  * Greif et al. (2016) Daniel Greif, Maxwell F. Parsons, Anton Mazurenko, Christie S. Chiu, Sebastian Blatt, Florian Huber, Geoffrey Ji, and Markus Greine r.  Site-resolved imaging of a fermionic Mott insulator.  _Science_ , 351(6276):953–957, 2016.
  * Rigol et al. (2003) M Rigol, Alejandro Muramatsu, GG Batrouni, and RT Scalettar.  Local quantum criticality in confined fermions on optical lattices.  _Physical review letters_ , 91(13):130403, 2003\.
  * Rigol and Muramatsu (2004) Marcos Rigol and Alejandro Muramatsu.  Quantum Monte Carlo study of confined fermions in one-dimensional optical lattices.  _Physical Review A_ , 69(5):053612, 2004.
  * Schneider et al. (2008) U Schneider, L Hackermuller, S Will, Th Best, Immanuel Bloch, Th A Costi, RW Helmes, D Rasch, and A Rosch.  Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice.  _Science (New York, N.Y.)_ , 322(5907):1520–1525, 2008.
  * Cloizeaux (1964) Jacques Des Cloizeaux.  Analytical Properties of n -Dimensional Energy Bands and Wannier Functions.  _Physical Review_ , 135(3A):A698–A707, August 1964.  ISSN 0031-899X.  doi: 10.1103/PhysRev.135.A698.
  * He and Vanderbilt (2001) Lixin He and David Vanderbilt.  Exponential decay properties of Wannier functions and related quantities.  _Physical Review Letters_ , 86(23):5341–5344, June 2001.  ISSN 0031-9007, 1079-7114.  doi: 10.1103/PhysRevLett.86.5341.
  * Sachdev (1999) Subir Sachdev.  Quantum phase transitions.  _Physics world_ , 12(4):33, 1999.
  * Maluckov et al. (2013) Aleksandra Maluckov, Goran Gligorić, Ljupčo Hadžievski, Boris A Malomed, and Tilman Pfau.  High-and low-frequency phonon modes in dipolar quantum gases trapped in deep lattices.  _Physical Review A_ , 87(2):023623, 2013.
  * Prezhdo and Kisil (1997) Oleg V. Prezhdo and Vladimir V. Kisil.  Mixing Quantum and Classical Mechanics.  _Physical Review A_ , 56(1):162–175, July 1997\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.56.162.
  * Caro and Salcedo (1999) J. Caro and L. L. Salcedo.  Impediments to mixing classical and quantum dynamics.  _Physical Review A_ , 60(2):842–852, August 1999\.  ISSN 1050-2947, 1094-1622.  doi: 10.1103/PhysRevA.60.842.
  * Oppenheim et al. (2020) Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda, and Zachary Weller-Davies.  Objective trajectories in hybrid classical-quantum dynamics.  _arXiv:2011.06009 [hep-th, physics:quant-ph]_ , November 2020.
  * M. and I. (1983) Abramowitz M. and Stegun I.  _Handbook of Mathematical Functions_.  Dover, New York, 1983.  Entries 13.6.38 and 13.5.16.