# Pairing phenomena and superfluidity of atomic Fermi gases in a two-
dimensional optical lattice: Unusual effects of lattice-continuum mixing

Lin Sun  Department of Physics and Zhejiang Institute of Modern Physics,
Zhejiang University, Hangzhou, Zhejiang 310027, China  Jibiao Wang  School of
Materials and New Energy, South China Normal University, Shanwei, Guangdong
516625, China  Xiang Chu  Department of Physics and Zhejiang Institute of
Modern Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China  Qijin
Chen<EMAIL_ADDRESS>Hefei National Laboratory for Physical Sciences at
Microscale and School of Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China  Shanghai Branch, CAS Center for Excellence
and Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
Department of Physics and Zhejiang Institute of Modern Physics, Zhejiang
University, Hangzhou, Zhejiang 310027, China

(August 27, 2024)

###### Abstract

We study the superfluid behavior of ultracold atomic Fermi gases with a short
range attractive interaction in a two-dimensional optical lattice (2DOL) using
a pairing fluctuation theory, within the context of BCS-BEC crossover. We find
that the mixing of lattice and continuum dimensions leads to exotic phenomena.
For relatively large lattice constant $d$ and small hopping integral $t$, the
superfluid transition temperature $T_{c}$ exhibits a remarkable reentrant
behavior as a function of the interaction strength, and leads to a pair
density wave ground state, where $T_{c}$ vanishes, for a range of intermediate
coupling strength. In the unitary and BCS regimes, the nature of the in-plane
and overall pairing changes from particle-like to hole-like, with an
unexpected nonmonotonic dependence of the chemical potential on the pairing
strength. The BEC asymptotic behaviors exhibit distinct power law dependencies
on the interaction strength compared to cases of pure 3D lattice, 3D
continuum, and 1DOL. These predictions can be tested in future experiments.
Ann Phys (Berlin) 2022, 2100511

##  I Introduction

Ultracold Fermi gases have become a powerful platform for investigating
condensed matter physics over the past decade, due to their multiple
experimentally tunable control parameters [1, 2]. They can exhibit a perfect
crossover from a BCS type of superfluidity to Bose-Einstein condensation (BEC)
of atom pairs, as the pairing strength is tuned from weak to strong via a
Feshbach resonance [3]. In particular, they can be put in an optical lattice
with a tunable geometry so that rich physics may emerge [4, 5, 6]. In a three
dimensional (3D) lattice, it is known that the superfluid transition
temperature $T_{c}$ scales as $-t^{2}/U$ in the BEC regime, due to “virtual
ionization” in the pair hopping process [7, 8], where $U<0$ is the onsite
attractive interaction, and $t$ is the hopping integral. This is to be
contrasted with the 3D continuum case, in which $T_{c}$ approaches
asymptotically $0.218T_{F}$ in the BEC regime, with $T_{F}$ being the Fermi
temperature. Given the sharp contrast between the 3D continuum and the 3D
lattice cases, usual behaviors may be expected when the spatial dimensions
contain both lattice and continuum. Recently, as such a lattice-continuum
mixed system, the superfluid behavior of Fermi gases in a 1D optical lattice
(1DOL), in the context of BCS-BEC crossover, has been studied. Interesting
effects of lattice-continuum mixing have indeed been found. For example, in
the population balanced case, the behavior of $T_{c}$ exhibits lattice effect;
$T_{c}$ decreases with increasing coupling strength in the BEC regime, with a
scaling of $T_{c}=\pi an/2m$, where $a$ is the $s$-wave scattering length, $n$
the fermion density and $m$ the fermion mass [9]. In contrast, in the presence
of population imbalance [10], $T_{c}$ often approaches a constant BEC
asymptote, mimicking its counterpart in the 3D continuum.

Two dimensional optical lattices (2DOL) have also been studied theoretically
[11, 12, 13]. They corresponds to a 2D array (2D lattice in the $xy$ plane) of
1D tubes (1D continuum alone the $z$ direction) in experiment, with lattice
constant $d$ and hopping integral $t$, subject to a short-range pairing
interaction $U(<0)$. 2DOL of 6Li in the (quasi-)1D limit has been realized
experimentally [14, 15, 16]. None of these theoretical and experimental
studies paid attention to or was aware of possible lattice-continuum mixing
effects in a more general parameter range. Most of them have focused on the 1D
or quasi-1D physics, including the Fulde-Ferrell-Larkin-Ovchinnikov states
(FFLO) state [17, *LO] in the presence of a population imbalance. Dimensional
crossover was also studied [16] in a population imbalanced Fermi gas of 6Li
atoms, with a focus on how phase separation evolves with $t$ and the
interactions. Apparently, the lattice-continuum effects may not show up for
any arbitrary choices of $d$ and $t$ as well as interaction strength.

In this paper, we investigate the superfluid behavior of a balanced two-
component Fermi gas in a 2DOL. We find a much stronger effect of lattice-
continuum mixing than in the 1DOL case, besides the wide spread pseudogap
phenomena [1]. In particular, two lattice dimensions provide a tighter
confinement in the momentum space. Via increasing lattice constant $d$ and
decreasing hopping integral $t$, such that the Fermi surface becomes open (in
contrast to a 3D closed ellipsoid), the effective number density in the
lattice dimensions can go above half filling, and may turn the in-plane and
overall pairing from particle-like into hole-like in the BCS and unitary
regimes, with chemical potential $\mu$ increasing with increasing pairing
strength, while it remains particle-like in the BEC regime. While the decrease
of $T_{c}$ in the BEC regime manifests strong lattice effects, we find that
for large $d$ and small $t$, $T_{c}$ exhibits a re-entrant behavior in the
regime of intermediate pairing strength; a pair density wave (PDW) emerges as
the ground state where $T_{c}$ vanishes. In the BEC regime, the momentum space
becomes quasi-1D, so that $T_{c}$ approaches
$\frac{\sqrt{6}ntd^{4}}{m}|U|^{-1}$ asymptotically, while $\mu\sim-U^{2}$ is
dominated by a quadratic dependence on $U$, with the pairing gap
$\Delta\sim|\mu|^{3/4}\sim|U|^{3/2}$. These behaviors are distinct from their
counterpart in 3D lattice, 3D continuum or 1DOL.

It is known that the superfluid phase in Fermi gases in 3D optical lattices
has yet not been accessible experimentally, due to the difficulty of deep
cooling. On the contrary, superfluidity in 1D or 2D optical lattices have been
realized long ago, e.g., in the early phase separation observation [19].

##  II Theoretical Formalism

###  II.1 Pairing ﬂuctuation theory

To focus on the effects of lattice-continuum mixing, we shall focus on the
2DOL without the complication of population imbalance. Under the one-band
tight-binding approximation for the lattice dimensions in the $xy$ plane, the
dispersion of noninteracting atoms is given by
$\xi_{\mathbf{k}}=\epsilon_{\mathbf{k}}-\mu\equiv{k_{z}}^{2}/2m+2t[2-\cos(k_{x}d)-\cos(k_{y}d)]-\mu$.
Following our recent studies [20, 21, 9, 10], we take $t$ to be small in our
calculation, under the constraint $2mtd^{2}<1$, so that it is physically
accessible. The critical coupling strength, which corresponds to a diverging
$s$-wave scattering length $a$ in the unitary limit in the presence of the
lattice, is given by
$U_{c}=-1/\sum_{\textbf{k}}1/2\epsilon_{\textbf{k}}=-0.16072\sqrt{2m}/\sqrt{t}d^{2}$.
Note that a big difference between 2DOL and 1DOL as well as 3D continuum is
that $U_{c}=0$ in the contact potential limit for the latter two cases. As a
consequence, $1/k_{F}a$ as defined via the Lippman-Schwinger equation is upper
bounded in the present case. Thus 2DOL is closer to the 3D lattice case in
this sense. In comparison with the pure 3D lattice case, however, the momentum
distribution will spread to large $k_{z}$ (in the BEC case or for small $t$
and large $d$), leading to a quasi-1D momentum space, with the kinetic energy
distributed mainly along the $k_{z}$ direction. For 3D lattice, which has a
finite momentum space defined by the first Brillouin zone (BZ), there is a
finite filling factor for a given fermion density. However, the overall
filling factor is zero for 2DOL. Nevertheless, as we show below, one can tune
$t$ and $d$ such that the effective filling factor within the two lattice
dimensions varies between 0 and 1.

With the modified atomic dispersion $\xi_{\mathbf{k}}$ for 2DOL, we follow the
pairing fluctuation theory previously developed for the pseudogap physics in
the cuprates [22], which has been extended to address the BCS-BEC crossover in
ultracold atomic Fermi gases [1]. This theory goes beyond the BCS mean-field
approximation by self-consistently including finite momentum pairing
correlations in the single particle self energy. Thus the self energy
$\Sigma(K)=\Sigma_{sc}(K)+\Sigma_{pg}(K)$ contains two parts, where the Cooper
pair condensate contribution $\Sigma_{sc}(K)=-\Delta_{sc}^{2}G_{0}(-K)$
vanishes above $T_{c}$, and finite momentum pair contribution
$\Sigma_{pg}(K)=\sum_{Q}t_{pg}(Q)G_{0}(Q-K)$ exists both above and below
$T_{c}$. Here $G_{0}$ and $G$ are the non-interacting and full Green’s
functions, respectively, $\Delta_{sc}$ the order parameter and
$t_{pg}(Q)=U/[1+U\chi(Q)]$ the pairing $T$ matrix, with pair susceptibility
$\chi(Q)=\sum_{K}G_{0}(Q-K)G(K)$. We take $\hbar=k_{B}=1$ and use four momenta
$K\equiv(\omega_{n},\mathbf{k})$, $Q\equiv(\Omega_{l},\mathbf{q})$,
$\sum_{Q}\equiv T\sum_{l}\sum_{\mathbf{q}}$, etc, where $\omega_{n}$
($\Omega_{l}$) is the odd (even) Matsubara frequency, following the notations
of Ref. [22] 111We also drop the volume $V$, which can be set to unity and
will be canceled out in the end..

We now recapitulate the main result of this theory. In the superfluid phase,
one can define the pseudogap via $\Delta_{pg}^{2}=-\sum_{Q}t_{pg}(Q)$, so that
the total gap $\Delta$ is given by
$\Delta^{2}=\Delta_{sc}^{2}+\Delta_{pg}^{2}$, which leads to the self energy
$\Sigma(K)\approx-\Delta^{2}G_{0}(-K)$, and the BCS form of the full Green’s
function
$G(K)=\frac{u_{\textbf{k}}^{2}}{i\omega_{n}-E_{\textbf{k}}}+\frac{v_{\textbf{k}}^{2}}{i\omega_{n}+E_{\textbf{k}}}\,,$
where $u_{\textbf{k}}^{2}=(1+\xi_{\textbf{k}}/E_{\textbf{k}})/2$,
$v_{\textbf{k}}^{2}=(1-\xi_{\textbf{k}}/E_{\textbf{k}})/2$, and
$E_{\textbf{k}}=\sqrt{\xi_{\textbf{k}}^{2}+\Delta^{2}}$. Then we have the
number equation,

$n=\sum_{\textbf{k}}\Big{[}\Big{(}1-\frac{\xi_{\textbf{k}}}{E_{\textbf{k}}}\Big{)}+2\frac{\xi_{\textbf{k}}}{E_{\textbf{k}}}f(E_{\textbf{k}})\Big{]}\,,\\\
$ (1)

where $f(x)$ is the Fermi distribution function. At $T\leq T_{c}$, the
Thouless criteria for pairing instability, $U^{-1}+\chi(0)=0$, leads to the
gap equation,

$0=\frac{1}{U}+\sum_{\textbf{k}}\Big{[}\frac{1-2f(E_{\textbf{k}})}{2E_{\textbf{k}}}\Big{]}\,.$
(2)

To evaluate the pseudogap, one can Taylor expand $t_{pg}^{-1}(Q)$ on the real
frequency axis, after analytical continuation,
$t_{pg}^{-1}(\Omega,\textbf{q})\approx
a_{1}\Omega^{2}+a_{0}(\Omega-\Omega_{\textbf{q}}+\mu_{p})$, with
$\Omega_{\textbf{q}}=B{q_{z}}^{2}+2t_{B}[2-\cos(q_{x}d)-\cos(q_{y}d)]$, and
$\mu_{p}=0$ at $T\leq T_{c}$. Here $B=1/2M$, with $M$ being the effective pair
mass in the $z$ direction, and $t_{B}$ the effective pair hopping integral in
the $xy$ plane. We stress that the $a_{1}$ term serves as a small quantitative
correction in the BEC regime, where $a_{1}T_{c}\ll a_{0}$. Then this
$T$-matrix expansion leads to the pseudogap equation

$|a_{0}|\Delta_{pg}^{2}=\sum_{\textbf{q}}\frac{b(\tilde{\Omega}_{\textbf{q}})}{\sqrt{1+4\dfrac{a_{1}}{a_{0}}\Omega_{\textbf{q}}}}\,,$
(3)

where $b(x)$ is the Bose distribution function and
$\tilde{\Omega}_{\textbf{q}}=[\sqrt{a_{0}^{2}+4a_{1}a_{0}\Omega_{\textbf{q}}}-a_{0}]/2a_{1}$
is the pair dispersion, which reduces to
$\tilde{\Omega}_{\textbf{q}}=\Omega_{\textbf{q}}$ when $a_{1}T_{c}/a_{0}\ll
1$, e.g., in the BEC regime. In this case, one can see that the pair density
is roughly given by $n_{p}=a_{0}\Delta^{2}$

Equations (1)-(3) form a closed set of self-consistent equations, which can be
used to solve for ($\mu$, $\Delta_{pg}$, $T_{c}$) with $\Delta_{sc}=0$. For
our numerics, we define Fermi momentum $k_{F}=(3\pi^{2}n)^{1/3}$ and Fermi
energy $E_{F}\equiv k_{B}T_{F}=\hbar^{2}k_{F}^{2}/2m$, as for the 3D continuum
with the same density $n$ and atomic mass $m$.

###  II.2 Asymptotic behaviors in the BEC limit

The asymptotic behavior at $T_{c}$ in the BEC limit can be solved
analytically, with $\mu\rightarrow-\infty$, so that
$f(E_{\mathbf{k}})=f(\xi_{\mathbf{k}})=0$. From the gap equation, we obtain

$\mu=-\frac{m}{8d^{4}}U^{2}+\frac{3n}{4}|U|\,.$ (4)

The leading order quadratic dependence of $\mu$ on $U$ is related to the
quasi-one dimensionality caused by the restricted momentum space for
($k_{x}$,$k_{y}$) within the first BZ. This should be contrasted to the 3D
lattice case, for which $\mu\sim U$ in the BEC limit 222For 1DOL and 3D
continuum, $U$ is renormalized down to zero in the contact limit so that an
appropriate parameter is scattering length $a$. We have $\mu\sim-e^{d/a}$ and
$\mu\sim-1/a^{2}$ for the two cases, respectively.. From the number equation,
we get

$\Delta=\left(\frac{8d^{2}n}{\sqrt{2m}}\right)^{1/2}|\mu|^{3/4}\,.$ (5)

This $\Delta\sim|\mu|^{3/4}\sim|U|^{3/2}$ scaling behavior should be
contrasted to $\Delta\sim\mu\sim|U|$ for 3D lattice, $\Delta\sim|\mu|^{1/2}$
for 1DOL (quasi-2D system) [9], and $\Delta\sim|\mu|^{1/4}$ for 3D continuum
[1].

For the effective pair mass in the continuum dimension and the effective pair
hopping integral in the lattice dimensions, one can easily obtain

$B=\frac{1}{4m},\quad\mbox{and}\quad t_{B}=\frac{3t^{2}}{4|\mu|},$ (6)

which yields

$n\approx\frac{\sqrt{T_{c}}}{\sqrt{\pi B}d^{2}}Li_{1/2}(e^{-4t_{B}/T_{c}})$
(7)

in the BEC regime via the pseudogap equation (3), where
$Li_{s}(z)=\sum_{n=1}^{\infty}(z^{n}/n^{s})$ is the polylogarithm function.
Note here the $a_{1}$ term is negligible, so that $n_{p}=a_{0}\Delta^{2}=n/2$.
Thus we obtain the BEC asymptotic solution

$T_{c}\approx 2nd^{2}\sqrt{Bt_{B}}\approx\frac{\sqrt{6}ntd^{4}}{m}|U|^{-1}$
(8)

using the lowest order expansion, $Li_{1/2}^{-2}(1-z)\approx z/\pi$ for
$z\rightarrow 0$. This indicates that $T_{c}$ scales inversely proportional to
$U$ in the deep BEC regime, similar to the 3D lattice case.

##  III Results and Discussions

Shown in the inset of Fig. 1(a) is a typical phase diagram of Fermi gases in
2DOL, with $t/E_{F}=0.05$ and $k_{F}d=2$, which satisfies $2mtd^{2}=0.2<1$. A
semi-log scale is used to show the full interaction range. The black solid and
red dashed curves are $T_{c}$ and the pair formation temperature $T^{*}$
(which is approximated by the mean-field solution for $T_{c}$), respectively,
so that the yellow shaded area indicates the superfluid phase. A pseudogap
state exists between $T^{*}$ and $T_{c}$. As usual, $T_{c}$ follows the BCS
solution in the weak-coupling regime. Then a pseudogap emerges as $|U|$
increases, which suppresses $T_{c}$ relative to $T^{*}$. After reaching a
maximum around unitarity at $U=U_{c}$, $T_{c}$ starts to decrease due to a
shrinking Fermi surface. A minimum is reached when $\mu=0$, where the Fermi
surface shrinks to zero and essentially all fermions have paired up. Beyond
this point, as the pairing becomes stronger, the system enters the bosonic
regime, with $\mu<0$, and $T_{c}$ first rises due to shrinking pair size and
hence decreasing pair mass, and then decreases towards the BEC asymptote, as
governed by the lattice effect [7, 8].

We note that a constant Hartree energy, $\Sigma_{\text{HF}}=Un/2$, has been
absorbed in our chemical potential $\mu$ 333See Eq. (2.4) of Ref. [32]. The
actual chemical potential is given by $\mu^{\prime}=\mu+Un/2$. Nevertheless,
we emphasize that the vanishing of Fermi surface is defined by $\mu=0$ rather
than $\mu^{\prime}=0$. 444In principle, $\mu$, when positive, can be measured
experimentally by detecting $k_{\mu}=\sqrt{2m\mu}$ (in the $z$ direction) on
the Fermi surface directly via measurement of the fermion momentum
distribution, e.g., using the time-of-flight imaging technique.

Figure 1: Phase diagram and effect of varying $d$ and $t$: Behavior of $T_{c}$
versus $U/U_{c}$ with (a) fixed $t/E_{F}=0.1$ for varying $k_{F}d$, and (b)
fixed $k_{F}d=2$ for varying $t/E_{F}$, as labeled. The insets of (a) and (b)
show typical phase diagrams without and with a PDW phase, respectively.

To demonstrate the effect of varying $d$ and $t$, we plot in Fig. 1 $T_{c}$
versus $U/U_{c}$ with (a) fixed $t/E_{F}=0.1$ but for varying $k_{F}d$ from
0.5 through 3, and (b) fixed $k_{F}d=2$ but for varying $t/E_{F}$ from 0.01
through 0.2. For fixed $t$ in panel (a), the maximum $T_{c}$,
${T_{c}}^{\text{max}}$, increases with $d$. At the same time, the entire
$T_{c}$ curve also expands horizontally, as $d$ increases. Since the lattice
momentum ($k_{x},k_{y}$) is restricted to the first BZ, fermions almost fully
occupy the $xy$-plane states when $t$ is small [9]. As $d$ increases, the BZ
shrinks, and more fermions have to occupy higher $k_{z}$ state, so that $\mu$
is pushed up, leading to higher $T_{c}$. Similarly in panel (b), as $t$
decreases with fixed $d$, ${T_{c}}^{\text{max}}$ decreases and moves toward
the BEC side of unitarity, as if the entire $T_{c}$ curve is expanding
horizontally toward the BEC regime. For fixed $d$, more fermions will occupy
the first BZ in the $xy$ plane, as $t$ decreases. This will remove the high
$k_{z}$ fermions, and decrease $\mu$. Meanwhile, a small $t$ causes difficulty
for pair hopping and thus suppresses $T_{c}$, making the system more 1D.

Remarkably, we can see that $T_{c}$ exhibits a reentrant behavior for large
$d$ and small $t$. This is illustrated more clearly as an example in the inset
of Fig. 1(b) for $k_{F}d=3$ and $t/E_{F}=0.05$. For a range of intermediate
coupling strength, $T_{c}$ shuts off completely before it recovers at stronger
couplings in the BEC regime, and PDW states emerge where $T_{c}$ vanishes,
which is also found in dipolar Fermi gases within our pairing fluctuation
theory [27].

Figure 2: Evolution of (a) $\mu$ and (b) the coefficients $a_{0}$ and $a_{1}$
(inset) at $T_{c}$ as a function of $U/U_{c}$ for $t/E_{F}=0.05$ and
$k_{F}d=3$. $T=0$ is used in the PDW state. Also shown is the 3D plot of the
corresponding Fermi surface.

Shown in Fig. 2 are the behaviors of (a) $\mu$ and (b) the coefficients
$a_{0}$ and $a_{1}$ (inset) as a function of $U/U_{c}$ for $t/E_{F}=0.05$ and
$k_{F}d=3$. It is remarkable that $\mu$ increases first as the pairing becomes
stronger, which is distinct from the 3D continuum and 3D lattice cases below
half filling. Even more remarkable is that $a_{0}<0$ before $\mu$ reaches its
maximum near $U/U_{c}=1.38$, even though the monotonic behavior of $a_{1}$ is
rather conventional. For larger $U/U_{c}$, $a_{0}$ becomes positive, and then
starts to decrease with $U/U_{c}$ in the BEC regime, where $\mu$ becomes
negative. The _unusual_ behaviors of $\mu$ and $a_{0}$ can be explained by the
Fermi surface, as shown in the inset. It has open ends, in contrast to a
closed ellipsoid for cases of large $t$ and small $d$. As is well known, in
the 3D lattice case above half filling, the Fermi surface becomes hole-like,
for which $a_{0}$ is negative and the chemical potential $\mu$ increases as
pairing strength increases. This can be easily understood via a particle-hole
transformation. Therefore, the sign of $a_{0}$ determines whether the fermion
pairs are particle-like or hole-like. The negative sign has to do with the
negative quasiparticle velocity on the hole-like Fermi surface. For the
present 2DOL case shown in Fig. 2, the effective filling factor in the first
BZ of the $xy$ plane is above 1/2, so that the contributions from lattice
dimensions dominate the sign of $a_{0}$ in the unitary and weak-coupling
regimes, even though the pairing in the $z$-direction always remains particle-
like. In the BEC regime, the gap becomes large so that the contributions of
the lattice dimensions spread across the entire BZ and become less dominant.
As a result, the continuum dimension will dominate the sign of $a_{0}$ in the
BEC regime. For hole-like pairing (as shown in the yellow region in the inset
of Fig. 3), we have $a_{0}<0$ and thus $n_{p}<0$, as required by the fact
$\mu$ is larger than its noninteracting counterpart, via the general relation
$n_{p}=n/2-\sum_{\mathbf{k}}f(\xi_{\mathbf{k}})$.

Figure 3: Behavior of (a) $B$ and (b) $t_{B}$ as well as $2n_{p}/n$ (inset) as
a function of $U/U_{c}$, calculated with the same parameters as in Fig. 2. The
yellow and grey shaded areas indicate hole-like pairing (with $a_{0}<0$) and
PWD state, respectively.

In Fig. 3, we show the corresponding behaviors of (a) the inverse pair mass
$B$ and (b) the pairing hopping integral $t_{B}$ as well as pair fraction
$2n_{p}/n$ (inset) as a function of $U/U_{c}$ for $t/E_{F}=0.05$ and
$k_{F}d=3$, calculated at $T_{c}$. We set $T=0$ for the PDW states. The
behavior of $B$ is rather conventional, with a minimum around $\mu=0$. In
contrast, $t_{B}$ is negative in the PDW state, leading to a vanishing
$T_{c}$, at a intermediate coupling strength, $3.19\leq U/U_{c}\leq 3.85$.
Upon the sign change of $t_{B}$ from positive to negative, the minimum of
$\tilde{\Omega}_{\textbf{q}}$ will literally move from $\mathbf{q}=0$ to
$\mathbf{q}=(\pi/d,\pi/d,0)$, within the lowest order tight-binding
approximation for the in-plane pair dispersion,
$\tilde{\Omega}_{\textbf{q}=(q_{x},q_{y},0)}$, as shown by the $U/U_{c}=3.5$
case in Fig. 4 (along the $q_{x}$ direction). This corresponds to a first
order phase transition from a homogeneous superfluid to a PDW ground state.
555When higher order wavevectors are included in the pair dispersion, the
minimum may be shifted to a different location. The physical origin of the PDW
states has to do with the strong inter-pair repulsive interaction for
relatively large pair size and moderately strong pairing interaction, which
leads to Wigner crystallization in the $xy$ plane. It remains unclear and will
be left for future studies whether the PDW phase can sustain superfluidity and
thus becomes a supersolid. Note that $\tilde{\Omega}_{\textbf{q}}$ becomes
gapped when $a_{0}<0$, which reduces the pseudogap.

Figure 4: Pair dispersion $\tilde{\Omega}_{\textbf{q}=(q_{x},0,0)}$ along the
$q_{x}$ direction for different $U/U_{c}$, as labeled, calculated with the
same parameters as in Fig. 2. For $U/U_{c}=3.5$, the $\Gamma$ point,
$(q_{x},q_{y},q_{z})=0$, becomes a local maximum of
$\tilde{\Omega}_{\textbf{q}=(q_{x},0,0)}$ in the $q_{x}$ and $q_{y}$
directions, signaling an instability of the conventional pair condensation at
$q=0$, which hence leads to a PDW state.

Figure 5: Temperature dependence of the gaps and the order parameter at and
below $T_{c}$ at unitarity, $U/U_{c}=1$, for $k_{F}d=2$ and $t/E_{F}=0.05$.
The order parameter $\Delta_{sc}$ decreases with $T$ and necessarily closes at
$T_{c}$. At very low $T$, $\Delta_{pg}^{2}\propto n_{\text{pair}}\propto
T^{3/2}$ so that $\Delta_{pg}\propto T^{3/4}$, which increases with $T$. Note
that there is already a rather large (pseudo)gap at $T_{c}$ in the unitary
regime.

.

Figure 6: Comparison between full numerical solutions (black solid) and BEC
asymptotic behaviors (red dashed lines) of (a) $-\mu$, (b) $\Delta$,(c)
$t_{B}$ and $B$ (inset), and (d) $T_{c}$ as a function of $U/U_{c}$ on a log-
log scale, for $t/E_{F}=0.05$ and $k_{F}d=2$. The green dotted line contains
higher order contributions in the expansion of $Li_{1/2}^{-2}(1-z)$

.

Next, we show in Fig. 5 the typical behaviors of the order parameter
$\Delta_{\text{sc}}$ and the gaps $\Delta$ and $\Delta_{\text{pg}}$ as a
function of temperature at unitarity. They are qualitatively similar to their
counterpart in the 3D continuum and 3D lattice cases. As $T$ increases from
zero, the order parameter decreases and vanishes at $T_{c}$, while the
pseudgap increases from zero and reaches its maximum at $T_{c}$. The increase
of $\Delta_{\text{pg}}$ reflects the nature that it results from finite
momentum pairing, whose population depends on thermal activation. The combined
total gap $\Delta$ follows the BCS mean-field-like behavior, and would
continue to exist up to $T^{*}$ far above $T_{c}$. As a function of
interaction strength, as $T_{c}$ decreases rapidly with decreasing $U/U_{c}$
towards the BCS regime, all gaps decrease and the ratio
$\Delta_{\text{pg}}(T_{c})/\Delta(0)$ decreases and becomes essentially zero
in the extreme BCS limit. In contrast, with increasing $U/U_{c}$ towards the
BEC regime, this ratio approaches unity, while all gaps increase. In this
case, $\Delta$ becomes essentially temperature independent, and system can be
regarded as a composite Bose gase.

Finally, we show in Fig. 6 the BEC asymptotic behavior (red dashed lines) of
(a) $-\mu$, (b) $\Delta$, (c) $t_{B}$ and $B$ (inset) and (d) $T_{c}$ as a
function of $U/U_{c}$ on a log-log scale for $t/E_{F}=0.05$ and $k_{F}d=2$,
obtained from Eqs. (4–6) and (8). For comparison, also plotted are the fully
numerical results (black solid lines) from Eqs. (1)-(3). It is obvious that
our asymptotic solution works well for $U/U_{c}\gtrsim 10$. Importantly, we
see that $T_{c}$ is governed by $\sqrt{Bt_{B}}$ in the BEC regime, as in Eq.
(8). A better asymptotic expression for $T_{c}$, shown as the green dotted
line in Fig. 6(d), can be obtained by including higher order effect via
fitting $Li_{1/2}^{-2}(1-z)$ with a power law in the range of $z\in[0,0.01]$,
which yields $T_{c}=2.22(nd^{2}\sqrt{B})^{0.964}t_{B}^{0.518}$.

It should be noted that the lattice constant $d$ is governed by the laser
wavelength, such as 532 nm or 1064 nm, which usually cannot be tuned
continuously in a typical laboratory. However, one can tune the density $n$
such that the dimensionless parameter $k_{F}d$ can be varied continuously. As
for the parameters $t$ and $U$, normally, one can tune the ratio of $U/t$ by
changing the lattice depth, as described in Ref. [29]. Since here $t$ is
measured in terms of the Fermi energy $E_{F}$, therefore, one can tune
$t/E_{F}$ by the density $n$ as well. We note, however, $k_{F}$ and $E_{F}$
are locked together via $E_{F}=\hbar^{2}k_{F}^{2}/2m$. Nevertheless, there is
already a large room for tuning these parameters experimentally. More
sophisticated devices, such as a free electron laser [30] or a laser that
allows contnuous wavelength tuning [31], may be used to allow independent
continuous tuning of $d$. In this case, all parameters can be tuned
independently.

##  IV Conculsions

In summary, we study the superfluid transition of a Fermi gas in a 2DOL in the
context of BCS-BEC crossover, and pay close attention to the exotic effect of
lattice-continuum mixing. We find that, for relatively large $d$ and small $t$
in a 2DOL, a PDW ground state emerges and $T_{c}$ vanishes at intermediate
pairing strength. In the BCS and unitary regimes, the nature of the in-plane
and overall pairing may change from particle-like to hole-like, with a
negative coefficient $a_{0}<0$, and the chemical potential $\mu$ may first
increase with increasing pairing strength. In the BEC regime,
$T_{c}\approx\frac{\sqrt{6}ntd^{4}}{m}|U|^{-1}$, with $\mu\sim-U^{2}$ and
pairing gap $\Delta\sim|\mu|^{3/4}\sim|U|^{3/2}$. These asymptotic scaling
behaviors are qualitatively distinct from their counterpart in the 1DOL, 3D
continuum or 3D lattice cases. Our theoretical predictions can be tested in
future experiments.

###### Acknowledgements.

This work was supported by the NSF of China (Grant No. 11774309 and No.
11674283), and the NSF of Zhejiang Province of China (Grant No. LZ13A040001).

## References

  * Chen  _et al._ [2005] Q. J. Chen, J. Stajic, S. N. Tan, and K. Levin, BCS–BEC crossover: From high temperature superconductors to ultracold superfluids, Phys. Rep. 412, 1 (2005).
  * Bloch  _et al._ [2008] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  * Chin  _et al._ [2010] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  * Bloch [2005] I. Bloch, Ultracold quantum gases in optical lattices, Nat. Phys. 1, 23 (2005).
  * Köhl  _et al._ [2005] M. Köhl, H. Moritz, T. Stöferle, K. Günter, and T. Esslinger, Fermionic atoms in a three dimensional optical lattice: Observing Fermi surfaces, dynamics, and interactions, Phys. Rev. Lett. 94, 080403 (2005).
  * Schneider  _et al._ [2008] U. Schneider, L. Hackermüller, S. Will, T. Best, I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch, and A. Rosch, Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice, Science 322, 1520 (2008).
  * Nozières and Schmitt-Rink [1985] P. Nozières and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: from weak to strong coupling superconductivity, J. Low Temp. Phys. 59, 195 (1985).
  * Chen  _et al._ [1999] Q. J. Chen, I. Kosztin, B. Jankó, and K. Levin, Superconducting transitions from the pseudogap state: d-wave symmetry, lattice, and low-dimensional effects, Phys. Rev. B 59, 7083 (1999).
  * Wang  _et al._ [2020a] J. B. Wang, L. F. Zhang, Y. Yu, C. H. Lee, and Q. J. Chen, Superfluidity and pairing phenomena in ultracold atomic Fermi gases in one-dimensional optical lattices. I. Balanced case, Phys. Rev. A 101, 053617 (2020a).
  * Wang  _et al._ [2020b] J. B. Wang, L. F. Zhang, Y. Yu, C. H. Lee, and Q. J. Chen, Superfluidity and pairing phenomena in ultracold atomic Fermi gases in one-dimensional optical lattices. II. Effects of population imbalance, Phys. Rev. A 101, 053618 (2020b).
  * Parish  _et al._ [2007] M. M. Parish, S. K. Baur, E. J. Mueller, and D. A. Huse, Quasi-one-dimensional polarized Fermi superfluids, Phys. Rev. Lett. 99, 250403 (2007).
  * Roscher  _et al._ [2014] D. Roscher, J. Braun, and J. E. Drut, Inhomogeneous phases in one-dimensional mass-and spin-imbalanced Fermi gases, Phys. Rev. A 89, 063609 (2014).
  * Zhao and Liu [2008] E. H. Zhao and W. V. Liu, Theory of quasi-one-dimensional imbalanced Fermi gases, Phys. Rev. A 78, 063605 (2008).
  * Liao  _et al._ [2010] Y.-a. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, Spin-imbalance in a one-dimensional Fermi gas, Nature 467, 567 (2010).
  * Pagano  _et al._ [2014] G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Schäfer, H. Hu, X.-J. Liu, J. Catani, C. Sias, M. Inguscio, _et al._ , A one-dimensional liquid of fermions with tunable spin, Nat. Phys. 10, 198 (2014).
  * Revelle  _et al._ [2016] M. C. Revelle, J. A. Fry, B. A. Olsen, and R. G. Hulet, 1D to 3D crossover of a spin-imbalanced Fermi gas, Phys. Rev. Lett. 117, 235301 (2016).
  * Fulde and Ferrell [1964] P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev. 135, A550 (1964).
  * Larkin and Ovchinnikov [1965] A. I. Larkin and Y. N. Ovchinnikov, Inhomogeneous state of superconductors, Sov. Phys. JETP 20, 762 (1965), [Zh. Eksp. Teor. Fiz. 47, 1136 (1964)].
  * Partridge  _et al._ [2006] G. B. Partridge, W. Li, R. I. Kamar, Y. A. Liao, and R. G. Hulet, Pairing and phase separation in a polarized Fermi gas, Science 311, 503 (2006).
  * Zhang  _et al._ [2017] L. F. Zhang, Y. M. Che, J. B. Wang, and Q. J. Chen, Exotic superfluidity and pairing phenomena in atomic Fermi gases in mixed dimensions, Sci. Rep. 7, 12948 (2017).
  * Zhang  _et al._ [2020] L. F. Zhang, J. B. Wang, Y. Yu, and Q. J. Chen, Ultra high temperature superfluidity in ultracold atomic Fermi gases with mixed dimensionality, Sci. China - Phys. Mech. Astron. 63, 227421 (2020).
  * Chen  _et al._ [1998] Q. J. Chen, I. Kosztin, B. Jankó, and K. Levin, Pairing fluctuation theory of superconducting properties in underdoped to overdoped cuprates, Phys. Rev. Lett. 81, 4708 (1998).
  * Note [1] We also drop the volume $V$, which can be set to unity and will be canceled out in the end.
  * Note [2] For 1DOL and 3D continuum, $U$ is renormalized down to zero in the contact limit so that an appropriate parameter is scattering length $a$. We have $\mu\sim-e^{d/a}$ and $\mu\sim-1/a^{2}$ for the two cases, respectively.
  * Note [3] See Eq. (2.4) of Ref. [32].
  * Note [4] In principle, $\mu$, when positive, can be measured experimentally by detecting $k_{\mu}=\sqrt{2m\mu}$ (in the $z$ direction) on the Fermi surface directly via measurement of the fermion momentum distribution, e.g., using the time-of-flight imaging technique.
  * Che  _et al._ [2016] Y. M. Che, J. B. Wang, and Q. J. Chen, Reentrant superfluidity and pair density wave in single-component dipolar Fermi gases, Phys. Rev. A 93, 063611 (2016).
  * Note [5] When higher order wavevectors are included in the pair dispersion, the minimum may be shifted to a different location.
  * Jaksch  _et al._ [1998] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold bosonic atoms in optical lattices, Phys. Rev. Lett. 81, 3108 (1998).
  * Madey  _et al._ [2016] J. Madey, M. O. Scully, and P. Sprangle, The free electron laser: conceptual history, Physica Scripta 91, 083003 (2016).
  * Zhang  _et al._ [2005] X. Zhang, A. Liu, C. Lu, and D. Tang, Continuous wavelength tuning in micromachined littrow external-cavity lasers, IEEE Journal of Quantum Electronics 41, 187 (2005).
  * Chen [2000] Q. J. Chen,  _Generalization of BCS theory to short coherence length superconductors: A BCS-Bose-Einstein crossover scenario_ , Ph.D. thesis, University of Chicago (2000), available as arXiv:1801.06266.