⇦Chinese Physics Letters, 2020, Vol. 37, No. 5, Article code 053702Express Letter Unusual Destruction and Enhancement of Superfluidity of Atomic Fermi Gases by Population Imbalance in a One-Dimensional Optical Lattice * Qijin Chen (陈启瑾)1,2,3**, Jibiao Wang (王继标)4**, Lin Sun (孙琳)2, Yi Yu (余祎)5 Affiliations 1Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315 2Department of Physics and Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027 3Synergetic Innovation Center of Quantum Information and Quantum Physics, Hefei 230026 4Laboratory of Quantum Engineering and Quantum Metrology, School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082 5College of Chemical Engineering, Zhejiang University of Technology, Hangzhou 310014 Received 31 March 2020, online 20 April 2020 *Supported by the National Natural Science Foundation of China (Grant Nos. 11774309 and 11674283) and the Natural Science Foundation of Zhejiang Province of China (Grant No. LZ13A040001).
**Corresponding author. Email: qchen@uchicago.edu; wangjibiao@gmail.com Citation Text: Chen Q J, Wang J B, Sun L and Yu Y 2020 Chin. Phys. Lett. 37 053702    Abstract We study the superfluid behavior of a population imbalanced ultracold atomic Fermi gases with a short range attractive interaction in a one-dimensional (1D) optical lattice, using a pairing fluctuation theory. We show that, besides widespread pseudogap phenomena and intermediate temperature superfluidity, the superfluid phase is readily destroyed except in a limited region of the parameter space. We find a new mechanism for pair hopping, assisted by the excessive majority fermions, in the presence of continuum-lattice mixing, which leads to an unusual constant Bose-Einstein condensate (BEC) asymptote for $T_{\rm c}$ that is independent of pairing strength. In result, on the BEC side of unitarity, superfluidity, when it exists, may be strongly enhanced by population imbalance. DOI:10.1088/0256-307X/37/5/053702 PACS:37.10.Jk, 67.85.-d, 74.25.Dw, 03.75.Ss © 2020 Chinese Physics Society Article Text Ultracold atomic Fermi gases have been an ideal system for quantum simulation and quantum engineering, due to their multiple tunable parameters.[1,2] Using a Feshbach resonance, one can vary the effective pairing strength from the weak coupling Bardeen-Cooper-Schrieffer (BCS) limit to strong pairing Bose-Einstein condensate (BEC) limit. As another widely explored parameter,[3–9] population imbalance leads to Fermi surface mismatch and thus renders pairing more difficult, causing suppressed superfluid transition temperature $T_{\rm c}$ or complete destruction of superfluidity at high population imbalances.[5] Among other tunable parameters is the geometry of the system; one can put the Fermi gas in an optical lattice, such as a one-dimensional (1D) optical lattice (OL), which we shall explore here. Unlike the widely studied 3D continuum or 3D lattice (for which each lattice site contains 1 or 2 fermions), 1DOL is distinct in that it is a lattice-continuum mixed system with each lattice "site" now containing many fermions. Such a layered Fermi gas system has not been properly studied in the literature, with and without a population imbalance. Population imbalance $p$ has been widely known to suppress or destroy superfluidity. Indeed, in a 3D homogeneous system, superfluidity is completely destroyed at $T=0$ in the unitary and BCS regimes,[5,10,11] leaving only possible intermediate temperature superfluids (ITSF). Nonetheless, in the BEC regime, stable superfluid exists even with very high $p$, and all minority fermions are paired up. This has been naturally understood as a consequence of vanishing Pauli blocking effect in the deep BEC regime, where the distribution of the constituent fermions in a Cooper pair spreads out over the entire momentum space and thus pairs can happily coexist with excessive fermions. In this Letter, we show that the pairing and superfluid behavior of a Fermi gas, when subject to a short-range attractive interaction ($U < 0$) in 1DOL with $p\ne 0$, is very different due to the lattice-continuum mixing. We find that superfluidity may be readily destroyed by population imbalance, except for a very restricted parameter range (away from small $t$, large $p$, and large lattice constant $d$), where population imbalance gives rise to an extra mechanism for pair hopping. When a BEC superfluid does exist, this leads to an unusual constant BEC asymptote for $T_{\rm c}$, and then can substantially increase $T_{\rm c}$ on the BEC side of unitarity as compared to the balanced case, in which $T_{\rm c}$ decreases with interaction strength. In addition, not all minority fermions are paired up in the BEC limit. We demonstrate that these unusual behaviors are associated with the mixing of a 2D continuum plane and a discrete lattice dimension, which leads to a constant ratio of $\mathit{\Delta}^2/\mu$, unlike in a 3D continuum or 3D lattice. This mixing enables enhanced pair hopping processes assisted by excessive fermions. There have seemingly been many theoretical studies on Fermi gases in optical lattices.[12–21] However, most studies have used the chemical potentials and magnetization as control variables and are thus limited to the weak and intermediate pairing strength regimes. In a 3D attractive Hubbard model with[22] and without[12] population imbalance, superfluid in the deep BEC regime exists only at low fillings. On the other hand, 1DOL of $^6$Li has been realized experimentally with and without population imbalance.[23,24] However, its phase diagram is yet to be explored.[25] We emphasize that 1DOL is fundamentally a 3D system, albeit anisotropic. It cannot be compared with a genuine 2D or 1D lattice, which has usually no more than 2 fermions per site, and does not support true long-range order as we study here. Here we use a previously developed pairing fluctuation theory.[1,11,26] It goes beyond the BCS mean-field treatment by self-consistently including finite momentum pairing in the self energy, which thus contains two parts, $\Sigma(K) = \Sigma_{\rm sc}(K) + \Sigma_{\rm pg}(K)$, where $\Sigma_{\rm sc}(K) = -\mathit{\Delta}_{\rm sc}^2G_0(-K)$ and $\Sigma_{\rm pg}(K) = \sum_Q t_{\rm pg}(Q)G_0(Q-K)$, corresponding to the contributions of the Cooper pair condensate and finite momentum pairs, respectively. We shall follow the notations of Ref. [26], such that $\hbar=k_{\rm B}=1$, and four momenta $K\equiv (\omega_n, {\boldsymbol k})$, $Q\equiv (\Omega_l, {\boldsymbol q})$, $\sum_Q\equiv T\sum_l\sum_{\boldsymbol q}$, etc. Here $G_0(K)$ is the non-interacting Green's function, $t_{\rm pg}(Q)$ the $T$ matrix, $\mathit{\Delta}_{\rm sc}$ the order parameter, and $\omega_n$ ($\Omega_l$) the odd (even) Matsubara frequency. The finite momentum pairing directly leads to the presence of a pseudogap when it becomes strong. This theory has been applied to 3D homogeneous and trapped Fermi gases,[11,27,28] as well as on a 3D or quasi-2D lattice,[12,19] and has been used by other groups.[22,29–31] Now we adapt this theory for 1DOL by modifying the noninteracting atomic dispersion into $\xi_{{\boldsymbol k}\sigma}=\epsilon_{\boldsymbol k}-\mu_\sigma \equiv {\boldsymbol k}_\parallel^2/2\,m + 2t[1-\cos (k_zd)]-\mu_\sigma$, where ${\boldsymbol k}_\parallel=(k_x,k_y)$ is the in-plane momentum, and $\mu_\sigma$ the chemical potential for spin $\sigma=\uparrow,\downarrow$. This one-band lattice dispersion is justified when the band gap in the $z$ direction is tuned to be much greater than the Fermi energy in the $xy$ plane. The derivation of our self-consistent equations is otherwise the same, so that we shall present the result directly, with an emphasis on the unusual new findings caused by population imbalance and the lattice-continuum mixing. In the superfluid phase, we define the pseudogap via $\mathit{\Delta}_{\rm pg}^2 = -\sum_Q t_{\rm pg}(Q)$, so that the total gap $\mathit{\Delta}$ is given by $\mathit{\Delta}^2 = \mathit{\Delta}_{\rm sc}^2 + \mathit{\Delta}_{\rm pg}^2$, which leads to the self energy $\Sigma_\sigma(K) \approx -\mathit{\Delta}^2 G_{0,\bar{\sigma}}(-K)$, and the full Green's function $$ G_{\sigma}(K)=\frac{u_{\boldsymbol k}^{2}}{i\omega_{n}-E_{{\boldsymbol k}\sigma}}+\frac{v_{\boldsymbol k}^{2}}{i\omega_{n}+E_{{\boldsymbol k}\bar{\sigma}}}\,,~~ \tag {1} $$ where $u_{\boldsymbol k}^{2}=(1+\xi_{\boldsymbol k}/E_{\boldsymbol k})/2$, $v_{\boldsymbol k}^{2}=(1-\xi_{\boldsymbol k}/E_{\boldsymbol k})/2$, $E_{{\boldsymbol k}\uparrow}=E_{\boldsymbol k}-h$, $E_{{\boldsymbol k}\downarrow}=E_{\boldsymbol k}+h$, and $E_{\boldsymbol k}=\sqrt{\xi_{\boldsymbol k}^{2}+\mathit{\Delta}^{2}}$, $\xi_{\boldsymbol k}=\epsilon_{\boldsymbol k}-\mu$, $\mu=(\mu_{\uparrow}+\mu_{\downarrow})/2$, $h=(\mu_{\uparrow}-\mu_{\downarrow})/2$. Then we have the number equations, $$\begin{align} n=&\sum_{\boldsymbol k}\Big[\Big(1-\frac{\xi_{\boldsymbol k}}{E_{\boldsymbol k}}\Big) +2\bar{f}(E_{\boldsymbol k})\frac{\xi_{\boldsymbol k}}{E_{\boldsymbol k}}\Big]\,,~~ \tag {2} \end{align} $$ $$\begin{align} p n=&\sum_{\boldsymbol k}\Big[f(E_{{\boldsymbol k}\uparrow})-f(E_{{\boldsymbol k}\downarrow})\Big]\,,~~ \tag {3} \end{align} $$ where $p=(n_\uparrow-n_\downarrow)/n$, $\bar{f}(x)=[f(x+h)+f(x-h)]/2$, and $f(x)=1/(e^{x/T}+1)$. We have the following gap equation with pair chemical potential $\mu_{\rm p}=0$ in the superfluid phase, $$ \frac{m}{4\pi a}=\sum_{\boldsymbol k}\Big[\frac{1}{2\epsilon_{\boldsymbol k}}-\frac{1-2\bar{f}(E_{\boldsymbol k})}{2E_{\boldsymbol k}}\Big]+a_{0}\mu_{\rm p}\,,~~ \tag {4} $$ where the interaction $U$ has been replaced by the $s$-wave scattering length $a$ via $U^{-1}=m/4\pi a-\sum_{\boldsymbol k}1/2\epsilon_{\boldsymbol k}$. Here a finite $\mu_{\rm p}$ extends this equation into the non-superfluid phase. We caution that the parameter $a$ does not necessarily yield the experimentally measured scattering length, which is better reflected by an effective scattering length $a_{\rm eff}$ such that $1/a_{\rm eff}=\sqrt{2\,mt}d/a$ (see Supplementary Secs. I, II and Fig. S1). The coefficient $a_0$ is determined via Taylor expanding $t_{\rm pg}^{-1}(Q)$ on the real frequency axis, $t_{\rm pg}^{-1}(\Omega,{\boldsymbol q})\approx a_{1}\Omega^{2}+a_{0}(\Omega-\Omega_{\boldsymbol q}+\mu_{\rm p})$, with $\Omega_{\boldsymbol q}=B_{\parallel}{\boldsymbol q}_{\parallel}^{2} + 2t_{\rm B}[1-\cos(q_{z}d)]$. Here $B_{\parallel}=1/2M_\parallel$, with $M_\parallel$ being the effective pair mass in the $xy$-plane, and $t_{\rm B}$ is the effective pair hopping integral. Then we have the pseudogap equation $$ a_0\mathit{\Delta}_{\rm pg}^{2}=\sum_{\boldsymbol q}\frac{b(\tilde{\Omega}_{\boldsymbol q})}{\sqrt{1+4\dfrac{a_{1}}{a_{0}}(\Omega_{\boldsymbol q}-\mu_{\rm p})}}\,,~~ \tag {5} $$ with $b(x)=1/(e^{x/T}-1)$ and pair dispersion $\tilde{\Omega}_{\boldsymbol q}=\{\sqrt{a_{0}^{2}[1+4a_{1}(\Omega_{\boldsymbol q}-\mu_{\rm p})/a_{0}]}-a_{0}\}/2a_{1}$, which reduces to $\tilde{\Omega}_{\boldsymbol q}=\Omega_{\boldsymbol q}-\mu_{\rm p}$ when $a_1/a_0 \ll 1$, e.g., in the BEC regime. Equations (2)-(5) form a closed set of self-consistent equations, which will be solved for ($\mu_{\uparrow}$, $\mu_{\downarrow}$, $\mathit{\Delta}_{\rm pg}$, $T_{\rm c}$) with $\mathit{\Delta}_{\rm sc}=0$, and for ($\mu_{\uparrow}$, $\mu_{\downarrow}$, $\mathit{\Delta}$, $\mathit{\Delta}_{\rm pg}$) in the superfluid phase. For our numerics, we consider $p>0$, and define Fermi momentum $k_{\rm F}=(3\pi^{2}n)^{1/3}$ and Fermi energy $E_{\rm F}\equiv k_{\rm B}T_{\rm F}=\hbar^{2}k_{\rm F}^{2}/2m$.[32] The asymptotic solution in the BEC limit, $\mu \rightarrow -\infty$, can be obtained analytically fully for $p=0$ or partially for $p>0$. For $p=0$, $\mu_\sigma = \mu$. However, for $p>0$, we have $\mu_\uparrow > 0$ at $T_{\rm c}$ throughout the BCS–BEC crossover, and $\mu_\downarrow = 2\mu -\mu_\uparrow$. This is self-consistently justified by the solution that $T_{\rm c} \ll \mathit{\Delta} \ll |\mu|$ in the BEC regime. Therefore, in the BEC limit, $f(E_{\boldsymbol k}^\downarrow) = f(\xi_{\boldsymbol k}^\downarrow) = 0$ for all $p$, but $f(E_{\boldsymbol k}^\uparrow) = f(\xi_{\boldsymbol k}^\uparrow) = 0$ only for $p=0$. From the number equations, we obtain $$ \mu = -te^{d/a} + 2\,t + \frac{2\pi dn_\uparrow}{m}\,,~~ \tag {6} $$ dominated by the leading two-body term. The exponential dependence of $\mu$ on $d/a$ results from the quasi-two dimensionality since $|k_z| \le \pi/d$. In the $t\rightarrow 0$ 2D limit, one finds ${d}/{a} \approx \ln\frac{|\mu|}{t}$, which diverges logarithmically. To leading order corrections in powers of $1/\mu$, we have $$\begin{alignat}{1} (1-p)n =& -\frac{m\mathit{\Delta}^2}{4\pi \mu d} -\frac{np\mathit{\Delta}^2}{2\mu^2},\quad{\rm or}~~ \tag {7} \end{alignat} $$ $$\begin{alignat}{1} \mathit{\Delta} =& \sqrt{\frac{4\pi|\mu|d(1-p)n}{m}} \left(1-\frac{\pi dnp}{\mu m} \right)\,.~~ \tag {8} \end{alignat} $$ At $T_{\rm c}$, $\mathit{\Delta}_{\rm pg} = \mathit{\Delta}$. Note that $\mathit{\Delta}^2/\mu$ approaches a constant in the BEC limit, in contrast to its counterpart in a 3D homogeneous case, where $\mathit{\Delta}\sim |\mu|^{1/4}$ so that $\mathit{\Delta}^2/\mu\rightarrow 0$. For $p=0$, one can easily obtain $$ B_\parallel = \frac{1}{4\,m}, \quad t_{\rm B} = \frac{t^2}{2|\mu|} \approx \frac{t}{2} e^{-d/a},~~ \tag {9} $$ which yields $$ T_{\rm c} \approx \dfrac{\pi a n}{2\,m} = \dfrac{k_{\rm F}a}{3\pi}T_{\rm F}~~ \tag {10} $$ in the BEC regime via the pseudogap equation (5). Now for $p>0$, one has to solve for $\mu_\uparrow$ and $T_{\rm c}$ numerically, since $\mu_\uparrow > 0$. We have $E_{\boldsymbol k}^\uparrow \approx \xi_{\boldsymbol k}^\uparrow + \frac{4\pi d n_\downarrow}{m}$, then Eq. (3) becomes $$ pn = \sum_{\boldsymbol k} f\Big(\xi_{\boldsymbol k}^\uparrow + \frac{4\pi d n_\downarrow}{m}\Big) .~~ \tag {11} $$ In the BEC regime, $T_{\rm c}$ is controlled by the inverse $T$-matrix expansion. The coefficients $a_0$ (and pair density $n_{\rm p}$) and $a_1$ are given by $$\begin{alignat}{1} n_{\rm p} = a_0 \mathit{\Delta}^2 =& n_\downarrow -\sum_{\boldsymbol k}\bigg[f(\xi_{\boldsymbol k}^\uparrow)-f\Big(\xi_{\boldsymbol k}^\uparrow + \frac{4\pi d n_\downarrow}{m}\Big)\bigg],~~ \tag {12} \end{alignat} $$ $$\begin{alignat}{1} a_1\mathit{\Delta}^2 =& A + \frac{n_\downarrow}{4|\mu|} (1 + B)\,.~~ \tag {13} \end{alignat} $$ For the inverse pair mass, we have $$ B_\parallel = \frac{1}{4\,m} + \delta B_\parallel\,,\quad t_{\rm B} = \frac{t^2}{n_{\rm p}}\bigg[C + \frac{n_\downarrow}{2|\mu|}(1 -D)\bigg].~~ \tag {14} $$ Here $A$, $B$, $C$, $D$ and $\delta B_\parallel $ depend on $\mu_\uparrow$ and $T$ only. They can be readily obtained via the inverse $T$-matrix expansion (see Supplementary Sec. III for concrete expressions). For $p=0$, $\mu_\uparrow \rightarrow -\infty$ so that $f(\xi_{\boldsymbol k}^\uparrow)$, as well as $A$, $B$, $C$, $D$ and $\delta B_\parallel $ all vanish. Then we recover $n_{\rm p} = n_\downarrow = n/2$, $a_1\mathit{\Delta}^2 = -n/8\mu$ and Eq. (9). Of paramount importance is that population imbalance leads to these extra terms in $a_0\mathit{\Delta}^2$, $a_1\mathit{\Delta}^2$, $B_\parallel$ and $t_{\rm B}$, which are associated with the excessive majority fermions via the Fermi functions. Equation (14) suggests that the pair motion in the $z$ direction for $p\ne 0$ is now strongly enhanced by these fermions as an extra pair hopping mechanism; a minority fermion may hop to the next site by exchanging its majority partner to one that is already there, which is guaranteed by the existence of a transverse continuum dimension, since the "site" is actually a 2D plane. Obviously, this extra mechanism will be dominant over the usual virtual pair unbinding–rebinding process[33] in the BEC regime. Indeed, the pair hopping integral $t_{\rm B}$ and thus $T_{\rm c}$ approach constant BEC asymptotes rather than decreasing with $1/k_{\rm F}a$. Furthermore, equation (12) indicates that $n_{\rm p} < n_\downarrow$ for $p\ne 0$, namely, not all minority fermions are paired in the BEC limit. In the deep BEC regime, equation (6) determines $\mu$, and equation (8) yields the gap $\mathit{\Delta}$, for given $1/k_{\rm F}a$. Then $\mu_\uparrow$ and $T_{\rm c}$ can be obtained via solving Eq. (5) (with $\mathit{\Delta}_{\rm pg}=\mathit{\Delta}$) along with Eq. (11), with the help of Eqs. (12) and (13).[34] Finally, $\mu_\downarrow = 2\mu - \mu_\uparrow$. Shown in Fig. 1 the zero $T$ phase diagram for $(t/E_{\rm F},k_{\rm F}d)= (0.2,0.5)$. These parameters allow a relatively large polarized superfluid (pSF) phase (yellow shaded region) in the BEC regime. A normal phase lies to the left of the black solid line of $T_{\rm c}^{\rm MF}=0$. The red line is given by the instability condition $\partial^2 \Omega/\partial\mathit{\Delta}^2=0$ against phase separation, following Refs.  [5,10], where $\Omega$ is the thermodynamic potential. The blue $T_{\rm c}=0$ curve is determined by $t_{\rm B}=0$. A possible pair density wave (PDW) state emerges in the dotted region where $t_{\rm B}$ becomes negative.[35] The rest space has an unstable mean-field solution of pSF at $T=0$. The grey and brown shaded regions allow ITSF; the former has a lower $T_{\rm c}$, whereas the latter does not but is unstable at low $T$, as shown in the inset for $1/k_{\rm F}a = 2$. To compare with the 3D continuum case,[10] we label the top axis with $1/k_{\rm F}a_{\rm eff}$. Apparently, the position of the normal/unstable boundary is roughly the same, but stable pSF solution no longer exists here for high $p\gtrsim 0.5$. Instead, a PDW phase emerges.
cpl-37-5-053702-fig1.png
Fig. 1. Zero temperature phase diagram in the $p $–$ 1/k_{\rm F}a$ plane, for $t/E_{\rm F}=0.2$ and $k_{\rm F}d = 0.5$. The orange dashed $\mu=0$ line separates the fermionic and bosonic regimes. Here "Normal" denotes normal Fermi gases. Besides the pSF (yellow shaded) and PDW (dotted region), and normal phases, the mean-field superfluid solution between the black (solid) and red (solid and dashed) curves is unstable at $T=0$. The grey and brown areas admit ITSF. An example in the brown area is shown in the inset for $1/k_{\rm F}a=2$. The green dashed and blue dotted lines show less restrictive instability conditions, with $n_{\rm s}/m$ being superfluid density. The top axis is labeled with $1/k_{\rm F}a_{\rm eff}$.
Shown in Fig. 2 is $T_{\rm c}$ versus $1/k_{\rm F}a$ for varying imbalance $p$, as labeled, for a physically accessible case with $t/E_{\rm F}=0.2$ and $k_{\rm F}d = 0.5$. The pSF phase exists only for $p\lesssim 0.5$. The $T_{\rm c}$ curves turn back on the BCS side, leading to an upper and lower $T_{\rm c}$ for a given $1/k_{\rm F}a$, exhibiting typical ITSF behavior in the BCS and unitary regimes, similar to the 3D homogeneous case.[5,10] Below the lower $T_{\rm c}$, phase separation, FFLO and/or PDW states may occur. The $T_{\rm c}$ solution in the yellow shaded region is unstable, corresponding to ITSF in the brown shaded area in Fig. 1. Besides the unpaired normal phase to the left, a pseudogap phase exists above (the upper) $T_{\rm c}$. As $1/k_{\rm F}a$ increases, $T_{\rm c}$ approaches a constant BEC asymptote. This should be contrasted with the dashed $p=0$ curve, for which $T_{\rm c} $ decreases with $1/k_{\rm F}a$ following Eq. (10). Therefore, relative to the $p=0$ case, imbalance may substantially raise $T_{\rm c}$ on the BEC side of unitarity. The plot of $T_{\rm c}$ versus $p$ in the inset for $1/k_{\rm F}a =2$ and $10$ shows an enhancement for $p\lesssim 0.1$ and $p\lesssim 0.3$, respectively.
cpl-37-5-053702-fig2.png
Fig. 2. Behavior of $T_{\rm c}$ versus $1/k_{\rm F}a$ with $t/E_{\rm F}=0.2$ and $k_{\rm F}d = 0.5$ for varying $p$, as labeled. $T_{\rm c}$ is unstable in the yellow shaded region. Also plotted for comparison is the balanced $T_{\rm c}$ curve (dashed line). Shown in the inset is $T_{\rm c}$ versus $p$ for $1/k_{\rm F}a = 2$ (red) and 10 (blue lines), manifesting enhancement of $T_{\rm c}$ by imbalance.
Our calculations reveal that as $d$ increases, the $T=0$ pSF phase shrinks quickly. For $k_{\rm F}d=1$, the upper pSF phase boundary (blue curve) in Fig. 1 moves down to $p\sim 0.11$. And for $d=2$, the pSF phase disappears completely. The grey shaded ITSF phase extends to $1/k_{\rm F}a = +\infty$ at low $p$, with both an upper and lower $T_{\rm c}$. Similar reduction of the superfluid phase can be achieved by decreasing $t$. Shown in Fig. 3 is a plot of $T_{\rm c}$ similar to Fig. 2 except now with increased lattice constant, $k_{\rm F}d = 2$. The superfluid phase exists only for relatively low $p$, exhibiting typical ITSF, with the lower $T_{\rm c}$ extending to $k_{\rm F}a = +\infty$ for $p \lesssim 0.0085$; both the upper and lower $T_{\rm c}$'s approach a constant BEC asymptote for these low $p$. For larger $p \gtrsim 0.009$, there is no superfluid in the deep BEC regime. The superfluid phase shrinks to zero as $p$ increases beyond about 0.135 at $1/k_{\rm F}a \approx -0.7$.
cpl-37-5-053702-fig3.png
Fig. 3. Behavior of $T_{\rm c}$ versus $1/k_{\rm F}a$ with $t/E_{\rm F}=0.2$ and $k_{\rm F}d = 2$ for varying $p$, as labeled. Also plotted for comparison is the balanced $T_{\rm c}$ curve (dashed line). The arrow points to the $p$ increasing direction.
Finally, we examine in Fig. 4 the asymptotic behavior of various quantities versus $1/k_{\rm F}a$ at $T_{\rm c}$ in the BEC regime for $t/E_{\rm F} =0.25$, $k_{\rm F}d=2$ and $p=0.01$. The solid and dashed lines represent the fully numerical and the BEC asymptotic solutions, respectively. Figure 4(a) demonstrates that the asymptotic solutions for $\mu$ and (pseudo)gap $\mathit{\Delta}$ given by Eqs. (6) and (8) work very well for $1/k_{\rm F}a > 2$. Figures 4(b)–4(d) presents $T_{\rm c}$, $a_0\mathit{\Delta}^2$, $a_1\mathit{\Delta}^2$, $\mu_\uparrow$, as well as $B_\parallel$, $B_z = t_{\rm B}d^2$ and $\mu_\uparrow$. They all quickly approach their BEC asymptotes for $1/k_{\rm F}a > 3$. In particular, the constant asymptote for $t_{\rm B}$ confirms that the excessive fermion assisted pair hopping dominates in the BEC regime. As shown in Fig. 4(c), $n_{\rm p}=a_0\mathit{\Delta}^2 < n_\downarrow $; only part of minority fermions form pairs.
cpl-37-5-053702-fig4.png
Fig. 4. Behavior of (a) $\mu$, $\mathit{\Delta}$, (b) $T_{\rm c}$, coefficients (c) $a_0$, $a_1$, (d) $B_\parallel$, $B_z$ and $\mu_\uparrow$, as a function of $1/k_{\rm F}a$ for $p=0.01$, $t/E_{\rm F}=0.25$ and $k_{\rm F}d=2$. For comparison, also plotted are the BEC asymptotes (dashed lines), as well as $n_\downarrow$. The energy unit is $E_{\rm F}$, and $2m=1$.
We have studied various situations for a big range of $(t,p,d)$ and found that the superfluidity can be easily destroyed by large $d$ and small $t$. Reducing $t$ in Figs. 2 and 3 may shrink the pSF phase quickly, as shown in Supplementary Fig. S3. Overall, in the multidimensional $(t,d,p,T)$ phase space, especially in the BEC limit, the superfluid phase exists only for small and intermediate $d$, small $p$, relatively large $t$ and intermediate (and low) $T$. To understand the destruction of superfluidity at large $d$ and small $t$, we note that when $d$ is large, more fermions will occupy the high $k_\parallel$ states. In addition, a small $t$ may further force the lattice band to be fully occupied, so that the Fermi surface becomes nearly dispersionless as a function of $k_z$. This makes it extremely hard to accommodate the excessive majority fermions, which will necessarily have to occupy high $k_\parallel$ states at a high energy cost. Furthermore, since $|k_z|\le \pi/d$, the Pauli blocking effect can no longer be eliminated in the $z$ direction in the BEC regime,so that $t_{\rm B}$ may be quickly suppressed to zero as $d$ and $p$ increase and/or $t$ decreases at low $T$. The enhancement and destruction of superfluidity are easily testable in future experiments. The enhancement also suggests that a small imbalance is beneficial for achieving superfluidity experimentally. Lastly, we mention that we do not consider the possible quasi-superfluid state of the Berezinskii-Kosterlitz-Thouless (BKT) type[36,37] even in the quasi-2D cases. Here not only the superfluidity but also pairing are made unstable by population imbalance when the true superfluidity is destroyed. It is yet to be studied whether a BKT-type superfluid is possible in such a case. References BCS–BEC crossover: From high temperature superconductors to ultracold superfluidsMany-body physics with ultracold gasesFermionic Superfluidity with Imbalanced Spin PopulationsPairing and Phase Separation in a Polarized Fermi GasStability conditions and phase diagrams for two-component Fermi gases with population imbalanceImbalanced Feshbach-resonant Fermi gasesTrapped fermions across a Feshbach resonance with population imbalanceSuperfluid stability in the BEC-BCS crossoverStability Criteria for Breached-Pair SuperfluidityIntermediate-Temperature Superfluidity in an Atomic Fermi Gas with Population ImbalanceTheory of superfluids with population imbalance: Finite-temperature and BCS-BEC crossover effectsSuperconducting transitions from the pseudogap state: d -wave symmetry, lattice, and low-dimensional effectsHigh-Temperature Superfluidity of Fermionic Atoms in Optical LatticesUltracold quantum gases in optical latticesFermionic Atoms in a Three Dimensional Optical Lattice: Observing Fermi Surfaces, Dynamics, and InteractionsTwo-Component Fermi Gas on Internal-State-Dependent Optical LatticesFormation of Molecules near a Feshbach Resonance in a 1D Optical LatticeFermion pairing with spin-density imbalance in an optical latticeSuperfuid-insulator transitions at noninteger filling in optical lattices of fermionic atomsTheory of ultracold atomic Fermi gasesStable Fulde-Ferrell-Larkin-Ovchinnikov pairing states in two-dimensional and three-dimensional optical latticesThe spin-imbalanced attractive Hubbard model in d=3: Phase diagrams and BCS–BEC crossover at low fillingSpin-Imbalanced Quasi-Two-Dimensional Fermi GasesAtom Pairing in Optical SuperlatticesPairing Fluctuation Theory of Superconducting Properties in Underdoped to Overdoped CupratesPseudogap phenomena in ultracold atomic Fermi gasesSuperfluidity in atomic Fermi gasesPairing Gap and In-Gap Excitations in Trapped Fermionic SuperfluidsSuperfluid shells for trapped fermions with mass and population imbalanceBCS-BEC crossover at finite temperature in spin-orbit-coupled Fermi gasesBose condensation in an attractive fermion gas: From weak to strong coupling superconductivityReentrant superfluidity and pair density wave in single-component dipolar Fermi gasesOrdering, metastability and phase transitions in two-dimensional systems
[1] Chen Q J, Stajic J, Tan S N and Levin K 2005 Phys. Rep. 412 1
[2] Bloch I, Dalibard J and Zwerger W 2008 Rev. Mod. Phys. 80 885
[3] Zwierlein M W, Schirotzek A, Schunck C H and Ketterle W 2006 Science 311 492
[4] Partridge G B, Li W, Kamar R I, Liao Y A and Hulet R G 2006 Science 311 503
[5] Chen Q J, He Y, Chien C C and Levin K 2006 Phys. Rev. A 74 063603
[6] Radzihovsky L and Sheehy D E 2010 Rep. Prog. Phys. 73 076501
[7] Yi W and Duan L M 2006 Phys. Rev. A 73 031604(R)
[8] Pao C H, Wu S T and Yip S K 2006 Phys. Rev. B 73 132506
[9] Forbes M M, Gubankova E, Liu W V and Wilczek F 2005 Phys. Rev. Lett. 94 017001
[10] Chien C C, Chen Q J, He Y and Levin K 2006 Phys. Rev. Lett. 97 090402
[11] Chen Q J, He Y, Chien C C and Levin K 2007 Phys. Rev. B 75 014521
[12] Chen Q J, Kosztin I, Jankó B and Levin K 1999 Phys. Rev. B 59 7083
[13] Hofstetter W, Cirac J I, Zoller P, Demler E and Lukin M D 2002 Phys. Rev. Lett. 89 220407
[14] Bloch I 2005 Nat. Phys. 1 23
[15] Köhl M, Moritz H, Stöferle T, Günter K and Esslinger T 2005 Phys. Rev. Lett. 94 080403
[16] Cazalilla M A, Ho A F and Giamarchi T 2005 Phys. Rev. Lett. 95 226402
[17] Orso G, Pitaevskii L P, Stringari S and Wouters M 2005 Phys. Rev. Lett. 95 060402
[18] Koponen T, Kinnunen J, Martikainen J P, Jensen L M and Törmä P 2006 New J. Phys. 8 179
[19] Chien C C, He Y, Chen Q J and Levin K 2008 Phys. Rev. A 77 011601
[20] Giorgini S, Pitaevskii L P and Stringari S 2008 Rev. Mod. Phys. 80 1215
[21] Cai Z, Wang Y and Wu C 2011 Phys. Rev. A 83 063621
[22] Cichy A and Micnas R 2014 Ann. Phys. 347 207
[23] Ong W, Cheng C, Arakelyan I and Thomas J E 2015 Phys. Rev. Lett. 114 110403
[24] Kangara J, Cheng C, Pegahan S, Arakelyan I and Thomas J E 2018 Phys. Rev. Lett. 120 083203
[25]An optical lattice in a theory paper in the literature often refers to a pure lattice in the context of a Hubbard model. Namely, a 1DOL means a simple 1D atomic chain. This is different from the 1DOL we study here.
[26] Chen Q J, Kosztin I, Jankó B and Levin K 1998 Phys. Rev. Lett. 81 4708
[27] Chen Q J and Wang J B 2014 Front. Phys. 9 539
[28] Yu Y and Chen Q J 2010 Physica C 470 S900
[29] Kinnunen J, Rodriguez M and Törmä P 2004 Science 305 1131
[30] Lin G D, Yi W and Duan L M 2006 Phys. Rev. A 74 031604(R)
[31] He L, Huang X G, Hu H and Liu X J 2013 Phys. Rev. A 87 053616
[32]Considering changing $t$ and $d$, here we define $k_{\rm F}$ and $T_{\rm F}$ as given by a homogeneous, unpolarized, noninteracting Fermi gas with the same total number density $n$ in 3D.
[33] Nozières P and Schmitt-Rink S 1985 J. Low Temp. Phys. 59 195
[34]Note that Eq. (5) depends on the product $a_0\mathit{\Delta}^2$ and the ratio $a_0/a_1$, but not on $\mathit{\Delta}$ separately.
[35] Che Y M, Wang J B and Chen Q J 2016 Phys. Rev. A 93 063611
[36]Berezinskii V L 1972 Sov. Phys.-JETP 34 610
[37] Kosterlitz J M and Thouless D J 1973 J. Phys. C 6 1181