⇦Chinese Physics Letters, 2016, Vol. 33, No. 8, Article code 080303⇨ A Realistic Model for Observing Spin-Balanced Fulde–Ferrell Superfluid in Honeycomb Lattices * Bei-Bing Huang(黄备兵)** Affiliations Department of Physics, Yancheng Institute of Technology, Yancheng 224051 Received 27 March 2016 ^*Supported by the Natural Science Foundation of Jiangsu Province under Grant No BK20130424, and the National Natural Science Foundation of China under Grant No 11547047.
^**Corresponding author. Email: hbb4236@ycit.edu.cn Citation Text: Huang B B 2016 Chin. Phys. Lett. 33 080303 Abstract The combination of spin–orbit coupling (SOC) and in-plane Zeeman field breaks time-reversal and inversion symmetries of Fermi gases and becomes a popular way to produce single plane wave Fulde–Ferrell (FF) superfluid. However, atom loss and heating related to SOC have impeded the successful observation of FF state until now. In this work, we propose the realization of spin-balanced FF superfluid in a honeycomb lattice without SOC and the Zeeman field. A key ingredient of our scheme is generating complex hopping terms in original honeycomb lattices by periodical driving. In our model the ground state is always the FF state, thus the experimental observation has no need of fine tuning. The other advantages of our scheme are its simplicity and feasibility, and thus may open a new route for observing FF superfluids. DOI:10.1088/0256-307X/33/8/080303 PACS:03.75.Ss, 05.30.Fk, 67.85.-d © 2016 Chinese Physics Society Article Text The inhomogeneous superfluid,[1] including Fulde–Ferrell (FF)[2] and Larkin–Ovchinnikov (LO)[3] superfluid, is a fascinating state proposed to understand the spin-imbalanced fermionic superfluid. In fermionic gases in the absence of spin–orbit coupling (SOC) with large spin imbalance, space inversion symmetry ensures that Cooper pairings with momentum ${\boldsymbol Q}$ and $-{\boldsymbol Q}$ coexist. As a result the LO phase is more favorable than single plane wave FF phase.[4] The experimental realization of SOC[5-8] dramatically changes this scenario. SOC mixes different spin components and makes the pairings possible on the same Fermi surface even if the spin imbalance exists. Once an in-plane Zeeman field is imposed, which breaks the time-reversal and space inversion symmetries simultaneously, the FF phase rather than the LO phase will have a larger phase space.[9-22] However, the realization of SOC requires near resonant Raman laser and inevitably brings some practical experimental problems such as atom-loss and heating.[23] Due to these obstacles, conclusive evidence of FF states is still elusive.

Fig. 1. (a) A schematic diagram of the honeycomb lattice. Here ${\boldsymbol a}_1$ and ${\boldsymbol a}_2$ are basis vectors for the unit cell, $t_{\delta}e^{i\theta_{\delta}}$ ($\delta=1,2,3$) are the complex hopping terms along three nearest neighbor directions $\delta_{1,2,3}$. The parameters we assume $\theta_1=\theta_2\ne \theta_3$ and $t_1=t_2\ne t_3$ can be obtained by periodical driving honeycomb lattice along the $x$-direction in a specified way. (b) The periodical inertial force induced by driving. An exact period $T$ is shown.

Comparing conventional spin-imbalanced fermionic gases with those with SOC and in-plane Zeeman field, the key ingredient for realizing FF phase is simultaneous breaking of time-reversal and inversion symmetries of the Fermi surface. In this Letter, we suggest an experimentally feasible model without SOC and Zeeman field in the honeycomb lattice to realize single plane wave FF state. Due to the absence of Zeeman field in our model, the FF state proposed here is spin balanced. We consider a system in a honeycomb lattice. A honeycomb lattice consists of two triangular sublattices denoted by A and B as shown in Fig. 1. The Hamiltonian we study is $$\begin{align} H=\,&-\sum_{i\delta\sigma}t_{\delta}[e^{i\theta_{\delta}}a_{i\sigma}^† b_{i+\delta\sigma} +e^{-i\theta_{\delta}}b_{i+\delta\sigma}^†a_{i\sigma}]\\ &-U\sum_i[a_{i\uparrow}^†a_{i\downarrow}^†a_{i\downarrow} a_{i\uparrow} +b_{i\uparrow}^†b_{i\downarrow}^†b_{i\downarrow} b_{i\uparrow}]\\ &-\mu \sum_{i\sigma}(a_{i\sigma}^†a_{i\sigma} +b_{i\sigma}^†b_{i\sigma}),~~ \tag {1} \end{align} $$ where $a_{i\sigma}$ and $b_{i\sigma}$ denote the fermionic annihilator operators for the sublattices A and B with spin $\sigma=(\uparrow, \downarrow)$, respectively. We choose the basis vectors as ${\boldsymbol a}_1=(\frac{\sqrt{3}}{2}a,-\frac{a}{2})$, ${\boldsymbol a}_2=(0,a)$ and three nearest vectors connecting A–B sites as $\delta_1=\frac{a}{\sqrt{3}}(\frac{1}{2},\frac{\sqrt{3}}{2})$, $\delta_2=\frac{a}{\sqrt{3}}(\frac{1}{2},-\frac{\sqrt{3}}{2})$, $\delta_3=\frac{a}{\sqrt{3}}(-1,0)$ with $a$ being the length of the nearest A–A (B–B) sites. Here $U$ ($>0$) and $\mu$ are on-site attractive interaction strength and chemical potential, respectively. The hopping terms $t_{\delta}$ generally depend on the hopping directions in an anisotropic honeycomb lattice. A feature of our model is phase factors $e^{i\theta_{\delta}}$ accompanying every hopping. As demonstrated in the following, these phase factors can be induced by periodical driving. Without loss of generality in this work, for simplicity, we assume $\theta_1=\theta_2\ne\theta_3$, $t_1=t_2=t$ and $t_3=\lambda t$, where $\lambda>0$ is the anisotropy parameter. This configuration of phase factors is equivalent to $\theta_1=\theta_2=0\ne\theta_3$ by a gauge transformation $a_i\rightarrow a_i e^{i\theta_1}$. Usually in some applications sublattice symmetry of honeycomb lattice can be broken by adding such a term $\propto\sum_{i\sigma}(a_{i\sigma}^†a_{i\sigma}-b_{i\sigma}^† b_{i\sigma})$ into the Hamiltonian. In this study we ignore it due to the fact that this term directly gaps out the Dirac point Fermi surface at half-filling. The model Eq. (1) with all $\theta_{\delta}=0$ and $U=0$ has been realized in experiments and is used to investigate moving and merging of Dirac points by controlling anisotropic parameter $\lambda$.[24] When $0 < \lambda < 2$ two Dirac points move in the opposite direction and merge into a semi-Dirac point at $\lambda=2$, which are featured by a linear-quadratic dispersion relation along different directions. When $\lambda>2$ the system opens a gap and becomes an insulator, in which the low energy excitation is the massive Dirac fermion.[25]

Fig. 2. The upper branch of a single particle spectrum for different anisotropy $\lambda$ and phase factor $\theta_3$. The parameters are $\lambda=0.5$, $\theta_3=0$ in (a) and $\theta_3=\pi/3$ in (b), $\lambda=1.5$, $\theta_3=0$ in (c) and $\theta_3=\pi/3$ in (d). By contrast, the phase factor $\theta_3$ only displaces Dirac points.

The nonzero $\theta_3$ should bring about multiple impacts on the properties of the system. Here we only list two points related to our contents. For real hopping terms the Hamiltonian has a pseudospin symmetry at half-filling,[26] which ensures the degeneracy of the superfluid and charge-density wave (CDW) for an attractive interaction. The complex hopping terms break this symmetry, thus the instability of CDW is very weak in our system. To investigate effects on the single-particle spectrum of nonzero $\theta_{\delta}$, we transform the Hamiltonian into momentum space and obtain $$\begin{alignat}{1} E_{\boldsymbol k}=\,&\pm t\Big[2+\lambda^2+2\cos{k_ya}\\ &+4\lambda\cos{\frac{k_ya}{2}} \cos\Big(\frac{\sqrt{3}}{2}k_xa-\theta_3\Big)\Big]^{1/2}.~~ \tag {2} \end{alignat} $$ We find that $\theta_3$ cannot create/annihilate Dirac points by itself but uniformly displaces them as long as $\lambda < 2$. The stability of Dirac points in the presence of $\theta_3$ is protected by the hidden symmetry advocated by Hou et al.[27] Additionally, $\theta_3$ also breaks (conserves) time-reversal and inversion symmetries of the system in $x$- ($y$-) direction due to $E_{-k_x,k_y}\ne E_{k_x,k_y}$ ($E_{k_x,-k_y}= E_{k_x,k_y}$) (see Fig. 2), which is vital to realize the FF state in our system. The attractive interaction $-U$ supports pairings among fermions. Due to two separated Fermi surfaces near the half-filling and symmetry breaking caused by $\theta_3$, in principle pairings can happen in the same Fermi surfaces (LO phase) or between two Fermi surfaces (FF phase). There are two factors to impair the stability of LO state. It is easily seen that in the LO phase, pairing momenta have two components for an arbitrary Fermi surface in contrast to the FF phase which only has a nonzero $x$ component due to $E_{k_x,-k_y}= E_{k_x,k_y}$. Extra pairing momentum will increase oscillating behavior of order parameters and kinetic energy of the system. On the other hand in the LO phase, two Fermi surfaces are completely decoupled especially near the half-filling, as a result only half the total density of state determines order parameters for every Fermi surface and the condensation energy is lowered. Thus we only consider the FF state with pairing momentum along the $x$-direction. Defining mean-field order parameters ${\it \Delta}_a=U\sum_{\boldsymbol k} < a_{{\boldsymbol Q}/2-{\boldsymbol k},\downarrow}a_{{\boldsymbol Q}/2+{\boldsymbol k},\uparrow}>={\it \Delta} e^{i\varphi}$, ${\it \Delta}_b=U\sum_{\boldsymbol k} < b_{{\boldsymbol Q}/2-{\boldsymbol k},\downarrow}b_{{\boldsymbol Q}/2+{\boldsymbol k},\uparrow}>={\it \Delta} e^{-i\varphi}$ with FF pairing momentum ${\boldsymbol Q}=(Q_x,0)$, the mean-field Hamiltonian can be arranged into $H_{\rm mf}=\sum_k[\psi_k^†H_{\rm BdG}(k)\psi_k-2\mu]+N(|{\it \Delta}_a|^2+|{\it \Delta}_b|^2)/U$, with $\psi_k^†=(a_{{\boldsymbol Q}/2+{\boldsymbol k},\uparrow}^†, a_{{\boldsymbol Q}/2-{\boldsymbol k},\downarrow},b_{{\boldsymbol Q}/2+{\boldsymbol k},\uparrow}^†, b_{{\boldsymbol Q}/2-{\boldsymbol k},\downarrow})$ and $$\begin{alignat}{1} &H_{\rm BdG}(k)\\ =\,&\left(\begin{matrix} -\mu & -{\it \Delta}_a & -D({\boldsymbol k}) & 0 \\ -{\it \Delta}_a^* & \mu & 0 & D^*(-{\boldsymbol k}) \\ -D^*({\boldsymbol k}) & 0 & -\mu & -{\it \Delta}_b \\ 0 & D(-{\boldsymbol k}) & -{\it \Delta}_b^* & \mu \end{matrix}\right),~~ \tag {3} \end{alignat} $$ where $D({\boldsymbol k})=\sum_{\delta}t_{\delta}e^{i\theta_{\delta}}e^{i({\boldsymbol Q}/2+{\boldsymbol k})\cdot {\boldsymbol \delta}}$, and $N$ is the number of unit cell. The Free energy for every unit cell at zero temperature $$\begin{align} F=\,&\sum_{\boldsymbol k} \{-2\mu+\sum_{i=1}^4\epsilon_i({\boldsymbol k}){\it \Theta}[-\epsilon_i({\boldsymbol k})] \} \\ &+\frac{|{\it \Delta}_a|^2+|{\it \Delta}_b|^2}{U}+\mu n, \end{align} $$ where ${\it \Theta}(x)$ is the Heaviside step function, $\epsilon_i({\boldsymbol k})$ ($i=1,\ldots,4$) are the eigenvalues of $H_{\rm BdG}(k)$, whose exact expressions are too complex to be presented here, and $n$ is the particle density for every unit cell. Here in view of the absence of CDW, particles uniformly distribute in two sublattices and we anticipate that two order parameters have equal magnitude ${\it \Delta}$ but different phase $\varphi$. The order parameters, the relative phase, chemical potential and FF pairing momentum are determined by requiring $\partial F/\partial {\it \Delta}=0$, $\partial F/\partial \varphi=0$, $\partial F/\partial n=\mu$, and $\partial F/\partial Q_x=0$. For model Eq. (1) at half-filling with $\theta_{\delta}=0$, the system shows the semimetal-superfluid quantum phase transition when the interaction strength $U$ increases to a critical value $U_{\rm c}/t\approx2.13$ at the mean-field level.[28] The existence of this phase transition depends on vanishing density of state on the Dirac point energy. As nonzero $\theta_{\delta}$ cannot annihilate the Dirac points, this quantum phase transition still happens in our model. In this study, we do not involve this phase transition by setting $U/t=4$. By minimizing the free energy for fixed anisotropy $\lambda$ and the particle number density $n$, we always have $Q_x\ne 0$ for $\theta_3\ne 0$. To further check stability of the FF phase, we also compare the free energy of the FF phase with that of the BCS phase by imposing ${\boldsymbol Q}=0$. Figure 3(a) shows this difference for $\theta_3=\pi/6$ and explicitly demonstrates that the FF phase is the true ground state. Since the mean-field Hamiltonian transforms $H_{\rm mf}(\mu)\rightarrow H_{\rm mf}(-\mu)$ under particle-hole transformation $a_{i\sigma}\rightarrow a_{i\overline{\sigma}}^†$ and $b_{i\sigma}\rightarrow -b_{i\overline{\sigma}}^†$ with $\overline{\sigma}=-\sigma$, the free energy has a salient symmetry about half-filling. In Fig. 3(b) we also plot chemical potential as the function of the particle number density for different $\lambda$ and $\theta_3$, as the compressibility $\kappa^{-1}\propto \partial \mu/\partial n$ is positive. This also illustrates stability of the FF phase. Thus we arrive at our main conclusion that in the presence of complex hopping terms in honeycomb lattices with on-site attractive interaction, BCS state destabilizes and gives way to the FF state.

Fig. 3. (a) The difference of free energy between FF and BCS phases for different filling $n$ and anisotropy $\lambda$. (b) Chemical potential from self-consistent calculation as the function of filling for different anisotropies $\lambda=0.5$ (square), $\lambda=0.8$ (circle), $\lambda=1.0$ (right triangle), $\lambda=1.2$ (up triangle), $\lambda=1.5$ (left triangle) and $\lambda=1.8$ (down triangle).

In the following we investigate the impact of $\theta_3$ on system properties. Since $\theta_3$ only displaces the Dirac points and does not change density of state on the Fermi surface, it should not affect magnitude of order parameter and chemical potential. This expectation is consistent with numerical results (see Figs. 4(a) and 4(b)). Interestingly the relative phase $\varphi$ and the FF pairing momentum $Q_x$ show linear behaviors about $\theta_3$ (see Figs. 4(c) and 4(d)), independent of particle number density and anisotropy. We find that this problem can be illustrated easily in pseudospin language. To use pseudospin, the first step we need to carry out is transforming the $-U$ model Eq. (1) into the $+U$ model, and this can be completed by a partial particle–hole transformation.[29] In this transformed model, the half-filling condition is always satisfied. Thus for small $t/U$, we obtain the pseudospin exchange model in the second-order perturbative theory as follows: $$\begin{align} H_{\rm se}=\sum_{i,\delta}H_{i,\delta}-2\mu\sum_i [T_{ia}^z+T_{ib}^z],~~ \tag {4} \end{align} $$ with $$\begin{align} H_{i,\delta}=\,&\frac{4t_{\delta}^2}{U}T_{ia}^zT_{i+\delta b}^z- \frac{4t_{\delta}^2}{U}\cos{2\theta_{\delta}}[T_{ia}^xT_{i+\delta b}^x+T_{ia}^yT_{i+\delta b}^y]\\ &-\frac{4t_{\delta}^2}{U}\sin{2\theta_{\delta}}[T_{ia}^xT_{i+\delta b}^y-T_{ia}^yT_{i+\delta b}^x],~~ \tag {5} \end{align} $$ and pseudospin operators $T_{ia}^x=\frac{1}{2}(a_{i\downarrow}a_{i\uparrow} +a_{i\uparrow}^†a_{i\downarrow}^†)$, $T_{ia}^y=\frac{1}{2i} (a_{i\uparrow}^†a_{i\downarrow}^† -a_{i\downarrow}a_{i\uparrow})$, $T_{ia}^z=\frac{1}{2}(a_{i\sigma}^†a_{i\sigma}-1)$, $T_{ib}^x=\frac{1}{2}(b_{i\downarrow}b_{i\uparrow} +b_{i\uparrow}^†b_{i\downarrow}^†)$, $T_{ib}^y=\frac{1}{2i}(b_{i\uparrow}^†b_{i\downarrow}^† -b_{i\downarrow}b_{i\uparrow})$, $T_{ib}^z=\frac{1}{2}(b_{i\sigma}^†b_{i\sigma}-1)$. In the model Eq. (3), the chemical potential becomes the Zeeman field in the $z$-direction. When $\mu=\theta_{\delta}=0$, this model is reduced to an isotropic antiferromagnetic model by an extra transformation $b_{i\downarrow}\rightarrow -b_{i\downarrow}$. If all pseudospins are parallel with the $z$-axis ($x$–$y$ plane), this corresponds to the CDW (superfluid) state, while other directions for pseudospins signify mixing of CDW and superfluid. For nonzero $\theta_{\delta}$, $H_{\rm se}$ becomes anisotropic, meanwhile degeneracy between CDW and superfluid is broken as mentioned above.

Fig. 4. The impact of phase factor $\theta_3$ on magnitude of order parameter ${\it \Delta}$ (a), chemical potential $\mu$ (b), FF pairing momentum $Q_x$ (c) and relative phase $\varphi$ (d). In (a) and (b) different lines correspond to different fillings and anisotropies. These two figures signify that ${\it \Delta}$ and $\mu$ do not depend on the $\theta_3$. In (c) and (d), all results from different fillings and anisotropies collapse to the same data points and signify $Q_x$ and $\varphi$ only depend on $\theta_3$. Two red solid lines in (c) and (d) correspond to the results from pseudospin analysis.

The evolution of superfluid state of model Eq. (1) can be conveniently analyzed in the classical limit of large pseudospin. Replacing pseudospin operators by vectors ${\boldsymbol S}_{ia}=S(\sin{\phi_a}\cos{\varphi_{ia}}$, $\sin{\phi_a}\sin{\varphi_{ia}}, \cos{\phi_a}$), ${\boldsymbol S}_{ib}=S(\sin{\phi_b}\cos{\varphi_{ib}}, \sin{\phi_b}\sin{\varphi_{ib}}, \cos{\phi_b})$ with $\varphi_{ia,b}=\mp\varphi-{\boldsymbol Q}\cdot {\boldsymbol r}_i$, the classical energy of every unit cell is $$\begin{align} \frac{E_{\rm cl}}{tS^2} =\,&-\frac{2\overline{\mu}}{S}[\cos{\phi_a}+\cos{\phi_b}] +\frac{4}{\overline{U}}(2\\ &+\lambda^2)\cos{\phi_a}\cos{\phi_b}\\ &-\frac{4}{\overline{U}} \sin{\phi_a}\sin{\phi_b}\Big\{2\cos{\Big[\frac{Q_x a}{2\sqrt{3}} -2\varphi\Big]}\cos{\frac{Q_ya}{2}}\\ &+\lambda^2\cos{\Big[-\frac{Q_xa}{\sqrt{3}} -2\varphi+2\theta_3\Big]}\Big\}, \end{align} $$ with $\overline{\mu}=\mu/t$, $\overline{U}=U/t$. The minimization for $Q_x$, $Q_y$ and $\varphi$ leads to $Q_xa=4\theta_3/\sqrt{3}$, $Q_y=0$, $\varphi=\theta_3/3$. This result is completely consistent with the mean-field numerical result (see solid lines in Figs. 4(c) and 4(d)). Thus the classical energy becomes $$\begin{align} \frac{E_{\rm cl}}{tS^2} =\,&-\frac{2\overline{\mu}}{S}[\cos{\phi_a}+\cos{\phi_b}]\\ &+\frac{4}{\overline{U}}(2+\lambda^2)\cos{[\phi_a+\phi_b]} \end{align} $$ and does not depend on $\theta_{\delta}$. This also illustrates independence of order parameters and chemical potential of $\theta_{\delta}$ in Figs. 4(a) and 4(b). Generally pseudospin analysis only adapts to strong coupling. However for our case $U/t=4$, strong coupling results can be applied. The minimization for $\phi_a$ and $\phi_b$ leads to $\phi_a=\phi_b=0$ for $\overline{\mu}>4S(2+\lambda^2)/\overline{U}$ and $\phi_a=\phi_b=\pi$ for $\overline{\mu} < -4S(2+\lambda^2)/\overline{U}$. Otherwise $\cos{\phi_a}=\cos{\phi_b}=\overline{\mu} \overline{U}/[4S(2+\lambda^2)]$. Here $\phi_a=\phi_b$ signifies uniform distribution of particle density and absence of CDW. The model Eq. (1) is available experimentally starting from an isotropic honeycomb lattice. To obtain complex hopping terms, we shake the lattice in a particular way specified in Ref. [30] (see Fig. 1(b)). If the shaking period is the shortest timescale, in the co-moving frame a static effective Hamiltonian can be obtained from the Floquet theory.[31] Compared with the undriven model, the hopping terms in the effective Hamiltonian are renormalized to different values depending on relative direction between hopping and shaking.[30] If shaking is along the $x$-direction, we will obtain $\theta_1=\theta_2\ne\theta_3$ and $t_1=t_2 \ne t_3$ required in this study. Here we should mention that in the experiment $\theta_1$, $\theta_2$, $\theta_3$ are not equal to zero simultaneously. However our conclusions calculated by requiring $\theta_1=\theta_2=0\neq \theta_3$ can be directly used to illustrate the experiment as long as the corresponding observables are invariant under the gauge transformation $a_i=a_ie^{i\theta_1}$. In particular, this gauge transformation cannot qualitatively change the pairing property of the ground state. At the exact half-filling, the Fermi surface of model Eq. (1) is composed of two separated Dirac points for any $\theta_3$ as long as $\lambda < 2$. For nonzero $\theta_3$ time-reversal and inversion symmetries are broken and pairings between two Dirac points definitely carry nonzero momentum, realizing the FF state. Away from the half filling, the point Fermi surface evolves into Fermi lines, reflecting the general characteristic of Fermi surface of the two-dimensional system. As demonstrated, the system remains as the FF state. From this analysis we realize that the FF state in model Eq. (1) does not result from the Dirac point structure of the Fermi surface, but from breaking of time-reversal and inversion symmetries. Thus the mechanism realizing FF state in model Eq. (1) can be extended to other lattices as long as the related symmetries are broken. A spin-balanced FF state without SOC was also proposed recently by a one-dimensional moving lattice to couple $s$ and $p_x$ bands of cubic optical lattices.[32] In this method the frequency difference of moving lattice must match the band gap between $s$ and $p_x$ bands. To a certain degree, this requires finer tuning than shaking in our model. In addition for a large coupling strength the ground state becomes a BCS superfluid eventually. However in our model, pseudospin analysis suggests that the ground state is still the FF state for arbitrary strong coupling. In conclusion, we have proposed the realization of a spin-balanced single plane wave FF state in honeycomb lattices with complex hopping terms. It is found that complex hopping terms induce the finite momentum Cooper pairing. Different from some popular suggestions associated with SOC and in-plane Zeeman field, our scheme is much simpler and more feasible experimentally, thus may open a new route for observing FF superfluids. References Inhomogeneous superconductivity in condensed matter and QCDSuperconductivity in a Strong Spin-Exchange FieldDetecting the Elusive Larkin-Ovchinnikov Modulated Superfluid Phases for Imbalanced Fermi Gases in Optical LatticesCollective Dipole Oscillations of a Spin-Orbit Coupled Bose-Einstein CondensateSpin-Injection Spectroscopy of a Spin-Orbit Coupled Fermi GasSpin-Orbit Coupled Degenerate Fermi GasesExperimental realization of two-dimensional synthetic spin–orbit coupling in ultracold Fermi gasesRoute to observable Fulde-Ferrell-Larkin-Ovchinnikov phases in three-dimensional spin-orbit-coupled degenerate Fermi gasesFFLO Superfluids in 2D Spin-Orbit Coupled Fermi GasesCompeting superfluid orders in spin-orbit-coupled fermionic cold-atom optical latticesUnconventional Superfluid in a Two-Dimensional Fermi gas with Anisotropic Spin-Orbit Coupling and Zeeman fieldsSuperfluid density and Berezinskii-Kosterlitz-Thouless transition of a spin-orbit-coupled Fulde-Ferrell superfluidTopological superfluids with finite-momentum pairing and Majorana fermionsTopological Fulde–Ferrell–Larkin–Ovchinnikov states in spin–orbit-coupled Fermi gasesGapless Topological Fulde-Ferrell Superfluidity in Spin-Orbit Coupled Fermi GasesPairing symmetry, phase diagram, and edge modes in the topological Fulde-Ferrell-Larkin-Ovchinnikov phaseGapless topological Fulde-Ferrell superfluidity induced by an in-plane Zeeman fieldThree-dimensional spin–orbit coupled Fermi gases: Fulde–Ferrell pairing, Majorana fermions, Weyl fermions, and gapless topological superfluidityFulde–Ferrell superfluidity in ultracold Fermi gases with Rashba spin–orbit couplingInhomogeneous Fulde-Ferrell superfluidity in spin-orbit-coupled atomic Fermi gasesExotic pairing states in a Fermi gas with three-dimensional spin-orbit couplingSpin–orbit coupling in quantum gasesCreating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb latticeSimulation and Detection of Dirac Fermions with Cold Atoms in an Optical LatticePseudospin symmetry and new collective modes of the Hubbard modelHidden symmetry and protection of Dirac points on the honeycomb latticeBCS-BEC Crossover on the Two-Dimensional Honeycomb LatticeThermodynamic properties of the extended Hubbard model with strong intra-atomic attraction and an arbitrary electron densityTunable Gauge Potential for Neutral and Spinless Particles in Driven Optical LatticesEffective time-independent description of optical lattices with periodic drivingFulde-Ferrell Superfluids without Spin Imbalance in Driven Optical Lattices

[1]	Casalbuoni R and Narduli G 2004 Rev. Mod. Phys. 76 263
[2]	Fulde P and Ferrell R A 1964 Phys. Rev. 135 A550
[3]	Larkin A I and Ovchinnikov Y N 1964 Zh. Eksp. Teor. Fiz. 47 1136
[4]	Loh Y L and Trivedi N 2010 Phys. Rev. Lett. 104 165302
[5]	Zhang J Y et al 2012 Phys. Rev. Lett. 109 115301
[6]	Cheuk L W et al 2012 Phys. Rev. Lett. 109 095302
[7]	Wang P et al 2012 Phys. Rev. Lett. 109 095301
[8]	Huang L et al 2016 Nat. Phys. 12 540
[9]	Zheng Z et al 2013 Phys. Rev. A 87 031602
[10]	Zheng Z et al 2014 Sci. Rep. 4 6535
[11]	Xu Y, Qu C, Gong M and Zhang C 2014 Phys. Rev. A 89 013607
[12]	Wu F, Guo G C, Zhang W and Yi W 2013 Phys. Rev. Lett. 110 110401
[13]	Cao Y et al 2015 Phys. Rev. A 91 023609
[14]	Qu C et al 2013 Nat. Commun. 4 2710
[15]	Zhang W and Yi W 2013 Nat. Commun. 4 2711
[16]	Cao Y et al 2014 Phys. Rev. Lett. 113 115302
[17]	Chan C F and Gong M 2014 Phys. Rev. B 89 174501
[18]	Hu H et al 2014 Phys. Rev. A 90 033624
[19]	Liu X J, Hu H and Pu H 2015 Chin. Phys. B 24 050502
[20]	Hu H and Liu X J 2013 New J. Phys. 15 093037
[21]	Liu X J and Hu H 2013 Phys. Rev. A 87 051608
[22]	Zhou X F, Guo G C, Zhang W and Yi W 2013 Phys. Rev. A 87 063606
[23]	Galitski V and Spielman I B 2013 Nature 494 49
[24]	Tarruell L et al 2012 Nature 483 302
[25]	Zhu S L, Wang B and Duan L M 2007 Phys. Rev. Lett. 98 260402
[26]	Zhang S 1990 Phys. Rev. Lett. 65 120
[27]	Hou J M and Chen W 2015 Sci. Rep. 5 17571
[28]	Zhao E and Paramekanti A 2006 Phys. Rev. Lett. 97 230404
[29]	Robaszkiewicz S, Micnas R and Chao K A 1981 Phys. Rev. B 23 1447
[30]	Struck J et al 2012 Phys. Rev. Lett. 108 225304
[31]	Hemmerich A 2010 Phys. Rev. A 81 063626
[32]	Zheng Z et al 2016 Phys. Rev. Lett. 116 120403