Fermionic superfluidity and spontaneous superflows in optical lattices

We study superfluidity of strongly repulsive fermionic atoms in optical lattices. The atoms are paired up through a correlated tunneling mechanism, which induces superfluidity when repulsive nearest-neighbor interactions are included in the Hubbard model. This paired superfluid is a metastable state which persists for a long time as the pair-broken process is severely suppressed. The mean-field phase diagram and low energy excitations are investigated in a square lattice system. Intriguingly, spontaneous superflows may appear in the ground state of a triangular optical lattice system due to antiferromagnetic frustration.


I. INTRODUCTION
Recently, the strongly correlated properties of ultracold atoms in optical lattices have attracted great interests in physicists [1][2][3]. The high tunability of the interaction strength between atoms, as well as the easy manipulation of the optical lattices makes the system realistically viable to implement a quantum simulator [4]. The quantum atomic gases in optical lattices make it possible to build a model system so that we can explore the correlated properties in many-body physics such as superconductivity, quantum magnetism, quantum criticality, etc, and examine the related theoretical models [5][6][7].
A recent experiment [8] showed that a couple of strongly repulsive atoms occupying the same site of an optical lattice can be stabilized by damping the single particle tunneling. The lifetime of the pair increases significantly with the on-site repulsion U of the Hubbard model, which is quite intriguing as intuitively an attractive force between particles is required to obtain a bound state. In the presence of a periodic spatial potential, the energy of a particle does not vary continuously but is restricted to particular ranges of values. A pair of strongly repelling particles can be stable because if it fell apart, the two isolated atoms would ensure kinetic energies that fall in a forbidden band [9]. Another experiment directly observed that for a pair of strongly repulsive atoms in the optical lattice, single-particle tunneling is severely suppressed by the requirement of energy conservation while atom-pair co-tunneling is permitted through the secondorder quantum process [10]. Although these experiments were performed on bosonic atoms, it is conceivable that they can be applied to fermionic atoms since no quantum statistics is involved.
The formation of metastable atom pair with repulsive interactions was first proposed by A.F. Andreev in his study of diffusion of impurities in quantum crystal [11,12]. The quantum liquid of repulsive particle pairs in optical lattices has been discussed by several authors [13][14][15][16][17]. In a recent paper, Rosch et al studied metastable superfluidity of repulsive fermionic atoms in optical lattices [18]. Other authors attempted to explore the possibility of counterflow superfluidity and to con-trol spin exchange interaction of two-species ultracold atoms in a commensurate optical lattice [19][20][21][22]. In the fermionic superconductivity and superfluidity, a central concept is pairing. In Bardeen-Cooper-Schrieffer (BCS) theory, electrons pair up by an attractive interaction mediated by phonons of the underlying crystals. The attraction between ultracold fermionic atoms is provided by Feshbach resonances. Another pairing mechanism comes from correlated hopping, which occurs in fermionic tightbinding models. It has been suggested as a possible explanation for the high-T c superconductivity [23][24][25].
In this paper we explore the fermionic superfluid (SF) in a system consisting of repulsive atom-pairs in optical lattices. We deal with a system partially filled by couples of spin-up and spin-down fermionic atoms. The lattice site is either occupied by two atoms or empty in the one-band Hubbard model. In the limit of the strong on-site repulsion U ≫ t, the pair-breaking tunneling is suppressed as is revealed by the experiment [8]. On the other hand, the atom-pairs may transport across the lattice through the second-order quantum transition [10]. As a result, the atoms always move in pairs. The effective Hamiltonian is obtained through the quantum perturbation theory, which is mapped to an anisotropic antiferromagnetic (AF) model instead of the usual ferromagnetic model in the Bose-Hubbard model. The system exhibits superfluidity when the nearest-neighbor (NN) repulsive interactions are included. This paired superfluid is a metastable state because the pair-breaking process is severely suppressed. We investigate the mean-field (MF) phase diagram and low energy excitations for a square lattice system. It exhibits a gapless mode in the SF state and a gapful mode in the solid state.
The AF feature of the effective Hamiltonian may lead to an interesting phenomenon when the underlying optical lattice is triangular. As well-known, the ground state has a long-range 120 0 Néel order. The variation of the azimuthal angles between the NN spins corresponds to the phase modulation of the superfluid state, which leads to a spontaneous superflow in the ground state due to AF frustration. As a result, the system exhibits a pattern of convection consisting of vortex-antivortex pairs. The paper is organized as follows: In Sec.II we obtain an effective Hamiltonian for strong on-site interactions by the second order quantum perturbation approximation. In Sec.III we demonstrate the superfliudity of the paired fermionic atoms by mapping the system to the pseudospin-1/2 antiferromagnetic model. The phase diagram and low energy excitations are calculated in the MF approximation. In Sec. IV we explore the possible phenomenon of spontaneous superflows of the paired atoms in a triangular lattice. A summary is included in Sec.V.

II. EFFECTIVE HAMILTONIAN
We focus on the partial pair-filling system with ν < 1. For sufficient low temperature and strongly repulsive onsite interaction, the atoms will be confined to the lowest band which is described by the Hubbard model [5], is the annihilation (creation) spin-σ fermionic operator, n i = n i↑ + n i↓ is the number operator with n iσ = f † iσ f iσ , and t σ is the tunneling matrix element. U > 0 is the on-site interacting energy and V > 0 is the NN interaction. In this work, we confine our discussions to the case of U ≫ t σ , V .
Since the pair-breaking processes are suppressed, the single particle hoping is eliminated in the second-order quantum perturbation theory. The on-site Hubbard term is considered as the the unperturbed Hamiltonain H 0 . The hopping term is treated as the perturbation H 1 , which should be calculated to the second order of t σ /U to avoid pair-breaking. The NN interaction term commutes with the Hubbard term and will be included in the effective Hamiltonian later. Using a generalization of the Schriffer-Wolf transformation [26], where [S, H 0 ] = −H 1 and S † = −S. Suppose |α are the degenerate paired states of the unperturbed H 0 with energy E 0 and |β are the pair-breaking intermediate states It should be emphasized that in the initial states |α all atoms are paired up in the lattice sites while in the intermediate states |β only one pair of atoms is breaking. We have α|H 1 |α ′ = β|H 1 |β ′ = 0 and α|S|β = α|H 1 |β /U . Disregarding the intermediate states with more breaking atom-pairs which will involve higher order of t σ /U , the the second order quantum perturbation Hamiltonian is then An alternative method of the degenerate quantum perturbation theory can be found in Refs. [19,27]. The effective Hamiltonian is then, where z is the number of the NN sites. In the secondorder quantum perturbation, the intermediate virtual state |β that breaks the atom-pair has a lower energy (−U ) than the initial and final states |α , which induce an attractive NN interaction. In order to prevent the atom-pairs from congregation, a moderate repulsive NN interaction V is introduced in the original Hamiltonian (1) to overcome the induced attractive interaction. The first term describes the pair-hopping, implying that the spin-up and spin-down atoms transport together across the lattice. The composite object behaves like a hardcore bosonic molecule because the fermion pairs always hop together to their nearest neighboring site and for each site only one pair of atoms is allowed. It should be noted that the pairing in our work is of s-wave type, where only the lowest single-particle band is considered in the Hubbard model.
To study the MF properties of the system, it is convenient to map the effective Hamiltonian (4) to the spin representation [19,28] The NN interaction V also acts as an external magnetic field exerting on the pseudo-spins.
In contrast to the ferromagnetic model [29,30] in the usual hardcore Bose-Hubbard model, the effective Hamiltonian (5) represents an anisotropic AF model, where there are several competitive phases dependent on the pair-filling ν as well as V and t σ . Rosch et al obtained a ferromagnetic model by making a particle-hole transformation for the down spins [18]. At the MF level, we minimize the energy at fixed z-polarization or pairfilling. Suppose the classical spins S i are in the X-Z plane with an angle θ i to the z-axis, then a bipartite structure with sublattices A and B is employed to describe the possible periodicity in the ground state [37][38][39]. The candidate states include an easy-plane AF phase (θ A = −θ B ) or paired SF with a non-vanishing order parameter f i↑ f i↓ = 0, and a canted AF phase (cos θ A = cos θ B ), which is actually a checkboard solid with a non-vanishing f i↑ f i↓ in one sublattice while a vanishing f i↑ f i↓ in another sublattice. In addition, there is a phase separation (PS) regime caused by the attractive NN interactions. The easy-plane ferromagnetic phase (θ A = θ B ) is proved to have higher energy than the easy-plane AF phase and does not appear in this system.

III. MEAN-FIELD RESULTS IN A SQUARE LATTICE
We are now ready to explore the superfluidity of fermionic atoms in a square optical lattice (z = 4). Hereafter we use the units of U = 1. Figure 1 displays the V − ν phase diagram for hopping integrals t ↓ = 0.1. We take, e.g., the ratio t ↑ /t ↓ = 1.1. Other choice of t ↑ /t ↓ does not alter the conclusion qualitatively. We compare the mean-field energies of each candidate phases to determine the ground state. Generally, the canted canted AF phase with cos θ A = cos θ B = 0, 1 is a supersolid. But in the square lattice we find θ B = 0 or π, implying the supersolid order degenerates to an ordinary solid. We will reexamine this issue through the low energy excitations. The solid phase takes place in the regime of 0.4 < ∼ ν < ∼ 0.6. For the SF order, there is a π-phase difference between the two sub-lattices (canted AF order). The phase diagram is symmetrical with respect to the pair-filling ν = 0.5 which results from particle-hole symmetry of the effective Hamiltonian (4).
We discuss the superfluidity in terms of the condensate The SF phase has a uniform condensate density ρ s = ν(1 − ν) for a given pair-filling, independent of the value of V . In Fig. 2, we plot the condensate density for the SF state (dashed curve) as well as the solid state (solid curves) versus the filling ν for V = 0.1. At the MF level, the condensate density vanishes in one sublattice while does not vanish in another sublattice. This indicates that this phase is a usual checkboard solid instead of a supersolid. More accurate calculations such as quantum Monte Carlo sim- ulations have demonstrated that supersolid states indeed do not exist in a square lattice system [31,32]. The superfluidity can also be investigated through the low-energy excitations, which provides an accurate probe for the nature of the quantum phase. We study the lowenergy excitations by introducing pseudo-spin operators a † i (a i ) for sublattice A and b † i (b i ) for sublattice B, respectively. In this case, a † i = S + i = f † i↓ f † i↑ , such as for sublattice A, is a composite bosonic operator. After making a rotation to align the local spins along the z-direction, we obtain the spin-wave type Hamiltonian in the momentum space as, where γ k = (cos k x +cos k y )/2, the renormalized parameterst = 4t ↑ t ↓ /U ,Ũ = 4[V − (t 2 ↑ + t 2 ↓ )/U ],H z = 2(zV − µ) and the summation for momentum k is restricted to a half of the Brillouin zone. H linear includes the linear term in operators a k and b k . With H linear = 0, θ A and θ B are determined and the above MF result is recovered. This Hamiltonian (6) can be diagonalized in terms of the Bogoliubov transformation to obtain the low-energy excitation spectrum. For the SF state we have,  Figure 3 exhibits the excitation energies versus the momentum along the x-direction for t ↓ = 0.1, t ↑ = 1.1t ↓ , and V = 0.1. For the superfluid state in Fig.3(a) with ν = 0.2, the energy spectrum is linear at small momentum which reveals a gapless mode. At larger momenta, a energy dip plays the role of a roton as in the 4 He superfluid. The excitations of the solid state in Fig.3(b) with ν = 0.45 reveals a gapful mode. No gapless low energy excitation mode is found in this parameter regime. It justifies that this phase is a usual checkboard solid rather than a supersolid.

IV. SPONTANEOUS SUPERFLOW IN A TRIANGULAR LATTICE
We now study an intriguing phenomenon of spontaneous superflow of the fermionic superfluidity for a triangular optical lattice system. Although there are some debates on the possible disordered ground state in a triangular AF model because of the geometric frustration as well as quantum fluctuations, the current consensus is that the ground state has a long-range Néel order [33][34][35][36]. We consider the classical spins S i and employ the 120 0 XY-Néel order for three sublattices A, B, and C. Let θ A , θ B and θ C be the corresponding polar angles which reflect the spatial density variations [37][38][39], the MF energy of the system is written as where µ is the Lagrangian multiplier that controls the total pair-filling. In formula (8), the 2π/3 azimuthal angle difference between spins in the three sublattices has been incorporated.
The MF energy should be minimized with respect to the angles θ A , θ B , and θ C at a given polarization S z = ν − 1 2 . It has a form similar to that in Ref. [40] except the first term may become negative. In that case, the system is phase separated. Generally, there is a supersolid phase with θ A = θ B = θ C . For V < (t ↑ + t ↓ ) 2 /U , the groundstate is a uniform superfluid. We focus on the uniform superfluid phase (θ A = θ B = θ C = cos −1 (ν − 1 2 )), which is implemented at a moderate NN interaction V > ∼ (t 2 ↑ + t 2 ↓ )/U . We explore the implications of this 120 0 Néel state and its possible consequence in the paired superfluid. According to the spin mapping f i↑ f i↓ = S − i = 1 2 sin θ i e −iφi , the paired superfluid has an order parameter phase as that of the azimuthal angle φ i of the spin. Therefore, the 2π/3 azimuthal angle difference of the spins implies a ∆φ ij = 2π/3 phase difference between neighboring sites of the superfluid. The Néel order of the antiferromagnetic model thus corresponds to a periodical phase modulation in the superfluid. In the theory of the Josephson tunneling, two weakly connected superfluids or superconductors will induce a current as a result of their phase difference as J ∝ sin ∆φ. Consequently, the ground state superfluid spontaneously flows along the link of the neighboring sites, as is shown in Fig.4(a). In the triangular lattice system, the superfluid flows form a closed ring-like vortex. Figure 4(b) schematically displays a regular convection pattern of superflows which contains a sequence of the vortex-antivortex pairs. Similar cellular superflows and periodic textures were suggested in the 3 He − A superfluid when a perpendicular magnetic field is applied to a sample slap [41]. The coupling between the superfluid velocity and the orbital axis favors spontaneous superflows. Early theoretical discussions of possible superflow in solid 4 He involved quantum tunneling through ground-state vacancies, as well as Bose-Einstein condensation and quantum exchanges within the lattice [12,42,43]. In 2004, Kim and Chan reported the observation of the unusual superflow without resistance from frictional forces in crystalline helium [44,45]. This remarkable finding has now been confirmed [46][47][48]. The latest experiments indicate that, rather than being an intrinsic property of a perfect quantum solid, superflows owe their existence to macroscopic defects or extended disorder in the structure of solid helium.
In a mismatched Josephson junction of ultracold fermionic atomic gases, M. L. Kulić [49] proposed an oscillating superfluid amplitude inside the weak link and as a result the so-called π-junction. If the junction is a part of the closed ring then spontaneous and dissipationless superfluid current can flow through the ring.

V. SUMMARY AND DISCUSSIONS
We have studied the superfluidity of strongly repulsive fermionic atoms from a correlated pairing mechanism. The superfluid is a metastable state with the optical lattice sites either doubly occupied or empty. The composite objects transport in the optical lattice through the second quantum processes via virtual pair-breaking states. It exhibits superfluidity below a critical temperature. Phase diagrams and low-energy excitations in the square optical lattice system are investigated. Due to the AF frustration, the correlated pairs may result in an appealing spontaneous superflow phenomenon in the triangular optical lattice system. Some authors explored the possibility of formation of the non-s-wave BEC through Feshbach resonance in a nonzero angular momentum channel on a lattice with double occupation [50][51][52]. Varying the detuning of nons-wave resonance can lead to various quantum phase transitions between the phases: S-wave BEC, non-S-wave BEC, conventional Mott insulator and orbital Mott insulator (with broken lattice symmetries). This becomes possible when the atoms are confined in the p-orbital Bloch band of an optical lattice rather than the usual sorbital band. The new condensate simultaneously forms an order of transversely staggered orbital currents, reminiscent of orbital antiferromagnetism or d-density wave in correlated electronic systems but different in fundamental ways.
The NN interaction depends on the overlap of the Wannier functions between the NN sites. A moderate value of V ∼ (t 2 ↓ + t 2 ↑ )/U is sufficient to create the superfluidity in the system. An alternative way of generating NN interaction by the long-range dipolar interaction is also possible [53,54]. In order to detect the superfluidity of the correlated pairs, photoassociation spectroscopy may be used [55]. Interference of matter waves released from the lattice has been used to probe the superfluidity of single atom condensation [56]. By tuning the interaction from repulsive into attractive, the fermionic atom pairs are converted into molecules. The sharp peaks will appear in the interference pattern of the released bosonic molecules due to the presence of a SF fraction.