Synthetic topological Kondo insulator in a pumped optical cavity

Motivated by experimental advances on ultracold atoms coupled to a pumped optical cavity, we propose a scheme for synthesizing and observing the Kondo insulator in Fermi gases trapped in optical lattices. The synthetic Kondo phase arises from the screening of localized atoms coupled to mobile ones, which in our proposal is generated via the pumping laser as well as the cavity. By designing the atom-cavity coupling, it can engineer a nearest-neighbor-site Kondo coupling that plays an essential role for supporting topological Kondo phase. Therefore, the cavity-induced Kondo transition is associated with a nontrivial topological features, resulting in the coexistence of the superradiant and topological Kondo state. Our proposal can be realized with current technique, and thus has potential applications in quantum simulation of the topological Kondo insulator in ultracold atoms.


Introduction
The experimental realization and manipulation of ultracold atoms provide a versatile platform with feasible controllability to simulate the many-body physics in strongly correlated systems [1].
Taking advantages of current technique, theoretic and experimental investigations on ultracold atoms advance in creating artificial gauge fields [2,3,4,5,6] and a variety of optical lattices [7,8,9,10], which opens the way to explore and predict unconventional properties of strongly correlated systems, such as the topological the Kondo insulator [11,12].
The Kondo phase arises from the screening of localized electrons hybridized with mobile ones, forming a so-called Kondo insulator [13]. It captures many unusual properties of strongly correlated systems such as the heavy fermion materials. With coexistence of the spin-orbital coupling, it brings rich physics with the discovery of the topological Kondo insulator [14,15,16,17]. Recent studies of ultracold atoms shed light on the realization of the Kondo insulator by proposing a variety of schemes that are widely based on the optical coupling [18], the Bloch-orbital hybridizations [19], and the orbital Feshbach resonance [20,21,22,23,24,25]. Especially in the earlier work [19], artificial gauge fields generated by the laser-induced Raman coupling make it possible to detect the topological Kondo insulator. On the other hand, the study of Fermi gases in an optical cavity [26,27,28,29,30,31] offers another framework for engineering artificial gauge fields [32,33,34,35,36,37]. It predicts the existence of the topological superradiant states [38,39,40]. This motivates us to find an alternative scheme to synthesize a topological Kondo insulator by implementing the cavity field.
Here in this paper, we present a proposal for realizing the synthetic topological Kondo insulator in a pumped optical cavity. The paper is organized as follows. In Section 2, we start with the Hamiltonian describing atoms coupled to a pumped optical cavity, and design it to obtain the Kondo lattice Hamiltonian. In Section 3, we show the phase transition by changing experimental controllable parameters, and show the topological features of the superradiant Kondo phase. The extension of our proposal to a higher-dimensional case are discussed in Section 4. In Section 5, we summarize the work. The details of the effective Hamiltonian and the slave boson approach, which we use to study the Kondo phase, are formulated in Appendix A and Appendix B.

Effective Hamiltonian
We consider ultracold fermionic atoms trapped in a one-dimensional (1D) optical lattice oriented in x direction. The atomic level structure is illustrated in Figure. 1(a). In practice, for 6 Li alkali atoms as an example, we can choose two nuclear spin states with |F, m F = |1/2, 1/2 and |1/2, = −1/2 as |g, ↑↓ , and |3/2, 3/2 and |3/2, 1/2 as |e, ↑↓ , respectively. The spinful atoms are initially prepared in |g with double filling in each site. The experimental setup for our proposal is sketched in Figure. 1(b). When placed into a high-finesse cavity, the Raman transition between |g, σ and |e, σ (σ =↑, ↓) is driven by a plane-wave pumping laser h(r) with frequency ω p and linear polarization in accompany with a single-mode standing-wave cavity field η(x) with frequency ω c and σ − polarization. The selection rule in the Raman transition can suppress the unwanted atomic transitions. The optical cavity is oriented in the x axis, while the pumping laser is placed in the z plane. In order to realize the Raman transition by using the same laser and cavity fields, we introduce an AC-Stark shift individually to |g, ↓ , which can be generated by a far-detuned laser. This system is described by a Hamiltonian composed of three terms, The first term H A describes the atom subsystem, Here describes free atoms confined in the lattice. {ψ λσ , ψ † λσ } are the annihilation and creation field operators for the level λ = g/e, respectively. V λ (x) = V λ sin 2 (k L x) is the optical lattice trap potential where k L = π/d with d as the lattice constant. ∆ aσ is the detuning between |g, σ and |e, σ . For simplicity without loss of generality, we assume ∆ a↑ = ∆ a↓ = ∆ a . In the whole paper we set = 1. The second term H C describes the cavity subsystem, Here {a, a † } are the annihilation and creation operators for cavity fields, respectively. ∆ c is the cavity-pump detuning. The last term H I describes the interaction between the atom and cavity subsystems, Here g(x) denotes the atom-cavity coupling mode, which is originated from h(r) and η(x). The term H.c. stands for Hermitian conjugation. The details of the Hamiltonian (1) are given in Appendix A. From the Hamiltonian (1), taken the cavity decay into considerations, the Heisenberg equation for the cavity fields a is written as where κ is the decay rate. Hereafter we make the mean-field approximation a = α.
As known in previous works [31], the system undergoes a superradiant transition charactered by α, which is driven by tuning experimental parameters, e.g. the pumping laser amplitude Ω p . When α is nonzero, the system is noted as the superradiant state. Due to the cavity decay, the system is indeed a non-equilibrium one. For steady cavity field, α satisfies ∂ t α = 0, and thus we obtain Here L is the length of the 1D lattice. Inserting α back into Eq.(1), the Hamiltonian is recast asĤ In order to derive a Hubbard Hamiltonian to quantum simulate the Kondo physics, we use the tight-binding approximation and expand ψ λσ (x) in terms of Wannier wave functions W λ (x). Besides, the following two manipulations are required: (i) atoms in |g are deeply trapped in the lattice to make the tunneling amplitude t g ≪ t e . This can be achieved by means of two groups of counter-propagate lasers to create optical lattices separately trapping |g and |e with different depth. (ii) Via Feshbach resonance, we introduce strong repulsive interaction between |g with opposite spins in the same site. For example of 6 Li [41], we can make broad Feshbach resonance between |g, ↑ and |g, ↓ . By contrast in the same parameter region, the scattering between |e, ↑ and |e, ↓ is off resonance, hence their interaction is ignorable. After accomplishing the above manipulations, |g can simulate the localized fermions in Kondo problems, while |e simulates the conduction fermions. The effective Hamiltonian is then expressed as where ij denotes the summation between nearest-neighbor sites, {λ jσ , λ † jσ } are the annihilation and creation operators for atoms on j-th site, and U is the repulsive interaction strength. The synthetic Kondo coupling [12], we know that, aiming to synthesize the gapped Kondo state, i.e. the Kondo insulator, it is required that the tunneling of the localized fermions hosts an opposite sign compared with the conduction fermions. Unfortunately, this requirement is not easily satisfied, because the signs of the tunneling for the lowest Bloch bands, i.e. t g and t e , are usually the same. On the other hand, if we invoke the following transformatioñ into the Hamiltonian (9), the atom fieldsg and e will host tunneling with opposite signs. However, the new operator representation may lead to a staggered Kondo coupling (V ij e ijπ e † iσg jσ + H.c.). Next, we design the pumping laser and cavity mode to eliminate the staggered phase.

Synthetic Kondo coupling
The synthetic Kondo coupling V ij = αK ij is generated by the pumping laser as well as the atom-cavity coupling. It is obviously proportional to the superrandiance order α, hence works only in the superrandiant state. As we focus on the physics of the 1D lattice system along the x axis, the plane-wave mode of the pumping laser can be approximately treat as a constant strength h(r) = Ω p . From [12], we know the on-site Kondo coupling does not introduce topological nontrivial properties to the system. On the other hand, the beyond-on-site components with odd parity will exhibit features of spin-orbital couplings, and host the possibility to harbor a topological Kondo state. For this sake, we design the standing-wave mode of the cavity as Ω(x) = Ω c sin(k c x) where Ω c is the atomcavity coupling strength and k c is the cavity mode momentum. In this way, the on-site component in J ij vanishes while the nearest-neighbor-site component dominates. When we tune k c to match k L /2, it will impose a phase e −ijπ into J ij , eliminating the staggered phase e ijπ generated by the new operator representation (10). Thus we can obtain .Ω is the coupling strength between |g and |e , and its detailed form is given in Appendix A.
Based on the above designs, the final expression of the effective Hamiltonian is written as where V ij = ±αKδ i,j±1 . At the limit t g ≪ t e and U → ∞, the Hamiltonian (11) describes the Kondo lattice model associated with the off-site Kondo coupling [12].

Slave boson method
We employ the slave boson method [13] to explore the possible Kondo phase arising from our model Hamiltonian (11). In the slave boson method, the localized fermion operatorg are decomposed into a fermion operatorĝ associated with an auxiliary boson operatorb,g In this paper, we make the mean-field approximation to the boson operatorb and recognize b as the order parameter of the emergent Kondo phase. We focus on the results at zero-temperature limit, because the mean-field method can capture the qualitative and the topological features of the lattice system. After processing the standard approach (see Appendix B), we can get the effective action Here τ ≡ it is the imaginary time. β ≡ 1/(k B T ) with temperature T and we set the Boltzmann constant k B = 1 in the whole paper. H eff is expressed as [42] H with E 0 ≡ j λ j (|b j | 2 − 1) + ∆ c |α| 2 . λ j is the Lagrange multiplier introduced by the number conservation on each site, and we can see ∆ a − λ j characterizes the effective chemistry potential of the localizedĝ fermions. The ground state at zero temperature limit can be determined by self-consistently minimizing the free energy F with respect to the order parameter b and λ (See Appendix B): (i) When b is nonzero, the system is a Kondo state, and moreover is a superradiant Kondo state if α = 0. (ii) When b vanishes, the system is a normal gas state.

Phase transition and topological features
In Figure. 2, we plot the Kondo order b and the superradiance order α with respect to K at zero temperature limit. Experimentally, K can be tuned by the pumping laser strength Ω p and the configuration of the optical cavity. The gap of the Kondo phase is the manifest signature distinct from the normal gas state, resulting in a Kondo insulator [13]. It originates from the screening of localizedĝ fermions coupled to the mobile e fermions, and is evaluated by the nonzero order parameter b. In Figure. 2 we can see there exist two states as the ground state of the lattice system. When K exceeds a threshold, the system simultaneously processes two types of phase transitions: the Kondo transition from the normal gas state to the gapped Kondo phase, and the superradiant transition from the normal gas state to the superradiant state. This is because in Section 2.2 we have know the Kondo coupling works only in a superradiant state. In another words, the atom-cavity coupling is responsible to the Kondo screening. As both the two order parameters exhibit discontinuous evolutions across the phase transition, it yields that both the Kondo and superradiant transitions are of first order. What interests us more here is whether the cavity-induced superradiant Kondo phase is a topological nontrivial state. To analyze its possible topological properties, we make a Fourier transformation to the effective Hamiltonian (14). In the base Ψ k = (ĝ k↑ ,ĝ k↓ , e ↑ , e ↓ ) T , it is written as where ǫ g (k) = 2t g cos(kd) + λ − ∆ a , ǫ e (k) = −2t e cos(kd), V k = −i2αK sin(kd), and I 2×2 is the 2×2 identity matrix. We treat b as a real number here, since its phase can be rotated off in H eff and does not change the physics. The Hamiltonian (15) can be decompose into two parts: where we have used the Campbell-Baker-Hausdorff expansion. It is easy to demonstrate [H 0 , H 1 ] = 0 since H 0 behaves like an identity matrix, and then we obtain H eff = H 1 . Therefore, the lattice system described by H eff shares the same topological properties from the eigenstates of H 1 [43,44]. We can see that the Hamiltonian H 1 respects the particle-hole symmetry: ΞH 1 (k)Ξ † = −H 1 (−k), where Ξ = σ x K ⊗ I 2×2 and K is the complex conjugate operator. It indicates the superradiant Kondo phase belongs to a D topological class [45], which is characterized by a Z 2 topological invariant [46]. In Figure. 3(a), we display the Bogoliubov-de Gennes (BdG) quasiparticle spectrum (see Appendix B) for the superradiant Kondo phase. Distinct from the trivial Kondo phase, inside the band gap, the superradiant Kondo phase hosts edge states whose wave functions are localized on the chain ends, as shown in Figure. 3(b). This is the key signature as a topological insulator for the superradiant Kondo phase. The lattice model has no spin hybridization and is spin degenerate. Therefore, the edge states are four-fold degenerate. They are protected by the particle-hole symmetry discussed above.

Discussions
Our proposal is readily extended to a two-dimensional (2D) case. It can be realized by engineering a 2D optical lattice and two optical cavities in the x-y plane. We assume the cavity modes η 1 (x) = Ω c sin(k L x/2) and η 2 (y) = Ω c sin(k L y/2). Then the Kondo coupling originated from the Raman transition can give rise to a Chern Kondo insulator state, which has also been proposed via Bloch-orbital hybridizations in ultracold Fermi gases [19]. It is noted that, in the 2D extension of our proposal, the emergency of the Chern Kondo insulator state is still ascribed to the cavity field.

Conclusions
In summary, we propose a scheme for synthesizing the Kondo insulator in ultracold Fermi gases placed in a pumped optical cavity. The synthetic Kondo coupling originates from the Raman transition, which in our proposal is driven by the pumping laser as well as the cavity field. Distinguished from the trivial Kondo phase, the atom-cavity coupling gives rise to a synthetic nearest-neighbor-site Kondo coupling, and is the key ingredient for supporting the topological superradiant Kondo phase with edge states gapped from the bulk. Our proposal is simple and reliable based on current experimental technique, and can provide a versatile platform for quantum simulating and studying the many-body physics and topological phases of the Kondo insulator.
The Hamiltonian H ′ (x) is rewritten as Adiabatically eliminating |s , we obtain the final form of the effective Hamiltonian in Section 2.1: where g(x) = h(r)η(x)/∆ =Ω sin(k L 2/x) withΩ ≡ Ω p Ω c /∆. In Section 2.1, we set ∆ c = −δ. ∆ a can be changed by δ associated with an AC-Stark shift generated by a far-detuned auxiliary laser. The parameters we choose in the phase diagram are accessible in practice. They can be estimated as follows. The numeric calculation via the maximally localized Wannier functions gives the tunneling t ≡ t e ≈ 0.143E R with V e = 2E R , and t g ≈ 0.0308E R ≈ 0.215t with the trap depth V g = 8E R Here E R ≡ 2 /2md 2 is the lattice recoil energy. When tuning ∆ ≈ 300E R , Ω p ≈ 100E R and Ω c ≈ 600t, it gives K ≈ 12.03t. Therefore, the parameter regions displayed in Figure 2 are available in real ultracold Fermi gases.

Appendix B. Slave boson approach
In the slave boson method, the localized fermion operatorg is written as a decomposition of a fermion operatorĝ as well as an auxiliary boson operatorb, Hereĝ andb satisfy the single occupancy constraint on each site, Then by excluding the U-term, the Kondo lattice Hamiltonian (9) can now be recast as We make the mean-field approximation to the boson operator The effective action is written as S = dτ L ≡ dτ (L 0 + L 1 + L 2 ) with Here L 1 is introduced by the constraint from the particle number conservation (B.2) with the Lagrange multiplier λ j , and L 2 is induced from photons. The effective Hamiltonian is represented by combiningH and L 1 + L 2 where E 0 ≡ j λ j (|b j | 2 − 1) + ∆ c |α| 2 . The Lagrange action is then written as L = jσ (ĝ † jσ ∂ τĝjσ + e † jσ ∂ τ e jσ ) − H eff . H eff can be diagonalized into a quadratic form by employing the Bogoliubov-de Gennes (BdG) transformation Here E η is the BdG quasiparticle spectrum. The coefficients u η ≡ (u η 1 , · · · , u η 2L ) T and v η ≡ (v η 1 , · · · , v η 2L ) T obey the BdG equation expressed as where [ĥ gσ ] ij = t g δ i,j±1 + (λ j − ∆ a )δ ij , [ĥ eσ ] ij = −t e δ i,j±1 , andv ij = ±αKb * j δ i,j±1 . The mean-field variables b j ≈ b and λ i ≈ λ are determined by minimizing the free energy [42] F = E 0 − 1 β k,α ln [1 + e −βEα(k) ] , (B.11) where E α (k) are eigenvalues of H BdG in the momentum space, and β ≡ 1/T . This is achieved by self-consistently solving the following equations ∂F ∂θ = 0 (θ = b, λ) (B.12) as well as the steady-state condition of the cavity (see Eq.(6)) which in the tight-binding model is formulated as ,σ e jσ − ĝ † j−1,σ e jσ ) . (B.13)