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Realizing and Detecting the Haldane's Quantum Hall effect with Ultracold Atoms

We design an ingenious scheme to realize the Haldane's quantum Hall model without Landau level by using ultracold atoms trapped in an optical lattice. Three standing-wave laser beams are used to construct a wanted honeycomb lattice, where different on-site energies in two sublattices required in the Haldane's model can be implemented through tuning the phase of one of the laser beams. The staggered magnetic field is generated from the Berry phase associated with the atom moving in a region with other three standing-wave laser beams. Moreover, we establish a relation between the Hall conductivity and the equilibrium atomic density upon turning on a stimulated uniform magnetic field, which enables us to detect the topological Chern number with the density profile measurement technique that is typically used in ultracold atoms experiments.

The quantum Hall effect (QHE) [1] in two-dimensional electron systems is one of the most peculiar quantummechanical phenomena observed in nature. The QHE is usually associated with a uniform external magnetic field, which splits the electron energy spectrum into discrete Landau levels (LLs). When the Fermi energy lies in the gap between two LLs, the Hall conductivity in units e 2 /h is accurately quantized to an integer. The precise quantization of the Hall conductivity was explained by Laughlin [2] based upon a gauge invariance argument, which is fundamental to the picture of edge states proposed by Halperin [3]. On the other hand, Thouless, Kohmoto, Nightingale and Nijs (TKNN) [4] interpreted the Hall conductivity as the topological Chern number of the U (1) bundle over the magnetic Brillouin zone of the bulk states.
Twenty years ago, Haldane showed in principle that a QHE may also result from breaking of time-reversal symmetry without any net magnetic flux through a unit cell of a periodic two-dimensional (2D) system, where the electron states retain their usual Bloch state character [5]. In his work, Haldane constructed a tight-binding model on a honeycomb lattice including a complex second nearest-neighbor hopping integral. The honeycomb lattice consists of two triangular sublatticesĀ andB with different on-site energies M and −M , as shown in Fig. 1a. For M = 0, the inversion symmetry is broken and the lattice possesses the point group C 3υ symmetry. A periodic vector potential A(r) is applied to the lattice, given that the total magnetic flux through each unit cell vanishes, i.e., the first-neighbor hopping integral t is unaffected. The second-neighbor hopping integral t ′ acquires a Peierls phase factor exp(ie A · dr/h), where the integration is along the hopping path. The Hamiltonian of the model is written as where a i and b i are the annihilation operators on site R i in sublatticesĀ andB, respectively. ϕ jl is the accumulated Peierls phase from site j to its second neighbor l, which is assumed to take the form ϕ jl = ±ϕ. The hopping directions for which ϕ jl = +ϕ are shown in Fig. 1a. The most interesting and unique feature of the model lies in that the phase of the system can be changed from a normal insulator to a Chern insulator by the simulation of parity anomaly [5,6,7,8,9,10,11]. However, it is extremely hard to realize the Haldane's model experimentally in ordinary condensed matter systems because of the unusual staggered magnetic flux assumed in the model.
On the other hand, the technology of ultracold atoms in an optical lattice provides a perspective approach to explore rich fundamental phenomena of condensed matter physics [12,13,14]. In particular, how to realize the QHE with cold atoms has attracted considerable interest [15,16,17,18,19]. Nevertheless, the atomic QHE has not been observed yet, mainly due to challenges in both the realization and detection of the atomic Hall effects. Although an effective magnetic field for neutral atoms can be simulated either by rotating the atoms [20] or by laser-induced Berry phases [21,22,23,24], the strong magnetic field region required for QHE has not been reached yet in experiments. For the rotating method, the system is close to the point at which the centrifugal potential cancels the external harmonic trap, and the atoms may fly apart at the rotation speed required by QHE [16,19]. For the laser-induced Berry phase approach, the cold atoms moving in a spatially varying laser field feel an effective gauge potential [21,22,23,24], but the region of the strong uniform field is rather small for two typical counterpropagating Gaussian laser beams. In addition, the detection method for cold atoms is very different from that for condensed matter systems; especially the widely used technique for QHE based on the transport measurements is not workable for atomic QHE. In this Letter, we design an ingenious scheme to realize the Haldane's quantum Hall model without LLs by using ultracold atoms trapped in an optical lattice. We work out a distinct method to construct the honeycomb lattices that have different on-site energies by three standing-wave laser beams. Although it is still hard to achieve a strong homogenous magnetic field required by the conventional QHE in atomic system, we may evade this bottleneck since LLs are not necessary in this unconventional QHE. We elaborate that the staggered magnetic field, which is hard to generate in condensed matter systems, may be rather easy to set up by other three standing-wave laser beams. In this scenario, different on-site energies in two sublattices can be easily adjusted through tuning the phase of one of the laser beams, and thus the whole phase diagram [5] including the exotic topological phase transition predicted by Haldane may be revealed experimentally. Furthermore, based on (2 + 1)-dimensional relativistic quantum mechanics calculations, we establish a direct relationship between the Hall conductivity and the equilibrium atomic density, such that the famous topological Chern number may be experimentally detected with the standard density profile measurement used in atomic systems [12].
Let us first consider single component fermionic atoms (e.g., 40 K, 6 Li, etc.) in a 2D honeycomb lattice [14,25], which can be realized by three detuned laser beams. A detuned standing-wave laser beam will create a potential in the form V 0 sin 2 (k L 0 · r), where V 0 is the potential amplitude and k L 0 is the wave vector of the laser. To generate the honeycomb lattice with different on-site energies in sublatticesĀ andB, the three laser beams with the same wave length but different polarizations are applied along three different directions: e y and √ 3 2 e x ± 1 2 e y , respectively. The potential is thus given by The potential contours are plotted in Fig. 1b and Fig. 1c. Under these conditions, atoms are trapped at the minima of the potential, forming a honeycomb lattice. An amazing feature here is that the different siteenergies of sublatticesĀ andB is controllable by the phase of laser beam χ. For instance, we get exactly the honeycomb lattice with the same on-site energies (M = 0) for χ = 2 3 π, as shown in Fig.1b, while the two sublattices have different on-site energies (M = 0) for χ = 2π 3 . Now we elaborate how to simulate the staggered magnetic field in the Haldane's model. Since the net flux per unit cell vanishes, the vector potential applied to the lattice must be periodic. Such magnetic fields can be created by Berry phase induced from two opposite-travelling standing-wave laser beams [22]. For the two laser beams with Rabbi frequencies Ω 1 = Ω 0 sin(yk L 2 + π 4 )e ixk L 1 and Ω 2 = Ω 0 cos(yk L 2 + π 4 )e −ixk L 1 , the effective gauge potential is generated as A 1 (r) =hk L 1 sin(2yk L 2 )e x [22]. Here, k L 1 = k L cos θ and k L 2 = k L sin θ with k L the wave vector of the laser and θ the angle between the wave vector and the e x axis. We emphasize that the choice of wave vector k L 2 of the laser beams must be a multiple of 2 √ 3π 3a in order to be commensurate with the optical lattice. We take k L 2 = 2 √ 3π 3a . The Peierls phases for the nearest-neighbor hopping in Fig. 1a are ϕ 12 = ϕ 61 = −ϕ 34 = −ϕ 45 = ϕ 0 and ϕ 23 = ϕ 56 = 0. For the next-nearest-neighbor hopping integrals, which are integrated on a period of the vector potential, the corresponding accumulated phases are ϕ 13 = ϕ 24 = ϕ 46 = ϕ 15 = 0, and Since the lattice has the symmetry of point group C 3v , the vector potential A 1 is rotated by ± 2 3 π to obtain the other two vector potentials. Then the total accumulated phases along the nearest-neighbor directions are found to cancel out because of the symmetry of honeycomb lattice. However, the total accumulated phases for the next-nearest-neighbor hopping along the arrowed directions of the dashed lines in Fig. 1a are just ϕ. Therefore, the total vector potential and magnetic field can be written as The contours of the magnetic field are plotted in Fig. 1d, in which the red lines indicate where the magnetic field vanishes. The total magnetic flux through each hexagon vanishes, as the Peierls phase accumulated along its edges is zero. As a consequence, the total Hamiltonian of this cold atomic system can be described by Eq. (1). With Fourier transformation a j = 1 √ N k e ik·Rj a k and b j = 1 √ N k e ik·Rj b k , the Hamiltonian of the system can be written by using "spinors" (a k , b k ) t as where h 0 (k) = 2t ′ cos ϕ i cos(k·a i ), h 1 (k) = t i cos(k· s i ), h 2 (k) = t i sin(k · s i ), and h 3 (k) = M + 2t ′ sin ϕ i sin(k · a i ). s i (i = 1, 2, 3) are the three vectors pointing from aB site to its three nearest neighbors. The vector h(k) = [h 1 (k), h 2 (k), h 3 (k)] is an effective magnetic field of the "spinors". The energy spectra are E = h 0 (k) ± |h(k)| and the energy gap is |h 3 3a (1, 0). If h 3 = 0, the conduction band touches the valence band at two Dirac points K + and K − . The famous TKNN index or Chern number for this system is given by C = 1 8π 1BZ d 2 kǫ µνĥ · (∂ kµĥ × ∂ kνĥ ) [26] withĥ(k) as the unit vector of h(k). It is demonstrated that the gauge invariance C can only take integer values and the quantum Hall conductivity is proportional to the Chern number [4]. For the Haldane's model, different phases of the system can be characterized by different values of C. The phase diagram is depicted in Fig. (2a) [5], in which the solid line is the critical boundary between the normal insulator with C = 0 and the Chern insulator with C = ±1. It is notable that the phase ϕ = π tan θ can be controllable by simply choosing the laser angle θ, while the energy difference M between sublattice can be tuned by the phase of laser beams. With such controllability, it is promising to realize the exotic topological phase transition between different Chern numbers in Fig. 2a.
We now turn to establish a direct connection between the topological Cherm number and the atomic density, noting that the latter can be detected with density profile measurements typically used in atomic systems. We first develop a Green's function method to calculate the atomic density. The system is actually described by a Dirac-like Hamiltonian which can be obtained by expanding Eq.(3) around two Dirac points K ± = ± 4π 3a (1, 0). By the substitution of k → K ± + p, we have is the Fermi velocity and m ± = M ± 3 √ 3t ′ sin ϕ. Under unitary transformation σ 2 H − σ 2 , we can write the Hamiltonian in a more symmetric form where the notation m = ±m ± is introduced for simplicity (i.e., we temporally omit subscripts ± below). It is observed that the conductivity σ xy (Chern number) of the system is related to the atomic density ρ according to the Streda formula σ xy = ∂ρ/∂B| µ,T once an additional uniform magnetic field B is applied. Such a magnetic field can be simulated by rotating the optical lattice at a constant frequency ω = eB/2m. We choose the vector potential as A 0 = A 1 = 0 and A 2 = Bx. By using the substitution p → p + eA with e > 0, Eq.(4) can be solved in the real space. The eigenenergies of the Hamiltonian can be obtained as [5] E n = −msgn(eB) n = 0 ± m 2 + 2nhv 2 F |eB| n = 1, 2, 3... , and the degeneracy of each LL is |B|/φ 0 per unit area with φ 0 the flux quantum. The density ρ in terms of the Green's function for the Dirac Hamiltonian is given by where / D = γ τ D τ , with D 0 =h∂ 0 − µ, D 1 =hv F ∂ 1 , D 2 = hv F ∂ 2 + iev F Bx, γ 0 = σ 3 , γ 1 = σ 2 , and γ 2 = −σ 1 in Euclidean space. From the standard Green function approach, the atomic density is explicitly obtained as where Θ stands for the unit step function and int[x] means the largest integer less than x. The second term of Eq. (7) is the atomic density induced into vacuum as µ → 0 by the uniform magnetic field. It is of parity anomaly since its corresponding Hall current is independent of the magnetic field after multiplying the drift velocity E/B, where E is the electric field. The total atomic density is given by the sum of the densities of the two components ρ = ρ + + ρ − . At µ = 0, the Hall conductivity at B = 0 can be obtained from the density by using the Streda formula as σ xy = C e 2 h , where is the Chern number. To show how to detect the Chern number of the system, we consider a finite magnetic field B. The calculated density ρ in unit of |eB|/φ 0 is plotted as a function of the normalized chemical potential µ/|m + | (for |m − | = 0.5|m + | and ehv 2 F B = 4|m + |) in Fig. 2. It is essential that the spatial density profile ρ(r) is uniquely determined by the function ρ(µ/|m + |) in the local density approximation, which is typically well satisfied for trapped fermions. Figures 2b-2e stand for four different cases. The plateaus in the atomic density have one-to-one correspondence to the plateaus in the Hall conductivity due to the finite magnetic field B > 0. We here focus on µ = 0, which is of our main interest. For m + < 0 < m − , which corresponds to C = −1, the atomic density ρ = −B/φ 0 < 0, as shown in Fig. 2b. For m + > 0 > m − , which corresponds to C = 1, the density ρ = B/φ 0 > 0, as shown in Fig. 2c. For the other two cases m + < m − < 0 and m + > m − > 0, m + and m − have the same sign, corresponding to C = 0, and the density ρ = 0, as seen from Figs. 2d and 2e. Therefore, a simple direct relation between the Chern number and the equilibrium atomic density is established as The important relation (8) actually provides us a feasible way to experimentally detect the Chern number C in different phases. In the absence of B, the density of the cold atoms at µ = 0 is first measured, which is denoted as ρ 0 . Then the optical lattice is rotated to generate the effective uniform magnetic field B, and the new density of the cold atoms ρ 1 is measured. If ρ 1 > ρ 0 , the system is in a Chern insulator phase with Chern number C = 1. If ρ 1 < ρ 0 , the system is still a Chern insulator with C = −1. However, if ρ 1 = ρ 0 , the system behaves like a normal insulator with Chern number C = 0. Since the density difference is actually quantized in units B/φ 0 , the above method could be rather robust.
Finally, we briefly address an alternative approach to realize the Haldane's QHE. The fermions we discussed are in the s-band of the honeycomb lattice. As for fermions in the porbital bands, a Haldane's quantum Hall model without LLs can also be implemented by rotating each optical lattice site around its own center [27]. Nevertheless, how to detect such p-band QHE is still desirably awaited.
In summary, we have shown that the Haldane's QHE model can be realized by using ultracold atoms in an optical lattice. We have established a relationship between the Hall conductivity and the equilibrium atomic density, which provides a feasible way to experimentally detect the Chern number C in different phases. Note added -After this work was completed we became aware that a relation between the density profile and the Hall conductivity in conventional QHE was also addressed in Ref. [28]. We thank Dr. H. Zhai for bringing our attention to that work.