Bosonic fractional quantum Hall conductance in shaken honeycomb optical lattices without flat bands

The fractional quantum Hall (FQH) effect is one of the most fascinating phenomena in past decades Stormer et al. (1999), where multiple many-body ground states share one integer Chern number and effectively each of the band obtains a fractional number to characterize the conductivity. In previous studies Stormer et al. (1999); Cooper (2020); Wen (2017), it was proved that the FQH effect can be achieved either in fermions or bosons. However, due to the fermionic nature of electrons in conventional materials, the FQH effect has only been observed experimentally in fermions. Thus, this leaves an open question about how to realize the bosonic FQH effect experimentally, for example, to realize and probe the Hall conductance corresponding to one half.

With the development of quantum simulations Bloch et al. (2008); Cooper et al. (2019), the platforms of ultracold atoms now provide good opportunities to study and simulate strong-correlated many-body systems Sørensen et al. (2005); Hafezi et al. (2007); Cooper and Dalibard (2013a); Wang et al. (2011); Neupert et al. (2011); Hudomal et al. (2019); Sun et al. (2011); Tang et al. (2011); Grushin et al. (2014); Cooper and Dalibard (2013b), particularly for bosonic FQH effect. There are many pioneer experiments in realizing non-trivial Chern numbers or large synthetic gauge fields Miyake et al. (2013); Aidelsburger et al. (2013); Jotzu et al. (2014); Aidelsburger et al. (2015, 2011); Wintersperger et al. (2020); Beeler et al. (2013); Stuhl et al. (2015); Lin et al. (2009) in order to reach regimes of strong correlations. Inspired by previous studies of FQH effect in “Haldane-like” models Wang et al. (2011); Neupert et al. (2011); Haldane (1988); Jotzu et al. (2014), we find that the bosonic FQH conductance can be achieved and experimentally initialized in shaken honeycomb optical lattices without synthetic magnetic fields. This scheme relies on the Feshbach resonance at s-wave scatterings Chin et al. (2010) which does not require special long-range interactions.

Meanwhile, compared to previous studies Wang et al. (2011); Neupert et al. (2011); Sun et al. (2011); Tang et al. (2011) requiring a flat band where the band gap is much larger than the band width, our scheme is realized with the nearest-neighbor-hopping model and does not require a very flat band. Our band gap is almost the same as the band width. This avoids special designs for high-order hoppings for far-apart lattice sites to flatten the bands, and reduces the complexity of Hamiltonian engineering. Furthermore, we find that the states with FQH conductance can be prepared by adiabatically turning on the shaking of static optical lattices which is topologically trivial. This simplifies the procedures to initialize the FQH states in optical lattices. We believe this scheme will inspire new opportunities to experimentally study the bosonic FQH effect in an easier way.

The honeycomb optical lattice is formed by the interference of three red-detuned lasers at the same frequency, which have a relative angle at 120 degrees and are in one incident plane (Fig. 1a). The polarizations of lattice beams are parallel to the incident plane. The dipole potential $V_{op}$ of the optical lattice can be written in the form of

where $V_{D}$ is the trap depth in our definition, $k=2\pi/\lambda$ is the wave vector, and $\lambda$ is the wavelength of lattice lasers. This gives a lattice spacing $a=\lambda/2\sqrt{3}$, corresponding to a hexagon whose side length is $\lambda/2\sqrt{3}$. When cold atoms are trapped in honeycomb lattices, they can be described by a tight-binding model, whose Hamiltonian $H_{0}$ is $H_{0}/\hbar=-\sum_{\langle i,j\rangle}t_{0}\hat{c}^{\dagger}_{i}\hat{c}_{j}$ where $t_{0}$ is the nearest-neighbor hopping amplitude; $\hat{c}_{i}^{\dagger}$ ($\hat{c}_{i}$) is the creation (annihilation) operator on lattice cite $i$; $\langle\cdot\rangle$ corresponds to the summation of all nearest-neighbor sites.

In order to create non-trivial topological bands in tight-binding honeycomb lattice, we apply a periodic modulation on the phases of the lattice beams to break the time reversal symmetry, where phases of each laser beam are modulated according to $\phi_{1}=\phi_{A}\cos(\Omega t+\pi/2)$, $\phi_{2}=\phi_{A}\cos(\Omega t-\pi/6)$, and $\phi_{3}=\phi_{A}\cos(\Omega t+5\pi/6)$. This creates a shaken lattice of which each site $i$ follows a circular motion $\vec{r}_{i}(t)$ where $\vec{r}_{i}(t)=\vec{r}_{i,0}-A[\cos(\Omega t)\hat{x}+\sin(\Omega t)\hat{y}]$ and $A$ is orbital radius of the circular trajectory. The lattice shaking alternates the original Hamiltonian $H_{0}$ into a time-dependent form $\hat{H}^{\prime}(t)$, and

where $\langle i,j\rangle$ corresponds to a pair of nearest-neighbor lattice sites, $z_{ij}=m_{a}\Omega A\rho_{ij}/\hbar$, $m_{a}$ is the mass of an atom, and $\rho_{ij}e^{i\phi_{ij}}=(\vec{r}_{i,0}-\vec{r}_{j,0})\cdot(\hat{x}+\hat{y}e^{-i\pi/2})$. Here $z_{ij}$ is the ratio of $m_{a}\Omega^{2}A\rho_{ij}$ to $\hbar\Omega$, where the numerator is the product of the centrifugal shaking force $m_{a}\Omega^{2}A$ and the distance $\rho_{ij}$, and the denominator is the Floquet energy $\hbar\Omega$.

A periodic Hamiltonian is decomposed into a Fourier transformation that $\hat{H}^{\prime}(t)=\sum_{l\in\mathbb{Z}}\hat{H}_{l}e^{il\Omega t}$. When $\Omega$ is large, we obtain an effective time-indepedent Floquet Hamiltonian $\hat{H}_{fl}$ based on high frequency expansion method Goldman and Dalibard (2014); Eckardt and Anisimovas (2015); Rahav et al. (2003); Bukov et al. (2015). We rewrite the effective Floquet Hamiltonian $\hat{H}_{fl}$ based on the nearest-neighbor (NN), next-nearest-neighbor (NNN), next-next-nearest-neighbor (NNNN), and next-next-next-nearest-neighbor (NNNNN) hoppings, i.e.

where $\langle\cdot\rangle$, $\langle\cdot\rangle_{2}$, $\langle\cdot\rangle_{3}$, and $\langle\cdot\rangle_{4}$ correspond to the summations of the NN, NNN, NNNN, and NNNNN sites. We plot $\tilde{t}_{m}$ in Fig. 1c and the detailed formula of $\tilde{t}_{m}$ can be found in the appendix. The effective NNN hopping amplitude has an imaginary part, which breaks the time-reversal symmetry, opens the band gap, and gives a non-zero Chern number to each band. The band flatness ratio is usually used to characterize the potential of many-body topology Parameswaran et al. (2013), where the ratio is defined by the band gap divided by the band width of the ground band. In Fig. 1, we present the flatness ratio, hopping strength and energy bands under different parameters. Usually a flat band is required to show the dominated bosonic FQH effect Wang et al. (2011); Neupert et al. (2011), while flattening the band is a challenge in ultracold-atom experiments since the intrinsic long-range hoppings are strongly suppressed for remote lattice sites. In our scheme, the flatness ratio is less than 2 where the gap is near the same as the band width. We find it is still suitable to realize the FQH conductance of 1/2.

Now we switch our description from single-body physics to many-body physics. To achieve the fractional conductance, we need both strong on-site interactions and finite nearest-neighbor interactions. This can be realized by conventional s-wave Feshbach resonance Chin et al. (2010). For a given trap depth $V_{D}$ in Eq. 1, we calculate the Wannier functions of each site by the methods in Ref. Marzari et al. (2012) (See Appendix B for more details). The on-site interaction $U$ is obtained by the integral $U={4\pi\hbar^{2}a_{s}\over m_{a}}\int d^{3}\bm{r}w^{\dagger}(\bm{r})w^{\dagger}(\bm{r})w(\bm{r})w(\bm{r})$ where $a_{s}$ is the scattering length and $m_{a}$ is the mass of one atom. $w(\bm{r}-\bm{r}_{i})$ is the Wannier function centered at the position $\bm{r}_{i}$. Usually in Bose-Hubbard models, we only keep the on-site interactions and ignore the NN interaction $V_{1}$. However, once the scattering length $a_{s}$ approaches to infinity under Feshbach resonance, the NN interaction $V_{1}$ becomes significant with a form of Zhai (2021)

Here $\bm{a}$ corresponds to the relative vector between two nearest-neighbor sites.

In rubidium 85, there is a Feshbach resonance at 155.3 G Courteille et al. (1998); Claussen et al. (2003) with a width around 11 G and a background scattering length -441$a_{0}$ where $a_{0}$ is the Bohr radius. Considering that magnetic-field fluctuations can be controlled within 1 mG in most of labs, the scattering length can be tuned up to 10000$a_{0}$ with a relative uncertainty less than 0.25%. This offers the opportunity that the on-site interaction reaches the hard-core regime while the nearest-neighbor interaction is still important in the tight-binding model. In Table 1, we list the hopping coefficients and interaction strength versus $V_{D}$ in unshaken lattices. At the condition of $a_{s}=10000a_{0}$ and $V_{D}=28E_{r}$ ($E_{r}={\hbar^{2}k^{2}\over 2m_{a}}$), the original $t_{0}$ is $0.031E_{r}$ and the on-site interaction $U$ is $88E_{r}$, which satisfies $U\gg t_{0}$. $U$ strongly repulses any doublons in one site. Meanwhile, the NN interaction $V_{1}$ is $0.56\hbar t_{0}$ and is significant compared with the hopping amplitudes.

In the strong-correlation regime, the single-particle band description cannot capture the actual physics. Therefore, using the twisted boundary condition, we write out the many-body Floquet Hamiltonian $\hat{H}_{fl-many}$ of hard-core bosons with the nearest-neighbor interaction $V_{1}$ under the trap depth $V_{D}=28E_{r}$. Here the twisted boundary condition is $\psi(\bm{r_{i}}+L_{m}\hat{\bm{m}})=e^{i\theta_{x}}\psi(\bm{r_{i}})$ and $\psi(\bm{r_{i}}+L_{n}\hat{\bm{n}})=e^{i\theta_{y}}\psi(\bm{r_{i}})$ where $L_{m}$($L_{n}$) is the lattice size along $m$- ($n$-) axis and the axes are marked in Fig. 1a. At $\theta_{x}=\theta_{y}=0$, the twisted boundary condition is the same as the periodic boundary condition. $\hat{H}_{fl-many}$ is a giant-sized sparse matrix (see Appendix E for more information). Then we apply the exact diagonalization to calculate the lowest four energy levels and their corresponding many-body bands in the case of 6 bosons in 24 lattice sites ($\hat{H}_{fl-many}$ has a size around $10^{5}\times 10^{5}$), where the bands are characterized by $\theta_{x}$ and $\theta_{y}$ instead of the quasi-momenta $k_{x}$ and $k_{y}$.

By scanning the Floquet parameter $z_{0}$ and NN interaction $V_{1}$, there are some regions that the lowest two many-body bands cross each other while they are away from the third band. In Fig. 2a, we present the energy difference between the second and third energy states ($E_{3}$-$E_{2}$) under the twisted boundary condition. Here we take the inherent long-range hopping beyond the tight-binding models ($t_{1}$ and $t_{2}$) into accounts in calculating the 0th-order effective Hamiltonian, and neglect them in higher-order expansions. We mark the candidates for the FQH conductance at 1/2 in Fig. 2a according to the gap opening. In Fig. 2c, we plot the many-body bands versus $\theta_{x}$ and $\theta_{y}$ in this phase regime. It shows that two lowest bands cross each other and are away from the third band.

To further verify the conductance, we calculate the total Chern number $\mathcal{C}$ of the lowest two bands. Here $\mathcal{C}$ equals ${1\over 2\pi}\sum_{K=1,2}\int d\theta_{x}d\theta_{y}F_{xy,K}(\theta_{x},\theta_{y})$ where $F_{xy,K}(\theta_{x},\theta_{y})=\textrm{Im}(\langle\frac{\partial\psi_{K}}{\partial\theta_{y}}|\frac{\partial\psi_{K}}{\partial\theta_{x}}\rangle-\langle\frac{\partial\psi_{K}}{\partial\theta_{x}}|\frac{\partial\psi_{K}}{\partial\theta_{y}}\rangle)$ is the Berry curvature of the $K$-th state in the ground state manifold. We divide the whole $\theta_{x}$-$\theta_{y}$ space into 20$\times$20 pieces, and calculate the energies and the discrete Berry curvatures Fukui et al. (2005) in Fig. 2c and d. The two lowest bands share one integer Chern number $\mathcal{C}=-1$ together, corresponding to FQH conductance of 1/2. We calculate cases in lattices of different sizes to show robustness of our scheme against the finite size effect (Fig. 2b), while the size of $\hat{H}_{fl-many}$ increases dramatically ($\hat{H}_{fl-many}$ has a size of $10^{8}\times 10^{8}$ for 36 sites).

The tight-binding Hamiltonian in Eq. 2 only considers the contributions from s-bands, while a large scattering length may cause the s-band to mix with higher-excited bands. We follow the same treatment in Ref Viebahn et al. (2021) which successfully describes the interband transitions in optical lattices. We build a model with two sites, two particles, and both s-bands and p-bands to estimate the contributions from higher bands. The detailed numerical calculations and analyses are listed in the appendix C, and only the conclusion is presented in the main text. The dominant interband transitions induced by a large scattering length is that two neighboring particles in s-bands will scatter to p-bands together due to collisions. The coupling matrix element of this process is less than 0.1$E_{r}$, but the energy detuning is more than 10$E_{r}$. This suggests that there will be a suppressing factor more than $10^{-4}$ for higher-band mixture, which is negligible in our systems.

Besides the phenomena of higher-band mixture due to a large scattering length, the Floquet modulation may also cause the transitions to the excited bands. A particle at site-$i$ may absorb $\Delta n_{p}$ Floquet ”photons”, and be excited to a higher band. The effective coupling strength $Q_{\Delta n_{p}}$ to excite a particle to higher bands via a resonant $\Delta n_{p}$-photon process is (see Appendix D for detailed calculations)

Here $B_{\Delta n_{p}}$ is a coefficient whose magnitude has weak dependence on $\Delta n_{p}$ and we list the detailed form in Appendix D. The threshold $z_{0,th}$ is a dimensionless quantity which is approximately the photon number $\Delta n_{p}$ divided by the Euler’s number $e$. In the case of $V_{D}=28E_{r}$, it requires at least 28 photons to excite a particle to the higher bands so $z_{0,th}$ is around 10, while $z_{0}$ in our proposal is 2.3. Then $(\frac{z_{0}}{z_{0,th}})^{\Delta n_{p}-1}$ in the equation provides a factor of $10^{-19}$. Therefore, the interband heating caused by Floquet modulation is negligible in our scenario.

Another potential question about our model is whether it’s appropriate to derive effective Hamiltonian by the high-frequency-expansion method. To eliminate this concern and prove that we can prepare such FQH states adiabatically in cold-atom platforms, in the following calculations we apply the original time-dependent Hamiltonian, which contains intrinsic long-range hoppings and does not include Floquet treatments. The original time-dependent Hamiltonian is in the form of

Here $t_{0}$, $t_{1}$, $t_{2}$ and $t_{3}$ are intrinsic NN, NNN, NNNN, and NNNNN hopping coefficients for static honeycomb optical lattices (Table I). The on-site interaction $U$ does not appear in the expression since we limit the Hilbert space into the states where each site can be filled with one particle at most. $z_{ij}$ and $\phi_{ij}$ have the same meaning as those in Eq. 2 and have different values for different hoppings.

Therefore, we simulate how to prepare the target FQH state, which is the ground state of the effective Hamiltonian under the twisted boundary condition, with FQH conductance of 1/2 . Initially, the lattice is static and unshaken, and we start with the ground state $|\psi(z_{0}=0)\rangle$ under twisted boundary conditions. Then we fix the shaking frequency $\Omega$ at 0.108 and linearly ramp the shaking amplitude $A$ to increase $z_{0}$ from 0 to 2.3 in $C$ Floquet-modulation cycles, and we calculate the fidelity, which is the module square of inner products, between the time-evolved state and the target FQH state.

In Fig. 3a, we plot the fidelity versus $z_{0}$ and different ramping rates (or total modulation cycles $C$). The fidelity at the end of ramping reaches over 85% for both cycles $C=300$ and $400$, and approaches to a constant while $C>400$. We find if we ignore $\tilde{t}_{3}$ (NNNNN hoppings) in the effective Hamiltonian, the fidelity drops below 80%. Although $\tilde{t}_{3}$ is only 1/30 of other major terms, it hurts the calculations of fidelities. It suggests that the ground state of the effective Hamiltonian is not identical to the experimentally-prepared state by adiabatic ramping, and this causes the fidelity to fall below 100%.

Following the discussion of FQH conductance in Ref. Niu et al. (1985), the ground state and the first excited state cross each other in the many-body energy spectrum while they are isolated from the higher excited states. Besides this level isolation, one of the 1/2-FQH states evolves into the other one when the boundary condition $\theta_{x}$ is changed by $2\pi$. Then, $4\pi$ instead of 2$\pi$ becomes the period for the state to evolve back to itself. In Fig. 3b, we show that the states of the effective Hamiltonian have a $4\pi$-period and the level isolation.

Therefore, for the adiabatic-evolved states without tight-binding approximations, we apply the same arguments to prove the FQH conductance. We use State I to denote the adiabatic-evolved state during the preparation. We tune $\theta_{x}$ by $2\pi$ very slowly, and State I evolves and cannot come back to itself since the gap is closed during the change of boundary conditions. We extract the orthogonal component at $2\pi$ change and refer it as State II. This state maintains a good fidelity (always above 75%) with the first excited state of the effective Hamiltonian.

We change $\theta_{x}$ for multiple $2\pi$ (Fig. 3c) and calculate the overall fidelity within this manifold of State I and II for the actual scenario. In Fig. 3c, we show how the fidelity changes while the boundary condition is being varied (green belts). When $\theta_{x}$ leaves 0 or $2\pi$, the overlap between State I and State II decreases since the state is evolving. When $\theta_{x}$ reaches integer times of $2\pi$, the fidelity comes back again. After a few $2\pi$, the fidelity is stably oscillating and does not decay while $\theta_{x}$ is varied. The shape of belts is due to rapid Floquet modulations. Within each modulation, the fidelity is oscillating and the long-time-scale fidelity behaves as an area instead of a line. Here the fidelity does not exactly come back to 1 and we believe this is mainly due to the numerical errors of the giant matrix and data storage.

To convince this point, we numerically calculate another two cases for direct comparisons. One case is for the effective Hamiltonian under Floquet theory (red dotted line), and another is for a topological-trivial Hamiltonian (blue solid line). For the effective Hamiltonian, the change of fidelity behaves very similar to the case of adiabatic-evolved state. It is robust and does not decay while we continue changing $\theta_{x}$. Since we analytically solve this model, the ideal case should be that the fidelity reaches 1 again when $\theta_{x}$ is integer times of $2\pi$. The fidelity below 1 is due to the issues of numerical errors of a large-size matrix. The same reason applies to the case of the adiabatic-evolved state. Therefore, the robustness of the fidelity in the manifold of State I and State II proves the properties of FQH conductance. For the case of the topological-trivial Hamiltonian, we find the fidelity is decreasing while we change $\theta_{x}$. It indicates that the wave function is leaking out from this manifold. There is no gap separating the energy-lowest two states with the other bands, which cannot support FQH conductance.

In conclusion, we find a scheme realizing the bosonic FQH conductance of 1/2 in optical lattices without flat band. By circularly shaking the optical lattice and applying Feshbach resonance, the system displays a fractional quantum Hall conductance of 1/2. We show that the state with this conductance can be experimentally prepared by adiabatic turning on the lattice shaking. It provides a convenient and robust way to investigate the bosonic FQH effect Kjäll and Moore (2012); Wen (1990); Estienne et al. (2015); Repellin and Goldman (2019); Price and Cooper (2012); Račiūnas et al. (2018) in cold-atom experiments.

We acknowledge financial supports from the National Natural Science Foundation of China under grant no. 11974202, and 92165203.