1. Introduction

Identifying ways to search and probe topological optical lattices and thus topological quantum matter is a quest of major relevance in ultracold atoms [1,2]. Over the past decade, much progress has been witnessed in realizing the topological phases in cold atomic systems, ranging from the Su-Schrieffer-Heeger model [3], the Hofstadter model [4–6] to the Haldane model [7]. Important cases are the quantum spin Hall insulators (QSHIs) that arise in spin-orbit coupled systems, where the existence of helical edge states produces a spin current along the edges of a strip. Realizing the QSHI for ultracold gases was suggested in early theoretical works [8,9]. The QSHI would be implemented by engineering the hyperfine structure of the ultracold atoms to synthesize the non-Abelian gauge to mimic spin-orbit coupling [9]. On the experimental side, the only realized examples are cases where independent quantum Hall (QH) insulators of opposite Chern number have been paired to form a single system [5]. The system thereby is protected by a $\mathbb {Z}$ topological index. Despite these advances, optical lattices featuring QSHI with spin-orbit interaction have so far been lacking to our knowledge.

Along this line, chiral ladder systems for ultracold atoms constitute one of timely topics of engineering the topological quantum matter with synthetic gauge and synthetic dimensions [10]. They represent a simple yet effective platform to study exotic quantum phases of ultracold atoms, given that ultracold atoms in optical lattices naturally realize such a strip geometry [11]. Apart from the ladder structure in the real dimension [11], the internal degrees of freedom of atoms such as the hyperfine states [12,13] and clock states [14–16], and the external degrees of freedom such as the momentum states [17,18] and lattice orbitals [19,20], can be exploited to fabricate the chiral ladders in the synthetic dimensions. To date, an intense theoretical and experimental investigation has revealed rich topological features of the chiral ladders [10]. For example, the chiral ladder systems in synthetic dimension have led to visualization of the chiral edge states related to the QH phase [12,13].

Motivated by these recent experimental realizations of ultracold atomic ladders immersed in synthetic gauge potentials [11–13], here we propose a chiral-ladder realization of the topological bands of QSHIs. To be specific, we utilize the leg degree of freedom to synthesize the pseudo-spin-orbit coupling and Zeeman splitting. By exposing it to a 1D bichromatic optical potential along the direction of legs, we develop a model of bichromatic chiral ladder. We analyze the energy bands of this bichromatic chiral ladder, revealing the quantum spin Hall phase. We further highlight the emergence of distinct topological phases by varying the parameters of the ladder systems. Our study is of direct experimental relevance for laboratories where ultracold gases in a chiral ladder geometry are realized [11–13,15–20], hence providing a realistic way to achieve topological quantum matter of ultracold atoms.

The rest of the paper is organized as follows. In Sec. 2 we construct the model of bichromatic chiral ladder. In Sec. 3 we calculate the energy bands and investigate the topological properties of the model, and finally summarize our results in Sec. 4.

2. Model of bichromatic chiral ladder

Our starting point is the ladder geometry of optical lattices for noninteracting spinless atoms that has been synthesized experimentally [11]. The access of QSHE at first requires the identification of two degrees of freedom representing the two spin states. We resort to a two-leg ladder threaded by a uniform artificial magnetic field [11], the so-called chiral ladder, to encode the two degrees of freedom and enable their mutual interactions. As sketched in Fig. 1(a), the chiral ladder consists of a two-leg strip with intra- and inter-leg hoppings $J$ and $K$. Each plaquette encloses a net gauge flux $2\phi$. According to the Peierls substitution, the Landau gauge adopted here will imprint a phase factor $\pm i \phi$ on the hoppings along the legs ($\pm$ for the A- and B-legs, respectively). In addition, we superimpose an energy offset $2\Delta$ between the legs. Physically, this offset corresponds to a deep double-well configuration oriented along the $x$ direction. The overall ladder Hamiltonian in real space reads:

 

Fig. 1. (a) Schematic representation of the two-leg ladder. This optical lattice has a double-well structure in the $x$ direction, while unlimitedly extends in the $y$ direction. $A$ and $B$ (the red and blue colors) label the two different degrees of freedom of legs, denoting the two different pseudo-spin states. $J$ and $K$ are the hopping amplitudes along the legs and rungs, respectively. An artificial magnetic flux $2\phi$ penetrates the plaquettes. An additional potential difference $2\Delta$ is imposed between the legs. (b)-(d) Band structures of the chiral ladders for various $K$’s and $\Delta$’s. Energy is set in units of $2J$. The color of the lines specifies the spin magnetization of the Bloch state. The magnetic flux is set as $2\phi =0.6\pi$. Other parameters: (b)$K=0,\; \Delta =0$ (c) $K=1,\; \Delta =0$, (d) $K=1,\; \Delta =1$.

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A precise connection can be made between a spin-orbit coupled chain and the chiral ladder. Following the interpretation of Ref. [21], one can think of Eq. (1) as a 1D optical lattice with pseudo-spins represented by $A$- and $B$-legs. Keeping this in mind, we introduce the spinor operator $\Psi _{n}=\left (c_{n,A}, \; c_{n,B}\right )^T$ and rearrange our model in the spinor space as follows:

To realize the topological bands of chiral ladders, we propose to impose a bichromatic optical superlattice $V(y)=V_p \cos ^2(k_p y)+V_a \cos ^2(k_a y+\theta )$ along the $y$ direction. This can be created by superimposing on a primary, deep lattice an auxiliary, weak lattice [22]. $V_p$ and $V_a$ separately denote the depth of the primary and auxiliary lattices. $k_p$ and $k_a$ are the two lattice wave numbers. The variable phase $\theta$ accounts for the relative position of the lattices. The bichromatic modulation is applied equally on the A- and B-legs (the shaded circle of Fig. 1(a)). When the primary lattice depth is sufficiently large, the wavefunctions can be expanded by a set of Wannier states $\mathit {w}(y)$ of the lowest band of the unperturbed primary lattice [23]. In the tight binding limit, this expansion leads to the onsite potential energies as follows $V(n)=\Lambda \cos (2\pi \beta n+\theta )$ [22]. Here, the modulation strength $\Lambda$ is parameterized by the depth of the auxiliary lattice, $\beta$ equals to the ratio $k_a/k_p$, and $\theta$ the phase offset. Summing up, the modulated optical ladder can be described by a tight-binding Hamiltonian

3. Quantum spin Hall phase and topological phase diagram

In order to identify the topological nature of the BCL structure, we now establish the connection of the 1D BCL to a 2D spin-orbit coupled square lattice pierced by a magnetic field. The 2D Hamiltonian can be found by using the approach of dimensional extension [24–28]. To be specific, given the parameter $\theta$ is cyclically varied in $[0,\; 2\pi ]$, it can be regarded as a quasi-momentum $k_z$ along a virtual coordinate $z$. After making the substitutions of $J \rightarrow t_{y}$, $\theta \rightarrow k_{z}$, and $\Lambda \rightarrow 2 t_z$ and relabeling the spinor as $\Psi ^{\phantom \dagger }_{n,k_z}$, the present 1D model can be converted into a 2D Hamiltonian in a mixed momentum-position representation $\left (n, \; k_z\right )$,

Having precisely mapped the 1D BCL onto the analogous 2D system, we at this point move to the topological phases of the BCLs. Herein, we assume a rational $\beta$, i.e., $\beta =p/q$ with $p,\; q$ being coprime integers. The length of unit cell of the ladder turns out to be $q$. Inserting the Bloch waves $\Psi _{n+q}=e^{i k_y} \Psi _n$ in Eq. (4) yields the band structures $E=E (k_y, \theta )$ of the modulated ladder, together with the associated eigenvectors, that are defined by $\theta$. Figure 2(a) displays the energy bands for $\beta =1/3$. To quantify the topological properties of our system, we calculate the spin-up (spin-down, respectively) gap Chern numbers defined in the parameter space $(k_y, \theta )$. In Fig. 2(a) we have labeled the spin gap Chern numbers. We note that for the middle gap the Chern numbers are given by $(C_{\uparrow }, \; C_{\downarrow })=(-1, \; +1)$. This leads to the nontrivial topological index $\frac {1}{2} (C_{\uparrow }-C_{\downarrow })=-1$ which signals the emergence of QSHI. It should be noted that the time-reversal symmetry is broken in the analogous 2D Hamiltonian, and thus this phase corresponds to the time-reversal-symmetry-broken QSHI phase [31]. In the meanwhile, the Chern numbers of the first gap are $(C_{\uparrow }, \; C_{\downarrow })=(0, \; -1)$. This indicates a spin-filtered QH phase [32,33]. A similar result holds for the third gap, but with $(C_{\uparrow }, \; C_{\downarrow })=(+1, \; 0)$.

 

Fig. 2. Energy spectrum for $2\phi =0.6\pi$, $\Lambda =1.5$, $\beta =1/3$, $\Delta =1$, and $K=0.1$. The energy is set in units of $J$. (a) Energy spectrum $E=E (k_y, \theta )$ for the infinite BCL. The integers near the graph label the spin Chern numbers of different bandgaps. (b) Energy spectrum obtained from a finite ladder. As a function of $\theta$, the spectrum is composed of the bulk bands (solid lines) and dispersion curves that traverse the gaps (dashed lines). $\blacksquare$ ($\bullet$) is the markers for the states at the same Fermi energy.

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The definite topological property in a finite system is the emergence of gapless edge states as the phase $\theta$ varies. In Fig. 2(b), illustrating the spectrum of a finite ladder, the dispersion curves of the additional states (dashed lines) are clearly superimposed on the bulk bandgaps. At a given Fermi level $E_{\textrm {Fermi}}=0$, gapless states, labeled as A, B, C and D, emerge in the middle bandgap. States A and C are localized near $n=L$, while B and D are localized near $n=0$ (Figs. 3(a)–3(d)). The slope of dispersion curves in Fig. 2(b) determines that the two states A and C are counterpropagating in the analogous 2D square lattice. Meanwhile, states A and C are almost fully spin-down and spin-up polarized, respectively (Figs. 3(a), 3(c)). Therefore, these two pair of mid-gap states form the helical edge states, indicating the QSHI phase of the middle gap. On the other hand, when the Fermi energy is adjusted inside the lowest gap, two propagating states traverse the bulk gap (E and F in Fig. 2(b)). E and F are characterized by the single spin-component which are localized at the opposite edges (Fig. 3(e), 3(f)). In the analogous 2D system the excitations E and F constitute a pair of chiral edges states with single spin-component. This is associated with the spin-filtered QH phase [32,33]. We emphasize that the pseudospin components in our proposal are manifested as the leg degree of freedom. As a result, these edge states are localized at the extremities of the left- and right-legs, respectively. This will give convenience to the direct observations of the topological phases.

 

Fig. 3. Mode amplitudes of the gapless states in Fig. 2(b). (a-d) correspond to the states marked by $\blacksquare$, while (e,f) to the states $\bullet$. The spin component $\Psi _{n\uparrow }$ (respectively, $\Psi _{n\downarrow }$) is represented in red (respectively, blue).

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To further show the rich topological phases in the BCL, we explore the phase diagram through tuning the synthetic Hamiltonian. Since changing the energy offset between legs is readily accessible to the experiment, we compute the phase diagram as a function of the $\Delta$. Figure 4 illustrates the diversity of the topological phases. The bulk is insulating in the white regions and classified into distinct phases: QH ($C_{\uparrow }=-1, C_{\downarrow }=-1$) or ($C_{\uparrow }=+1, C_{\downarrow }=+1$), QSH ($C_{\uparrow }=-1, C_{\downarrow }=+1$), spin-filtered QH ($C_{\uparrow }=\pm 1, C_{\downarrow }=0$) or ($C_{\uparrow }=0, C_{\downarrow }=\pm 1$), and ordinary insulator ($C_{\uparrow }=0, C_{\downarrow }=0$). Take $E_{\textrm {Fermi}}=0$ and $E_{\textrm {Fermi}}=-1$ for example. With the increase of $\Delta$ the excitation of $E_{\textrm {Fermi}}=0$ will undergo the regimes of metal, QSH insulator, and ordinary insulator successively. On the other hand, for the $E_{\textrm {Fermi}}=-1$ the energy offset can turn a QH phase into a metal, and then a spin-filtered QH phase. Therefore, by manipulating $\Delta$ and $E_{\textrm {Fermi}}$ the BCL can host rich topological phases.

 

Fig. 4. $E_{\textrm {Fermi}}$-$\Delta$ phase diagram. Here, the bandgaps (bands) are designated by the white (shaded) regions, respectively. The pairs of integers indicate the Chern numbers of the bandgaps for spin up and down, distinguishing the topological regimes of the model.

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4. Discussions and conclusions

One of the key features of our scheme is that the spinless ultracold atoms are utilized and the synthetic spin-orbit textures are built from the leg degree of freedom which is coupled to the Abelian gauge field. Neither internal states (e.g., hyperfine states) nor a non-Abelian gauge is used in our setting. Besides that, since the “spin" itself is synthesized in real space [11], our method allows for a real-space-resolved detection of the topological phases, instead of spin-resolved techniques. This character will facilitate greatly the direct observations of the spin-orbit coupled topological phases. On the other hand, due to the versatility of the optical ladders, we remark that our results are also applicable to the structures prepared in the artificial dimensions [12–20].

As an alternative to access the higher-dimensional Hamiltonians that host topological phases, the topological pumping in lower dimensional systems provides an additional avenue towards studying topological states of matter [24,34–38]. The pumping experiments of ultracold atoms have been demonstrated in optical superlattices [35,36]. Therefore, by adiabatically and periodically varying a set of BCLs’ parameters, e.g., the relative phase of the bichromatic lattices, one can drive versatile quantized transports during each cycle, such as charge [34] and spin pumping [39]. The implementation of pumping in our BCL system will arouse the interest from the experimental side.

In conclusion, we have proposed a system of bichromatic chiral ladder for studying the topological bands for ultracold atoms, utilizing the concepts of synthetic spin-orbit coupling and Zeeman splitting. We have demonstrated that the quantum spin-Hall phase can emerge within this setup. In the meanwhile, the bichromatic chiral ladders can produce rich topological phases via tuning the system parameters. We conclude that the bichromatic chiral ladders hence constitute a surprisingly simple yet versatile scenario to explore synthetic topological quantum matter for the ultracold atoms. Our proposal to engineer topological quantum matter is of direct experimental relevance in ultracold atoms [11–13,19]. Taking this work as a basis, we believe that turning on the interactions will trigger the studies on the exotic topological many-body states in atomic chiral ladders [40–42].