Devil's staircases in synthetic dimensions and gauge fields

We study interacting bosonic or fermionic atoms in a high synthetic magnetic field in two dimensions spanned by continuous real space and a synthetic dimension. Here the synthetic dimension is provided by hyperfine spin states, and the synthetic field is created by laser-induced transitions between them. While the interaction is short-range in real space, it is long-range in the synthetic dimension in sharp contrast with fractional quantum Hall systems. Introducing an analog of the lowest-Landau-level approximation valid for large transition amplitudes, we derive an effective one-dimensional lattice model, in which density-density interactions turn out to play a dominant role. We show that in the limit of a large number of internal states, the system exhibits a cascade of crystal ground states, which is known as the devil's staircase, in a way analogous to the thin-torus limit of quantum Hall systems.

Figure 1

(a) A system of atoms confined in continuous 1D space and having $N_y$ hyperfine spin states labeled by $y=1,2,\cdots ,N_y$ [see Eq. (1)]. Transitions with a spatially dependent phase factor $e^{ibx}$ are induced between these internal states. For an interval of length $\mathrm{\Delta }x$, a magnetic flux of $b\mathrm{\Delta }x$ pierces between the neighboring internal states. (b) By performing the Fourier transformation in the $y$ direction, the kinetic part of the Hamiltonian decouples into components with different values of the wave number $k_y=2\pi n_y/N_y$ in the $y$ direction [see Eqs. (7) and (8)]. For each $k_y$, an effective cosine potential emerges with shifted locations of minima, allowing a description in terms of Wannier orbitals. These cosine potentials may also be viewed as spatially dependent optically dressed levels. The Wannier orbitals are labeled by a serial number $m$ [see Eq. (10)] from left to right, forming a 1D lattice.

Figure 2

Processes in the effective Hamiltonian [see Eqs. (14) and (18)]: (i) hopping ${\widehat{a}}_{m+N_y}^{†}{\widehat{a}}_m$, (ii) a density-density interaction ${\widehat{n}}_m{\widehat{n}}_{m^{\prime }}$, and (iii) correlated hopping ${\widehat{n}}_{m^{\prime }}{\widehat{a}}_{m+N_y}^{†}{\widehat{a}}_m$. Shaded and empty circles indicate the presence and absence of a particle, respectively.

Figure 3

GS phase diagram of the effective Hamiltonian (22). This effective Hamiltonian consists only of density-density interactions and gives an appropriate description of the original system in the limit of a large number of internal states. We consider the regime $\ell /a<{\sigma }_0\simeq 1.957$, where the condition of convexity is satisfied and therefore a complete devil's staircase appears. The number associated with each region indicates the filling factor $\nu =N/N_{\varphi }$, and we focus on the case of $0\le \nu \le 1/2$.

Figure 4

Competition of crystal states at (a) $\nu =1/2$ and (b) $\nu =1/3$. For $\nu =1/q$, we plot the energy of the crystal state $\overline{(q-d)(q+d)}$ (with $d=1,\cdots ,q-1$) per particle relative to that of the state $\overline{q}$, as given by Eq. (26).