Emergent Devil’s Staircase without Particle-Hole Symmetry in Rydberg Quantum Gases with Competing Attractive and Repulsive Interactions

The devil’s staircase is a fractal structure that characterizes the ground state of one-dimensional classical lattice gases with long-range repulsive convex interactions. Its plateaus mark regions of stability for specific filling fractions which are controlled by a chemical potential. Typically, such a staircase has an explicit particle-hole symmetry; i.e., the staircase at more than half filling can be trivially extracted from the one at less than half filling by exchanging the roles of holes and particles. Here, we introduce a quantum spin chain with competing short-range attractive and long-range repulsive interactions, i.e., a nonconvex potential. In the classical limit the ground state features generalized Wigner crystals that—depending on the filling fraction—are composed of either dimer particles or dimer holes, which results in an emergent complete devil’s staircase without explicit particle-hole symmetry of the underlying microscopic model. In our system the particle-hole symmetry is lifted due to the fact that the staircase is controlled through a two-body interaction rather than a one-body chemical potential. The introduction of quantum fluctuations through a transverse field melts the staircase and ultimately makes the system enter a paramagnetic phase. For intermediate transverse field strengths, however, we identify a region where the density-density correlations suggest the emergence of quasi-long-range order. We discuss how this physics can be explored with Rydberg-dressed atoms held in a lattice.

Figure 1

(a) Level scheme. Two ground states, $|0⟩$ and $|1⟩$, are coupled coherently through a Raman process. The $|1⟩$ state is weakly dressed to two Rydberg states, $|R_1⟩$ and $|R_2⟩$, by blue- and red-detuned lasers. (b) Interaction potential between two atoms in the Rydberg dressed $|1⟩$ state. Varying the laser parameters and the lattice spacing $a$, we achieve a competing interaction where the NN interaction is attractive and the longer range interactions are repulsive. The blue-detuned laser induces the two-atom resonant excitation at a distance $R_{\text{res}}$ (the dashed line). (c) Tuning the competing interaction by changing the laser detuning ${\mathrm{\Delta }}_2$. In the example, two Rydberg states of Rubidium atoms, $|R_1⟩=|60S⟩$ and $|R_2⟩=|70S⟩$, are considered. The dispersion constants of the respective Rydberg states are $C_6^{(1)}=140.4\mathrm{GHz}\mu {\mathrm{m}}^6$ and $C_6^{(2)}=882.3\mathrm{GHz}\mu {\mathrm{m}}^6$. Laser parameters are given in the following. In (b), we assume ${\mathrm{\Delta }}_2=250\mathrm{MHz}$. Other parameters used in plotting (b) and (c) are ${\mathrm{\Omega }}_1=30\mathrm{MHz}$, ${\mathrm{\Delta }}_1=-300\mathrm{MHz}$, ${\mathrm{\Omega }}_2=22\mathrm{MHz}$, and $a=2\mu \mathrm{m}$ (see the text for details).

Figure 2

Emergent devil’s staircase. (a) Staircase pattern of the filling fraction $f$ versus NN interaction strength $W$. The particle part ($f<1/2$) and the hole part ($f>1/2$) of the staircase are indicated by the brown and blue shaded areas and merge at filling fraction $f=1/2$. (b) Magnified view of the dimer particle staircase. The brown curve is obtained with the effective dimer particle theory of Eq. (2). The black curve (in all panels) is the numerically calculated staircase taking into account any rational filling fractions $f=p/q$, with $q\le 20$, where the range of the interaction has been cut off at $k\le 1000$ lattice sites. (c) At large filling fraction ($f>1/2$), the main plateaus (black) correspond to arrangements of a single kind of $R$-mers. The small intermediate plateaus (green) contain mixtures of $R$-mers and ($R+1$)-mers. The red stars correspond to the predicted transition points given by Eq. (4). (d) A unified picture for the staircase structure at large fillings is obtained by employing an effective dimer hole theory of Eq. (5), which agrees excellently with the numerical data.

Figure 3

Quantum melting of the dimer crystal with increasing transverse field strength $g$. (Left panel) Transverse magnetization $\sum _i⟨{\widehat{s}}_i^x⟩/L$ as a function of $W$ and $g$, where the simulation was done for $L=60$ lattice sites with open boundary conditions and the range of the interactions has been truncated to ten lattice sites. The four largest stability regions $f=1$, $f=3/5$, $f=1/2$, $f=2/5$ are marked by dashed lines; see also Fig. 2. (Right panel) The iDMRG calculations of the density-density correlation function $g^{(2)}(k)$ versus $k(\ge 1)$ for three different values of $g=(0.05,0.15,0.3)$ at $W=-0.91$, corresponding to the points $A$, $B$, and $C$ in the left panel (see the text for details).