Bosons with long-range interactions on two-leg ladders in artificial magnetic fields

Motivated by experiments exploring the physics of neutral atoms in artificial magnetic fields, we study the ground state of bosons interacting with long-range dipolar interactions on a two-leg ladder. We focus on the limit where the number of particles per site is large and the interactions are weak. Using two complementary variational approaches, we find rich physics driven by the long-range forces. Generically, long-range interactions tend to destroy the Meissner phase in favor of modulated density wave phases. For example, nearest-neighbor interactions produce an interleg charge density wave phase, where the total density remains uniform but the density on each leg of the ladder is modulating in space, out of phase with one another. Next-nearest-neighbor interactions lead to a fully modulated biased ladder phase, where all of the particles are on one leg of the ladder. This state simultaneously breaks $Z_2$ reflection symmetry and $\mathrm{U}\left(1\right)$ translation symmetry. For values of the flux near $\varphi =\pi$, we find a switching effect for arbitrarily weak interactions, where the density is modulated but the chiral current changes sign on every plaquette. Arbitrarily weak attractive interactions along the rungs destroy the Meissner phase completely, in favor of a modulated density wave phase. Remarkably, varying the rung to ladder hopping produces a cascade of first-order transitions between modulated density wave states with different wave vectors, which manifests itself as discrete jumps in the chiral current. Polarizing the dipoles along the ladder direction yields a region of phase space where a stable biased ladder phase occurs even at arbitrarily weak rung hopping. We discuss the experimental consequences of our work and draw connections between our work and recent experiments on cold atoms in synthetic dimensions.

Figure 1

Dipolar Bose gas on a two-leg ladder. The arrows on the sites of the ladder correspond to dipoles, aligned in the direction of the external field indicated by $B$. The external field could be an electric field in the case of polar molecules or a magnetic field for magnetic atoms such as Dy and Er. The hopping along the $x$ direction is denoted by $J$. An artificial flux threads the system, which implies that hopping in the $y$ direction picks up a position-dependent phase $e^{il\varphi }$. The local on-site interaction is denoted by $U$ and is always assumed to be repulsive, $U\ge 0$. Additionally, there is a nearest-neighbor interaction along $x$ (ladder) and $y$ (rung) denoted $V_x$ and $V_y$, respectively, and a next-nearest-neighbor interaction $V_{\text{NNN}}$. The sign and magnitude of $V_x,V_y$, and $V_{\text{NNN}}$ can be tuned by tilting the external field relative to the plane of the ladder. Higher-order contributions to the dipolar interaction are significantly smaller in magnitude and do not have much effect on the overall phase diagrams that we present.

Figure 2

Repulsive dipoles. (a) Global phase diagram showing the density difference between left and right legs of the ladder as a function of $V/J$ and $K/J$ at $U=0$ and $\varphi =\pi /2$. Dipolar interactions push the Meissner phase to stronger rung-ladder coupling and give rise to an interleg CDW phase, where the relative densities on each leg of the ladder modulate out of phase with one another in real space, as shown in (c). At intermediate rung hopping and weak dipolar interactions, a biased ladder (BL) phase is present. (b) Real-space density profiles in the left (solid line), right (dashed line), and total density (dotted line) for large next-nearest-neighbor interactions. All of the particles reside on only one leg of the ladder, producing a fully modulated biased ladder phase. (d) Real-space current profile at $\varphi =0.9\pi$, showing switching of the sign of the chiral current. Dhar et al. [29] refer to this phase as a chiral superfluid. (e) Chiral current at $\varphi =0.9\pi ,U=0,V/J=1$, and $K/J=1.5$.

Figure 3

Dipoles pointing along the $y$ (rung) direction. (a) Phase diagram of the attractive rung, dipolar bosonic ladder for $U=0$, and $\varphi =\pi /2$ as a function of $V$ and $K/J$. A modulated density wave phase (with $\gamma =\pi /4)$ is found throughout the entire phase diagram. The density plot shows evolution of $k_0$ with $V/J$ and $K/J$, which approaches zero as $K$ becomes large, but $\gamma$ remains finite. The inset shows real-space total density at two different values of $K$, at fixed $V$. (b),(c) Evolution of the chiral current and $k_0$, respectively, for fixed $V$, as a function of $K/J$, showing jumps. The discrete jumps indicate first-order transitions between CDW phases with different incommensurate wavelengths and is reminiscent of the devil's staircase pattern found in dipolar Mott insulators and superfluids in 1D [19, 21]. For comparison, the evolution of $k$ for purely short-range interactions is also shown (dashed line), revealing a single first-order transition from the modulated density phase to the Meissner phase. (d) Schematic showing the real-space density and current profile in the two-leg ladder. Increasing $K/J$ increases the wavelength of the modulations (shaded region). Arrows indicate the direction of the chiral current. In this picture, “vortices” sit in the regions between the shaded areas, where the chiral current reverses direction. Therefore, the vortex density decreases as the $K/J$ increases (cf. Ref. [32], Fig. 5). Both variational approaches never find a transition to a homogeneous density (Meissner) phase at any $K$.

Figure 4

Dipoles pointing along the $x$ (ladder) direction. Density plot showing the density difference between the two legs of the ladder normalized to the total density for attractive ladder interactions: $V_x=-2V,V_y=V$, and $U/J=1$. For finite $V/J$, there is a transition from an interleg CDW to a biased ladder phase at arbitrarily small $K/J$, which occupies a large portion of the phase diagram. Increasing $K/J$ leads to a direct transition from the biased ladder to the Meissner phase. Increasing the dipolar interactions further leads to a collapse of the gas, signaled by the appearance of imaginary frequencies at long wavelengths.