Quantum phases of interacting bosons on biased two-leg ladders with magnetic flux

The realization and detection of various phase transitions in interacting two-leg bosonic flux ladders are at the frontier of the present theoretical and experimental research in condensed-matter physics. We develop a biased two-leg Bose-Hubbard flux ladder model and first achieve the ground states and phase transition conditions of this system by the mean-field approach and variational analysis. Rich phases are revealed, including two different kinds of plane-wave phase, i.e., plane wave I (characterized with one single energy minimum of the energy band) and plane wave II (characterized with two nondegenerate local energy minima of the energy band), and interestingly the asymmetry vortex phase with the breaking of both $Z_2$ reflection symmetry and translational symmetry of space, which is characterized by the vortex with imbalanced density distributions on two legs. Moreover, the corresponding quantum phases can also be distinguished by the order parameters and the excitation spectra of the plane-wave phase intuitively. Furthermore, we demonstrate that this biased ladder system also makes it possible to realize a different dynamical asymmetric vortex state, which shows a dynamical supersolidlike property.

Figure 1

Band structures of the noninteracting ladder system (a) $\tilde{\omega }=0$ and (b) $\tilde{\omega }=0.5$ for different rung-to-leg coupling ratios $\tilde{K}$.

Figure 2

Individual particle densities and currents for different phases. For the biased ladder $\tilde{\omega }=0.2$, there are two kinds of plane-wave phase (a) PWI with $\tilde{K}=1.5$, (c) PWII with $\tilde{K}=0.75$, and a superposition state (e) AVP with $\tilde{K}=0.65$. For the unbiased ladder $\tilde{\omega }=0$, we show the (b) Meissner phase with $\tilde{K}=1.5$, (d) BLP with $\tilde{K}=1$, and (f) vortex phase with $\tilde{K}=0.65$. The colorbar represents the particle density, blue arrows denote local currents, and the length of the arrow indicates the current strength, which is normalized to the local current of the Meissner phase. The other parameters are $\tilde{U}=0.4$ and $\varphi =\pi /2$.

Figure 3

Phase diagrams in the $\tilde{K}\text{−}\varphi$ plane with different atomic interactions (a) $\tilde{U}=0.2$, (b) $\tilde{U}=0.5$, and (c) $\tilde{U}=0.8$; phase diagrams in the $\tilde{K}\text{−}\tilde{U}$ plane with different energy biases (d) $\tilde{\omega }=\pm 0.1$, (e) $\tilde{\omega }=\pm 0.4$, and (f) $\tilde{\omega }=\pm 0.6$; and phase diagrams in the $\tilde{\omega }\text{−}\tilde{K}$ plane with different atomic interactions (g) $\tilde{U}=0.2$, (h) $\tilde{U}=0.6$, and (i) $\tilde{U}=1$. The region I($\pm$) and region II($\pm$) correspond to two kinds of plane-wave phase PWI and PWII, respectively. The region III($\pm$) corresponds to the AVP.

Figure 4

Chiral current $j_c$ (red solid circle line), average rung current $\mathrm{avg}|j_{\mathrm{rung}}|$ (black solid square line), and density imbalance $\mathrm{\Delta }n$ (blue solid triangle line) for different quantum phases; cut through the phase diagram of the $\tilde{K}\text{−}\tilde{U}$ plane ($\varphi =\pi /2$) at $\tilde{\omega }=0.1$ with (a) $\tilde{U}=0.3$ and (b) $\tilde{U}=0.6$ [see Fig. 3], respectively; cut through the phase diagram of the $\tilde{K}\text{−}\varphi$ plane ($\tilde{U}=0.8$) at $\tilde{K}=1.5$ with (c) $\tilde{\omega }=0.1$ and (d) $\tilde{\omega }=0.2$ [see Fig. 3], respectively; and slopes of the chiral current $\partial j_c/\partial \varphi$ (red line) and density imbalance $\partial \mathrm{\Delta }n/\partial \varphi$ (blue line) vs $\varphi$ with (e) $\tilde{\omega }=0.1$ and (f) $\tilde{\omega }=0.2$, respectively.

Figure 5

Rotonlike minimum softening and energetic instability for decreasing (a) rung-to-leg coupling ratio $\tilde{K}$ and (c) energy bias $\tilde{\omega }$, and phase boundary of the AVP with different (b) energy biases $\tilde{\omega }$ and (d) atomic interactions $\tilde{U}$. The other parameter is $\varphi =\pi /2$.

Figure 6

Time evolution of the sum density obtained by the Runge-Kutta numerical simulation of the GP equation (2) with different energy bias (a) $\tilde{\omega }=0.1$ and (c) $\tilde{\omega }=0.4$. (b, d) Corresponding density distributions and local current configurations for variational results at different times $t=200$, 150, 100, 50, and 0. The colorbar represents the particle density. The other parameters are $\tilde{U}=0.8, \tilde{K}=0.6$, and $\varphi =\pi /2$. (e, f) The velocity of the vortex for different rung-to-leg coupling ratios $\tilde{K}$ vs (e) the atomic interaction and (f) the energy bias. Solid lines correspond to the variational results of Eq. (19), and symbols correspond to numerical simulation results of the GP equation (2).