Frustrated Bose ladder with extended range density-density interaction

TL;DR

This work investigates how extended-range density-density interactions influence the frustrated two-leg Bose ladder with ring exchange, focusing on the stabilization of the d-wave correlated Bose liquid (DBL) and the emergence of a density-modulated s-wave paired (DMSP) phase. Using DMRG simulations, the authors map out SF, DBL, and DMSP phases via observables such as the momentum distribution $n(q)$, two-particle pairing $P_2$, density-wave order $O_{DW_2}$, and twist stiffness $O_{twist}$. They find that extended interactions not only promote DBL but also shift its onset to smaller $K$, and that a rung-blockade limit can realize DBL even more readily; they also identify the DMSP as a distinct intermediate phase with coexisting density modulation and pairing features. The results have experimental relevance for dipolar bosons and synthetic-dimension ladders, where nonlocal interactions and rung constraints naturally arise, offering new avenues to observe DBL-like metallic bosonic behavior in controlled settings.

Abstract

When hard-core bosons on a two-leg ladder get frustrated by ring exchange interactions, the elusive d-wave Bose liquid (DBL) can be stabilized, a bosonic analog of a correlated metal. Here, we analyze the effect of extended Hubbard interactions on the DBL phase. Strikingly, these interactions are found to act in favor of the exotic Bose liquid. This observation is of immediate relevance for physical systems in which non-local exchange processes occur as a consequence of extended-range density-density interactions. Our observation also helps to achieve DBL physics in a synthetic-dimension ladder, where on-site interactions translate into non-local interactions along a synthetic rung. In this context, we also consider the extreme limit, in which the local hardcore constraint is elevated to an effective rung blockade. In addition to the enhancement of DBL physics due to extended-range density-density interactions, we also find signatures of an interesting intermediate phase between the superfluid and the DBL regime. This phase, labeled as the density modulated s-wave paired (DMSP) phase, combines features of density wave and s-wave pairing. Our results offer new insights into the physics of frustrated bosons by highlighting the influence of density-density interaction and rung-blockade.

Figures (9)

• Figure 1: We base our work on the two-leg ladder model shown above. Apart from horizontal hopping $t$ and ring exchange $K$, nearest neighbor density-density interactions are also taken into account. These are shown as: $V$ for the horizontal in-chain interaction, $V_d$ represents the diagonal ones, and $U_p$ models the vertical interaction strength.
• Figure 2: Different phases as a function of $K$ are shown at $V=0$. We used $L_x=36$. The rightmost ($i.e.$ DBL) shaded region has double peaks in momenta distribution. The left and middle regions cannot be distinguished based on momenta peaks. One has to look into two different order parameters $O_{twist}$ and $O_{DW_2}$. The middle part shows density wave modulation, and the left part shows finite SF stiffness.
• Figure 3: We show (a) momentum distribution and (b) pair correlation for the DBL phase with $L_x=48$, at $K=2.0$ and $V=0.0$. In (c), a schematic representation for the choice of diagonals is presented, which has been followed for the computation of $P_2$. In panel (a), we have indicated the position of the $q_x= \pm \pi n_f$ by dashed line, showing that momenta peaks exactly coincide with the filling. Also, the pair correlation oscillates with a period of $1/n_f=4$ sites.
• Figure 4: We show the properties of the DMSP at $K=1.2$ and $V=0.0$, with $L_x=48$. (a) The momentum distribution has a single peak. (b) The two-particle diagonal pairing shows s-wave-like behavior. The $dia-perp$ and $dia-para$ configurations (see Fig. \ref{['fig:DBL']}(c)) are denoted as $perp$ and $para$, respectively. The curve marked as $avg$ shows the mean value of these two configurations, which is $P_2^{avg}$. The same is shown in the inset on a $log-log$ scale to visualize the power scaling. However, the single-particle density (c) shows density wave ordering.
• Figure 5: (a) The $O_{DW_2}$ becomes a faithful marker to scrutinize the effect of $V$, as a function of $K$. This quantity is only non-zero in DMSP that appears between SF and DBL. To begin with, for all values of $V$ shown in the plot, the intermediate DMSP sector shifts to a lower $K$, suggesting an early onset of the neighboring DBL phase with increasing $V$. (b) The onset of DBL with increasing $V$ can be explicitly seen, if $n(q_x,0)$ is plotted as a function of $V$. The results are at $K=1.2$. We see the appearance of DBL peaks staring from a DMSP phase at $V=0$ (Fig. \ref{['fig:DMSP']}).