Persisting Meissner state and incommensurate phases of hard-core boson ladders in a flux

The phase diagram of a half-filled hard-core boson two-leg ladder in a flux is investigated by means of numerical simulations based on the density matrix renormalization group (DMRG) algorithm and bosonization. We calculate experimentally accessible observables such as the momentum distribution, as well as rung current, density wave, and bond-order wave correlation functions, allowing us to identify the Mott Meissner and Mott vortex states. We follow the transition from commensurate Meissner to incommensurate vortex state at increasing interchain hopping until the critical value [Piraud et al. Phys. Rev. B 91, 140406 (2015)] above which the Meissner state is stable at any flux. For flux close to $\pi$, and below the critical hopping, we observe the formation of a second incommensuration in the Mott vortex state that could be detectable in current experiments.

Figure 1

Left panel: Phase diagram of (1) as a function of flux per plaquette $\lambda$ and $\mathrm{\Omega }/t$. The phase boundary between the Meissner and the vortex phase is shown by the black solid line, displaying the persistence of the Meissner phase [30] above the threshold $\mathrm{\Omega }>{\mathrm{\Omega }}_c$ for all fluxes $\lambda$, except $\lambda =\pi$. For comparison the red-dashed line shows the boundary ${\lambda }_c^{\left(0\right)}\left(\mathrm{\Omega }\right)$ for the noninteracting system. In the shaded area a second incommensuration appears. In the green area (region II) extra peaks at $k=\pi$ develop in the FT of the rung current correlations and they become the dominant correlations in the blue region (III). The double line (green vs dark red) at $\lambda =\pi$ represents the transition to a localized phase. In the right panel we show intensity plots of $n\left(k\right)$ versus $\lambda$ and $k$. In panel (a), at $\mathrm{\Omega }/t=1.75$ the system is always in the MM phase; indeed, only a single maximum at $k=0$ is visible for all $\lambda$. At $\lambda =\pi ,n\left(k\right)=1$ indicating the formation of a fully localized state (dark red solid line). In panel (b) and (c), for $\mathrm{\Omega }/t=1$ and $0.25$, respectively, the transition from MM to the MV with two maxima symmetric around $k=0$ is shown.

Figure 2

We show FT of correlation functions as from DMRG simulation for $L=64$ at $\lambda =\pi /4$ for two different values of the $\mathrm{\Omega }/t=0.0625$ and 1, respectively, in the vortex (black solid line) and Meissner phases [dashed (red) line]. Panel (a) shows the FT of the rung-current correlation function $C\left(k\right)$, panel (b) the spin correlation functions $S\left(k\right)$, and panel (c) the charge bond-order correlation function $S_{\mathrm{BOW}}^c$. In panel (d) the spin resolved momentum distribution is shown, with $n_{-\sigma }\left(k\right)=n_{\sigma }(-k)$.

Figure 3

We show FT of correlation functions as from DMRG simulations for $L=64$ at $\lambda =\pi$ at various $\mathrm{\Omega }/t$. Panel (a) shows the FT of the rung-current correlation function $C\left(k\right)$, panel (b) the spin correlation function $S\left(k\right)$, and panel (c) the charge bond-order correlation function $S_{\mathrm{BOW}}^c$. In panel (d) the spin resolved momentum distribution is shown. Dotted (red) curves are for $\mathrm{\Omega }/t=1.75$ in the fully localized state, while black solid, dashed (blue), gray (green) solid, and dot-dashed (magenta) lines are, respectively, for $\mathrm{\Omega }/t=1.5,1.25,1$, and $0.5$.

Figure 4

We show FT of correlation functions as from DMRG simulations for $L=64$ at $\lambda =3\pi /4$ at various $\mathrm{\Omega }/t$. Panel (a) shows the FT of the rung-current correlation function $C\left(k\right)$, panel (b) the spin correlation function $S\left(k\right)$, and panel (c) the symmetric bond-order correlation function $S_{\mathrm{BOW}}^c$. In panel (d) the chain resolved momentum distribution is shown, with $n_{-\sigma }\left(k\right)=n_{\sigma }(-k)$. Dotted (red) curves are for $\mathrm{\Omega }/t=1.5$ in the Meissner phase, while black solid, dashed (blue), and dot-dashed (magenta) lines are, respectively, for $\mathrm{\Omega }/t=1.25,1$, and $0.625$.