Vortex lattice melting in a boson ladder in an artificial gauge field

We consider a two-leg boson ladder in an artificial U(1) gauge field and show that, in the presence of interleg attractive interaction, the flux induced vortex state can be melted by dislocations. For increasing flux, instead of the Meissner to vortex transition in the commensurate-incommensurate universality class, first, an Ising transition from the Meissner state to a charge density wave takes place, then, at higher flux, the melted vortex phase is established via a disorder point where incommensuration develops in the rung current correlation function and in momentum distribution. Finally, the quasi-long-range ordered vortex phase is recovered for sufficiently small interaction. Our predictions for the observables, such as the spin current and the static structure factor, could be tested in current experiments with cold atoms in bosonic ladders.

Figure 1

Phase diagram at $n=0.5$ for a fixed value of interchain hopping $\mathrm{\Omega }/t=0.5$ as a function of the applied flux $\lambda$ and as a function of the strength of the interchain interaction from DMRG simulations for $L=64$ in PBC. The area under the dashed black line shows the stability region for the Meissner phase, the dark-red region between the solid and the dashed black lines shows the region of stability for the Meissner-CDW phase. The light-blue area under the dotted line displays the stability region for the vortex phase and the shaded blue area is the one where the second incommensuration occurs. The light-green region above the solid and dotted lines is where the melted vortex phase is stable. The dot-dashed curve is the Lifshitz curve where the occurrence of the melted vortex phase is detected in the correlation function for rung current (see text for explanations).

Figure 2

Upper panel shows the spin current $J_s$ as a function of the applied flux, red line is only a guide to eye. Panels below show $S^c\left(k\right)$ (red solid line), $S^s\left(k\right)$ (dark-green thick solid line), $C\left(k\right)$ (black dashed line), $n\left(k\right)$ (blue dotted line), where the argument of this last quantity has been shifted by $\pi$. Left, center, and right panels show these quantities for the cases indicated by the points $\mathrm{A},\mathrm{B}$, and $\mathrm{C}$ in the upper panel, respectively corresponding to cases where the system is in the Meissner phase, in the M-CDW phase, and in the melted vortex phase. Data shown are from DMRG simulations in PBC for $L=64$.

Figure 3

In the main panel, red and black solid dot data show the central charge $c$ as a function of $\lambda$ while red and black solid lines are the numerical derivative of $J_s$ with respect to $\lambda$, respectively for $U_{\perp }/t=-1.5$ and $U_{\perp }/t=-2.0$ at $n=0.5$ and $\mathrm{\Omega }/t=0.5$. Data shown in the main panel for the central charge are extracted from the Von Neumann entropy computed with DMRG in PBC discarding size effects using $m=600$ states in simulations. In the inset, we show the central charge $c$ at $U_{\perp }/t=-1.5$ for different sizes namely $L=64$, 48, and 32, fitted with Lorentzian curves whose spread reduces on increasing the size and whose maximum is at $\lambda =0.15\pi$.

Figure 4

In the left panel, we show the value of the $S^c(k=\pi /2)$, i.e., where the correlation function develops a peak on entering the Meissner-CDW, as a function of $\lambda$ for $U_{\perp }=-1.5, n=0.5$, and $\mathrm{\Omega }=0.5/t$. Red dots are data from simulation at $L=64$ in PBC. The black dashed curves are fits using $f\left(\lambda \right)=a/{(\lambda -{\lambda }_{\mathrm{cdw}})}^{1/4}+b$. On the right panel, we have plotted the $ln{S}^c(k=\pi /2)$ as a function of the logarithm of the deviation from the critical lambda, to find the $1/4$ exponent as the slope of the linear function. We have found in both cases that the slope $m=0.24\left(1\right)$ is in agreement with an Ising transition.

Figure 5

Upper panel shows the spin current $J_s$ as a function of strength of interchain interaction, solid red line is only a guide to eye. Panels below show $S^s\left(k\right)$ (dark-green thick solid line), $C\left(k\right)$ (black dashed line), $S^c\left(k\right)$ (red solid line), and $n\left(k\right)$ (blue dotted line) where the argument of this last quantity has been shifted by $\pi$. Left, center, and right panels show these quantities for the cases indicated by the points $\mathrm{A},\mathrm{B}$, and $\mathrm{C}$ in the upper panel, respectively corresponding to cases where the system is in the melted vortex phase and in the vortex phase C. Data shown are from DMRG simulations in PBC for $L=64$.

Figure 6

In the main panel, we show the central charge as a function of the inverse of the size $L$. The upper curve is for $\lambda =0.15\pi$, the critical value inferred from fitting the finite size and number of states $m_s$ central charges as a function of $\lambda$ (see Fig. 3), while the lower curve is for $\lambda =0.75\pi$ well inside the melted vortex phase. The value of the central charge $c\left(L\right)$ has been extracted fitting Von Neumann entropy from simulation data at fixed $L$ and $m_s$ as a function of $f\left(x\right)=ln(L/\pi sin(\pi x/L\left)\right)$ so that the central charge is related to the slope of the fitting straight line. Residual dependence from $m_s$ has been removed assuming a dependence of the form $c\left(m_s\right)=a+b/m_s^2$. In the inset we show two typical extrapolations from the raw data Von Neumann entropy at $m_s=600$. The upper curve refers to data from the PBC-DMRG simulation for the case $\lambda =0.15\pi$ for $L=64$, while the lower curve is for $\lambda =0.75\pi$ at $L=48$. The $m_s$-extrapolated value for the cases plotted in the inset panel are shown in the main panel by the red and black open dots, respectively.