Miscible-immiscible quantum phase transition in coupled two-component Bose-Einstein condensates in one-dimensional optical lattices

We study the miscible-immiscible quantum phase transition in a linearly coupled binary Bose-Hubbard model in one dimension that can describe the low-energy properties of a two-component Bose-Einstein condensate in optical lattices. With the quantum many-body ground state obtained from the density matrix renormalization group algorithm, we calculate the characteristic physical quantities of the phase transition controlled by the linear coupling between the two components. Furthermore we calculate the Binder cumulant to determine the critical point and construct the phase diagram. The strong-coupling expansion shows that in the Mott insulator regime the model Hamiltonian can be mapped to a spin-$1/2$ $\mathit{XXZ}$ model with a transverse magnetic field.

Figure 1

Mean occupation number for spin-up bosons $\langle N_{\uparrow }\rangle$ (red circles) and spin-down bosons $\langle N_{\downarrow }\rangle$ (black triangles) on the $j\text{th}$ site throughout a 200-site lattice with an open boundary condition (a) below, (b) near but still below, and (c) above the critical point in the MI regime. The interaction strengths are $U=5$ and $U_{\uparrow \downarrow }=2U$. The linear coupling strengths are (a) $\Omega =0.18$, (b) $\Omega =0.214$, and (c) $\Omega =0.248$.

Figure 2

The absolute occupation imbalance $|\langle \Delta N\rangle |$ on the middle site $L/2$ of lattices having 50 (blue circles), 100 (red squares), 150 (cyan diamonds), and 200 (orange triangles) sites as a function of the coupling coefficient $\Omega$, when $U=5$ and $U_{\uparrow \downarrow }=2U$. The green crosses are the data for an infinite lattice from iDMRG calculations. The dashed lines are used to guide the eyes. The solid black line is drawn with parameters from a finite-size scaling about the critical point with exponent $\beta =1/8$.

Figure 3

The correlation function $C\left(\ell \right)$ calculated in the bulk of a lattice with 200 sites with 60 sites cut off at both ends. The blue crosses (red pluses) are the values of the correlation function between the zeroth site and the $\ell \text{th}$ site when $\Omega =0.192$ ($\Omega =0.248$), which is below (above) the critical point. The curves are the fittings with respect to a sum of two exponential functions. The interaction strengths are $U=5$ and $U_{\uparrow \downarrow }=2U$.

Figure 4

The longer (blue diamonds) and shorter (green squares) correlation lengths extracted by fitting the correlation function with a sum of two exponential functions for a lattice of 200 sites. The longer correlation length shows divergence in a certain region of the coupling coefficient $\Omega$. The black dashed line shows the fitting curve with the critical exponent $\nu$ as an independent variable and the red solid line shows the fitting curve with the plausible fixed exponent $\nu =1$ (see text). The interaction strengths are $U=5$ and $U_{\uparrow \downarrow }=2U$.

Figure 5

The Binder cumulant $U_L$, Eq. (12), as a function of the coupling coefficient $\Omega$ for lattices of $L=50$ (blue dash-dotted curve), 100 (green dotted curve), 150 (red dashed curve), and 200 (black solid curve), when $U=5$ and $U_{\uparrow \downarrow }=2U$. The inset zooms into the region where four curves cross each other, near ${\Omega }_c=0.2153$.

Figure 6

The phase diagram in the space of the coupling parameter $\Omega$ and the on-site intracomponent interaction $U$. The intercomponent interaction $U_{\uparrow \downarrow }=2U$. The border between the superfluid and the MI will be updated in upcoming works. The red circles are the data points of the border between miscible and immiscible phases determined by the Binder cumulant for lattices of $L=50$ and $L=100$. The dotted curve connecting the red circles interpolates the data points (red circles). The solid curve is attained by fitting the data points with $C/U$ for $U>10$.

Figure 7

The entanglement entropies up to half of a 200-site lattice, below (red circles), near (blue triangles), and above (black squares) the critical point in the MI regime. The interaction strengths are $U=5$ and $U_{\uparrow \downarrow }=2U$.

Figure 8

Correlation length vs number of states for various values of the coupling coefficient $\Omega$ around the critical point in the MI regime. The other parameters are $U=5$ and $U_{\uparrow \downarrow }=2U$.

Figure 9

The correlation length obtained from iDMRG calculations for a translationally invariant infinite lattice with the increasing size of the number of states: $m=50$ (green triangles), $m=300$ (red circles), and $m=\infty$ (black circles). The data for $m=\infty$ are obtained by extrapolation. The error bars show the uncertainty in the extrapolation. The other parameters are $U=5$ and $U_{\uparrow \downarrow }=2U$.

Figure 10

The mean occupation number and its derivative with respect to $\Omega$ for symmetric states and antisymmetric states around the critical point when $U=5$ and $U_{\uparrow \downarrow }=2U$. The black solid curves are the fitting curves for the derivatives to a logarithmic function. The arrows point to the corresponding $y$ axis of the mean occupation number or its derivative.