Berezinskii-Kosterlitz-Thouless Transition of Two-Component Bose Mixtures with Intercomponent Josephson Coupling

We study the Berezinskii-Kosterlitz-Thouless (BKT) transition of two-component Bose mixtures in two spatial dimensions. When phases of both components are decoupled, half-quantized vortex-antivortex pairs of each component induce two-step BKT transitions. On the other hand, when phases of both components are synchronized through the intercomponent Josephson coupling, two species of vortices of each component are bound to form a molecule, and, in this case, we find that there is only one BKT transition by molecule-antimolecule pairs. Our results can be tested by two weakly connected Bose systems such as two-component ultracold diluted Bose mixtures with the Rabi oscillation, and multiband superconductors.

Figure 1

Profile of the relative phase $\mathrm{\Delta }\phi$ for (a) the vortex molecule $[1,1{]}_{r_0}$ and the vortex-molecule pair $[1,1{]}_{\delta }\stackrel{r_0}{-}[-1,-1{]}_{\delta }$ with $\delta =0.5r_0$. Closed (open) red and blue circles represent positions of vortices of $\{1,0\}$ and $\{0,1\}$ ($\{-1,0\}$ and $\{0,-1\}$), respectively. Black solid lines show kinks having the relative phase $\mathrm{\Delta }\phi =\pi$. Both panels are obtained by numerically minimizing the Hamiltonian [Eq. (2)] with fixing positions of vortices.

Figure 2

Superfluid number density ${\rho }_{\mathrm{s}}$ for (a): $n_1/n_2=0.5$ and $q=0$, (b): $n_1/n_2=0.5$ and $q=0.1g_{1}n$, (c): $n_1/n_2=1$ and $q=0$, and (d): $n_1/n_2=1$ and $q=0.1g_{1}n$ as a function of the temperature $T/T_{\mathrm{BKT}}^{(0)}$, where $T_{\mathrm{BKT}}^{(0)}$ is the BKT transition temperature for the single-component Bose system.. The system sizes are $L=64\xi$ (red lines), $L=96\xi$ (green lines), $L=128\xi$ (blue lines), and $L=160\xi$ (yellow lines). The BKT transition temperatures $T_{\mathrm{BKT}1}$, $T_{\mathrm{BKT}2}$ and $T_{\mathrm{BKT}}$ are shown as the thick solid lines. The thin dashed lines in the all panels show the relation ${\rho }_{\mathrm{s}}/T=2m/(\pi {\hslash }^2)$. The thin dotted lines in panels (a) and (c) show relations $({\rho }_{\mathrm{s}}-0.57n)/T=2m/(\pi {\hslash }^2)$ and ${\rho }_{\mathrm{s}}/T=4m/(\pi {\hslash }^2)$, respectively. The thick dashed line in panel (b) and dotted lines in panels (b) and (d) show the crossover temperatures $T_{\text{vortex}}$ and $T_{\mathrm{kink}}$, respectively (see Fig. 3).

Figure 3

Arrhenius plots for number densities $⟨{\rho }_{\mathrm{v}1}⟩$ and $⟨{\rho }_{\mathrm{v}2}⟩$ of vortex-antivortex pairs for first (red lines) and second (green lines) components, and length density $⟨{\rho }_{\mathrm{k}}⟩$ of kinks (black lines) for (a): $n_1/n_2=0.5$ and $q=0.1g_{1}n$ and (b): $n_1/n_2=1$ and $q=0.1g_{1}n$. The BKT transition temperature $T_{\mathrm{BKT}}$ are shown as solid lines. The dashed line in panel (a) and dotted lines in both panels show the crossover temperatures $T_{\text{vortex}}$ and $T_{\mathrm{kink}}$ above which $⟨{\rho }_{\mathrm{v}1}⟩$ and $⟨{\rho }_{\mathrm{k}}⟩$ have deviations larger than 1% from the Arrhenius relation, respectively.

Figure 4

Equilibrium snapshots of vortices and kinks with $L=64\xi$, $n_1/n_2=0.5$ and $q=0.1g_{1}n$ at (a): $T=T_{\text{vortex}}$ and (b): $T=T_{\mathrm{BKT}}$. Closed (open) red and blue circles represent positions of vortices for $\{1,0\}$ and $\{0,1\}$ ($\{-1,0\}$ and $\{0,-1\}$), respectively. Black solid lines show kinks. Green dashed closed lines in panel (b) denote molecule-antimolecule pairs.

Figure 5

Phase diagrams for the temperature $T$ and the Josephson coupling $q$ for (a): $n_1/n_2=0.5$ and (b): $n_1/n_2=1$. The red, purple, and blue lines at $q=0$ show phases for unbounded vortex of both components, unbounded vortex of the first component, and bounded vortex of both components, respectively. The black solid lines and filled circles show the BKT transition temperature $T_{\mathrm{BKT}}$. The dashed line with open circles in panel (a) and dotted lines with open squares in the both panels indicate the crossover temperatures $T_{\text{vortex}}$ and $T_{\mathrm{kink}}$.