Preemptive vortex-loop proliferation in multicomponent interacting Bose-Einstein condensates

We use analytical arguments and large-scale Monte Carlo calculations to investigate the nature of the phase transitions between distinct complex superfluid phases in a two-component Bose-Einstein condensate when a nondissipative drag between the two components is being varied. We focus on understanding the role of topological defects in various phase transitions and develop vortex-matter arguments, allowing an analytical description of the phase diagram. We find the behavior of fluctuation induced vortex matter to be much more complex and substantially different from that of single-component superfluids. We propose and numerically investigate a drag-induced “preemptive vortex loop proliferation” scenario. Such a transition may be a quite generic feature in many multicomponent systems where symmetry is restored by a gas of several kinds of competing vortex loops.

Figure 1

(Color online) Schematic phase diagram which follows from separations of variables for the model in Eqs. (1, 11), ${\rho }_1={\rho }_2=\rho$. The gray-shaded area is the forbidden regime ${\rho }_d>\rho /2$. The lines separating the various regions are (a) Solid (red) line: The proliferation of (1,0) and (0,1) vortices from an ordered background (i.e., large $\rho$) along the${\rho }_d=\rho -{\rho }_c$ line. (b) Dashed (blue) line: The proliferation of $\left(1,-1\right)$-vortices from an ordered background, along the ${\rho }_d=\rho /2-{\rho }_c/4$ line. c) Dashed-dotted vertical (black) line: The proliferation of (1,0) and (0,1) vortices from a background of proliferated $\left(1,-1\right)$ vortices along the $\rho =2{\rho }_c$ line. This vortex-matter phase diagram has the same topology as that obtained from the $J$-current model (Ref. 3). I, $\text{U}\left(1\right)\times \text{U}\left(1\right)$; II, U(1) symmetry in the phase sum; and III, a fully symmetric case.

Figure 2

(Color online) Schematic phase diagram with regions I, II, III, and IV for the model in Eqs. (1, 13) with ${\rho }_2=\alpha {\rho }_1>{\rho }_1=\rho$. For the purposes of illustration, we have taken $\alpha =1.2$. The gray shaded area is the forbidden parameter regime ${\rho }_d>\alpha \rho /\left(1+\alpha \right)$. The lines separating the various regions are obtained as follows. (a) Solid (red) lines: The proliferation of (0,1) vortices from an ordered background along the ${\rho }_d=\alpha \rho -{\rho }_c$ line, as well as the proliferation of (1,0) vortices from an ordered background along the ${\rho }_d=\rho -{\rho }_c$ line. (b) Dashed (blue) line: The proliferation of $\left(1,-1\right)$ vortices from an ordered background along the lines of ${\rho }_d=\left(1+\alpha \right)\rho /4-{\rho }_c/4$. (c) Dashed-dotted (black) line: The line of proliferation of individual vortices (1,0) or (0,1) in a background of proliferated vortices $\left(1,-1\right)$, as given by Eq. (16). When we cross this line from right to left, passing from region II to III, the stiffness of the clapping mode ${\theta }_1+{\theta }_2$ is destroyed by the proliferation of individual vortices.

Figure 3

(Color online) Plot of ${\rho }_{\text{clap}}\left({\rho }_I,{\rho }_{dI},\alpha \right)/{\rho }_c$ as a function of $\alpha$. The dashed (black) line shows that the remaining stiffness of the clapping mode [in the background of proliferated composite $\left(1,-1\right)$ vortices] is less than the critical coupling for vortex-loop proliferation in a parameter regime $2/3<\alpha <3/2$. This is the parameter regime where it is correct to limit oneself to the sector where the composite proliferated background vortices are of the type $\left(1,-1\right)$. For $\alpha <2/3$, the composite proliferated background vortices are of type $\left(-n,1\right)$, while for $\alpha >3/2$, the composite proliferated background vortices are of type $\left(1,-n\right)$. The fact that ${\rho }_{\text{clap}}\left({\rho }_I,{\rho }_{dI},\alpha \right)<{\rho }_c$ indicates that we are in a parameter regime where the full restoration of $\text{U}\left(1\right)\times \text{U}\left(1\right)$ symmetry proceeds from a preemptive vortex-loop proliferation phase transition, as explained in the text. The solid (red) line shows ${\rho }_{\text{clap}}\left({\rho }_I,{\rho }_{dI},\alpha \right)$ for $\alpha <2/3$ in a regime where $\left(2,-1\right)$ vortices trigger preemptive proliferation of all topological defects and $\alpha >3/2$ where $\left(1,-2\right)$ vortices initiate the phase transition into a fully symmetric state.

Figure 4

(Color online) Phase diagram and a set of helicity moduli for the model [Eq. (22)] with equal bare stiffnesses. The shaded region illustrates the forbidden parameter regime ${\rho }_d>\rho /2$. The helicity moduli are ${{\rm Y}}_1$, ${{\rm Y}}_2$, and ${{\rm Y}}_{-}$. The leftmost helicity moduli are measured for a drag ${\rho }_d=0.30\rho$, while the rightmost helicity moduli are for ${\rho }_d=0.39\rho$.

Figure 5

(Color online) The energy histograms for $\left(\rho ,{\rho }_d\right)\approx \left(0.60,0.20\right)$, with $\alpha =1$, i.e., in the preemptive region. A clear double peak structure is seen to develop, which is an indication of a first order transition. The areas under the histograms are normalized to 1.

Figure 6

(Color online) Phase diagram and a set of helicity moduli for the model [Eq. (22)] for $\alpha =1.1$. The shaded region illustrates the forbidden parameter regime ${\rho }_d>{\rho }_1{\rho }_2/\left({\rho }_1+{\rho }_2\right)$. The helicity moduli are ${{\rm Y}}_1$, ${{\rm Y}}_2$, and ${{\rm Y}}_{-}$. The leftmost helicity moduli are measured for a drag ${\rho }_d=0.25\rho$, while the rightmost helicity moduli are for${\rho }_d=0.39\rho$.

Figure 7

(Color online) The energy histograms for $\left(\rho ,{\rho }_d\right)\approx \left(0.60,0.22\right)$, with $\alpha =1.1$, i.e., in the preemptive region. A clear double peak structure is seen to develop, which is an indication of a first order phase transition. The areas under the histograms are normalized to 1.

Figure 8

(Color online) Phase diagram and a set of helicity moduli for the model Eq. (22), $\alpha =0.55$. The shaded region illustrates the forbidden parameter regime ${\rho }_d>{\rho }_1{\rho }_2/\left({\rho }_1+{\rho }_2\right)$. The helicity moduli are ${{\rm Y}}_1$, ${{\rm Y}}_2$, and ${{\rm Y}}_{1,-2}$. Here, the latter corresponds to ${{\rm Y}}_{-}$, with the difference that the ${\theta }_2$ phase is twisted twice as much as ${\theta }_1$. The leftmost helicity moduli are measured for a drag ${\rho }_d=0.32\rho$, while the rightmost helicity moduli are for${\rho }_d=0.336\rho$.

Figure 9

(Color online) The energy histograms for $\left(\rho ,{\rho }_d\right)\approx \left(1.77,0.58\right)$, with $\alpha =0.55$, i.e., in the preemptive region. A clear double peak structure is seen to develop, which is an indication of a first order transition. The areas under the histograms are normalized to 1.

Figure 10

(Color online) Same as Fig. 2, but now also including a negative drag coefficient ${\rho }_d$. In region I, the system features a broken $\text{U}\left(1\right)\times \text{U}\left(1\right)$ symmetry. In regions II, IV, and V, the system features a broken U(1) symmetry associated with the phase sum, one individual phase, and the phase difference, respectively. In region III, the system is in the fully symmetric state (see also Fig. 2).