Bosonic topological phase in a paired superfluid

We study an effective model of two interacting species of bosons in two dimensions, which is amenable to sign problem free Monte Carlo simulations. In addition to conventional ground states, we access a paired superfluid which is also a topological phase, protected by the remaining $U\left(1\right)\times {\mathbb{Z}}_2$ symmetry. This phase arises from the condensation of a composite object, the bound state of vortices and antivortices of one species, to a boson of the second species. We introduce a bulk response function, the Ising analog of the quantized Hall effect, to diagnose the topological phase. The interplay of broken symmetry and topology leads to interesting effects such as fractionally charged vortices in the paired superfluid. Possible extensions towards realistic models of cold atomic bosons are discussed.

Figure 1

Typical world-line configurations in the SPT phase. Blue lines and red lines correspond to the ${\mathbb{Z}}_2$ world lines and the vorticity current, respectively. Threading a unit of flux quantum shifts the total ${\mathbb{Z}}_2$ charge from the (a) even to the (b) odd sector.

Figure 2

Phase diagram of the coupled rotor model in Eq. (2) as a function of ${\alpha }_B$ and $\lambda$ for ${\alpha }_A=4$. The numerically computed critical points are marked by black circles. We probed $U\left(1\right)$ symmetry breaking by using the condensate fraction. The SPT phase, which is also a paired superfluid of type B rotors, is identified by measuring the topological response to an external flux. Red lines correspond to parameter cuts along which we perform finite size scaling scans in the labeled figures.

Figure 3

Single particle condensate fraction of type A rotors. (a) The condensate fraction, $C_1^A$, is nonvanishing only in phase II, where $\lambda$ and ${\alpha }_B$ are sufficiently large. (b) Ordering transition of type A rotors for $\lambda =2$ as a function of ${\alpha }_B$ and for system sizes $L=4,8,16,22$.

Figure 4

Single and pair condensate fraction of type B rotors. (a) For small values of $\lambda$ the condense fraction is finite and it vanishes at a critical coupling ${\lambda }_c\left({\alpha }_B\right)$. (b) The pair condensate function, $C_2^B$, remains finite even for $\lambda >{\lambda }_c\left({\alpha }_B\right)$ and sufficiently small ${\alpha }_B$. (c),(d) Finite size scaling analysis of the phases' transition for ${\alpha }_B=0.05$ as a function of $\lambda$ and for system sizes $L=4,8,16,32$.

Figure 5

Winding number distribution, $P\left(W_{\tau }\right)$, for ${\alpha }_B=0.05$. Below the critical coupling (red), $\lambda =1.22<{\lambda }_c$, we find a broad distribution corresponding to a superfluid. Above the critical coupling (blue), $\lambda =2.83>{\lambda }_c$, the distribution remains wide but the probability for odd number of particles vanishes, $P\left(W_{\tau }\right|W_{\tau }\text{is}\text{odd})=0$, indicating on a paired superfluid.

Figure 6

Finite size scaling analysis of total ${\mathbb{Z}}_2$ charge, ${\mathcal{C}}_{{\mathbb{Z}}_2}$, at the phase transition between phase I and the paired superfluid, for ${\alpha }_B=0.05$. ${\mathcal{C}}_{{\mathbb{Z}}_2}$ is a dimensionless observable and thus it is expected to obtain a universal value at the critical point. This behavior is clearly since in panel (a) where curves for different system size cross at the critical coupling, ${\lambda }_c=1.597\left(1\right)$, with small dependence on system size. (b) Finite sizes scaling curve collapse. The horizonal axis is rescaled according to Eq. (6).

Figure 7

Probability histogram of $\langle \left|J_A\right|\rangle$ for ${\alpha }_B=0.6$. (b) At the critical binding, ${\lambda }_c=1.16$, the histogram develops a double peak structure corresponding to coexistence of phase I and phase II and indicating on a first order phase transition. (a),(c) By contrast, slightly above and below the critical point the histograms consist of a single peak.

Figure 8

Total ${\mathbb{Z}}_2$ charge of type B rotors as a function $\lambda$ for $L=4,8,16,32, {\alpha }_B=0.05$ and for magnetic flux ${\mathrm{\Phi }}_A=0$ and ${\mathrm{\Phi }}_A=2\pi$. In phase I, $\lambda <\lambda , {\mathcal{C}}_{{\mathbb{Z}}_2}$ fluctuates and average to 1/2. For $\lambda >{\lambda }_c$, in the absence of an external magnetic flux ${\mathcal{C}}_{{\mathbb{Z}}_2}$ vanishes as expected in a paired superfluid. Threading a magnetic flux ${\mathrm{\Phi }}_A=2\pi$ shifts the ${\mathcal{C}}_{{\mathbb{Z}}_2}$ charge from the even to the odd sector of the theory, i.e., ${\mathcal{C}}_{{\mathbb{Z}}_2}=1$.

Figure 9

Single particle Green's function, $g_1^{A/B}$, evaluated along the edges of a cylindrical. The simulation parameters are $L_y=L_{\tau }=20,L_x=8, {\alpha }_B=0.05$, and $\lambda =2.5$. Solid lines are a numerical fit to the power law form in Eq. (7).

Figure 10

Combined MC move. A planar angle ${\theta }_A\left(r_i\right)$ is drowned randomly and a bond current loop of type B rotors belonging to the dual lattice and surrounding the bond $b\left(r_i\right)$ is constructed.