Quantum fluctuations of a resonantly interacting $p$-wave Fermi superfluid in two dimensions

Using the Gaussian pair fluctuation theory, we investigate quantum fluctuations of a strongly interacting two-dimensional chiral $p$-wave Fermi superfluid at the transition from a Bose-Einstein condensate (BEC) to a topologically nontrivial Bardeen-Cooper-Schrieffer superfluid. Near the topological phase transition at zero chemical potential, $\mu =0$, we observe that quantum fluctuations strongly renormalize the zero-temperature equations of state, sound velocity, pair-breaking velocity, and Berezinskii-Kosterlitz-Thouless (BKT) critical temperature of the Fermi superfluid, all of which can be nonanalytic functions of the interaction strength. The indication of nonanalyticity is particularly evident in the BKT critical temperature, which also exhibits a pronounced peak near the topological phase transition. Across the transition and towards the BEC limit we find that the system quickly becomes a trivial interacting Bose liquid, whose properties are less dependent on the interparticle interaction. The qualitative behavior of composite bosons in the BEC limit remains to be understood.

Figure 1

The third-order ladder diagram considered in the Gaussian pair fluctuation theory. The solid line with arrow represents the $2\times 2$ BCS Green's functions. The dashed line with two vertices indicates the interparticle interaction. There are four types of ladders, ${\mathrm{\Pi }}_{s{s}^{\prime }}$, depending on the choice of the interaction vertex: $s=+$ for ${\sigma }_{+}\mathrm{\Gamma }\left({\mathbf{k}}^{\prime }\right)$ and $s=-$ for ${\sigma }_{-}{\mathrm{\Gamma }}^{*}\left(\mathbf{k}\right)$.

Figure 2

The chemical potential $\mu$ as a function of the scattering energy $E_b$, calculated by using the mean-field theory (dashed line) and GPF theory (solid line with circles). The dot-dashed line shows the asymptotic behavior in the BEC limit predicted by the mean-field theory in Eq. (47), which corresponds to a constant molecular scattering length for composite bosons. In the main figure, we have subtracted the contribution from the two-body bound state with the binding energy ${\varepsilon }_B\equiv max(-E_b,0)$. The inset highlights the chemical potential near the topological phase transition (i.e., $\mu \sim 0$ or $E_b\sim 0$). Here, we take a cutoff momentum $k_0=30k_F$.

Figure 3

The pairing gap $\mathrm{\Delta }$ as a function of the scattering energy $E_b$, predicted by using the mean-field theory (dashed line) and GPF theory (solid line with circles). The inset highlights the kink in the pairing gap near the topological phase transition (i.e., $E_b\sim 0$). Here, we take a cutoff momentum $k_0=30k_F$.

Figure 4

The fraction of fermions $n_F/n$ (solid line) and Cooper pairs $n_B/n$ (dashed line), as a function of the scattering energy $E_b$. Here, we take a cutoff momentum $k_0=30k_F$.

Figure 5

The pressure (main figure) and total energy (inset), as a function of the scattering energy $E_b$, predicted by using the mean-field theory (dashed line) and GPF theory (solid line with circles). The pressure and energy are normalized with respect to the ideal gas values $P_0=n{\varepsilon }_F/2$ and $E_0=N{\varepsilon }_F/2$, respectively. Here, $N$ is the total number of particles. For the total energy, we have subtracted the contribution from the two-body bound state, $-N{\varepsilon }_B/2$. As before, we take a cutoff momentum $k_0=30k_F$.

Figure 6

The sound velocity and pair-breaking velocity, as a function of the scattering energy $E_b$, predicted by using the mean-field theory (dashed line or dash-dotted line) and GPF theory (solid line with circles or stars). As the scattering energy decreases (or the interaction strength increases), the critical velocity $v_c=max\{v_{pb},c_s\}$ slowly increases. Here, we take a cutoff momentum $k_0=30k_F$.

Figure 7

The BKT transition temperature, calculated by using the mean-field theory (dashed line) and GPF theory (solid line with circles), as a function of the scattering energy $E_b$. The inset shows the normal density fractions contributed by fermions, $n_{n,f}/n$, and by pairs, $n_{n,b}/n$, and the superfluid density fraction $n_s/n=1-n_{n,f}/n-n_{n,b}/n$ at zero scattering energy $E_b=0$. The cross point between the superfluid fraction $n_s/n$ and the line $16T/T_F$ determines the BKT temperature at $E_b=0$. Here, we take a cutoff momentum $k_0=30k_F$.

Figure 8

The chemical potential $\mu$ (with the two-body bound-state contribution subtracted) as a function of the scattering energy $E_b$, at two cutoff momenta $k_0=10k_F$ (dashed line with stars) and $k_0=30k_F$ (solid line with circles). The inset shows the chemical potential $\mu$ and pairing gap $\mathrm{\Delta }$ at the zero scattering energy $E_b=0$, as a function of the inverse cutoff momentum $k_0^{-1}$. All the results are predicted by using the GPF theory.