Tricritical Physics in Two-Dimensional $p$-Wave Superfluids

We study effects of quantum fluctuations on two-dimensional $p+ip$ superfluids near resonance. In the standard paradigm, phase transitions between superfluids and zero density vacuum are continuous. When strong quantum fluctuations near resonance are present, the line of continuous phase transitions terminates at two tricritical points near resonance, between which the transitions are expected to be first-order ones. The size of the window where first-order phase transitions occur is shown to be substantial when the coupling is strong. Near first-order transitions, superfluids self-contract due to phase separations between superfluids and vacuum.

Figure 1

(a) Schematics of superfluid phases in the $\mu$ vs $1/A$ plane. The horizontal axis is the inverse of $p$-wave scattering area $A$ measuring the detuning. At resonance, $1/A=0$. In the standard paradigm, continuous quantum phase transitions occur when $\mu$ reaches a critical value (bold solid line). Shaded areas represent weakly interacting regions. (b) Near resonance when strong quantum fluctuations are present, the actual transitions become first order (dashed line). Tricritical points (black dots) on each side of resonance terminate the lines of continuous transitions and separate them from the discontinuous ones. As a result, the weakly interacting regime (light gray) is shifted away from resonance while a strongly fluctuating regime (dark gray) emerges. In (c) and (d), we illustrate the density profile (solid curve) of superfluids in a trap (dashed curve) away and near resonance respectively. From the center towards the edge of the trap, the local chemical potential $\mu (r)$ follows the corresponding paths in (b).

Figure 2

Tricritical points for different $g_0$ are labeled by squares and dots on the BCS and BEC side, respectively. They are fitted to Eq. (3) represented by the straight lines. Phase transitions driven by chemical potentials are first order between the tricritical points, and continuous otherwise.

Figure 3

(a) Phase diagram in the $\mu$ vs $1/A$ plane. Vertical dashed line marks resonance. First order phase transition line ${\mu }_c(A)$ (lower dashed curve) is separated from continuous phase transition line (solid curve) by a tricritical point (dot) on each side of resonance. (The tricritical point on BCS side is outside the range of this plot.) The upper dotted-dashed curve represents the crossover ${\mu }^{*}(A)$ between strong and weak fluctuation regimes. Below the crossover line, quantum fluctuations dominate in the superfluid phase. (b) Phase diagram mapped to $n$ vs $1/A$ plane from (a) using equation of state (4). The dashed curve represents the critical density $n_c(A)$ in first-order transitions, below which superfluids are self-contracted liquidlike droplets with constant density $n_c$. Note that at a fixed low density as the scattering area is varied continuously, superfluids can undergo two consecutive transitions as illustrated by the horizontal trajectory.