Quantum phase diagram of the integrable $p_x+i{p}_y$ fermionic superfluid

We determine the zero-temperature quantum phase diagram of a $p_x+i{p}_y$ pairing model based on the exactly solvable hyperbolic Richardson-Gaudin model. We present analytical and large-scale numerical results for this model. In the continuum limit, the exact solution exhibits a third-order quantum phase transition, separating a strong-pairing from a weak-pairing phase. The mean-field solution allows to connect these results to other models with $p_x+i{p}_y$ pairing order. We define an experimentally accessible characteristic length scale, associated with the size of the Cooper pairs, that diverges at the transition point, indicating that the phase transition is of a confinement-deconfinement type without local order parameter. We propose an experimental measurement to detect the transition. We show that this phase transition is not limited to the $p_x+i{p}_y$ pairing model but can be found in any representation of the hyperbolic Richardson-Gaudin model and is related to a symmetry that is absent in the rational Richardson-Gaudin model.

Figure 1

Quantum phase diagram of the $p_x+i{p}_y$ or hyperbolic model in terms of the fermion density $\rho$ and the attractive coupling strength $g$. It shows two superfluid phases, strong-pairing (confined) and weak-pairing (deconfined) phases, separated by a third-order confinement-deconfinement QPT at vanishing chemical potential, $\mu =0$. Cooper pairs deconfine between the phase-transition line and the Moore-Read line (indicated by a dashed line). The latter line corresponds to $\rho =1-1/g$ and vanishing total energy, $E=0$, and represents a situation where all Cooper pairs are deconfined. The small symbols at $\rho =0.25$ (quarter filling) indicate the configurations that are displayed in detail in Figs. 3, 4.

Figure 2

Real and imaginary parts of the spectral parameters $E_{\alpha }$ as a function of $g=GL$ for ten pairons in a disk of dimension $L=40$ (quarter filling), with levels as specified in Table .

Figure 3

Pairon distribution for $L=504$ at quarter filling, $\rho =0.25$, for $g=0.5$, $g=1.5$, and $g=2.5$.

Figure 4

Momentum density profiles, $n_k=⟨c_k^{†}{c}_k⟩$, for the same parameter sets as in Fig. 3 (see symbols), compared to the mean-field solution for the same system (continuous lines).

Figure 5

Finite-size scaling of the energy density $\epsilon$ at quarter filling for several interaction strengths (corresponding to distinct regions of the phase diagram), both for the exact solution (symbols) as for the mean-field solution (lines).

Figure 6

Analytical results for the chemical potential $\mu$ as a function of $g$, for densities $\rho =0.05,0.15,\dots ,0.95$, in ascending order. The symbols mark the transition points at $g={\left(1-2\rho \right)}^{-1}$.

Figure 7

Analytical results for the pairing field $\Delta$ and its derivative $d\Delta /dg$ in the inset, as a function of $g$ for various densities.

Figure 8

Analytical results for the energy density $\epsilon$ of the integrable $p_x+i{p}_y$ model as a function of the coupling strength $g$ for representative densities $\rho$, in the thermodynamic limit. The symbol $\odot$ marks the QPT point while $⊡$ the Moore-Read line.

Figure 9

Higher order derivatives of the energy density $\epsilon$ as a function of $g$ for various densities. The small circles mark the transition point at $g={\left(1-2\rho \right)}^{-1}$.

Figure 10

Condensate fraction of pairs, $n_c$, as a function of $g$, for densities $\rho =0.05,0.10,\dots ,0.95$, in descending order. The circles mark the transition points at $g={\left(1-2\rho \right)}^{-1}$ while the triangular symbols mark the point where $n_c$ is maximal.

Figure 11

Root-mean-square radius, $r_{\mathsf{rms}}$, of the condensate wave function in units of $1/\sqrt{2\omega }$, at quarter filling. The dashed lines indicate the Moore-Read point $\left(\epsilon =0\right)$ and the phase-transition point $\left(\mu =0\right)$.

Figure 12

Contours used to evaluate the continuum limit.