Theory of Non-Hermitian Fermionic Superfluidity with a Complex-Valued Interaction

Motivated by recent experimental advances in ultracold atoms, we analyze a non-Hermitian (NH) BCS Hamiltonian with a complex-valued interaction arising from inelastic scattering between fermions. We develop a mean-field theory to obtain a NH gap equation for order parameters, which are different from the standard BCS ones due to the inequivalence of left and right eigenstates in the NH physics. We find unconventional phase transitions unique to NH systems: superfluidity shows reentrant behavior with increasing dissipation, as a consequence of nondiagonalizable exceptional points, lines, and surfaces in the quasiparticle Hamiltonian for weak attractive interactions. For strong attractive interactions, the superfluid gap never collapses but is enhanced by dissipation due to an interplay between the BCS-BEC crossover and the quantum Zeno effect. Our results lay the groundwork for studies of fermionic superfluidity subject to inelastic collisions.

Figure 1

Numerical solution of ${\mathrm{\Delta }}_0$ as a function of $\gamma /t$ obtained from the NH gap equation in Eq. (5) at $\beta \to \infty$ and $\mu =0$ for (a) $U_1/t=1.8$ and (b) $U_1/t=10$. We assume a cubic lattice with energy dispersion ${\epsilon }_{\mathit{k}}=-2t(\cos k_x+\cos k_y+\cos k_z)$, where $t$ is the hopping amplitude. The dashed lines denote the asymptotic behavior in the strong-dissipation limit. The insets show (a) an enlarged view near the origin (weak dissipation) and (b) that of the real part.

Figure 2

(a) Real and (b) imaginary parts of the quasiparticle energy spectrum $E_{\mathit{k}}=\pm \sqrt{{\epsilon }_{\mathit{k}}^2+{\mathrm{\Delta }}_0^2}$ (orange for the positive sign, blue for the negative sign) at the critical points. (c) Exceptional lines in two dimensions. (d) Exceptional surfaces in three dimensions. The energy dispersion is for a square lattice ${\epsilon }_{\mathit{k}}=-2(\cos k_x+\cos k_y)$ in (a), (b), and (c), and for a cubic lattice ${\epsilon }_{\mathit{k}}=-2(\cos k_x+\cos k_y+\cos k_z)$ in (d). The gap is set to ${\mathrm{\Delta }}_0=0.19i$ for (a), (b), and (c), and ${\mathrm{\Delta }}_0=0.4i$ for (d).

Figure 3

Phase diagram of the NH BCS model at $\beta \to \infty$ and $\mu =0$. The yellow region corresponds to the normal phase. The red region shows the metastable superfluid phase for which a non-trivial solution of the gap equation gives a local energy minimum. The blue region shows the superfluid phase corresponding to a nontrivial solution of the gap equation that gives an effective ground state. A region with small $U_1$ is not shown because of the limitation of numerical calculations.