Many-Body Characterization of Particle-Conserving Topological Superfluids

What distinguishes trivial superfluids from topological superfluids in interacting many-body systems where the number of particles is conserved? Building on a class of integrable pairing Hamiltonians, we present a number-conserving, interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain, that remains exactly solvable for periodic and antiperiodic boundary conditions. Our model allows us to identify fermion parity switches that distinctively characterize topological superconductivity (fermion superfluidity) in generic interacting many-body systems. Although the Majorana zero modes in this model have only a power-law confinement, we may still define many-body Majorana operators by tuning the flux to a fermion parity switch. We derive a closed-form expression for an interacting topological invariant and show that the transition away from the topological phase is of third order.

Figure 1

Third-order phase transition between the topological (weak-pairing) and trivial (strong-pairing) superconducting phases. The continuous line denotes the second order derivative of the ground-state energy density $e_0$, evaluated in the thermodynamic limit. Circles are the exact solution for $L=2048$, $N=512$, and antiperiodic boundary conditions. Top inset: Discontinuity in the third-order derivative. Bottom inset: Quantum phase diagram in the ($\rho$, $g$)-plane. Dashed and full lines represent, respectively, the Moore-Read ($g_M^{-1}=1-\rho$) and Read-Green ($g_c^{-1}=1-2\rho$) boundaries.

Figure 2

Ground state energies for the RGK chain (in units of $t_1\equiv 1$, for $t_2=0$) for even ($N=2M$) and odd ($N=2M\pm 1$) number of fermions, and with periodic ($\varphi =0$) or antiperiodic ($\varphi =2\pi$) boundary conditions. The main plot shows the odd-even difference as a function of the interaction strength $g$ for a finite system (data points, for $N=512$, $L=2048$) and in the thermodynamic limit (continuous lines). The topologically nontrivial state is entered for $g<g_c=2$. The insets show the even and odd energies themselves, for the finite system (lower two insets) and in the thermodynamic limit (upper two insets), illustrating the fermion parity switches ${\mathcal{P}}_N(\varphi )$.