Metal-insulator-superconductor transition of spin-3/2 atoms on optical lattices

We use a slave-rotor approach within a mean-field theory to study the competition of metallic, Mott-insulating, and superconducting phases of spin-3/2 fermions subjected to a periodic optical lattice potential. In addition to the metallic, the Mott-insulating, and the superconducting phases that are associated with the gauge symmetry breaking of the spinon field, we identify an emerging superconducting phase that breaks both roton and spinon field gauge symmetries. This superconducting phase emerges as a result of the competition between spin-0 singlet and spin-2 quintet interaction channels naturally available for spin-3/2 systems. The two superconducting phases can be distinguished from each other by quasiparticle weight. We further discuss the properties of these phases for both two-dimensional square and three-dimensional cubic lattices at zero and finite temperatures.

Figure 1

Zero-temperature phase diagram showing four different phases: M, metallic phase; MI, Mott-insulating phase; Z-SC, superconducting phase with nonzero quasiparticle weight; and SC, superconducting phase with zero quasiparticle weight. (a) Phase diagram for the two-dimensional square lattice. (b) Phase diagram for the three-dimensional cubic lattice. The metal phase is characterized by a global U(1) symmetry-broken state associated with the rotor degrees of freedom and both superconducting phases (SC and Z-SC) are characterized by the global U(1) symmetry-broken states associated with the spinon degrees of freedom. The Z-SC phase shows additional U(1) symmetry breaking associated with the rotor sector.

Figure 2

Zero-temperature physical properties as a function of $U_2$ at (a) $U_0=6t$ and (b) $U_0=2t$. (a) The quasiparticle weight $Z$ (green line) continuously and monotonically decreases from unity to zero, showing the metal to Mott insulator transition at $U_2\approx 3.8t$. For $3.8t\le U_2\le 6t$, both the quasiparticle weight and the pairing order parameter remain zero, showing the Mott-insulating phase. For $U_2>6t$, the singlet pairing order parameter $\mathrm{\Delta }$ (blue line) becomes finite, indicating the Mott insulator to superconductor transition. (b) Only the metallic phase exists for low values of $U_2$ showing nonzero $Z$ and zero $\mathrm{\Delta }$. For $2t\le U_2\le 3.3t$, the existence of the Z-SC phase is evident as both $Z$ and $\mathrm{\Delta }$ have nonzero values. Notice for both cases that, while $Q_{\theta }$ (gray line) is continuous at each phase boundary, $Q_f$ shows discontinuities. For clarity, the pairing order parameter $\mathrm{\Delta }$ has been increased by a factor of 5 in these figures.

Figure 3

Finite-temperature properties of (a) the metallic phase and (b) the Z-SC phase of fermions in a three-dimensional cubic lattice. (a) Quasiparticle weight $Z$ as a function of temperature $k_{B}T/t$ in the metallic phase. The interaction parameters were chosen as $U_2=2t$ and $U_0=4t$. The quasiparticle weight (green line) monotonically decreases from the ground-state metallic phase to the Mott-insulating phase at $k_{B}T/t\approx 0.58t$. Notice that the finite-temperature metal-insulator transition is first order, showing a discontinuity of $Z$ at the transition. At the transition, both $Q_{\theta }$ and $Q_f$ are finite and continuous. (b) Superconducting order parameter $\mathrm{\Delta }$ as a function of temperature $k_{B}T/t$ in the Z-SC phase close to the Z-SC–SC boundary. The interaction parameters were chosen as $U_2=1.5t$ and $U_0=0.5t$. While $\mathrm{\Delta }$ continuously decreases (blue line) from a finite value to zero, showing a second-order thermal phase transition, both $Q_f$ and $Q_{\theta }$ values remain almost constant as a function of temperature.