Quantum normal-to-inhomogeneous superconductor phase transition in nearly two-dimensional metals

In multi-band systems, electrons from different orbitals coexist at the Fermi surface. An attractive interaction among these quasi-particles gives rise to inter-band or hybrid pairs which eventually condense in a superconducting state. These quasi-particles have a natural mismatch of their Fermi wave-vectors, $\delta k_F$, which depends on the strength of the hybridization between their orbitals. The existence of this natural scale suggests the possibility of inhomogeneous superconducting ground states in these systems, even in the absence of an applied magnetic field. Furthermore, since hybridization $V$ depends on pressure, this provides an external parameter to control the wave-vectors mismatch at the Fermi surface. In this work, we study the phase diagram of a two-dimensional, two-band metal with inter-band pairing. We show that as the mismatch between the Fermi wave-vectors of the two hybrid bands is reduced, the system presents a normal-to-inhomogeneous superconductor quantum phase transition at a critical value of the hybridization $V_c=\Delta_0$. The superconducting ground state for $V<V_c$ is characterized by a wave-vector with magnitude $|\mathbf{q}_c|=q_c=2 \Delta_0/\bar{v}_f$. Here $\Delta_0$ is the superconducting gap in the homogeneous state and $\bar{v}_f$ the average Fermi velocity. We discuss the nature of the quantum critical point (QCP) at $V_c$ and obtain the associated quantum critical exponents.