Out-of-plane bond-order phase, superconductivity, and their competition in the $t$-$J_{\parallel }$-$J_{\perp }$ model: Possible implications for bilayer nickelates

Almost four decades of intense research have been invested to study the physics of high-T${}_c$ cuprate superconductors. The recent discovery of high-T${}_c$ superconductivity in pressurized bilayer nickelates and its potential similarities with cuprate superconductors may open a new window to understand this long-standing problem. We have studied the proposed bilayer $t$-$J_{\parallel }$-$J_{\perp }$ model [where $J_{\parallel }$ ($J_{\perp }$) is the in-plane (out-of-plane) magnetic exchange] in a large-$N$ approach on the basis of the path integral representation for Hubbard operators, which allows to obtain results at mean-field and beyond mean-field level. We find that $J_{\perp }$ is a candidate for triggering high superconducting $T_c$ values at quarter filling (hole doping $\delta =0.5$) of the $d_{x^2-y^2}$ orbitals. Beyond mean-field level, we find a new phase, an out-of-plane bond-order phase (z-BOP), triggered also by $J_{\perp }$. z-BOP develops below a critical temperature which decreases with increasing doping and vanishes at a quantum critical point below quarter filling. The occurrence of this phase and its competition with superconductivity leads to a superconducting dome-shaped behavior as a function of doping and as a function of $J_{\perp }$. Qualitative comparisons with the physics of cuprates and the recent literature on the new pressurized nickelates are given along the paper.