Pair-density-wave superconductivity and Anderson's theorem in bilayer nickelates

Abstract

The recent experimental observations of high temperature superconductivity in bilayer nickelate have attracted lots of attentions. Previous studies have assumed a mirror symmetry $\mathcal M$ between the two layers and focused on uniform and clean superconducting states. Here, we show that breaking this mirror symmetry via an applied displacement field can stabilize a pair-density-wave (PDW) superconductor, which is similar to the Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) state, but at zero magnetic field. Based on a mean-field analysis of a model of $d_{x^2-y^2}$ orbital with an effective inter-layer attraction, we demonstrate that the PDW phase is robust over a wide range of displacement field, interlayer hopping strengths, and electron fillings. Finally, we analyze disorder effects on interlayer superconductivity within the first Born approximation. Based on symmetry considerations, we show that pairing is weaken by disorders which break the mirror symmetry, even with unbroken time reversal symmetry. Our results establish bilayer nickelate as a tunable platform for realizing finite-momentum pairing and for exploring generalized disorder effects.

Figures (7)

• Figure 1: (a) Schematic illustration of a bilayer nickelate. Each layer forms a square lattice hosting the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals. In the effective one-orbital description of in $d_{x^2-y^2}$ orbital, we only keep inplane hopping $t$, and effective interlayer hopping $t_\perp$, generated as a higher-order process in the underlying two-orbital model (dashed arrows). An external displacement field $D$ breaks mirror symmetry between the two layers. (b) The $d_{z^2}$ orbital is nearly half-filled, while the $d_{x^2-y^2}$ orbital is close to quarter filling. The energy levels are shown schematically; in reality, the $d_{x^2-y^2}$ orbital forms a dispersive band with average filling $n=1-x$, where $x$ denotes hole doping. (c) Fermi-surface mismatch between the two layers leads to Cooper pairing with finite momentum $\delta k_F = k_{F,t} - k_{F,b}$. For finite $t_\perp$, the two layers are hybridized and $Q$ is generally defined in the band basis.
• Figure 2: (a) Band dispersion of the effective one-orbital model in Eq. \ref{['eq:one-orbital']}. We set $t=1$ and choose $t_\perp=0.2$ and $t_\perp=0.5$. The chemical potential is set to $\mu=-1.487$ and $\mu=-1.833$, respectively, to fix the average electron density of the $d_1$ orbital to $n_1=1-x$ with $x=0.5$. (b,c) Fermi surfaces for $t_\perp=0.2$ (red) and $t_\perp=0.5$ (blue). The interlayer hopping $t_\perp$ splits the two bands and induces a Lifshitz transition. For small $t_\perp$, two electron pockets appear near $\Gamma$, while increasing $t_\perp$ drives one of the electron pockets into a hole pocket at $M$. (d) For comparison, we show the Fermi surface of the two-orbital model in Eq. \ref{['eq:two-orbital']}, hosting the $\alpha$, $\beta$, and $\gamma$ pockets.
• Figure 3: (a) Zero-temperature mean-field phase diagram in the $(D, t_\perp)$ plane. We fix $J_\perp = 4$ and $x = 1/2$. As $D$ increases, the system undergoes consecutive transitions from a uniform superconducting (SC) phase to a pair-density-wave (PDW) phase, and finally to a normal state. (b, c) Mean-field solutions that minimize the free energy. For small $t_\perp$, the optimal pairing momentum $Q$ is close to $\delta k_F$, the Fermi-momentum difference along the $x$-direction as illustrated in Fig. \ref{['fig:2']}(b), while it deviates from this value as $t_\perp$ increases. The transition between the uniform SC and PDW phases is of first order.
• Figure 4: Filling dependence of the zero-temperature mean-field phase diagram, where we fix $t_\perp = 0.1$ and $J_\perp = 4$. The PDW phase appears over a wide range of filling $x$.
• Figure S1: (a) Zero-temperature mean-field phase diagram without a form factor, $t_\perp(\bm{k}) = t_\perp$. We fix $J_\perp = 4$ and $x = 1/2$. (b,c) Fermi surfaces without (b) and with (c) the form factor at $D=0$. In the absence of the form factor, the nesting vector is relatively well defined, which stabilizes the PDW phase, compared to Fig. \ref{['fig:3']}(a).