Excitonic instability and unconventional pairing in the nodal-line materials ZrSiS and ZrSiSe

We use a functional renormalization group (fRG) approach to investigate potential interaction-induced instabilities in a two-dimensional model for the Dirac nodal-line materials ZrSiS and ZrSiSe employing model parameters derived from ab initio calculations. Our results characterize the excitonic instability recently found in random-phase approximation for ZrSiS as an on-site spin-charge-degenerate exciton. Beyond this, we show that the fRG analysis produces an energy scale for the onset of the instability, in good agreement with the experimentally observed mass enhancement. Additionally, by exploring the parameter space of the model, we find that reducing the band splitting increases the instability scale and gives the chance to drive the system into an unconventional superconducting pairing state. The model parameters for the case of the structurally similar material ZrSiSe suggest the $d$-wave superconducting state as the leading instability with a very small critical scale.

Figure 1

Left panel: Bilayer square lattice with layers $A$ (bottom) and $B$ (top). The lattice constant is set to $a=1$. Right panel: Fermi-surface patching scheme within the BZ with the tight-binding parameters given in the text. Note that we have added a finite chemical potential of $\mu =0.1$ eV to exhibit the presence of two energy bands.

Figure 2

Fermi surfaces and effective interaction vertex near the critical scale for the appearing instabilities in eV. The axes of the vertices $V_{{\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_4}$, with the indicated layer combination, are labeled with the patch number (cf. Fig. 1). Wave vectors $k_1$ are depicted vertically and $k_2$ are depicted horizontally. We fix $k_3$ to be on patch 1. The insets show the Fermi surfaces for the chosen $\mathrm{\Delta }/\mathrm{eV}\in \{0.835,0.167,0.04\}$ (from left to right). Left panel: Excitonic instability for $\mathrm{\Delta }=0.835\mathrm{eV}$. Middle panel: $d$-wave pairing instability for $\mathrm{\Delta }=0.167\mathrm{eV}$. For the choice ${\alpha }_i=B$ the structure is weaker due to the smaller $U_{BB}$. Right panel: Antiferromagnetic instability for $\mathrm{\Delta }=0.04\mathrm{eV}$. Again, for the choice ${\alpha }_i=B$ the structure is weaker due to the smaller $U_{BB}$. The other layer combinations which remain zero in the flow are not shown.

Figure 3

Tentative phase diagram for the effective model of ZrSiS as a function of the on-site energy shift. The critical scales ${\mathrm{\Lambda }}_c$ can be interpreted as an estimate for the transition temperatures. Left inset: Evolution of ${\mathrm{\Lambda }}_c$ of the excitonic instability with small chemical potentials. The green line shows the full fRG results with all correlation channels active. The gray line corresponds to the crossed particle-hole diagrams, only. Right inset: ${\mathrm{\Lambda }}_c$ of the excitonic instability as a function of nearest-neighbor interaction.

Figure 4

Left panel: Tentative phase diagram of the effective model for ZrSiSe as a function of chemical potential $\mu$ with fixed on-site energy shift $\mathrm{\Delta }=0.41\mathrm{eV}$. The critical scales ${\mathrm{\Lambda }}_c$ serve as an estimate for the transition temperatures. For $\left|\mu \right|\lesssim 0.08$ eV no instability can be detected. Right panel: ${\mathrm{\Lambda }}_c$ vs $\mathrm{\Delta }$ with $\mu$ adjusted such that the Fermi level lies at the band crossing. The label dSC denotes a $d_{x^2-y^2}$-wave pairing instability, and AFM an antiferromagnetic instability.