Correlation effects and quantum oscillations in topological nodal-loop semimetals

We study the unique physical properties of topological nodal-loop semimetals protected by the coexistence of time-reversal and inversion symmetries with negligible spin-orbit coupling. We argue that strong correlation effects occur at the surface of such systems for relatively small Hubbard interaction $U$, due to the narrow bandwidth of the “drumhead” surface states. In the Hartree-Fock approximation, at small $U$ we obtain a surface ferromagnetic phase through a continuous quantum phase transition characterized by the surface-mode divergence of the spin susceptibility, while the bulk states remain very robust against local interactions and remain nonordered. At slightly increased interaction strength, the system quickly changes from a surface ferromagnetic phase to a surface charge-ordered phase through a first-order transition. When Rashba-type spin-orbit coupling is applied to the surface states, a canted ferromagnetic phase occurs at the surface for intermediate values of $U$. The quantum critical behavior of the surface ferromagnetic transition is nontrivial in the sense that the surface spin order parameter couples to Fermi-surface excitations from both surface and bulk states. This leads to unconventional Landau damping and consequently a naïve dynamical critical exponent $z\approx 1$ when the Fermi level is close to the bulk nodal energy. We also show that, already without interactions, quantum oscillations arise due to bulk states, despite the absence of a Fermi surface when the chemical potential is tuned to the energy of the nodal loop. The bulk magnetic susceptibility diverges logarithmically whenever the nodal loop exactly overlaps with a quantized magnetic orbit in the bulk Brillouin zone. These correlation and transport phenomena are unique signatures of nodal-loop states.

Figure 1

Schematic illustration of the noninteracting tight-binding model for nodal-loop semimetals on a tetragonal lattice. (a) Lattice structure and hopping terms; the thick black arrow indicate surface electric field which generates Rashba SOC denoted by ${\lambda }_R$. (b)–(d) Nodal loops projected onto the (001) surface BZ, with the shaded region indicating the drumhead surface states, (b) for $t_2<t_1$, (c) $t_2=t_1$, and (d) $t_2>t_1$.

Figure 2

Surface band structures of the noninteracting tight-binding model without surface SOC, (a) and (b), and with surface SOC, (c) and (d). (a) $t_2=0.75t_1$, and (b) $t_2=1.25t_1$; (c) $t_2=0.75t_1$, ${\lambda }_R=0.0625t_1$, and (d) $t_2=1.25t_1$, ${\lambda }_R=0.0625t_1$. The energy bands are plotted along the high-symmetry path marked by the thick black lines in Fig. 1.

Figure 3

Phase diagram of the NLSMs with Hubbard interactions in the $t_2-U$ parameter space: (a) without surface Rashba SOC, with the inset shows the local charge density distribution in the surface CDW phase when $t_2=1.25t_1$ and $U=0.5t_1$, and (b) with surface Rashba SOC, where ${\lambda }_{\mathrm{R}}=0.0625t_0$.

Figure 4

(a) Dispersion of the spin susceptibility $({\chi }_{zz}\left(q\right))$ for a 60-layer slab of nodal-loop metal with $t_2=t_1$ and $U=0.25t_1$. (b) The $U$ dependence of the surface spin fluctuations at $\mathrm{\Gamma }$ (denoted by ${\chi }_{zz}^{\mathrm{surf}}$) for different $t_2$ values.

Figure 5

Wavevector dependence of the surface-mode in-plane spin susceptibilities for different $t_2$ values at fixed $U=0.5t_1$, (a) ${\chi }_{xx}$, and (b) ${\chi }_{yy}$. The high-symmetry points $M_1=(\pi ,0)$, $M_2=(0,\pi )$, $X=(\pi ,\pi )$, and $\mathrm{\Gamma }=(0,0)$.

Figure 6

Schematic illustration of different types of electron-hole excitations that couple to surface spins. (a) When the surface bands are partially filled. (b) When the surface bands are (nearly) completely filled. The electron-hole excitations purely from the surface (bulk) states are denoted by “$s\text{−}s$” (“$b\text{−}b$”), while the process of creating a hole in the surface states and an electron in the bulk states is denoted by “$s\text{−}b$.”

Figure 7

Numerical calculations of the surface dynamical susceptibility of slightly hole-doped nodal-loop semimetals with partially filled surface bands: (a) frequency dependence at $q_{\parallel }=0$; and (b) wave-vector dependence at $\nu =0.008$. Note the horizontal axis in (b) is $1/q_x$.

Figure 8

Numerical calculations of the surface dynamical susceptibility of charge neutral nodal-loop semimetals with nearly completely filled surface bands: (a) the frequency dependence at $q_{\parallel }=0.4$, and (b) the wave-vector dependence at $\nu =0.025$.