Designing Three-Dimensional Flat Bands in Nodal-Line Semimetals

Electrons with large kinetic energy have a superconducting instability for infinitesimal attractive interactions. Quenching the kinetic energy and creating a flat band renders an infinitesimal repulsive interaction the relevant perturbation. Thus, flat-band systems are an ideal platform to study the competition of superconductivity and magnetism and their possible coexistence. Recent advances in the field of twisted bilayer graphene highlight this in the context of two-dimensional materials. Two dimensions, however, put severe restrictions on the stability of the low-temperature phases due to enhanced fluctuations. Only three-dimensional flat bands can solve the conundrum of combining the exotic flat-band phases with stable order existing at high temperatures. Here, we present a way to generate such flat bands through strain engineering in topological nodal-line semimetals. We present analytical and numerical evidence for this scenario and study the competition of the arising superconducting and magnetic orders as a function of externally controlled parameters. We show that the order parameter is rigid because the three-dimensional quantum geometry of the Bloch wave functions leads to a large superfluid stiffness in all three directions. Using density-functional theory and numerical tight-binding calculations, we further apply our theory to strained rhombohedral graphite and CaAgP materials.

Popular Summary

The recent discovery of unconventional superconductivity in a stacked, twisted pair of graphene sheets is a famous consequence of what is known as a flat energy band, in which the kinetic energy of the electrons becomes negligible, and their mutual interactions dominate. Materials with flat energy bands give rise to enhanced correlation effects and exotic phases of matter. So far, the experimental study of these intriguing effects has focused on 2D materials because of the lack of realistic proposals for their 3D counterparts. In this work, we present a viable approach to fill this gap to lift the study of flat-band physics into the third dimension.

We show theoretically how to use strain engineering to generate quasiflat 3D energy bands in nodal-line semimetals, which are materials whose valence and conduction bands cross to form closed loops. We unravel the underlying mechanism and investigate the competition of the arising superconducting and magnetic orders. The required strain profile can be realized, for instance, by bending the sample, which allows for in situ tuning of the emerging correlated phases and the transition temperatures. Moreover, we show that these systems support a nontrivial 3D quantum geometry giving rise to large superfluid stiffness and supercurrents along all directions. Finally, we identify rhombohedral graphite and CaAgP as promising material candidates to realize our proposal.

Our setup not only represents a 3D analog of the celebrated twisted bilayer graphene but also opens the door to tunable correlated phases in 3D materials. Our findings are also relevant for metamaterials and cold atomic gases, where the ingredients required for our proposal can be artificially engineered to exacting precision.

Figure 1

(a) Schematic representation of a nodal-line semimetal band structure. Here, $\mathbf{Q}$ is a vector pointing to a momentum on the nodal line such that $|\mathbf{Q}|=Q$ is the radius of a circular nodal loop. A Dirac cone is formed with respect to the perpendicular momentum components $q$ and $k_z$. (b) Example of a strain profile leading to the formation of a 3D flat band. The strain can be created, e.g., by bending the sample or growing it on a cylindrical surface. Here, $R$ is the radius of the cylinder in the middle of the sample. (c) Bulk-boundary correspondence of the zeroth PLL of the NLSM. Because of the strain, the radius of the nodal circle varies as a function of $z$, such that the radius of the momentum-space area of the drumhead surface states on the top surface $Q_{\mathrm{top}}$ is different from the bottom surface $Q_{\mathrm{bot}}$. Zeroth PLL bulk states appear in the momentum space region between $Q_{\mathrm{top}}$ and $Q_{\mathrm{bot}}$. (d) Spectrum of the model Hamiltonian $H_{\mathrm{tb}}$ [Eq. (1)] in the presence of strain [Eq. (2)] with parameters $t_1=0.25t$, $t_2=0.8t$, $L=1000$, and $R=8000$. For comparison, we also show the energies of the effective Hamiltonian $H_{\mathrm{eff}}$ (orange bands) constructed from the analytical low-energy solutions (see Supplemental Material [38]). (e) Numerical (blue) and analytical (orange) wave functions of the zeroth PLL at different momenta $k_x=k_i$ ($i=1$, 2, 3, 4) indicated in panel (d). The drumhead surface states appear at both surfaces for $|k_x|<Q_{\mathrm{top}},Q_{\mathrm{bot}}$. At $|k_x|=Q_{\mathrm{top}}$, the drumhead surface states from the top surface are deformed into the zeroth PLL bulk states and move towards the bottom surface with increasing $Q_{\mathrm{top}}<|k_x|<Q_{\mathrm{bot}}$. At $|k_x|=Q_{\mathrm{bot}}$, the PLL bulk state hybridizes with the drumhead surface state at the bottom surface. (f) Degeneracy of the flat band as a function of the height $L$ of the sample. Numerically, the degeneracy is calculated by integrating the density of states over an energy window ($|E|<{10}^{-3}t$). The model parameters $t_{1,2}$ are the same as in panel (d). Bold lines are the numerical results, and dashed lines correspond to the analytical formulas. In the presence of strain, the degeneracy grows linearly with $L$, demonstrating the existence of a 3D flat band with degeneracy $N_{\mathrm{flat}}$ proportional to the volume of the sample (green dashed line). In the absence of strain, the degeneracy of the zero-energy drumhead surface states saturates with increasing $L$, demonstrating that it is proportional to the area of the surface of the sample, $N_0$.

Figure 2

Phase diagram of the model Hamiltonian [Eq. (1)] in the presence of strain [Eq. (2)] and interactions [Eqs. (3) and (5)]. (a) Critical temperatures for magnetism [Eq. (4)] and superconductivity [Eq. (6)] as a function of the filling factor $C$ in the typical situation where the effective interaction strength for superconductivity is smaller than for magnetism $G_0=0.75V_0$. At half-filling $C=1$, the system realizes magnetic order, but doping away from half-filling leads to superconductivity, resembling the phase diagrams experimentally observed in cuprates and twisted bilayer graphene. (b) Phase diagram as a function of $C$ and $G_0/V_0$, assuming that the system realizes the phase with the larger critical temperature. The dashed horizontal line represents a path through the phase diagram corresponding to panel (a).

Figure 3

Three-dimensional flat bands in nodal-line semimetals CaAgP (first row) and rhombohedral graphite (second row). (a) Top and side views of the CaAgP crystal structure. Next to these, we indicate the shape and position of its nodal circle (red), which is centered at the $\mathrm{\Gamma }$ point of the hexagonal Brillouin zone. The beige hexagon represents the (001) surface Brillouin zone with the surface projection of the nodal loop. (b) Bulk bands of the two-band tight-binding model without strain along paths with fixed $k_z$. The bands only cross in the plane $k_z=0$. (c) Tight-binding spectrum of a strained (001) slab for $L=200$, $R=800$ with nearly flat bands. The inset shows the spatial profile of the flat-band wave function at the momentum indicated by the red dot. It is a superposition of a bottom-surface state and a (Gaussian) Landau-level state. (d) Top and side views of ABC-stacked (rhombohedral) graphite next to a sketch of its nodal lines. In contrast to CaAgP, there are several nodal lines that spiral around axes going through the K and K’ points of the hexagonal Brillouin zone. Their projections into the surface Brillouin zone form approximate nodal circles. (e) Bulk bands of the two-band tight-binding model without strain along paths with fixed $k_z$. The nodal spiral crosses the considered path for two different values of $k_z$. (f) Tight-binding spectrum of a strained (001) slab for $L=100$, $R=400$ with flat bands. The inset shows the spatial profile of the flat-band state at the red dot, which is again composed of a Landau-level and a bottom-surface state.