Unconventional Superconductivity and Density Waves in Twisted Bilayer Graphene

We study electronic ordering instabilities of twisted bilayer graphene around the filling of $n=2$ electrons per supercell, where correlated insulator state and superconductivity have been recently observed. Motivated by the Fermi surface nesting and the proximity to Van Hove singularity, we introduce a hot-spot model to study the effect of various electron interactions systematically. Using the renormalization group method, we find that $d$ or $p$-wave superconductivity and charge or spin density wave emerge as the two types of leading instabilities driven by Coulomb repulsion. The density-wave state has a gapped energy spectrum around $n=2$ and yields a single doubly degenerate pocket upon doping to $n>2$. The intertwinement of density wave and superconductivity and the quasiparticle spectrum in the density-wave state are consistent with experimental observations.

Popular Summary

Graphene, a single layer of carbon atoms, is a good conductor. However, a drastic change in electrical properties occurs in a stack of two graphene sheets when one layer is rotated relative to the other. Surprisingly, at a certain “magic” twist angle, adding or removing a few electrons per $\mathrm{10\hspace{0.17em}000}$ carbon atoms is found to change bilayer graphene from a conductor to an insulator or superconductor. Here, we theoretically study the nature and origin of these insulating and superconducting states.

We introduce a microscopic model to systematically study ordered states driven by electron interaction, whose effect is enhanced by the intrinsic energy-band spectrum of twisted bilayer graphene. Our theory predicts that the superconducting state has an unconventional pairing symmetry and identifies the insulating state as either a charge-density wave or a spin-density wave.

Our analysis of superconductivity and density waves provides an explanation consistent with the recent experimental observations in twisted bilayer graphene.

Figure 1

(a),(b) Two Fermi surfaces at the Van Hove energy from different valleys shown in red and blue, reproduced from the band structure calculation for twisted bilayer graphene with $\theta =2°$ [28]. Shaded areas are filled and VHS appear at the points, where two Fermi surfaces encircling $K$ and $K^{\prime }$ touch. Each Fermi surface has $C_3$ symmetry about $\mathrm{\Gamma }$, $K$, and $K^{\prime }$. The Fermi surfaces in (a) and (b) are related by $C_2$ rotation with respect to an in-plane axis along $\mathrm{\Gamma }\text{−}K$. (c),(d) Fermi surfaces slightly away from the Van Hove energy. The carrier density is lower than that at the Van Hove energy in (c) and higher in (d). DOS is larger near the VHS points (hot spots), where electron interaction predominates. We assign patches (circles with dashed lines) centered at hot spots. (e) Three inequivalent wave vectors ($Q^{+}$, $Q^{-}$, and $Q^{\prime }$), along with symmetry-related ones (not shown), connect various pairs of hot spots.

Figure 2

(a) Diagrammatic representation of scattering processes. The four diagrams describe the change of either saddle point or valley index, where solid and dashed lines correspond to electron propagators with different indices. (b) Nine momentum-conserving scattering processes out of 16 distinct scattering processes $g_{ij}$. Hexagons are the Brillouin zone boundaries and the dots are located at saddle points with two colors corresponding to the two valleys. There are three types in the interactions (from left to right): intravalley, intervalley density, and intervalley exchange interactions.

Figure 3

One-loop corrections to the coupling constants. The solid lines represent the fermion propagators and the wavy lines correspond to interactions. The leftmost diagram involves a particle-particle loop, and the other three diagrams have particle-particle loops.

Figure 4

Phase diagrams with the density-density interactions. (a),(b) Phase diagrams obtained by the RG analysis in the presence of the density-density interactions ($g_{14}=g_{24}=g_{44}=g_{22}=g_{32}=g_{42}$). Two parameters for nesting are varied in each phase diagram: (a) $d_{1-}$ and $d_{2-}$ ($d_{3-}=0$) and (b) $d_{1-}$ and $d_{3-}$ ($d_{2-}=0$). Colors represent critical RG scales for instabilities: warm (cool) colors correspond to high (low) energy and temperature. The solid lines represent phase boundaries, obtained in the range of $g_0{y}_c\le 15$. There are ${\mathrm{CDW}}^{\prime }$ or ${\mathrm{SDW}}^{\prime }$ and normal regions in the vicinity of $d_{1-}=0$ (not shown), but critical temperatures are extremely low for those instabilities ($g_0{y}_c>15$). (c),(d) Running coupling constants and (e),(f) susceptibilities for possible orders. We show the results for two cases with different strength of nesting: (c),(e) $d_{1-}=0.4$ and (d),(f) $d_{1-}=d_{3-}=0.25$. The vertical dashed lines indicate the positions where instabilities occur.

Figure 5

Effect of the exchange interactions. (a)–(c) Running coupling constants and (d)–(f) susceptibilities for possible orderings. We choose the strengths of density-density interactions $g_{14}=g_{24}=g_{44}=1$ (intravalley) and $g_{22}=g_{32}=g_{42}=1$ (intervalley). Fermi surface nesting is weak in (a) and (d) with $d_{1-}=0.1$ and strong in (b), (c), (e), and (f) with $d_{1-}=0.4$. Finite exchange interactions lift the degeneracies of susceptibilities, cf. Figs. 4 and 4. We choose the exchange interactions to be initially repulsive ($g_{11}=g_{31}=g_{41}=0.1$) in (a), (b), (d), and (e) and attractive ($g_{11}=g_{31}=g_{41}=-0.1$) in (c) and (f). The vertical dashed lines indicate the positions where instabilities occur.

Figure 6

(a) Reduced Brillouin zone in a triple-$Q$ density-wave state. We assume the three ordering vectors ${\mathit{Q}}_j$ ($j=1$, 2, 3), which are parallel to the $\mathrm{\Gamma }-M$ lines and satisfy $|{\mathit{Q}}_j|=Q^{-}=G/4$. (b) Energy spectrum in the ${\mathrm{CDW}}^{-}$ or collinear ${\mathrm{SDW}}^{-}$ state with order parameter $\mathrm{\Delta }=0.16D$. $D$ is the original conduction bandwidth. There are 32 bands in the reduced Brillouin zone and the seventeenth band from the bottom is colored in red. Filling of 16 bands corresponds to the filling $n=2$.

Figure 7

Simplified Fermi surface to show Fermi surface nesting with different wave vectors. (a),(b) Fermi surfaces for different valley degrees of freedom. Labels 1, 2, and 3 are the patch indices, assigned at the hot spots.

Figure 8

Diagrammatic representation of interactions and summation for the susceptibility. (a) Diagrams for interactions in the particle-particle channel $V^{\mathrm{pp}}$ and in the particle-hole channel $V^{\mathrm{ph}}$. The two-particle interactions are drawn by four-leg ladders. (b) Interactions for various orderings to the lowest order. (c) RPA-like resummation for susceptibilities.

Figure 9

SC pairing symmetries and order parameters at hot spots. Note that the order parameters are not normalized for each pairing.

Figure 10

RG flows on the $(g_{4+},g_{3+})$ plane without Fermi surface nesting other than the BCS channel. Black solid lines are the separatrix lines of the flows. One finds the same RG flow for $g_{4-}$ and $g_{3-}$.

Figure 11

Energy contour plots of the energy bands ${\epsilon }_{\mathit{k}}^{\tau }$, Eq. (e3). Panels (a) and (b) correspond to the energy dispersions of the two different valleys. The parameters are chosen as $t_0=5.4$, $t_1=1$, $t_2=-1$, $t_2^{\prime }=0.3$, and $t_3=0.6$. The Lifshitz transition and accompanying VHS are found at $E/t_1\approx 6.1$. The bandwidth is $D=7.2$.

Figure 12

Seven momentum-nonconserving scattering processes.