Chiral twist on the high-$T_c$ phase diagram in moiré heterostructures

We show that the large orbital degeneracy inherent in moiré heterostructures naturally gives rise to a “high-$T_c$” like phase diagram with a chiral twist—wherein an exotic quantum anomalous Hall insulator phase is flanked by chiral $d+id$ superconducting domes. Specifically, we analyze repulsively interacting fermions on hexagonal (triangular or honeycomb) lattices near Van Hove filling, with an $\text{SU}\left(N_f\right)$ flavor degeneracy. This model is inspired by recent experiments on graphene moiré heterostructures. At this point, a nested Fermi surface and divergent density of states give rise to strong (${ln}^2$) instabilities to correlated phases, the competition between which can be controllably addressed through a combination of weak coupling parquet renormalization group and Landau-Ginzburg analysis. For $N_f=2$ (i.e., spin degeneracy only), it is known that chiral $d+id$ superconductivity is the unambiguously leading weak coupling instability. Here we show that $N_f\ge 4$ leads to a richer (but still unambiguous and fully controllable) behavior, wherein at weak coupling the leading instability is to a fully gapped and chiral Chern insulator, characterized by a spontaneous breaking of time-reversal symmetry and a quantized Hall response. Upon doping this phase gives way to a chiral $d+id$ superconductor. We further consider deforming this minimal model by introducing an orbital splitting of the Van Hove singularities, and discuss the resulting RG flow and phase diagram. Our analysis thus bridges the minimal model and the practical moiré band structures, thereby providing a transparent picture of how the correlated phases arise under various circumstances. Meanwhile, a similar analysis on the square lattice predicts a phase diagram where (for $N_f>2$) a nodal staggered flux phase with “loop current” order gives way upon doping to a nodal $d$-wave superconductor.

Figure 1

Phase diagram for repulsively interacting fermions on hexagonal lattices near Van Hove filling with $\text{SU}\left(N_f\right)$ flavor symmetry, $N_f\ge 4$. A quantum anomalous Hall insulator dominates near Van Hove filling $\mu =0$. The state arises from a chiral $3Q$ loop current order with imaginary ordering at all nesting momenta. Upon doping this gives way to a chiral $d+id$ superconductor. The phase of the order parameter winds by $\pm 4\pi$ around the Fermi surface, where $\theta =\pm 2\pi /3$ is defined.

Figure 2

Setup of patch model. (a) Brillouin zone of hexagonal lattices (black solid) with inscribed Fermi surface (green dashed) at Van Hove filling. The patches are set on the Fermi surface corners ${\mathbf{M}}_{\alpha }$'s. Blanked and shaded regions represent opposite sides of Fermi level. (b) Reduced Brillouin zone in the loop current phase. (c) Four independent interactions in the low-energy theory. Solid and dashed lines indicate fermions at different patches. (d) Test vertices in (left) superconducting and (right) density wave channels, with (e) the corresponding susceptibilities.

Figure 3

Parquet RG results for two orbital hexagonal lattices with $N_f=4$ and $N_p=3$. Notice that only the regime below $d_1^{\text{max}}=1/2$ is accessible. (a) Critical interactions $G_i$'s. (b) Susceptibility exponents ${\alpha }_I$'s. The most negative exponent indicates the leading instability. While the $d$-wave superconductivity is leading in the low nesting regime, loop current state takes over at high nesting.

Figure 4

Evolution of susceptibilities under RG. The bare interactions $g_i=g_0$ with $g_0=0.1$ are chosen, and $d_1$ is set as constant throughout the RG flow. The leading susceptibility diverges at a scale $y_c\sim g_0^{-1}$, indicating the development of an instability. (a) At $d_1=0.3$, the leading instability occurs in the $d$-wave superconducting channel since only ${\chi }_{d\text{SC}}$ diverges. (b) At $d_1=0.5$, both ${\chi }_{\text{iCDW}}$ and ${\chi }_{d\text{SC}}$ are divergent. The loop current state is dominant since ${\chi }_{\text{iCDW}}$ grows more rapidly. Notice that at intermediate scales, the real FDW state has the largest susceptibility, but it does not represent the leading weak coupling instability as the problem flows to strong coupling.

Figure 5

Real-space configurations of quantum anomalous Hall insulator on the (a) triangular and (b) honeycomb lattices, showing pattern of fluxes through the real space quadrupled unit cell. The ratio of $-3\varphi$ and $\varphi$ fluxes is $1:3$ in each quadrupled unit cell, leading to a zero net flux.

Figure 6

Van Hove fermiology under orbital splitting. The originally degenerate saddle points at each ${\mathbf{M}}_{\alpha }$ separate in opposite directions, with the two colors indicating different orbitals. Blank and shaded regions represent opposite sides of Fermi level. (a) In the immediate vicinity of Van Hove singularities, the $X$-like saddle points serve as more appropriate descriptions. (b) The $K$-like saddle points may apply if the regime of interest is finitely away from the Van Hove singularities. (c) Inter- and intrapatch nesting momenta.

Figure 7

Two fixed trajectories arise under orbital splitting, guided by [(a) and (b)] inter- and [(c) and (d)] intraorbital flavor fluctuations, respectively. We set $d_1^{+}=d_1^{-}=d_1$ in the fixed trajectory analysis [(a) and (c)] for simplicity, and run the RG flows [(b) and (d)] with maximal nesting $d_1^{\pm }=1/2$. The irrelevant $g_{44}$ and ${\mathrm{PDW}}^{\prime }$ are neglected. [(a) and (b)] Starting with $g_{14}=g_{22}=g_{24}=g_{32}=g_{42}=0.1$, the interorbital flavor fluctuations guide the RG flow to the first fixed trajectory. $\mathrm{C}/{\mathrm{SDW}}_{+}^{-}$ grow first at intermediate scales. Approaching the critical scale $y_c$, the $d/p$-wave superconductivities diverge as the only leading instabilities. ${\mathrm{CDW}}_{-}^{+}$ remains next leading along the fixed trajectory. [(c) and (d)] Setting $g_{24}=0.2$ instead, the RG flow is turned to the second fixed trajectory by intraorbital flavor fluctuations. ${\mathrm{SDW}}_{\pm }^{+}$ grow and become the only leading instabilities.

Figure 8

Susceptibility exponents in all channels for $N_f=4$ on the square lattice $N_p=2$.