Parquet renormalization group analysis of weak-coupling instabilities with multiple high-order Van Hove points inside the Brillouin zone

We analyze the weak-coupling instabilities that may arise when multiple high-order Van Hove points are present inside the Brillouin zone. The model we consider is inspired by twisted bilayer graphene, although the analysis should be more generally applicable. We employ a parquet renormalization group analysis to identify the leading weak-coupling instabilities, supplemented with a Ginzburg-Landau treatment to resolve any degeneracies. Hence we identify the leading instabilities that can occur from weak repulsion with the power-law divergent density of states. Five correlated phases are uncovered along distinct stable fixed trajectories, including $s$-wave ferromagnetism, $p$-wave chiral/helical superconductivity, $d$-wave chiral superconductivity, $f$-wave valley-polarized order, and $p$-wave polar valley-polarized order. The phase diagram is stable against band deformations which preserve the high-order Van Hove singularity.

Figure 1

Tentative electronic phase diagram at the high-order Van Hove singularity in twisted bilayer graphene. The primary interactions are weakly repulsive, with the setup the same as in Fig. 6. Perturbatively (a) repulsive and (b) attractive intervalley exchange interactions are introduced as in Fig. 7. Five potential correlated phases are uncovered, including $s$-wave ferromagnetism ($s\mathrm{FM}$), $p$-wave chiral/helical superconductivity ($p\mathrm{C}$/HSC), $d$-wave chiral superconductivity ($d\mathrm{CSC}$), $f$-wave valley-polarized order ($f\mathrm{VP}$), and $p$-wave polar valley-polarized order ($p\mathrm{PVP}$).

Figure 2

High-order Van Hove singularity in twisted bilayer graphene and patch model. (a) Tight-binding honeycomb superlattice model for the nearly flat bands in twisted bilayer graphene [3]. The model contains the nearest- and the fifth-nearest-neighbor hoppings $t_{1,2}{\sum }_{{\langle ij\rangle }_{1,5},\tau =\pm }(c_{i\tau }^{†}{c}_{j\tau }+\text{H.c.})$, as well as an imaginary fifth-nearest-neighbor hopping $-i{t}_3{\sum }_{{\langle ij\rangle }_5,\tau =\pm }\tau c_{i\tau }^{†}{c}_{j\tau }+\text{H.c.}$ Here $c_{i\tau }$ denotes the fermion operator at the site $i$ in the $\tau =\pm$ valley. (b) Band structure with high-order Van Hove singularity at $t_1=1, t_2=0.15$, and $t_3=0.2720717$. Each figure illustrates the band structure from a valley. Note that the high-order Van Hove singularity occurs on a ‘critical surface’ in the three-dimensional phase spase spanned by $t_1, t_2$, and $t_3$. (c) The band structure (1) in the vicinity of a high-order saddle point $\mathbf{P}$ with $\kappa =0$. (d) The setup of patch model, where the patches are set at the high-order saddle points.

Figure 3

Feynman diagrams. (a) Six primary interactions without intervalley exchange. The solid and dashed lines describe the electrons from patches with different patch labels, while single and double lines characterize the electrons from different valleys. (b) The three interactions with intervalley exchange. (c) The test vertices in the particle-hole (first two) and particle-particle (last two) channels at momenta $\mathbf{0}$ (first and third) and ${\mathbf{Q}}^o$ (second and last). (d) Susceptibilities captured by the test vertices.

Figure 4

Particle-hole and particle-particle susceptibilities at the high-order Van Hove singularity. Here we assume maximal Fermi surface nesting at $\mathbf{q}={\mathbf{Q}}^o$ by setting $\kappa =0$. Each curve indicates a rescaled static susceptibility ${\tilde{\mathrm{\Pi }}}_{\mathbf{q}}^{\text{ph/pp}}={\mathrm{\Pi }}_{\mathbf{q}}^{\text{ph/pp}}/{\mathrm{\Pi }}^0$.

Figure 5

The susceptibilities of potential instabilities flow under RG. The maximal divergence occurs in the leading irreducible pairing channel at the critical scale $y=y_c$. Each figure illustrates the RG flow toward a stable fixed trajectory starting from weak repulsion, with (a) degenerate $s$-wave ferromagnetism/$f$-wave spin-valley-polarized order, (b) degenerate $d/p$-wave superconductivities, (c) $f$-wave valley-polarized order, or (d) $p$-wave valley-polarized order as the leading instability. The bare repulsions are set as ${\lambda }_{ij}=0.1$, except for (b) ${\lambda }_{24}=0.3$, (c) ${\lambda }_{42}=0.3$, and (d) ${\lambda }_{24}={\lambda }_{42}=0.3$.

Figure 6

Phase diagram of the potential instabilities. The color map indicates the critical scale $y_c=ln(\mathrm{\Lambda }/T_c)$, which is smaller in the phases and larger along the phase boundaries. In each two-interaction phase diagram, two of the interactions ${\tilde{\lambda }}_{ij}={\lambda }_{ij}/{\lambda }_0$ are varied, and the rest ones are set at constant repulsion ${\lambda }_{ij}={\lambda }_0=0.1$. The full set of two-interaction phase diagrams is demonstrated in Appendix pp4.

Figure 7

The breakdown of degeneracy between different instabilities by intervalley exchange. The bare values of the primary repulsions in (a) and (b) [(c) and (d)] are set the same as in Fig 5 [Fig 5]. Meanwhile, the intervalley exchange is taken as perturbative repulsion (attraction) ${\lambda }_{i1}=(-)0.01$ in (a) and (c) [(b) and (d)]. With repulsive intervalley exchange, (a) the $s$-wave ferromagnetism wins over the $f$-wave spin-valley-polarized order, while (c) the $p$-wave superconductivity beats the $d$-wave one. On the other hand, the $d$-wave superconductivity beats (b) the $f$-wave spin-valley-polarized order and (d) the $p$-wave superconductivity when the interalley exchange is attractive.

Figure 8

The phase distinction with (a) repulsive and (b) attractive intervalley exchange ${\lambda }_{i1}=\pm 0.01$. Here ${\tilde{\lambda }}_{24}={\lambda }_{24}/{\lambda }_0$ is varied, while the other primary interactions are constant ${\lambda }_{ij\ne 24}={\lambda }_0=0.1$. (a) With repulsive intervalley exchange, the $s$-wave ferromagnetism and the $p$-wave superconductivity form distinct phases separated by a peak of $y_c$. The three phases are (left) $f$-wave valley-polarized order, (center) $s$-wave ferromagnetism, and (right) $p$-wave superconductivity [Fig. 1]. (b) When the intervalley exchange is attractive, the right peak vanishes. This indicates that the $d$-wave superconductivity beats the $f$-wave spin-valley-polarized order and occupies the according phase regime [Fig. 1].

Figure 9

One-loop corrections to the interactions.

Figure 10

RG equations of the primary interactions without intervalley exchange (Part one). The red diagrams are those involve the intervalley exchange.

Figure 11

RG equations of the primary interactions without intervalley exchange (part two). The red diagrams are those involve the intervalley exchange.

Figure 12

RG equations of the interactions with intervalley exchange. The red diagrams are those involve the intervalley exchange.

Figure 13

The corrections to the test vertices under RG in the particle-hole [(a) and (b)] and particle-particle [(c) and (d)] channels with zero-momentum [(a) and (c)] and finite-momentum ${\mathbf{Q}}^o$ [(b) and (d)] pairings. The red diagrams are those involve the intervalley exchange. For the particle-hole channels [(a) and (b)], the diagrams with internal fermion loops are involved only for equal-spin pairings.

Figure 14

The full set of two-interaction phase diagrams.