Twisted Trilayer Graphene,
Quasiperiodic Superconductor

Quasiperiodic TBG

In quasiperiodic twisted bilayer graphene, we showed that critical filaments of multifractal wavefunctions thread through parameter space and enhance superconducting pairing. The key: in the chiral limit, the Dirac fermions simulate the surface of a 3D class-CI topological superconductor.

Zhang, JHW, Foster — PRB 111, 024207 (2025)

Magic-angle semimetals

Band nodes + incommensurate potentials generate eigenstate criticality — flat bands and multifractal wavefunctions at magic couplings.

Fu, König, JHW, Chou, Pixley — npj QM 5, 71 (2020)

TTG: a simulator of class AIII

The additional moiré pattern in TTG changes the symmetry class to AIII via non-Abelian gauge potentials, with criticality of the integer quantum Hall plateau transition at finite energies.

\(\Delta\) and DOS in quasiperiodic TBG. (a) Periodic: SC at magic angle only. (b) Quasiperiodic: SC enhanced broadly.

At the Anderson transition

At the Anderson transition, wavefunctions are multifractal: their spiky, self-similar density profiles enhance Cooper pairing by increasing effective interaction matrix elements.

Feigel'man et al., PRL 98, 027001 (2007); Zhang & Foster, PRB 106, L180503 (2022)

In TTG: an extended critical phase

\(|\psi_E(x,y)|^2\) in quasiperiodic TTG. Self-similar spatial structure.

In TTG, the effective topology drives this from a property of a phase transition into a spectrum-wide multifractal phase: all normal-state wavefunctions are critical (for small twist angles $\theta_{12}+\theta_{23}\approx 1.5^\circ$), not just those at isolated energies.

In the chiral limit (\(\alpha=0\), \(w_\text{AB}=110\) meV):

• SC order parameter \(\Delta\) is finite across a broad range of \(\theta_{23}\), not just at commensurate (magic) angles
• DOS and \(\Delta\) do not track each other — SC is not simply flat-band physics

The quasiperiodicity does not degrade phase coherence. The topology pins the normal-state conductivity to \(\sigma_n = 3e^2/\pi h\), so the stiffness suppression you would normally get from spatial inhomogeneity never kicks in.

Trivedi, Scalettar, Randeria, PRB 54, R3756 (1996)

(c) \(\Delta\) and DOS \(\nu(0)\) vs. \(\theta_{23}\)
(e) Superfluid stiffness \(D_s/\pi\); dashed: ideal Dirac

(d,f) Chiral, \(w_\text{AB}=2w_0\): \(\Delta\) uniform.
(g,h) Broken chiral: SC enhanced.

Doubling interlayer coupling (\(w_\text{AB}\!=\!2w_0\!=\!220\) meV, achievable with hydrostatic pressure):

• \(\Delta\) becomes uniform across twist angles — no fine-tuning needed
• Stiffness follows ideal Dirac: \(D_s/\pi \approx 3\Delta/\pi\)
• With broken chiral symmetry (\(\alpha\!=\!0.7\)), SC is further enhanced at incommensurate angles

The Kubo conductivity of the normal state makes the criticality visible:

• At incommensurate angles, \(\sigma_{xx}\) locks to the IQHT critical value \(\approx 0.6\, e^2/h\) across a wide energy window
• Spectrum-wide quantum criticality: not just at isolated energies (unlike the usual quantum Hall effect)
• At doubled coupling (\(w_\text{AB}\!=\!2w_0\)), criticality is stabilized even with broken chiral symmetry

KPM convergence distinguishes the regimes: critical states plateau with increasing polynomial order \(N_\mathcal{C}\), while ballistic states grow without saturating.

Normal-state \(\sigma_{xx}\) vs energy for various \(\theta_{23}\). Red/blue dashed: WZW and IQHT critical values. (e,f) KPM convergence at incommensurate angle.

(a) Real-space \(\tau_2\), collinear. (b) Momentum-space \(\tau_2^k\), non-collinear.

We use a collinear model: moiré reciprocal vectors aligned in direction, differing only in magnitude — this lets us reach much larger system sizes. The non-collinear model keeps the full directional mismatch but is more expensive computationally.

Non-collinear: Zhu, Carr, Massatt, Luskin, Kaxiras — PRL 125, 116404 (2020)

The IPR \(P_2 \sim L^{-\tau_2}\) maps out the critical phase. The collinear model gives real-space \(\tau_2\) (small = critical); the non-collinear gives momentum-space \(\tau_2^k\) (large = critical). The trends are inverted, but they identify the same critical and ballistic regions in twist-angle space.

• Both: broad triangular region of criticality away from commensurate lines where \(\tau_2 \approx 2\) (extended)
• The critical region is exactly where \(\sigma_{xx}\) locks to the IQHT value