Twisted Trilayer Graphene,
Quasiperiodic Superconductor
arXiv:2512.22340
Collaborators
Builds on our previous work on quasiperiodic twisted bilayer graphene:
Zhang, Wilson, Foster — PRB 111, 024207 (2025)
Twisted trilayer graphene is quasiperiodic
Three graphene layers with two independent twist angles \(\theta_{12}\) and \(\theta_{23}\) produce interfering moiré patterns.
Generically quasiperiodic — no translational symmetry on the moiré scale.
Yet experiments observe superconductivity in TTG:
- Uri et al., Nature 620, 762 (2023)
- Xia et al., arXiv:2509.03583 (2025)
- Hoke et al., arXiv:2509.07977 (2025)
when the wavefunctions are this inhomogeneous?
(a) Helical TTG. (b) Multifractal \(|\psi|^2\).
Previous work & the key insight
Quasiperiodic TBG
In quasiperiodic twisted bilayer graphene, we showed that critical filaments of multifractal wavefunctions thread through parameter space and enhance superconducting pairing.
Zhang, Wilson, 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, Wilson, Chou, Pixley — npj QM 5, 71 (2020)
The connection
In the chiral limit, TTG’s Dirac fermions couple via non-Abelian gauge potentials \(\Rightarrow\) the system simulates the surface of a 3D class-AIII topological superconductor.
\(\Delta\) and DOS in quasiperiodic TBG. (a) Periodic: SC at magic angle only. (b) Quasiperiodic: SC enhanced broadly.
The effective topology prevents Anderson localization → spectrum-wide quantum criticality
Multifractality enhances superconductivity
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 fine-tuned transition into a spectrum-wide multifractal phase: all normal-state wavefunctions are critical, not just those at isolated energies.
Pairing channel: we study intervalley \(s\)-wave as the simplest conventional scenario. The critical normal state is a single-particle effect, independent of pairing symmetry. Exotic nodal pairing remains an open direction.
Robust superconductivity across twist angles
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
Prediction: Pressure enhances SC
(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
Applying pressure to TTG should strengthen superconductivity and make it observable across a wider range of twist angles.
Accessible pressures: Carr et al. PRB 98 (2018); Yankowitz et al. Science 363 (2019)
Conductivity: fingerprint of criticality
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.
Seemingly counterintuitive: pressure reduces normal-state conductivity (more critical) while enhancing SC. But critical wavefunctions are known to boost Cooper pairing — what’s unique here is that the topology also gives you rigid stiffness.
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.
Universal across models
(a) Real-space \(\tau_2\) in the collinear model.
(b) Momentum-space \(\tau_2^k\) in the non-collinear model.
White dashed: commensurate lines.
The multifractal exponent \(\tau_2\) (from the IPR \(P_2 \sim L^{-\tau_2}\)) maps out the critical phase:
- Both models agree: broad triangular region of \(\tau_2 \lesssim 1.8\) (critical) away from commensurate lines where \(\tau_2 \approx 2\) (extended)
- The critical region coincides with where conductivity locks to the IQHT value
- The surface-state connection holds in both models — not an artifact of the collinear approximation
The experimental test: if you measure \(\sigma_{xx} \sim 0.6\,e^2/h\) across a range of energies, you are seeing the IQHT critical metal.
Summary & outlook
Future directions: Nodal / exotic pairing · TMD multilayers · Experimental verification via transport
Zhang, Zhu, Wilson, Foster — arXiv:2512.22340
Built on: Zhang, Wilson, Foster PRB 111, 024207 (2025)
Thank you!