Twisted Trilayer Graphene, Quasiperiodic Superconductor

Abstract

Twisted multilayer moiré materials are generically quasiperiodic on the moiré scale due to the interference of different misaligned moiré periodicities. Spatial inhomogeneities such as these can be detrimental to superconductivity; nonetheless, superconductivity has been observed in quasiperiodic twisted trilayer graphene (TTG). Here, we systematically study the superconducting properties of TTG. We reveal that an interplay between quasiperiodicity and topology drives TTG into a critical regime, enabling it to host superconductivity with rigid phase stiffness for a wide range of twist angles, rather than at a fine-tuned value. The criticality in the normal state is due to the Dirac fermions coupled by quasiperiodic tunneling simulating 3D topological superconductor surface states. This critical-metal regime is marked by multifractal wave functions across the spectrum and scale-invariant transport reminiscent of the integer quantum Hall plateau transition. We demonstrate this with large-scale wave function and Kubo conductivity calculations. These observations lead to a clear experimental implication: stronger interlayer coupling in TTG further stabilizes both the criticality and superconductivity, allowing superconductivity to be seen across a wider range of angles with experimentally accessible pressures.

Figures (5)

• Figure 1: Fractal wave functions and robust superconductivity in TTG.a, Illustration of helical twisted trilayer graphene with two independent twist angles $\theta_{12}$ and $\theta_{23}$. b, A typical quantum-critical, normal-state single-particle wave function in quasiperiodic TTG ($\theta_{12} =0.5^\circ$ and $\theta_{23}=0.925^\circ$). The self-similar structure in position space is induced by the quasiperiodicity. c, Spatial average of the superconductivity order parameter $\Delta$ and density of states near the Fermi energy $\nu(E = 0)$ in the chiral limit Tarnopolsky2019 with $w_\mathrm{AB}=w_0=110$ meV; d, Same as c, but with doubled interlayer coupling $w_\mathrm{AB}=2 w_0=220$ meV. e and f, Superfluid stiffness $D_s/\pi$ (red) in the chiral limit with $w_\mathrm{AB}=w_0$ and $w_\mathrm{AB}=2w_0$, respectively. The black dashed line shows $D_s/\pi = 3\Delta/\pi$, which is the result for an ideal Dirac superconductor without moiré potentials Kopnin2008Uchoa2009Kopnin2009Zhang2025. g and h, Superconducting order parameter for the realistic TTG model with broken chiral symmetry ($\alpha\ne 0$, see text) with $w_\mathrm{AB}=w_0$ and $w_\mathrm{AB}=2w_0$, respectively. $\Delta$ and $D_s/\pi$ are given in units of $v_F k_{\theta_{12}} \approx 124$ meV and the attractive interaction is set to be $U=4 v_F k_{\theta_{12}} =496$ meV. The linear system size is $L = 77$ for c--h, while the polynomial order $N_\mathcal{C}$ utilized in the kernel polynomial method is $2048$ for $\Delta$ and $1024$ for $D_s$ calculations SM.
• Figure 2: Normal-state conductivity in TTG.a--b, Conductivity $\sigma_{xx}$ of TTG in the chiral limit versus eigenstate energy $E$ with $w_\mathrm{AB}=w_0$ and $w_\mathrm{AB}=2w_0$, respectively. c--d, Including realistic chiral symmetry breaking $\alpha=0.7$. e--f, Conductivity evaluated with different KPM polynomial orders $N_\mathcal{C}$. The red and blue dashed lines show reference values for critical states predicted to occur in class AIII Ludwig1994Sbierski2020: WZW [zero energy, $3 e^2/(\pi h)$] and IQHT [finite energy, $\approx 0.6 e^2/h$]. The linear system size $L = 308$.
• Figure 3: The ratio of normal-state conductivity evaluated with $2N_\mathcal{C}$ and $N_\mathcal{C}$.a-b, The ratio $\sigma_{xx}^{2N_\mathcal{C}}/\sigma_{xx}^{N_\mathcal{C}}$ of TTG in the chiral limit with $w_\mathrm{AB}=w_0$ and $w_\mathrm{AB}=2w_0$, respectively. c-d, The ratio for TTG with chiral symmetry breaking. The ratio is expected to be $\approx 1$ (dashed line) for critical states, and deviates from $1$ for ballistic transport. The peaks in the ratio correspond to commensurate twist angles. Results are averaged over the energy window $v_F k_{\theta_{12}} (-1,1)$ and $L = 308$.
• Figure 4: Multifractal dimension in collinear and non-collinear model for TTG. Here we show results in the chiral limit with $w_\mathrm{AB}=w_0$. a, Real-space multifractal dimension $\tau_2$ for the $E = 0$ wave function in TTG evaluated with the collinear model. b, Momentum-space multifractal dimension $\tau_2^k$ in TTG evaluated with the non-collinear model. Small $\tau_2$ and large $\tau_2^k$ indicate a quantum-critical state. The multifractal dimensions in both models reflect the same criticality behavior in the normal state of TTG.
• Figure 5: Conductivity in units of $e^2/h$ obtained from the non-collinear model in the chiral limit for $w_\mathrm{AB} = 220$ meV with $\theta_{12}=0.7^\circ$ and varying $\theta_{23}$. The smearing width is 5 meV. The black dashed line is 0.6, which is the IQHT value.