Motivated by recent advances in the fabrication of twisted bilayers of 2D materials, we consider the low-energy properties of a twisted pair of two-dimensional nodal superconductors. We study both the cases of singlet and triplet superconductors. It is demonstrated that the Bogoliubov-de Gennes (BdG) quasiparticle dispersion undergoes dramatic reconstruction due to the twist. In particular, the velocity of the neutral massless Dirac excitations near the gap nodes is strongly renormalized by the interlayer hopping and vanishes at a ``magic angle'' where in the limit of a circular Fermi surface a quadratic band touching is formed. In addition, it is shown that the BdG disperion can be tuned with an interlayer displacement field, magnetic field, and current, which can suppress the velocity renormalization, create finite BdG Fermi surfaces, or open a gap, respectively. Finally, interactions between quasiparticles are shown to lead to the emergence of a correlated superconducting state breaking time-reversal symmetry in the vicinity of the magic angle. Estimates of the magic angle in a variety of nodal superconductors are presented, ranging from the cuprates to the organic and heavy fermion superconductors, all of which are shown to be promising for the experimental realization of our proposal.
We study a model of a stripe ordered doped antiferromagnet consisting of coupled Hubbard ladders which can be tuned from quasi-one-dimensional to two-dimensional. We solve for the magnetization and charge density on the ladders by Hartree-Fock theory and find a set of solutions with lightly doped ``spin-stripes'' which are antiferromagnetic and more heavily doped anti-phase ``charge-stripes''. Both the spin- and charge-stripes have electronic spectral weight near the Fermi energy but in different regions of the Brillouin zone; the spin-stripes in the ``nodal'' region, near (π/2,π/2), and the c…
In the magic angle twisted bilayer graphene (MATBG), one of the most remarkable observations is the $C_3$-symmetry-breaking nematic state. We identify that the nematicity in MATBG is the $E$-symmetry ferro bond order, which is the modulation of correlated hopping integrals owing to the $E$-symmetry particle-hole pairing condensation. The nematicity in MATBG originates from prominent quantum interference among $SU(4)$ valley+spin composite fluctuations. This novel "valley + spin fluctuation interference mechanism" is revealed by the density wave equation analysis for realistic multiorbital Hubb…
We investigate the correlated electrons in the magic-angle twisted bilayer graphene by using the slave-rotor mean-field theory. Owing to the extended figure of Wannier orbitals, we study the two-orbital cluster Hubbard model with spin-valley fourfold degeneracy, focusing around half filling of valence bands below the neutrality point. The theory predicts multiple Mott insulator phases at fractional fillings not only for integer charges per moir{é} site, and it demonstrates that long-range electron hopping is highly suppressed because multiple-charge excitations are induced. Furthermore, the Ke…
The ability to control the strength of interaction is essential for studying quantum phenomena emerging from a system of correlated fermions. For example, the isotope effect illustrates the effect of electron-phonon coupling on superconductivity, providing an important experimental support for the BCS theory. In this work, we report a new device geometry where the magic-angle twisted bilayer graphene (tBLG) is placed in close proximity to a Bernal bilayer graphene (BLG) separated by a 3 nm thick barrier. Using charge screening from the Bernal bilayer, the strength of electron-electron Coulomb …
In this work we address the re-entrance of magic-angle phenomena (band flatness and quantum-geometric transport) in twisted bilayer graphene (TBG) subjected to strong magnetic fluxes $\pm Φ_0$, $\pm 2 Φ_0$, $\pm 3 Φ_0$... ($Φ_0 = h/e$ is the flux quantum per moiré cell). The moiré translation invariance is restored at the integer fluxes, for which we calculate the TBG band structure using accurate atomistic models with lattice relaxations. Similarly to the zero-flux physics outside the magic angle condition, the reported effect breaks down rapidly with the twist. We conclude that the magic-ang…
Twisted bilayer graphene (TBG) develops large moiré patterns at small twist angles with flat energy bands hosting domes of superconductivity. The large system size and intricate band structure have however hampered investigations into the superconducting state. Here, using full-scale atomistic modelling with local electronic interactions, we find at and above experimentally relevant temperatures a highly inhomogeneous superconducting state with nematic ordering on both atomic and moiré length scales. The nematic state has a locally anisotropic real-valued d-wave pairing, with a nematic vector …
In the past two years, magic-angle twisted bilayer graphene has emerged as a uniquely versatile experimental platform that combines metallic, superconducting, magnetic and insulating phases in a single crystal. In particular the ability to tune the superconducting state with a gate voltage opened up intriguing prospects for novel device functionality. Here we present the first demonstration of a device based on the interplay between two distinct phases in adjustable regions of a single magic-angle twisted bilayer graphene crystal. We electrostatically define the superconducting and insulating …
It has recently been shown that superconductivity in magic-angle twisted trilayer graphene survives to in-plane magnetic fields that are well in excess of the Pauli limit, and much stronger than the in-plane critical magnetic fields of magic-angle twisted bilayer graphene. The difference is surprising because twisted bilayers and trilayers both support the magic-angle flat bands thought to be the fountainhead of twisted graphene superconductivity. We show here that the difference in critical magnetic fields can be traced to a $\mathcal{C}_2 \mathcal{M}_{h}$ symmetry in trilayers that survives …
Twisted bilayered graphenes at magic angles are systems housing long ranged periodicity of Moiré pattern together with short ranged periodicity associated with the individual graphenes. Such materials are a fertile ground for novel states largely driven by electronic correlations. Here we find that the ubiquitous Casimir force can serve as a platform for macroscopic manifestations of the quantum effects stemming from the magic angle bilayered graphenes properties and their phases determined by electronic correlations. By utilizing comprehensive calculations for the electronic and optical respo…
Twisted bilayer graphene (TBG) has been experimentally observed to exhibit almost flat bands when the twisting occurs at certain magic angles. In this letter, we report new results on the continuum model of twisted bilayer graphene and its electronic band structure. Under we show that in the approximation of vanishing AA-coupling, the magic angles (at which there exist entirely flat bands) are given as the eigenvalues of a non-hermitian operator, and that all bands start squeezing exponentially fast as the angle $θ$ tends to $0$. In particular, as the interaction potential changes, the dynamic…
Moiré systems displaying flat bands have emerged as novel platforms to study correlated electron phenomena. Insulating and superconducting states appear upon doping magic angle twisted bilayer graphene (TBG), and there is evidence of correlation induced effects at the charge neutrality point (CNP) which could originate from spontaneous symmetry breaking. Our theoretical calculations show how optical conductivity measurements can distinguish different symmetry breaking states, and reveal the nature of the correlated states. In the specific case of nematic order, which breaks the discrete rotati…
We present a theory of superconductivity in twisted bilayer graphene in which attraction is generated between electrons on the same honeycomb sublattice when the system is close to a sublattice polarization instability. The resulting Cooper pairs are spin-polarized valley-singlets. Because the sublattice polarizability is mainly contributed by interband fluctuations, superconductivity occurs over a wide range of filling fraction. It is suppressed by i) applying a sublattice polarizing field (generated by an aligned BN substrate) or ii) changing moiré band filling to favor valley polarization. …
Recent measurements of the resistivity in magic-angle twisted bilayer graphene near the superconducting transition temperature show two-fold anisotropy, or nematicity, when changing the direction of an in-plane magnetic field [Cao \textit{et al.}, Science \textbf{372}, 264 (2021)]. This was interpreted as strong evidence for exotic nematic superconductivity instead of the widely proposed chiral superconductivity. Counter-intuitively, we demonstrate that in two-dimensional chiral superconductors the in-plane magnetic field can hybridize the two chiral superconducting order parameters to induce …
Magic-angle twisted trilayer graphene (MATTG) recently emerged as a highly tunable platform for studying correlated phases of matter, such as correlated insulators and superconductivity. Superconductivity occurs in a range of doping levels that is bounded by van Hove singularities which stimulates the debate of the origin and nature of superconductivity in this material. In this work, we discuss the role of spin-fluctuations arising from atomic-scale correlations in MATTG for the superconducting state. We show that in a phase diagram as function of doping ($ν$) and temperature, nematic superco…
Twisted bilayers of nodal superconductors were recently proposed as a promising platform to host superconducting phases that spontaneously break time-reversal symmetry. Here we extend this analysis to twisted multilayers, focusing on two high-symmetry stackings with alternating ($\pm θ$) and constant ($θ$) twist angles. In analogy to alternating-twist multilayer graphene, the former can be mapped to twisted bilayers with renormalized interlayer couplings, along with a remnant gapless monolayer when the number of layers $L$ is odd. In contrast, the latter exhibits physics beyond twisted bilayer…
The velocities of the quasiparticles that form Cooper pairs in a superconductor are revealed by the upper critical magnetic field. Here we use this property to assess superconductivity in magic-angle twisted bilayer graphene (MATBG), which has been observed over a range of moiré band filling, twist angle, and screening environment conditions. We find that for pairing mechanisms that are unrelated to correlations within the MATBG flat bands, minima in an average Fermi velocity $v_F^* \equiv k_B T_c \ell_c /\hbar $, where $\ell_c$ is the magnetic length at the critical perpendicular magnetic fie…
Twisted bilayer graphene (tBLG) has recently emerged as a new platform for studying electron correlations, the strength of which can be controlled via the twist angle. Here, we study the effect of internal screening on electron-electron interactions in undoped tBLG. Using the random phase approximation, we find that the dielectric response of tBLG drastically increases near the magic angle and is highly twist-angle dependent. As a consequence of the abrupt change of the Fermi velocity as a function of wave vector, the screened interaction in real space exhibits attractive regions for certain t…
Moiré quantum matter has emerged as a novel materials platform where correlated and topological phases can be explored with unprecedented control. Among them, magic-angle systems constructed from two or three layers of graphene have shown robust superconducting phases with unconventional characteristics. However, direct evidence for unconventional pairing remains to be experimentally demonstrated. Here, we show that magic-angle twisted trilayer graphene (MATTG) exhibits superconductivity up to in-plane magnetic fields in excess of 10 T, which represents a large ($2\sim3$ times) violation of th…
Magic-angle twisted bilayer graphene (MA-tBLG) has appeared as a tunable testing ground to investigate the conspiracy of electronic interactions, band structure, and lattice degrees of freedom to yield exotic quantum many-body ground states in a two-dimensional Dirac material framework. While the impact of external parameters such as doping or magnetic field can be conveniently modified and analyzed, the all-surface nature of the quasi-2D electron gas combined with its intricate internal properties pose a challenging task to characterize the quintessential nature of the different insulating an…
We provide a mathematical account of the recent Physical Reviews Letter by Tarnopolsky--Kruchkov--Vishwanath. The new contributions are a spectral characterization of magic angles, its accurate numerical implementation and an exponential estimate on the squeezing of all bands as the angle decreases. Pseudospectral phenomena due to the non-hermitian nature of operators appearing in the model play a crucial role in our analysis.
Coulomb screening of superconductivity in magic-angle twisted bilayer graphene
School of Physics & Astronomy, University of Manchester（曼彻斯特大学物理与天文学学院）；National Graphene Institute, University of Manchester（曼彻斯特大学国家石墨烯研究所）；Department of Physics, Washington University in St. Louis（华盛顿大学圣路易斯分校物理系）；Department of Physics and Astronomy, University of Pennsylvania（宾夕法尼亚大学物理与天文学系）；Department of Physics, University of Antwerp（安特卫普大学物理系）；Henry Royce Institute for Advanced Materials, The University of Manchester（曼彻斯特大学亨利·罗伊斯先进材料研究所）；National Institute for Materials Science（国立材料研究所）；Department of Materials Science and Engineering, National University of Singapore（新加坡国立大学材料科学与工程系）；Department of Physics, National University of Singapore（新加坡国立大学物理系）
Topological flat bands, valley polarization, and interband superconductivity in magic-angle twisted bilayer graphene with proximitized spin-orbit couplings
The simultaneous co-existence and gate-tuneability of the superconducting (SC), magnetic and topological orders in magic angle twisted bilayer graphene (MATBG) open up entirely new possibilities for the creation of complex hybrid Josephson junctions (JJ). Here we report on the creation of gate-defined, magnetic Josephson junctions in MATBG, where the weak link is gate-tuned close to the correlated state at a moiré filling factor of ν=-2. A highly unconventional Fraunhofer pattern emerges, which is phase-shifted and asymmetric with respect to the current and magnetic field directions, and shows…
Simultaneous transport and tunneling spectroscopy of moiré graphene: Distinct observation of the superconducting gap and signatures of nodal superconductivity
Magic-angle twisted trilayer graphene (MATTG) recently exhibited robust superconductivity at a higher transition temperature (TC) than the bilayer version. With electric gating from both the top and bottom sides, the superconductivity was found to be closely associated to two conditions: the finite broken mirror symmetry and carrier concentrations between two to three carriers per moiré unite cell. Both conditions may be achieved by graphene-metal contacts where charge transfers and interfacial electric fields are generated to balance work function mismatch. In this study, we explore the super…
We revisit the localized Wannier state description of the twisted bilayer graphene, focusing on the chiral limit. We provide a simple method for constructing such 2D exponentially localized -- yet valley polarized -- Wannier states, centered on the sites of the honeycomb lattice, paying particular attention to maintaining all the unobstructed symmetries. This includes the unitary particle-hole symmetry, and the combination of $C_2\mathcal{T}$ and the chiral particle-hole symmetry. The $C_2\mathcal{T}$ symmetry alone remains topologically obstructed and is not represented in a simple site-to-si…
Recent experimental and theoretical investigations demonstrate that twisted trilayer graphene (tTLG) is a highly tunable platform to study the correlated insulating states, ferromagnetism, and superconducting properties. Here we explore the possibility of tuning electronic correlations of the tTLG via a vertical pressure. A full tight-binding model is used to accurately describe the pressure-dependent interlayer interactions. Our results show that pressure can push a relatively larger twist angle (for instance, $1.89^{\circ}$) tTLG to reach the flat-band regime. Next, we obtain the relationshi…
We model the newly synthesized magic-angle twisted bilayer-graphene superconductor with two $p_{x,y}$-like Wannier orbitals on the superstructure honeycomb lattice, where the hopping integrals are constructed via the Slater-Koster formulism by symmetry analysis. The characteristics exhibited in this simple model are well consistent with both the rigorous calculations and experiment observations. A van Hove singularity and Fermi-surface (FS) nesting are found in the doping levels relevant to the correlated insulator and unconventional superconductivity revealed experimentally, base on which we …
