MATBG as Topological Heavy Fermion: I. Exact Mapping and Correlated Insulators

Magic-angle ($\theta=1.05^\circ$) twisted bilayer graphene (MATBG) has shown two seemingly contradictory characters: the localization and quantum-dot-like behavior in STM experiments, and delocalization in transport experiments. We construct a model, which naturally captures the two aspects, from the Bistritzer-MacDonald (BM) model in a first principle spirit. A set of local flat-band orbitals ($f$) centered at the AA-stacking regions are responsible to the localization. A set of extended topological conduction bands ($c$), which are at small energetic separation from the local orbitals, are responsible to the delocalization and transport. The topological flat bands of the BM model appear as a result of the hybridization of $f$- and $c$-electrons. This model then provides a new perspective for the strong correlation physics, which is now described as strongly correlated $f$-electrons coupled to nearly free topological semimetallic $c$-electrons - we hence name our model as the topological heavy fermion model. Using this model, we obtain the U(4) and U(4)$\times$U(4) symmetries as well as the correlated insulator phases and their energies. Simple rules for the ground states and their Chern numbers are derived. Moreover, features such as the large dispersion of the charge $\pm1$ excitations and the minima of the charge gap at the $\Gamma_M$ point can now, for the first time, be understood both qualitatively and quantitatively in a simple physical picture. Our mapping opens the prospect of using heavy-fermion physics machinery to the superconducting physics of MATBG.Comment: New references are added. Discussions on band geometry, relevant experiments, and more terms in the exchange interactions are added to the supplementary material

Dynamical Symmetry Indicators for Floquet Crystals

Various exotic topological phases of Floquet systems have been shown to arise from crystalline symmetries. Yet, a general theory for Floquet topology that is applicable to all crystalline symmetry groups is still in need. In this work, we propose such a theory for (effectively) non-interacting Floquet crystals. We first introduce quotient winding data to classify the dynamics of the Floquet crystals with equivalent symmetry data, and then construct dynamical symmetry indicators (DSIs) to sufficiently indicate the "inherently dynamical" Floquet crystals. The DSI and quotient winding data, as well as the symmetry data, are all computationally efficient since they only involve a small number of Bloch momenta. We demonstrate the high efficiency by computing all elementary DSI sets for all spinless and spinful plane groups using the mathematical theory of monoid, and find a large number of different nontrivial classifications, which contain both first-order and higher-order 2+1D anomalous Floquet topological phases. Using the framework, we further find a new 3+1D anomalous Floquet second-order topological insulator (AFSOTI) phase with anomalous chiral hinge modes.Comment: Close to the published versio

Twisted bilayer graphene I. Matrix elements, approximations, perturbation theory and a k⋅pk\cdot p 2-Band model

We investigate the Twisted Bilayer Graphene (TBG) model to obtain an analytic understanding of its energetics and wavefunctions needed for many-body calculations. We provide an approximation scheme which first elucidates why the BM $K_M$-point centered calculation containing only $4$ plane-waves provides a good analytical value for the first magic angle. The approximation scheme also elucidates why most many-body matrix elements in the Coulomb Hamiltonian projected to the active bands can be neglected. By applying our approximation scheme at the first magic angle to a $\Gamma_M$-point centered model of 6 plane-waves, we analytically understand the small $\Gamma_M$-point gap between the active and passive bands in the isotropic limit $w_0=w_1$. Furthermore, we analytically calculate the group velocities of passive bands in the isotropic limit, and show that they are \emph{almost} doubly degenerate, while no symmetry forces them to be. Furthermore, away from $\Gamma_M$ and $K_M$ points, we provide an explicit analytical perturbative understanding as to why the TBG bands are flat at the first magic angle, despite it is defined only by vanishing $K_M$-point Dirac velocity. We derive analytically a connected "magic manifold" $w_1=2\sqrt{1+w_0^2}-\sqrt{2+3w_0^2}$, on which the bands remain extremely flat as $w_0$ is tuned between the isotropic ($w_0=w_1$) and chiral ($w_0=0$) limits. We analytically show why going away from the isotropic limit by making $w_0$ less (but not larger) than $w_1$ increases the $\Gamma_M$- point gap between active and passive bands. Finally, perturbatively, we provide an analytic $\Gamma_M$ point $k\cdot p$ $2$-band model that reproduces the TBG band structure and eigenstates in a certain $w_0,w_1$ parameter range. Further refinement of this model suggests a possible faithful $2$-band $\Gamma_M$ point $k\cdot p$ model in the full $w_0, w_1$ parameter range.Comment: 25+21 pages, 13+7 figures. Published versio

TBG II: Stable Symmetry Anomaly in Twisted Bilayer Graphene

We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with a lattice model. Previous works, e.g., [Phys. Rev. Lett. 123, 036401], [Phys. Rev. X 9, 021013], [Phys. Rev. B 99, 195455], and others [1-4] found that the two flat bands in TBG possess a fragile topology protected by the $C_{2z}T$ symmetry. [Phys. Rev. Lett. 123, 036401] also pointed out an approximate particle-hole symmetry ($\mathcal{P}$) in the continuous model of TBG. In this work, we numerically confirm that $\mathcal{P}$ is indeed a good approximation for TBG and show that the fragile topology of the two flat bands is enhanced to a $\mathcal{P}$-protected stable topology. This stable topology implies $4l+2$ ($l\in\mathbb{N}$) Dirac points between the middle two bands. The $\mathcal{P}$-protected stable topology is robust against arbitrary gap closings between the middle two bands the other bands. We further show that, remarkably, this $\mathcal{P}$-protected stable topology, as well as the corresponding $4l + 2$ Dirac points, cannot be realized in lattice models that preserve both $C_{2z}T$ and $\mathcal{P}$ symmetries. In other words, the continuous model of TBG is anomalous and cannot be realized on lattices. Two other topology related topics, with consequences for the interacting TBG problem, i.e., the choice of Chern band basis in the two flat bands and the perfect metal phase of TBG in the so-called second chiral limit, are also discussed.Comment: references adde

Spin Space Groups: Full Classification and Applications

In this work, we exhaust all the spin-space symmetries, which fully characterize collinear, non-collinear, commensurate, and incommensurate spiral magnetism, and investigate enriched features of electronic bands that respect these symmetries. We achieve this by systematically classifying the so-called spin space groups (SSGs) - joint symmetry groups of spatial and spin operations that leave the magnetic structure unchanged. Generally speaking, they are accurate (approximate) symmetries in systems where spin-orbit coupling (SOC) is negligible (finite but weaker than the interested energy scale); but we also show that specific SSGs could remain valid even in the presence of a strong SOC. By representing the SSGs as O($N$) representations, we - for the first time - obtain the complete classifications of 1421, 9542, and 56512 distinct SSGs for collinear ($N=1$), coplanar ($N=2$), and non-coplanar ($N=3$) magnetism, respectively. SSG not only fully characterizes the symmetry of spin d.o.f., but also gives rise to exotic electronic states, which, in general, form projective representations of magnetic space groups (MSGs). Surprisingly, electronic bands in SSGs exhibit features never seen in MSGs, such as nonsymmorphic SSG Brillouin zone (BZ), where SSG operations behave as glide or screw when act on momentum and unconventional spin-momentum locking, which is completely determined by SSG, independent of Hamiltonian details. To apply our theory, we identify the SSG for each of the 1604 published magnetic structures in the MAGNDATA database on the Bilbao Crystallographic Server. Material examples exhibiting aforementioned novel features are discussed with emphasis. We also investigate new types of SSG-protected topological electronic states that are unprecedented in MSGs

Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries

Adding magnetic flux to a band structure breaks Bloch's theorem by realizing a projective representation of the translation group. The resulting Hofstadter spectrum encodes the non-perturbative response of the bands to flux. Depending on their topology, adding flux can enforce a bulk gap closing (a Hofstadter semimetal) or boundary state pumping (a Hofstadter topological insulator). In this work, we present a real-space classification of these Hofstadter phases. We give topological indices in terms of symmetry-protected Real Space Invariants (RSIs) which encode bulk and boundary responses of fragile topological states to flux. In fact, we find that the flux periodicity in tight-binding models causes the symmetries which are broken by the magnetic field to reenter at strong flux where they form projective point group representations. We completely classify the reentrant projective point groups and find that the Schur multipliers which define them are Arahanov-Bohm phases calculated along the bonds of the crystal. We find that a nontrivial Schur multiplier is enough to predict and protect the Hofstadter response with only zero-flux topology