Ground State and Hidden Symmetry of Magic-Angle Graphene at Even Integer Filling

Abstract

In magic angle twisted bilayer graphene (TBG), electron-electron interactions play a central role, resulting in correlated insulating states at certain integer fillings. Identifying the nature of these insulators is a central question, and it is potentially linked to the relatively high-temperature superconductivity observed in the same devices. Here, we address this question using a combination of analytical strong-coupling arguments and a comprehensive Hartree-Fock numerical calculation, which includes the effect of remote bands. The ground state we obtain at charge neutrality is an unusual ordered state, which we call the Kramers intervalley-coherent (K-IVC) insulator. In its simplest form, the K-IVC order exhibits a pattern of alternating circulating currents that triples the graphene unit cell, leading to an “orbital magnetization density wave.” Although translation and time-reversal symmetry are broken, a combined “Kramers” time-reversal symmetry is preserved. Our analytic arguments are built on first identifying an approximateU(4)×U(4)symmetry, resulting from the remarkable properties of the TBG band structure, which helps select a low-energy manifold of states that are further split to favor the K-IVC state. This low-energy manifold is also found in the Hartree-Fock numerical calculation. We show that symmetry-lowering perturbations can stabilize other insulators and the semimetallic state, and we discuss the ground state at half-filling and give a comparison with experiments.

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Plain Language Summary

Twisted bilayer graphene, in which a one-atom-thick sheet of carbon atoms is placed askew atop another, has garnered a lot of attention in recent years as a platform for exploring novel electronic behaviors. When the relative twist angle between the two graphene layers is close to a “magic angle” of about one degree, electrons are robbed of much of their kinetic energy and experience strong interactions that drastically change the system’s overall properties, including an insulating behavior that is difficult to explain. Here, we numerically and analytically identify a symmetry breaking in twisted bilayer graphene that may be responsible for this insulating behavior.

When two sheets of graphene are stacked atop one another and twisted, the interplay between the honeycomblike arrangements of atoms in each sheet creates a moiré pattern, which provides a periodic arrangement of cells in which electrons can gather. The insulating behavior appears only when these cells are filled with an integer number of electrons. Our numerical and analytical work suggests that when these cells have an even number of electrons, a pattern of alternating circulating currents can arise, which leads to a magnetic density wave that breaks certain symmetries in the system but preserves others. This symmetry breaking, we argue, can explain the observed insulating behavior.

This insulator could serve as a “parent” state for superconducting phases observed in the same moiré devices, when, on average, there are a noninteger number of electrons per cell. It therefore provides an interesting starting point for studies of the origin of superconductivity in moiré systems. The newly identified symmetry breaking is subtle, so directly detecting it will be a challenge.