Correlated Insulators, Semimetals, and Superconductivity in Twisted Trilayer Graphene

Motivated by recent experiments indicating strong superconductivity and intricate correlated insulating and flavor-polarized physics in mirror-symmetric twisted-trilayer graphene, we study the effects of interactions in this system close to the magic angle, using a combination of analytical and numerical methods. We identify asymptotically exact correlated many-body ground states at all integer filling fractions $\nu$ of the flat bands. To determine their fate when moving away from these fine-tuned points, we apply self-consistent Hartree-Fock numerics and analytic perturbation theory, with good agreement between the two approaches. This allows us to construct a phase diagram for the system as a function of $\nu$ and the displacement field, the crucial experimental tuning parameter of the system, and study the spectra of the different phases. The phase diagram is dominated by a correlated semimetallic intervalley coherent state and an insulating sublattice-polarized phase around charge neutrality $\nu =0$, with additional spin polarization being present at quarter ($\nu =-2$) or three-quarter ($\nu =+2$) fillings of the quasiflat bands. We further study the superconducting instabilities emerging from these correlated states, both in the absence and in the additional presence of electron-phonon coupling, also taking into account possible Wess-Zumino-Witten terms. In the experimentally relevant regime, we find triplet pairing to dominate, possibly explaining the observed violation of the Pauli limit. Our results have several consequences for experiments as well as future theoretical work and illustrate the rich physics resulting from the interplay of almost-flat bands and dispersive Dirac cones in twisted-trilayer graphene.

Popular Summary

The remarkable physics of twisted bilayer graphene—two sheets of graphene stacked at slight angles to one another—has inspired many incarnations of twisted materials in the last few years. A recent notable example is mirror-symmetric twisted trilayer graphene, where three layers of graphene are stacked with alternating twist angles. It exhibits robust superconductivity, alongside various other correlated phases, and can be efficiently tuned with an electric field, making trilayer graphene an exciting platform for exploring strongly correlated physics. Here, we study the phase diagram of this system to better understand the nature of its correlated insulating, semimetal, and superconducting phases.

We numerically and analytically study the phase diagram as a function of the electric field, the type of electron tunneling between adjacent graphene layers, and the number of electrons per unit cell. We show that the ground state of the system in the absence of an electric field decouples into a product of the ground state of graphene and that of twisted bilayer graphene. Our results further establish the dominance of insulating and semimetallic phases in the presence of an electric field, which are unique to the trilayer system. We use our resulting phase diagram for the correlated normal states to constrain the form of the superconductor, considering different pairing mechanisms.

Our results provide a possible origin for several previously unexplained aspects of trilayer experiments and shed light on the energetics at play in the system. Furthermore, the properties of the correlated phases we find are expected to be useful for the interpretation of future experiments and, as such, help elucidate the complex physics in mirror-symmetric twisted trilayer graphene.

Figure 1

Schematic phase diagram for MSTG based on our analysis, as a function of displacement field $D_0$ and filling fraction $\nu$. Because of particle-hole symmetry of our model, the phase diagram depends only on $|\nu |$. We show the form of the dominant particle-hole instabilities, the intervalley-coherent (IVC) semimetallic phase, and a sublattice-polarized (SLP) insulator at $\nu =0$ (charge neutrality); the IVC and SLP state coexist with spin polarization (SP) around $|\nu |=2$. We discuss three different types of superconducting regimes labeled SC I–III. We indicate in black their respective spin structure for the expected sign of the intervalley Hund’s coupling $J_H$ resulting from Coulomb interactions alone (with opposite sign in gray).

Figure 2

(a) Schematic of the geometry of MSTG. The top and bottom layers shown in red and green, respectively, are aligned, while the middle layer (blue) is twisted at relative angle $\theta$ to the top and bottom layers. (b) Band structure of MSTG at $D_0=0$ (left) and $D_0>0$ (right) for $w_1=124\mathrm{meV}$ and twist angle $\theta =1.53°$ along the one-dimensional cut through the MBZ shown in (c). We represent the spin-degenerate bands in the two different valleys $\eta =\pm$ with solid ($\eta =+$) and dotted ($\eta =-$) lines. At $D_0=0$, the band structure is given by that of TBG and single-layer graphene, and we color the (two per spin and valley) quasiflat TBG bands red. For $D_0>0$, the bands of these two subsystems mix, but we continue to label the two bands per spin and valley closest to the Fermi as “TBG-like” bands indicated in red.

Figure 3

HF band structures in the limit $D_0=0$ (left) and nonzero $D_0$ (right) as obtained from Eq. (4.1) for the optimal $P_{\mathit{k}}$ in the respective symmetry-breaking channel defined in Table 2. We show only the states which are either ground states or subleading energy states for some region of our phase diagram, with a full set of band structures available in the Supplemental Material [59] Sec. C 3. Each band of the HF Hamiltonian is colored according to the expectation value of the mirror-symmetry operator within each band as a function of $\mathit{k}$, with the expectation value ranging from $-1$ (red) to $+1$ (blue).

Figure 4

Phases obtained by self-consistent HF method as a function of $w_0$ and $D_0$. Both the leading instability (red) and the subleading ones (black, in parentheses) are shown. The phases are identified by the symmetries they break (see Table 2), and representative band structures can be found in Fig. 3. We show only one state as a representative of each pair of Hund’s partners which are degenerate in our Hartree-Fock procedure. We label the nearly degenerate ${\mathrm{SLP}}_{+}$, ${\mathrm{SLP}}_{-}$, ${\mathrm{SSLP}}_{+}$, and ${\mathrm{SSLP}}_{-}$ phases with “SLP,” though note the preference in our numerics is for the ${\mathrm{SLP}}_{-}$ state from a Hartree contribution.

Figure 5

(a) Line cuts for fixed $w_0$ showing the energy of our self-consistent solutions relative to a SP state as a function of $D_0$. We use $w_0=124\mathrm{meV}$ and four bands per spin and valley, with dielectric constant $\epsilon =7$ and screening distance $d=40\mathrm{nm}$. Full energies as a function of $w_0$ and $D_0$ can be found in the Supplemental Material [59] Fig. S2. Note that the ${\mathrm{SLP}}_{\pm }$ and ${\mathrm{SSLP}}_{-}$ are not exactly degenerate, as we discuss in the main text, but are plotted as a single line here since the splitting is too small to be visible on the scale of the plot. (b) Matrix form of our converged ${\mathrm{IVC}}_{-}$ and ${\mathrm{SLP}}_{-}$ orders $P_{\mathit{k}}$ for $\mathit{k}$ near the $\mathrm{\Gamma }$ point and near the $K$ point. The matrix structure of $P_{\mathit{k}}$ is organized such that the largest block outlined in blue denotes spin flavor, the next largest block outlined in red denotes valley flavor, the green block denotes upper and lower bands in the continuum model, and the final two boxes denote the graphenelike and TBG-like band in the upper-band box and the TBG-like then graphenelike bands in the lower-band box.

Figure 6

HF band structures at $\nu =2$ for the spin-polarized ${\mathrm{SLP}}_{-}$ state (a) at $D_0=0$ and (b) $D_0>0$ and spin-polarized ${\mathrm{IVC}}_{-}$ state (c) at $D_0=0$ and (d) $D_0>0$.