Understanding symmetry breaking in twisted bilayer graphene from cluster constraints

Twisted bilayer graphene is an exciting platform for exploring correlated quantum phases, extremely tunable with respect to both the single-particle bands and the interaction profile of electrons. Here, we investigate the phase diagram of twisted bilayer graphene as described by an extended Hubbard model on the honeycomb lattice with two fermionic orbitals (valleys) per site. Besides the special extended cluster interaction $Q$, we incorporate the effect of gating through an on-site Hubbard-interaction $U$. Within quantum Monte Carlo, we find valence-bond solid, Nel-valley antiferromagnetic or charge-density wave phases. Further, we elucidate the competition of these phases by noticing that the cluster interaction induces an exotic constraint on the Hilbert space, which we dub the cluster rule, in analogy to the famous pyrochlore spin-ice rule. Formulating the perturbative Hamiltonian by projecting into the cluster-rule manifold, we perform exact diagonalization and construct the fixed-point states of the observed phases. Finally, we compute the local electron density patterns as signatures distinguishing these phases, which could be observed with scanning tunneling microscopy. Our paper capitalizes on the notion of cluster constraints in the extended Hubbard model of twisted bilayer graphene and suggests a scheme towards realization of several symmetry-breaking insulating phases in a twisted-bilayer graphene sheet.

Figure 1

Phase diagram in the $(Q/t,U/t)$ plane. The fixed-point states are sketched in the respective regions of the phase diagram. The $(U,Q)$ pairs mark the phase boundary obtained by extrapolating the QMC data to the thermodynamic limit. The error bars indicate the direction of a scan in the plane. The uncertainty is $\mathrm{\Delta }Q/t=1.0$ for horizontal and $\mathrm{\Delta }U/t=0.01$ for vertical scans. Some transition points in these phase diagrams are already known from the literature: the $U/t=0$ axis at $N_{\text{s}}=2$ was studied in Ref. [22], the $Q/t=0$ axes at $N_{\text{s}}=1$ and 2 were studied in Refs. [30, 31]. The squares identify the points of the phase diagram where we later show infinite-volume extrapolations of susceptibilities within QMC. The dashed violet lines indicate the possible paths in the phase diagrams induced by the dependence of the interaction parameters on the gate distance $d$ shown in Fig.  2. (a) The spinless $N_{\text{s}}=1$ case. (b) The spinful $N_{\text{s}}=2$ case. (c), (d) At $N_{\text{s}}=2$, distributions of the complex order parameter $\langle {\widehat{M}}_{{\mathit{K}}_{\text{M}}}\rangle =\text{Tr}\widehat{G}{\widehat{M}}_{{\mathit{K}}_{\text{M}}}$ for $Q/t,U/t=(84.4,0.24)$ and $(96.6,0.24)$, respectively. These parameters are marked as stars in panel (b). The distributions are characteristic for the (c) cVBS and (d) pVBS Kekuléé orders. (e) Experimental proposal to measure the STM pattern in TBlG with a single metallic gate and a layer of boron nitride.

Figure 2

Left axis: Ratio of the on-site potential correction $U$ to the cluster interaction $Q$ as a function of $d/L_{\text{M}}$, where $d$ is the distance to the metallic gates from the TBlG sample. Right axis: Dependence of the screened cluster interaction $Q\left(d\right)$ on the gate distance $d$. The details of both calculations are given in Appendix pp2. Solid (dashed) lines correspond to the setups with two (one) metallic gate(s).

Figure 3

(a) $N_{\text{s}}=1$, quadratic extrapolations of susceptibilities in the DSM, cVBS, and valley Nel phases corresponding to the parameters $(U/t,Q/t)=(0,20),(0,100)$, and $(2,80)$, respectively. The Kekuléé susceptibility in the cVBS phase and the Nel susceptibility in the valley Nel phase are divided by 5 at $N_{\text{s}}=1$, while the Kekuléé susceptibility in the cVBS phase as multiplied by 5 at $N_{\text{s}}=2$ for better general visibility. The nonzero extrapolations are denoted with bold lines. (b) Extrapolations in the $N_{\text{s}}=2$ case in the DSM, cVBS, and pVBS phases corresponding to the parameters $(U/t,Q/t)=(0,40),(0,60)$ and $(6,60)$, respectively.

Figure 4

Two moves in the cluster-rule perturbation theory. (a) Bond flip exchanging fermions of ${\xi }_1$ and ${\xi }_2$ on sites $i$ and $j$. (b) The hexagon flip process that moves three fermions with ${\xi }_1,{\xi }_2,{\xi }_3$ over three nonadjacent bonds in a hexagon $⎔$ in the same direction, such that the cluster rule is respected. (c) An example of a cluster-rule configuration for the $N_{\text{s}}=1$ case. Left (right) colored semicircles indicate presence of a spin up (down). Note that in each hexagon, there are exactly six fermions. The arrows along the bonds indicate the possible single-particle hops between adjacent sites that are allowed by the Pauli principle. The round arrows in the centers of the hexagon indicate possible (b) moves, namely, there are two (b) moves possible in the upper-right hexagon, one (b) move possible in the lower hexagon, and no moves possible in the upper-left hexagon.

Figure 5

(a) Spectrum obtained within exact diagonalization of the Hamiltonian Eq. (9) at $U/t=0.2$. Here, we plot the energy gap between an excitation (and mark its quantum numbers) and a featureless ground state (with quantum numbers $S=0, \mathit{k}=0, A_1$). The lowest excitations in selected symmetry sectors are connected with lines, and their quantum numbers are given. Other excitations are shown as simple black squares. (b) The case $U/t=-0.2$. (c) The resulting phase diagram with phase transitions pinned from the crossings of the excited states' energies.

Figure 6

Upper row: Local electron density $\rho \left(\mathit{r}\right)$ of the CDW and the ${\tau }^1, {\tau }^2$, and ${\tau }^3$ Nel orders, which could be measured using STM. The images show the local electron density in one of the graphene layers on the microscopic graphene scale. White dots show the positions of the sites of one of the graphene layers. Bottom row: Fourier transforms $\rho \left(\mathit{q}\right)$ of the respective STM images. The central peak at $\mathit{q}=\mathit{0}$ is removed for visibility. The six graphene Bragg peaks are circled; the additional peaks in the Fourier transform for the ${\tau }^1$ and ${\tau }^2$ orders are a result of the $\sqrt{3}\times \sqrt{3}$ translational symmetry breaking in the presence of intervalley coherence and correspond to momenta $\mathit{K}-{\mathit{K}}^{\prime }$.