Universal Features of Landau Fans of Twisted Bilayer Graphene with Large Superlattices

TL;DR

This paper identifies universal features in the Landau fan behavior of twisted bilayer graphene with large superlattices, suggesting a common underlying pattern influenced by Hubbard-like physics and symmetry breaking.

Contribution

It introduces a Hubbard-like model with mean fields to explain the universal Landau fan features across different TBG samples, highlighting the interplay of Mott physics and symmetry breaking.

Findings

Universal Landau fan dispersion patterns identified

Degeneracy reduction linked to insulating phases

Model explains sample-dependent behaviors

Abstract

Current experiments on different samples of twisted bilayer graphene (TBG) have found different sets of insulating phases. Despite this diversity, many features of these insulating phases appear to be universal. They include the dispersion of Landau fans away from charge neutrality, a reduced Landau fan degeneracy from the expected value at charge neutrality, and the further reduction of this degeneracy when crossing an insulating phase with odd number of electrons in the superlattice unit cell. We point out that all these behaviors as well as the ferromagnetic behavior observed in some of the insulating states suggest an underlying "ideal" pattern, with different part of it realized in the different samples in different experiments. We further show that such pattern can be accounted for by a Hubbard like model for the superlattice augmented with a set of chemical potential dependent mean fields that break the symmetry of the eight internal degrees of freedoms successively. The simultaneous importance of Mott like physics and mean field physics may be a general feature of twisted 2D electronic materials with large superlattices, not necessarily confined to graphene.

Full text

Twisted bilayer graphene (TBG) at a small twist angle is full of puzzles and surprises. Through twisting to the “magic” angle Cao et al. (2018a, b); Lu et al. (2019) or through out-of-plane pressure Yankowitz et al. (2019), superconductivity can be activated. At the same time, its normal state also displays a whole host of baffling phenomena which seem to be related to each other Cao et al. (2018a, b); Yankowitz et al. (2019); Sharpe et al. (2019); Polshyn et al. (2019); Lu et al. (2019). First of all, a sequence of insulating phases are found on both sides of charge neutrality (CN) at integer filling, i.e. integer number of electrons per superlattice unit cell. The sequence of insulating phases observed appears to be sample dependent Cao et al. (2018a, b); Yankowitz et al. (2019); Sharpe et al. (2019); Polshyn et al. (2019); Lu et al. (2019). Secondly, the sizes of the Fermi surfaces on different sides of each insulating phases revealed from the Landau fans are dramatically different. In all cases, the Landau fans disperse in a direction away from CN. Thirdly, the degeneracy of Landau fan near CN is reduced from the expected value of 8 (two spins, two valleys, and two layers) Cao et al. (2016) to 4 Cao et al. (2018a, b); Yankowitz et al. (2019); Lu et al. (2019); and continues to decrease to 1 as one passes through the successive insulating phases to the bottom and to the top of the flat band Yankowitz et al. (2019); Lu et al. (2019). In all cases, a change of degeneracy by a factor of 2 takes place whenever an insulating phase with odd number of electrons per superlattice unit cell is crossed. The recurring pattern and the systematic reduction of degeneracy of the Landau fan suggests that similar physical mechanisms are at work as the insulating phases are crossed successively. Even though the number of insulating states varies between samples, the features mentioned above appear to be universal.

The insulating phases are surely caused by interactions. The question is they are consequences of mean field physics or strong (Mott-like) correlations. The success of the continuum theory Bistritzer and MacDonald (2011) in predicting band flattening around $1.05^{\circ}$ shows some aspects of band theory are essential, even though it predicts a Landau fan degeneracy of 8, twice of the observed value. On the other hand, the emergence of insulating phases at integer number of electrons per superlattice unit cell is most efficiently accounted for by Mott physics. As a matter of fact, if one were to generate insulating phases in continuum theory through different types of density wave formation, one would obtain many insulating phases not necessarily at integer filling, due to different types of nesting vectors. Hence, while some aspects of the normal state are readily explained by continuum theory, others are easily explained by the opposite Mott physics. Neither viewpoint can provide simple explanation for the systematic reduction of Landau fan degeneracy.

There has been an avalanche of theoretical studies since the announcement of superconductivity in TBG. Current theoretical studies have spanned both viewpoints. In particular, Ref.Yuan and Fu (2018) and Lian et al. (2018) have addressed the issue of Landau fan degeneracy. In Ref.Yuan and Fu (2018), lattice distortion and coulomb interaction are suggested to be the cause for reducing the Landau fan degeneracy near CN from 8 to 4; and Mott physics is invoked for the formation of the insulating phases at carrier density $n=\pm n_{s}/2$, where $n_{s}=4$ electrons per superlattice unit cell. However, the reduction of Landau level degeneracy from 4 to 2 around the insulating phases at $|n|=n_{s}/2$ was not discussed. On the other hand, Ref.Lian et al. (2018) makes use of the continuum theory. The reduction of degeneracy from 8 to 4 is attributed to Zeeman effect, and its further reduction from 4 to 2 is found at sufficiently high magnetic field. This picture makes no reference to the insulating phases at $n=\pm n_{s}/2$ at zero field. So the suggested mechanism for degeneracy reduction is not tied to the physics of the insulators. It is not clear at the moment how the Landau fan degeneracy of the insulating phases at $n=n_{s}/4$ and $\pm 3n_{s}/4$ Lu et al. (2019); Yankowitz et al. (2019); Sharpe et al. (2019) are explained by these pictures.

Differences between the Hubbard models of large superlattices and those of simple lattices: If one were to derive a Hubbard model to describe the flat band physics of superlattices with large unit cells, one would expect the hopping integral between the Wannier states of the superlattice would have included some interaction effects. At the same time, there will also be residential interactions between Wannier states, such as charging energy within each superlattice unit cell (analogous to the usual Hubbard $U$), as well as effective interactions between different internal degrees of freedom (i.e. layer, valley, and spin). Since the spatial extent of the Wannier state includes hundreds of original unit cells, one expects these effective interactions will be chemical potential dependent. Since the charging energy is electrostatic in nature (in a dielectric medium determined by many-body physics), it will retain the usual charging form $q^{2}/2C$ within each unit cell, where $q$ is the charge and $C$ is the capacitance of the cell. On the other hand, the residual interaction between different degrees of freedom can be mean field like. The purpose of this paper is to show that the puzzling behavior of the Landau fans observed over the entire density regime (summarized below) can be captured by a generalized Hubbard model that includes a set of very simple chemical dependent mean fields.

Summary of experimental situation: Before proceeding, we first summarize the current experimental findings schematically in Fig. 1. This figure shows a particle-hole symmetric pattern. It is identical to the result in Ref.Yankowitz et al. (2019) except for the filling interval $-1/2<n/n_{s}<-1/4$. We shall use this figure to illustrate the key findings in different experiments, as well as the working of our model. For TBG, the flat band near CN can be viewed as tight binding model on the superlattice with electrons carrying 8 degrees of freedom: layer index up and down $(u,d)$, sublattice (or valley) index $(K,K^{\prime})$, and spin index $(\uparrow,\downarrow)$. The allowed number of electron ($q$) per superlattice site ranges from 8 to 0, where $q=8$ and $q=0$ correspond to the full band and the empty band. Charge neutrality (CN) is at $q=4$. The carrier density is $n=q-4$, and the filling is $n/n_{s}=(q/4)-1$, ($n_{s}=4$). The experimental findings are:

$({\bf 1})$ Location of the insulating phases: All insulating phases discovered so far occur at integer number of electrons per (superlattice) site, i.e. $q=$ integer. In Ref.Cao et al. (2018b, a), only the insulating phases at $q=0,2,4,6,8$ are observed. In the absence of magnetic field, insulating phases at all integer values of $q$ except for $q=3$ (or $n=-n_{s}/4$) are found Yankowitz et al. (2019); Lu et al. (2019). The $q=3$ insulating state is reported in Lu et al. (2019) when applying magnetic field. Another recent experiment (Ref.Sharpe et al. (2019)) has only found insulating state above charge neutrality, at $q=4,6,7$, besides the band insulator at $q=0$ and 8. In Fig.1, we show the case where insulating phases occur at all integer $q$.

$({\bf 2})$ Landau fan degeneracy ($G$) around CN: Current experiments show that the Landau fans emerge symmetrically on both sides of CN. In Ref.Cao et al. (2018b, a); Yankowitz et al. (2019); Lu et al. (2019), these Landau fans have degeneracy $G=4$ instead of 8 predicted by band theory Bistritzer and MacDonald (2011) (See Fig. 1). In Ref. Sharpe et al. (2019), this degeneracy is even reduced to 2. Despite this difference, the reduction of degeneracy from the expected value of 8 occurs in all current experiments.

$({\bf 3})$ Landau fans away from CN: As filling $n$ increases above CN, the Landau fan expands until it reaches an insulating phase. On the other side of the insulator, a new Landau fan emerges. The Landau fans on both sides of an insulating phase indicate a large Fermi surface at fillings below it and a small Fermi surface above it. An opposite situation is found for fillings below CNCao et al. (2018b, a); Yankowitz et al. (2019); Sharpe et al. (2019); Lu et al. (2019) (See Fig.1). That all Landau fans disperse away from CN, and the dramatic changes in Fermi surface area after crossing an insulating phase appear to be universal.

$({\bf 4})$ Degeneracy patterns away from CN : In Ref.Yankowitz et al. (2019), the degeneracy of the Landau fans $(G)$ varies as $4,2,2,1$ and $4,2,1$ as one sweeps through the metallic phases above and below CN. Moreover, the reduction of $G$ appears at insulating states with odd $q$’s. Fig.1 shows the particle-hole symmetric version of the finding of Ref.Yankowitz et al. (2019), with the pattern of $G$ (4,2,2,1) on both sides of CN. Experiments that observed fewer insulating states have fewer Landau fans. That $G$ is reduced by 2 across insulating phases with odd $q$, and remain constant across those with even $q$ also appears to be universal Cao et al. (2018b, a); Yankowitz et al. (2019); Lu et al. (2019); Sharpe et al. (2019).

$({\bf 5})$ **Magnetic field effects **: At sufficiently large magnetic fields, the degeneracy of Landau fans is found to reduce in the interval $(2<q<4)$ in Ref.Yankowitz et al. (2019).

$({\bf 6})$ Ferromagnetism: Anomalous Hall effect is found at $q=7$ state in Ref.Sharpe et al. (2019). It was taken as evidence of ferromagnetism.

$({\bf 7})$ Landau fans associated with non-integer $q$: In Ref. Yankowitz et al. (2019), a Landau fan is found to converge to $n=-2n_{s}/3$, corresponding to non-integer number of electrons in each superlattice unit cell.

We shall present below a simple model that can account for $({\bf 1})$ to $({\bf 6})$. In a separate paper Ho and Li (2019), we shall show that in the weak coupling, density wave states triggered by nesting effects can lead to Landau fans at non-integer $q$ as in $({\bf 7})$.

Our model: With our earlier discussions, we consider a Hubbard model consists of the usual hopping $\hat{T}$, a charging energy $H_{c}=q^{2}/2C=(e^{2}/2C)n^{2}\equiv Un^{2}$, and a set of mean fields that set in at different chemical potentials $\mu$, favoring certain structures of internal degrees of freedom. We shall first consider interactions that lead to a particle-hole symmetric Landau fan pattern shown in Fig. 1, and then show how to obtain the features $({\bf 1})$-$({\bf 6})$. We shall consider the $\mu$-dependent mean field $H_{L}$, $H_{V}$, and $H_{S}$ that favor specific configurations in layer, valley, and spin spaces separately. (We stress that such separation is chosen to simplify the discussions. Our results remain unchanged even when the mean fields couple different types of degrees of freedom.) We shall denote the states favored and disfavored in layer space ${\cal L}\equiv(u,d)$ as ${\cal U}$ and ${\cal D}$, those in valley space ${\cal V}\equiv(K,K^{\prime})$ as ${\cal K}$ and ${\cal K}^{\prime}$, and those in spin ${\cal S}\equiv(\uparrow,\downarrow)$ spaces as $\alpha$ and $\beta$. The pairs $({\cal U},{\cal D})$, $({\cal K},{\cal K}^{\prime})$, and $(\alpha,\beta)$ are rotations of $(u,d)$, $(K,K^{\prime})$, and $(\uparrow,\downarrow)$ respectively. These interactions are turned on as follows, and are shown in Fig. 2 :

$({\bf A})$ The charging energy $H_{c}$ is present for all $\mu$. The capacitance $C$ can be $\mu$-dependent. $H_{c}$ will produce Mott insulators at integer number per cell $q=1,2,\ldots,7$. It is invariant with respect to rotations in layer ${\cal L}$, valley ${\cal V}$ and spin ${\cal S}$ spaces, and is therefore 8-fold degenerate. The upper and lower bound of the chemical potential of the $q$-particle Mott insulator will be denoted as $\mu_{q}^{-}$ and $\mu_{q}^{+}$.

$({\bf B})$ A symmetry breaking field $H_{L}$ in the layer space ${\cal L}$ is also present for all $\mu$ favoring a layer state ${\cal U}$. $H_{L}$ is independent of valley and spin indices and is 4-fold degenerate. (A similar situation occurs in the weak coupling case. The coupling of unlike valley Dirac cones from different layers can produce a layer polarization that reduces the Landau fan degeneracy from 8 to 4, while opening a gap at $n=\pm n_{s}/2$ simultaneously.Ho and Li (2019)).

$({\bf C})$ As $\mu$ increases above CN, new mean fields are set in at the Mott phases with odd particle numbers $q$ per site. At the onset of the $q=5$ Mott state, $\mu^{-}_{5}$, a mean field $H_{V}$ favoring the valley state ${\cal K}$ is turned on as shown in Fig. 2, gradually establishing an ordering in state ${\cal K}$ within the insulating phase ($\mu^{-}_{5}<\mu<\mu^{+}_{5}$). Like $H_{c}$, $H_{V}$ does not affect the spin degrees of freedom and has a degeneracy of 2.

$({\bf D})$ As $\mu$ increases further to $\mu^{-}_{7}$, the onset of the $n=7$ Mott state, a mean field $H_{S}$ favoring the spin state $\alpha$ is turned on as shown in Fig. 2, gradually polarizing the electrons to spin $\alpha$ within the interval ($\mu^{-}_{7}<\mu<\mu^{+}_{7}$). (See Fig 2.)

$({\bf E})$ Similar symmetry breaking fields are switched on in a particle-hole symmetric manner for $\mu$ below CN, i.e. $-H_{V}$ is switched on at $\mu_{3}^{+}$ favoring the state ${\cal K^{\prime}}$ within the range $\mu_{3}^{-}<\mu<\mu_{3}^{+}$, and $-H_{S}$ is switched on at $\mu_{1}^{+}$ favoring the state $\beta$ within the range $\mu_{1}^{-}<\mu<\mu_{1}^{+}$ as shown in Fig. 2.

Although we have chosen to first break the valley symmetry and then the spin symmetry as $\mu$ moves away from CN, the order can be changed in either or both sides of CN without affecting our conclusions. We now show that the interactions $({\bf A})-({\bf E})$ lead to the pattern in Fig. 1 :

(I) We shall denote the Mott state with $q$ electrons per site as $|q\rangle$. Both (A) and (B) imply that the Mott states with $q\leq 4$ are make up of electrons in the ${\cal U}$ state. The creation operator for the ${\cal U}$ state and the orthogonal ${\cal D}$ state are denoted as $\hat{\cal U}^{\dagger}_{\ell}$ and $\hat{\cal D}^{\dagger}_{\ell}$, where $\ell$ labels the four states $(K\uparrow)$, $(K^{\prime}\downarrow)$, $(K\uparrow)$, $(K^{\prime}\downarrow)$. The Mott state at CN is $|4\rangle=\prod_{\bf R}|4\rangle_{\bf R}$, where $|4\rangle_{\bf R}=\prod_{\ell=1}^{4}\hat{\cal U}^{\dagger}_{\ell}({\bf R})|0\rangle$ is a valley and spin singlet. See Fig. 3. Its particle and hole excitations are the momentum ${\bf k}$ states ${\cal D}^{\dagger}_{\ell}({\bf k})|4\rangle$ and ${\cal U}_{\ell}({\bf k})|4\rangle$. Since $H_{L}$ is independent of $\ell$, these excitations are 4-fold degenerate and behave like a spin-3/2 particles. As $\mu$ moves away from CN, these “spin=3/2” particles gather into a growing Fermi surface , leading to an expanding Landau fan with $G=4$ on both sides of CN as shown in Fig.1 in the interval $-n_{s}/4<n<n_{s}/4$. It is useful to think of the Mott state with $q$ as $q=4+n$ state, where $n=-4,\ldots,+4$, with the state $|4\rangle$ being an inert background, since it is a valley and spin singlet.

(II) As $\mu$ approaches $\mu^{-}_{5}$ from below, one reaches the onset of the $q=5=4+1$ (or $n=n_{s}/4$ Mott state). The large Fermi sea of the “spin-3/2” fermions turns into a Mott insulator of spin-3/2 particles. (See Fig.3) Each site is in the state $|5,\ell\rangle_{\bf R}=\hat{D}^{\dagger}_{\ell}({\bf R})|4\rangle_{\bf R}$. In the absence of symmetry breaking fields in the space ${\cal V}\otimes{\cal S}$, super-exchange will generate a Heisenberg like Hamiltonian for this system. The Mott state is then $|5,\mu_{5}^{-}\rangle=\hat{O}^{\dagger}(\mu^{-}_{5})|4\rangle$, where $\hat{O}^{\dagger}(\mu^{-}_{5})=\sum_{[\ell]}\Psi_{\ell_{1},\ell_{2},..\ell_{N}}\prod_{{\bf R}_{i}}\hat{\cal D}^{\dagger}_{\ell_{i}}({\bf R}_{i})$, and $\Psi$ is the ground state of the relevant spin-3/2 Heisenberg Hamiltonian, which we shall take as a singlet in ${\cal V}\otimes{\cal S}$. However, as $\mu$ increases further, the mean field $H_{V}$ discussed in $({\bf C})$ grows. It favors the valley state ${\cal K}$, and hence reduces the degeneracy $\ell$ to $({\cal K},\sigma)$, $\sigma=\uparrow,\downarrow$. This process changes the spin-3/2 fermions into spin-1/2 fermions as $\mu$ sweeps through the insulator from below, $\mu^{-}_{5}<\mu<\mu^{+}_{5}$. The ground state near $\mu^{+}_{5}$ is $|5,\mu_{5}^{+}\rangle=\hat{O}^{\dagger}(\mu^{+}_{5})|4\rangle$, where $\hat{O}^{\dagger}(\mu^{+}_{5})=\sum_{[\sigma]}\Phi_{\sigma_{1},\sigma_{2},..\sigma_{N}}\prod_{{\bf R}_{i}}\hat{\cal D}^{\dagger}_{{\cal K}\sigma_{i}}({\bf R}_{i})$, and $\Phi$ is the ground state of the spin-1/2 Heisenberg model near $\mu^{+}_{5}$, which is a spin singlet. See Fig.2 and 3.

The electron excitation at $\mu_{5}^{+}$ is $\hat{\cal D}^{\dagger}_{{\cal K},\sigma}({\bf k})|5,\mu_{5}^{+}\rangle$, which is two fold degenerate and is a spin-1/2 fermion. As $\mu$ increases above $\mu^{+}_{5}$, these excitations form a growing Fermi surfaces starting from zero size at $\mu^{+}_{5}$, leading to a growing Landau fan dispersing towards the top of the band with $G=2$. In contrast, the Landau fan for $\mu$ just below $\mu_{5}^{-}$ is due to the electron excitations originated from the state $|q=4\rangle$, which has a large Fermi surface at $\mu_{5}^{-}$. So there is a great asymmetry of the size of Fermi surface on different sides of $|q=5\rangle$ as shown in Fig.1.

(III) As $\mu$ reaches $\mu^{-}_{6}$, the onset of the $q=6=4+2$ Mott state ($n=n_{s}/2$), the spin-1/2 excitations emerging from $|5\rangle$ are localized, turning into the Mott state $|6\rangle=\prod_{\bf R}\hat{\cal D}^{\dagger}_{{\cal K}\uparrow}({\bf R})\hat{\cal D}^{\dagger}_{{\cal K}\downarrow}({\bf R})|4\rangle$ (See Fig.3). Since this state is a valley ferromagnet and a spin singlet, its internal structure of this state is unchanged over the range $\mu^{-}_{6}<\mu<\mu^{+}_{6}$.

The electron excitation of this state is ${\cal D}^{\dagger}_{{\cal K}^{\prime}\sigma}({\bf k})|2\rangle$, which is again a spin-1/2 fermion. Hence, the degeneracy of the Landau fans remains $G=2$ on both sides of the (even) $q=6$ insulator, unlike the (odd) $q=5$. Yet in both cases, the size of Fermi surface jumps from large value to 0 as $\mu$ passes through an insulating gap from below as shown in See Fig.1.

(IV) As $\mu$ increases to $\mu^{-}_{7}$, the onset of the $q=7$ Mott state $|7\rangle$ (or $n=(3/4)n_{s}$), each site is a Fock state $|7,\sigma\rangle_{\bf R}={\cal D}^{\dagger}_{K^{\prime}\sigma}|6\rangle_{\bf R}$, which is a spin-1/2 particle. (See Fig.3). Again, super-exchange will generate an antiferromagnet Heisenberg Hamiltonian for this system. The Mott state at $\mu^{-}_{7}$ is then $|7,\mu_{7}^{-}\rangle=\hat{O}^{\dagger}|6\rangle$, $\hat{O}^{\dagger}=\sum_{[\sigma]}\Phi_{\sigma_{1},\sigma_{2},..\sigma_{N}}\prod_{{\bf R}_{i}}\hat{\cal D}^{\dagger}_{{\cal K}^{\prime}\sigma_{i}}({\bf R}_{i})$, where $\Phi$ is the antiferromagnet ground state. However, as mean field $H_{S}$ is turned on with increasing $\mu$ (see ${\bf D}$), the ground state turns into a ferromagnet with spin state $\alpha$, and $|7,\mu_{7}^{+}\rangle=\prod_{\bf R}{\cal D}^{\dagger}_{{\cal K}^{\prime},\alpha}({\bf R})|6\rangle$. The particle excitations are obtained by adding fermions in the orthogonal spin states, ${\cal D}^{\dagger}_{{\cal K}^{\prime},\beta}({\bf k})|7,\mu_{7}^{+}\rangle$. These excitations are spin, valley, and layer polarized. They then lead to a non-degenerate Landau fan growing out from $\mu_{7}^{+}$ with $G=1$, as shown in Fig.1.

(V) The situation below CN is the particle-hole mirror of $({\bf I})$ to $({\bf IV})$, with the creation of the ${\cal D}$ particle replaced by the removal of the ${\cal U}$ particles. Reasoning similar to $({\bf I})-({\bf IV})$ then lead to the insulating structure shown in Fig. 3, which leads to the degeneracy pattern $(4,2,2,1)$ on both sides of CN as shown in Fig 1.

Connection to experiments: We now relate the results $({\bf I})$ to $({\bf V})$ to experimental results $({\bf 1})$ to $({\bf 6})$:

$({\bf 1^{\prime}})$ Since the charging energy is sample dependent, the absence of Mott state at $q=3$ (or $n=-n_{s}/4$) in current experiments (discussed in $({\bf 1})$) can occur if the capacitance $C$ is very large in the range of $\mu$ for the $q=3$ state. It remains to be seen whether this insulating state remains absent in future experiments. The missing insulating phase at $n=\pm n_{s}/4$ and $n=\pm 3n_{s}/4$ in Ref. Cao et al. (2018b, a) could also be explained by the same reason. The absence of insulating state at densities below CN reported in Ref.Sharpe et al. (2019) will then correspond to a very large $C$ for all $\mu$ below CN.

$({\bf 2^{\prime}})$The observations in $({\bf 2})$-$({\bf 4})$: Our model will give rise to the observation in Ref.Yankowitz et al. (2019) provided the insulating phase at $q=3$ is eliminated (see $({\bf 1^{\prime}})$). The lower degeneracy of 2 around CN observed in Ref.Sharpe et al. (2019) can be obtained from our model if the valley polarization $H_{V}$ that favors a particular valley state ${\cal K}$ is present for all $\mu$, just like the layer polarization $H_{L}$. This will freeze both layer and valley degrees of freedom, leaving spin is the only degeneracy. Our model also shows that all the universal features in $({\bf 2})$-$({\bf 4})$.

$({\bf 3^{\prime}})$ In our model, the $q=7$ (or $n=3n_{s}/4$) Mott state is a spin ferromagnet. Even though our model need not be the the reason for the observed ferromagnetic state Ref.Sharpe et al. (2019), (See (6)), it shows a pathway to ferrmagnetism in the insulating phases. Should spin symmetry be broken before the valley symmetry as one moves away from CN, ferromagnetism will be found in more than one insulating states.

$({\bf 4}^{\prime})$ Zeeman Effect: The wavefunctions of the degenerate states in a Landau fan generally involve mixing of layer, valley, and spin degrees of freedom. A magnetic field will deform these wavefunctions, but need not split the degeneracy if the energy of symmetry mixing still dominates over the Zeeman energy. The critical magnetic field at which the splitting of Landau fan occurs is therefore a measure of the the strength of symmetry mixing.

Final remarks: While we have made the point that many puzzling features of twisted bilayer graphene can be explained by a simple model that includes both strong correlation and mean field physics, the specific features of graphene invoked are only related to its the degrees of freedom of the superlattice, which is quite minimal. Our formulation, as it stands, is not restricted to graphene. It will be interesting to test these results in other twisted 2D materials.

Acknowledgments: The work is supported by MURI Grant FP054294-D, the NASA Grant on Fundamental Physics 1541824, and the OSU MRSEC Seed Grant.