Twisted Bilayer Graphene

Twisted bilayer graphene (TBG) consists of two graphene monolayers rotated by a small relative angle. Near the first magic-angle regime, the moiré hybridization strongly suppresses the velocity and bandwidth of bands near charge neutrality. Coulomb, phonon, strain, disorder, and substrate energy scales can then compete on unusually equal terms.

The key competition is easy to picture: an electron can propagate within either layer or tunnel into the other. A small twist brings the two Dirac cones close in momentum, so tunneling mixes states that would otherwise have nearly the same energy. We will construct that coupling, explain what makes a band “magic,” and then ask what the resulting low-energy states can tell us about interactions. Superconductivity and correlated insulators occur in selected devices; neither follows from the twist angle alone.

Required background. Graphene supplies the valley-resolved operators and material perturbations. Moiré Superlattices supplies twist-dependent lengths and reciprocal mismatches. van der Waals Heterostructures supplies interlayer tunneling, alignment and gate control. Brillouin Zones supplies the momentum-space folding used to organize the continuum model.

Twist-controlled bilayer graphene

The continuum model starts from the two monolayer Dirac Hamiltonians and retains the slowly varying part of interlayer tunneling. It applies near the graphene valleys, at small twist, where the moiré period is much larger than the atomic spacing. Atomic reconstruction changes its parameters and interactions can change its bands, so these are successive physical approximations, not interchangeable descriptions.

Twist Angle and Structure

Angle is a field, not only a fabrication setting

In the idealized construction, layer 1 is rotated by $+\theta/2$ and layer 2 by $-\theta/2$. Their Dirac points in a fixed valley are separated by

$$\begin{aligned} k_\theta &= 2K\sin\!\left(\frac{|\theta|}{2}\right), \\ K &= \frac{4\pi}{3a}, \end{aligned}$$

where $a\simeq0.246\,\mathrm{nm}$ is graphene’s lattice constant. The corresponding moiré period and cell area are derived at Moiré Superlattices. Around $\theta=1.1^\circ$, they are approximately

$$L_M\simeq12.8\,\mathrm{nm}, \qquad A_M\simeq142\,\mathrm{nm}^2.$$

Let $\mathbf d(\mathbf r)$ be the additional relative in-plane displacement from the ideal rotated configuration. To first order in its small gradient, the symmetric part is relative strain and the antisymmetric part is an additional local rotation:

$$\begin{aligned} \varepsilon_{ij}^{\mathrm{rel}} &= \frac{1}{2} \left( \partial_i d_j+\partial_j d_i \right), \\ \delta\theta(\mathbf r) &= \frac{1}{2} \left( \partial_x d_y-\partial_y d_x \right). \end{aligned}$$

Raman spectroscopy, scanning probes, diffraction, and quantum oscillations weight this inhomogeneous field differently. A transport-inferred angle is therefore not automatically the local angle in the region controlling a tunneling spectrum or a weak superconducting link.

Interlayer adhesion also reconstructs the nominally rigid pattern. Near small twist, low-energy AB and BA regions expand, AA regions shrink, and strain concentrates near domain walls. Nam and Koshino (2017) show how this relaxation modifies the bands. In continuum models it commonly reduces the effective AA-like tunneling amplitude $w_0$ relative to the AB/BA-like amplitude $w_1$. Corrugation, pressure, dielectric environment, and microscopic parameterization all affect those numbers.

Filling conventions must be stated

Let $n$ be carrier density measured from charge neutrality. The dimensionless filling used most often in TBG is

$$\nu=nA_M.$$

With spin and valley unresolved, each isolated conduction or valence active band accommodates four carriers per moiré cell. Thus

$$\begin{aligned} \nu&=0 &&\text{at charge neutrality}, \\ \nu&=\pm4 &&\text{at the active-band edges}. \end{aligned}$$

The conduction and valence active bands together contain eight one-electron states per cell: two band labels times two spins times two valleys. Describing $\nu=\pm2$ as “half filling” can be convenient, but it is ambiguous unless the reference band and sign convention are named. Density offsets, trapped charge, quantum capacitance, and spatially varying $A_M$ must be included before an integer label is treated as a microscopic fact.

Continuum Hamiltonian

One valley couples three nearby momenta

Smooth moiré tunneling transfers momenta much smaller than the separation between graphene’s two valleys. To leading order, the valleys $\xi=\pm1$ can therefore be modeled independently, while spin is a spectator in the one-electron Hamiltonian. In the layer-sublattice basis, one common continuum convention is

$$\mathcal H_\xi(\mathbf r) = \begin{pmatrix} h_{\xi,+\theta/2} & T_\xi(\mathbf r) \\ T_\xi^\dagger(\mathbf r) & h_{\xi,-\theta/2} \end{pmatrix}.$$

For $\mathbf p=-i\hbar\nabla$, the rotated monolayer block is

$$h_{\xi,\phi}=v_F\left[\xi\sigma_x(R_{-\phi}\mathbf p)_x+\sigma_y(R_{-\phi}\mathbf p)_y\right].$$

The Pauli matrices act on sublattice. A different Bloch-phase, valley, or layer-rotation convention moves signs and phases among the terms; spectra are unchanged when the basis is transformed consistently.

The leading interlayer tunneling has three harmonics,

$$T_\xi(\mathbf r) = \sum_{j=1}^{3} T_{\xi j} e^{i\xi\mathbf q_j\cdot\mathbf r},$$

Choose the spatial axes and relative layer Bloch phases so that $\mathbf q_j=k_\theta R_{\varphi_j}(0,-1)$, with $\varphi_j=2\pi(j-1)/3$. Thus all three transfers have magnitude $k_\theta$ and differ by $120^\circ$. A compatible matrix convention is

$$T_{\xi j}=w_0\sigma_0+w_1\left[\cos\varphi_j\,\sigma_x+\xi\sin\varphi_j\,\sigma_y\right].$$

$w_0$ and $w_1$ are effective continuum parameters, not bare hopping integrals at one atomic registry. The differences $\mathbf q_i-\mathbf q_j$ generate the moiré reciprocal lattice. Expanding a Bloch state in the momentum-space network connected by those vectors turns the continuum problem into a matrix eigenvalue problem; convergence must be checked against the plane-wave cutoff.

The dimensionless coupling organizes the magic regime

For the minimal model, the principal dimensionless parameter is

$$\alpha = \frac{w_1} {\hbar v_F k_\theta}.$$

Reducing $\theta$ increases $\alpha$. For equal tunneling amplitudes $w_0=w_1$, the first momentum-shell truncation of Bistritzer and MacDonald (2011), neglecting the small rotation of the intralayer spinors, gives

$$\frac{v_*}{v_F} = \frac{1-3\alpha^2}{1+6\alpha^2} =1-9\alpha^2+O(\alpha^4).$$

The denominator comes from normalizing the hybridized state; retaining only the numerator gives the wrong weak-coupling coefficient. This finite-shell expression is not an exact full-continuum magic-angle condition. In the distinct chiral model $w_0=0$, Tarnopolsky, Kruchkov, and Vishwanath (2019) establish exactly flat bands at discrete couplings, with the first near $\alpha=0.586$. Realistic $w_0$, relaxation, strain, remote hopping, and interactions alter the dispersion and particle-hole asymmetry.

This is why the magic angle is not a material constant. A quoted value near $1.05^\circ$ or $1.1^\circ$ silently assumes values of $v_F$, $w_0$, $w_1$, structural relaxation, and a definition of “magic.” Velocity suppression, minimum bandwidth, maximum remote-band isolation, strongest correlations, and optimal superconductivity need not occur at precisely the same angle.

Top: the first momentum shell contains one two-component sublattice state in one layer and three in the other. The dashed links are moiré reciprocal vectors. Bottom: the signed coefficient $(1-3\alpha^2)/(1+6\alpha^2)$ in this eight-state approximation, with equal tunneling amplitudes and unrotated intralayer spinors. Its zero at $1/\sqrt3$ is an estimate from this truncation. The exactly flat chiral model has different tunneling parameters and a first magic coupling near $0.586$; the two results must not be equated.

Magic-Angle Flat Bands

Flatness is a scale hierarchy

A narrow band slows motion, but other energy scales determine what that slowdown makes possible:

Scale	Physical role
$W$	Active-band bandwidth: residual kinetic dispersion.
$\Delta_+$, $\Delta_-$	Remote-band gaps: help determine whether projecting onto the active bands is justified.
$E_C$	Screened Coulomb scale before including wavefunction form factors.
$\Gamma$, $k_BT$	Disorder broadening and thermal energy: can obscure a narrow band or an ordered-state gap.

A narrow $W$ does not imply that the active bands are isolated, nor that a local-orbital Hubbard model is controlled. The TBG wavefunctions carry layer, sublattice, valley, and momentum-space texture; their projected interaction matrix elements depend on form factors and on screening by gates and remote bands.

For an active-band projection, a representative interaction is

$$H_{\mathrm{int}} = \frac{1}{2A} \sum_{\mathbf q} V(\mathbf q) :\bar\rho(-\mathbf q)\bar\rho(\mathbf q):,$$

with projected density

$$\bar\rho(\mathbf q) = \sum_{\mathbf k,m,n,f} \Lambda_{mn}^{f}(\mathbf k,\mathbf q) c^\dagger_{\mathbf k+\mathbf q,mf} c_{\mathbf k,nf}.$$

Here $A$ is sample area, $m,n$ label active bands and $f$ labels spin and valley flavors; intervalley transfer is neglected. To make the overlap precise, write normalized Bloch states as $\psi_{\mathbf k n}(\mathbf r)=A^{-1/2}e^{i\mathbf k\cdot\mathbf r}u_{\mathbf k n}(\mathbf r)$ and define

$$\Lambda_{mn}^{f}(\mathbf k,\mathbf q) =\langle\psi_{\mathbf k+\mathbf q,mf}|e^{i\mathbf q\cdot\mathbf r}|\psi_{\mathbf k,nf}\rangle.$$

Momenta outside the mini Brillouin zone are folded back with the corresponding Bloch-state phase transformation. For transfers that do not require folding, this is the cell-averaged overlap of periodic parts. The colons remove the self-interaction; the neutralizing background also removes the uniform Hartree term. Discarding $\Lambda$ discards the quantum geometry and much of the sublattice structure that distinguish TBG from a featureless narrow band.

Flat bands are measured through complementary windows

Scanning tunneling spectroscopy can resolve narrow peaks, local gaps, symmetry breaking, and spatial inhomogeneity, but the tunneling matrix element and tip electrostatics affect intensity. Photoemission measures occupied dispersion and replicas but averages over a finite region and may operate at temperatures different from transport. Compressibility accesses $d\mu/dn$, while Landau fans and quantum oscillations constrain degeneracies and Fermi-surface areas. None alone reconstructs the complete interacting spectral function.

Interactions also reshape the bands. Hartree potentials depend strongly on filling and can broaden, invert, or pin portions of the active manifold; exchange can split flavors and alter topology. Agreement between a noninteracting continuum band and one spectrum at one filling is therefore not a license to reuse that band unchanged across the phase diagram.

Correlated Insulating States

Integer filling is a coordinate, not a diagnosis

The 2018 observation by Cao et al. of an insulating state near $\nu=-2$ established that a partially filled narrow manifold could become strongly resistive. Subsequent devices found insulating or semimetallic behavior at several integer fillings, often with Landau-fan resets, compressibility anomalies, or flavor-polarized states. The detailed sequence depends on twist inhomogeneity, strain, screening, substrate alignment, and magnetic or displacement fields.

Calling every such state a Mott insulator is too specific. Competing explanations include:

• interaction-driven flavor polarization that opens a Slater-like or exchange gap;
• intervalley-coherent, valley-polarized, spin-polarized, or sublattice-polarized order;
• a Chern insulator or other topological mass;
• charge order favored by nonlocal interactions;
• a one-electron gap enhanced by interactions;
• disorder localization or percolation through an inhomogeneous gap landscape.

The Mott Insulators page gives the general diagnostic standard. In TBG, the strongest case combines activated transport with a chemical-potential jump or incompressible interval, spectroscopy of a gap and spectral-weight transfer, a consistent broken-symmetry signature, and a model that includes the measured structure.

The useful thermodynamic quantity is $\partial\mu/\partial n$. At zero temperature, a charge gap appears as a jump in $\mu$ when $n$ crosses the insulating filling, or equivalently as a density plateau over a range of $\mu$. These descriptions concern different horizontal axes. The signed carrier density $n$ is measured relative to neutrality here; inserting it into $n^2\partial\mu/\partial n$ would spuriously force that “inverse compressibility” to vanish at neutrality. Report the derivative itself when using this carrier convention.

Negative $\partial\mu/\partial n$ can occur from exchange. Stability of the gated device depends on the total energy, including geometric capacitance and long-range electrostatics, so this electronic contribution alone does not establish an instability.

Flavor cascades expose interaction-driven reconstruction

In an approximate description, spin rotations and valley charge conservation produce a near-fourfold flavor space. As filling changes, exchange can favor sequential flavor polarization. Compressibility “sawteeth,” chemical-potential resets, changes in Landau-fan degeneracy, and tunneling spectra that reorganize at integer fillings support such cascade physics. Wong et al. (2020) directly resolve the spectroscopic reorganization across integer fillings.

Spin rotations and valley charge conservation are weaker assumptions than a full $SU(4)$ flavor symmetry. Lattice-scale intervalley scattering can violate valley conservation, whereas smooth strain or substrate potentials can select spatial and valley-dependent structures while preserving it. Zeeman coupling acts on spin; phonons and remote bands modify the interactions. Which symmetry survives depends on the terms retained. A flavor count inferred from a Landau fan is also not a direct image of an order parameter: magnetic breakdown, small pockets, and reconstruction can alter the fan.

Entropy supplies another diagnostic: Rozen et al. (2021) measure an excess electronic entropy and its suppression by an in-plane field. Re-entrant transport and thermodynamic behavior consistent with a Pomeranchuk-like effect indicate that some correlated states carry substantial spin or valley entropy. This supports local-moment-like physics in parts of the phase diagram, but it does not by itself prove a particular lattice model or zero-temperature order.

Superconductivity

What the experiments establish

Cao et al. (2018) first reported superconducting domes adjacent to a correlated state in near-magic TBG. Pressure, electrostatic tuning, and subsequent devices showed that superconductivity can also appear when a nearby correlated insulator is weak or absent. In particular, Saito et al. (2020) observe superconductivity surviving farther from the flat-band condition than the correlated insulators. The data therefore rule out the universal claim that superconductivity must arise by doping one fixed Mott parent.

Different measurements probe different parts of superconductivity. A resistance drop can reflect fluctuations or current redistribution; zero resistance establishes a low-resistance path but need not establish a homogeneous condensate. Critical-current and magnetic-field dependence, a pairing gap, Josephson response and finite phase stiffness add complementary constraints. Moiré Superconductivity develops these probes and their limitations.

For a homogeneous two-dimensional phase with stiffness $\rho_s$ in energy units, the ideal Berezinskii–Kosterlitz–Thouless jump is

$$k_BT_{\mathrm{BKT}} = \frac{\pi}{2} \rho_s\!\left(T_{\mathrm{BKT}}^{-}\right).$$

Finite area, disorder, and a distribution of local transition temperatures smear this relation. It is a stringent consistency test, not a universal fitting curve.

Pairing mechanism and gap structure remain open

Electronic fluctuations and phonons can both generate attractive channels, and screening changes their competition. Quantum geometry can contribute to phase stiffness even when the dispersion is almost flat; this is a statement about the response of a paired state, not a pairing interaction by itself. These distinctions separate three questions: why pairs form, what symmetry their gap has, and whether their phases remain coherent.

Tunneling and transport have reported anisotropic or nodal behavior in some devices, while other measurements can be fit more simply. Broken rotational symmetry may originate in the normal state, the superconducting state, strain, or their coupling. Pauli-limit comparisons are likewise not decisive without orbital depairing, $g$-factor, spin-orbit, and inhomogeneity analyses.

Chen et al. (2024), using micrometre-scale photoemission, observed nearly equally spaced flat-band replicas in superconducting, hBN-unaligned near-magic devices, consistent with strong coupling to an optical phonon. The same study explicitly did not establish that this coupling is the principal pairing mechanism. Tanaka et al. (2025), using kinetic-inductance measurements, found an anisotropic temperature dependence and a superfluid stiffness larger than a conventional-dispersion estimate, strengthening the case that quantum geometry matters. These are important constraints on a theory, not closure of the mechanism debate.

Topology and Valley Physics

Approximate symmetries protect the neutral Dirac points

Neglecting intervalley scattering gives an approximate valley $U(1)$ conservation law. Physical time reversal exchanges the two valleys. Within one valley, the antiunitary combination $C_{2z}\mathcal T$ leaves momentum fixed and, in a suitable spinless basis, squares to $+1$. Together with threefold rotation, it protects the active-band Dirac crossings at charge neutrality in the ideal continuum model.

The symmetry and Wannier-obstruction analysis of Zou et al. (2018) underlies this restriction: the two Dirac cones within one valley have the same chirality. Consequently, a two-band tight-binding model cannot simultaneously use exponentially localized Wannier orbitals and preserve all of the relevant valley, spatial, and antiunitary symmetries in an onsite form. For the ideal flat-band manifold, Po and collaborators (2019) construct larger symmetric models and show how a suitable set of additional trivial bands removes this Wannier obstruction. “Fragile” describes that dependence on the chosen band subspace; it does not mean any extra band will remove the obstruction or that a Chern band becomes trivial in the same way.

This statement is not “each unbroken TBG band has a nonzero Chern number.” With $C_{2z}\mathcal T$ intact, the valley bands contain protected Dirac crossings and do not form individually isolated Chern bands. The obstruction concerns the connected two-band subspace and its symmetry-compatible localization.

Breaking the protecting symmetry can reveal Chern bands

Alignment with hexagonal boron nitride can break $C_{2z}$ by making the two graphene sublattices inequivalent. The neutral Dirac points can then gap, and valley-resolved bands may acquire nonzero Chern numbers. The values and signs depend on mass conventions, relaxation, remote-band hybridization, and which active band remains isolated.

Interactions can select one valley or flavor and convert valley Berry curvature into a net orbital magnetization and Hall response. With the volume convention $\mathcal A=i\langle u\vert\nabla_{\mathbf k}u\rangle$ and $j_x=\sigma_{xy}E_y$, the strongest integer quantum anomalous Hall diagnosis requires

$$\sigma_{xy} = -C\frac{e^2}{h}, \qquad \sigma_{xx}\rightarrow0,$$

at zero applied magnetic field, together with reproducible magnetization reversal and a well-defined insulating gap. Hysteresis or a large anomalous Hall signal supports a time-reversal-breaking magnetic state, but does not by itself identify an orbital microscopic mechanism or establish exact topological quantization.

Near $\nu=3$, hBN-aligned near-magic devices have shown first a large hysteretic anomalous Hall effect and later an intrinsic quantized anomalous Hall state, as measured by Serlin et al. (2020). Magnetic-field studies have also exposed sequences of interaction-driven Chern insulators. These results establish that topology, valley polarization, and interactions can cooperate in TBG; they do not imply that every integer correlated state is topological.

Further questions

A quantitative account must explain why comparable devices can favor different flavor orders and superconducting states. Local strain, screening and substrate alignment provide physical ways to compare them. Pairing symmetry and microscopic attraction remain distinct questions, and remote bands can affect both interactions and topology. Quantum Matter Research follows the evolving research questions; Correlated Insulators in Moiré Systems compares the candidate insulating mechanisms across platforms.

Exercises

Exercise 1: density scale of a near-magic device

For graphene lattice constant $a=0.246\,\mathrm{nm}$ and twist angle $\theta=1.10^\circ$, use

$$L_M = \frac{a}{2\sin(\theta/2)}$$

to estimate $L_M$, $A_M=(\sqrt3/2)L_M^2$, the density for one carrier per cell, and the density at $\nu=4$.

Solution

With $\theta/2=0.55^\circ$,

$$\begin{aligned} L_M &\simeq \frac{0.246\,\mathrm{nm}} {2\sin(0.55^\circ)} \\ &\simeq 12.8\,\mathrm{nm}. \end{aligned}$$

Therefore

$$\begin{aligned} A_M &= \frac{\sqrt3}{2}L_M^2 \simeq 142\,\mathrm{nm}^2, \\ \frac{1}{A_M} &\simeq 7.0\times10^{11}\,\mathrm{cm}^{-2}, \\ n_{\nu=4} &= \frac{4}{A_M} \simeq 2.8\times10^{12}\,\mathrm{cm}^{-2}. \end{aligned}$$

These are ideal geometric values. A filling calibration should use the locally measured moiré area and electrostatic offsets.

Exercise 2: a model-dependent magic angle

In the chiral model $w_0=0$, take $w_1=110\,\mathrm{meV}$, $v_F=0.90\times10^6\,\mathrm{m\,s^{-1}}$, $a=0.246\,\mathrm{nm}$, and first-magic coupling $\alpha_1=0.586$. Using $k_\theta\simeq K\theta$ at small angle and $\hbar v_F\simeq0.592\,\mathrm{eV\,nm}$, estimate the corresponding $\theta$.

Solution

First,

$$K = \frac{4\pi}{3a} \simeq 17.0\,\mathrm{nm}^{-1}.$$

Solving $\alpha_1=w_1/(\hbar v_FK\theta)$ gives

$$\begin{aligned} \theta &\simeq \frac{0.110\,\mathrm{eV}} {(0.586)(0.592\,\mathrm{eV\,nm}) (17.0\,\mathrm{nm}^{-1})} \\ &\simeq 0.0186\,\mathrm{rad} \simeq 1.07^\circ. \end{aligned}$$

Changing $v_F$ or $w_1$ changes this estimate. That sensitivity is precisely why “the magic angle” must be accompanied by a parameter convention.

Exercise 3: count the active states

Explain why the combined valence and conduction active manifold contains eight states per moiré cell, yet the conventional filling range is $-4\leq\nu\leq4$.

Solution

There are two active band labels, one valence-like and one conduction-like. Each has two spins and two valleys, so

$$2_{\mathrm{band}} \times 2_{\mathrm{spin}} \times 2_{\mathrm{valley}} = 8$$

states per cell. Charge neutrality places the valence-like active band filled and the conduction-like active band empty. Removing all four valence states reaches $\nu=-4$; adding all four conduction states reaches $\nu=+4$. The eight-state manifold is therefore traversed over an eight-carrier interval centered at $\nu=0$.

Exercise 4: identify a thermodynamic charge gap

At zero temperature, a homogeneous device has fixed density $n_0$ while the chemical potential varies from $\mu_-$ to $\mu_+$. What is its charge gap? Sketch verbally the same behavior on a plot of $\mu$ versus $n$. Does this distinguish a Mott insulator from a band insulator?

Solution

The addition and removal thresholds are separated by $\Delta_c=\mu_+-\mu_-$. The density-versus-chemical-potential curve has a plateau at $n_0$; the chemical-potential-versus-density curve instead jumps from $\mu_-$ to $\mu_+$ as density crosses $n_0$. Finite temperature and disorder round these ideal features. Both a band insulator and a Mott insulator can have a charge gap. To distinguish them, compare the one-electron band structure with interaction-dependent spectroscopy, symmetry breaking and tuning of the gap.

Exercise 5: BKT consistency

Assuming the ideal jump relation, estimate the phase stiffness immediately below a transition at $T_{\mathrm{BKT}}=1.5\,\mathrm K$. Use $k_B=0.08617\,\mathrm{meV\,K^{-1}}$.

Solution

Rearranging the jump condition,

$$\begin{aligned} \rho_s(T_{\mathrm{BKT}}^-) &= \frac{2}{\pi}k_BT_{\mathrm{BKT}} \\ k_BT_{\mathrm{BKT}} &= 0.1293\,\mathrm{meV}, \\ \rho_s(T_{\mathrm{BKT}}^-) &= \frac{2}{\pi} (0.1293\,\mathrm{meV}) \\ &\simeq 0.082\,\mathrm{meV}. \end{aligned}$$

A measured stiffness far below this value would be inconsistent with a homogeneous ideal BKT transition at $1.5\,\mathrm K$. Finite-size and inhomogeneous broadening should be modeled before drawing a stronger conclusion.

Exercise 6: valley cancellation and anomalous Hall response

Suppose the isolated active band in valley $K$ has Chern number $+1$ and its time-reversed partner in $K'$ has Chern number $-1$. For one spin sector, compare both bands being completely filled with only the $K$ band being filled. Explain separately what time reversal implies for equal partial occupations.

Solution

Filling both complete bands gives

$$C_{\mathrm{tot}} = (+1)+(-1) = 0,$$

so their quantized charge Hall contributions cancel. Equal partial occupations related by time reversal also cancel the intrinsic Hall response, but one cannot then assign each contribution a full-band Chern integer. If interactions select one filled spin-resolved $K$ flavor and leave the corresponding $K'$ flavor empty, that contribution has

$$\sigma_{xy} = -\frac{e^2}{h}.$$

In this clean-band example, filling the band with the chemical potential in a spectral gap gives a quantized Hall response, provided no other occupied bands contribute a compensating Chern number. With disorder, a mobility gap containing localized bulk states can also support a Hall plateau; an empty spectral interval is not necessary. The plateau requires vanishing bulk dissipative conduction in the ideal zero-temperature limit; see Bellissard, van Elst and Schulz-Baldes (1994).

References

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