Correlated Insulators in Moiré Systems

A correlated insulator in a moiré system is a charge-incompressible or charge-localized phase at a commensurate moiré filling whose insulating behavior requires electron interactions beyond a calibrated independent-particle miniband description. This definition deliberately separates an observation from its interpretation. A resistance peak at an integer or fractional filling is evidence for inhibited transport; it is not yet proof of a thermodynamic gap, a Mott state, a Wigner crystal, or any particular order.

At one carrier per moiré cell, repulsion can suppress local charge fluctuations while leaving spin or valley degrees of freedom active. At one-third filling, longer-range repulsion can instead select a repeating pattern of occupied cells. These examples motivate the central question: does insulation arise from local charge constraints, spatial order, a reconstructed band structure, or a combination?

Required background. Moiré Superlattices supplies the unit-cell and filling scales. Flat Bands supplies the distinction between suppressed bandwidth and interaction-driven order. Mott Insulators supplies charge-incompressibility criteria and the tests that distinguish a Mott mechanism from other causes of insulating transport.

Interaction-driven insulation in moiré bands

We will first translate gate density into carriers per cell, then compare local constraints with enlarged-period charge order. Thermodynamic, transport and imaging measurements answer different parts of this question. The general charge-gap definition is developed in Mott Insulators; here we use it to interpret moiré devices.

Filling-Controlled Phases

Filling is a calibrated, signed quantity

For a device with local carrier density $n$, reference density $n_{\mathrm{ref}}$, and moiré-cell area $A_M$, define

$$\nu = \bigl(n-n_{\mathrm{ref}}\bigr)A_M.$$

The sign, reference, and active-manifold capacity must accompany the number. In one convention $\nu=0$ is charge neutrality; in another it is an empty conduction miniband or a filled valence miniband. A four-flavor manifold may be called full at $\nu=4$, whereas a normalized convention calls the same point filling one. “Half filling” is therefore ambiguous unless the number of active one-particle states per cell is stated.

For a triangular moiré lattice of period $L_M$, the cell area is $A_M=(\sqrt3/2)L_M^2$.

The inferred density of one carrier per cell is only as accurate as the local twist, strain, reconstruction, and electrostatic calibration. A small spatial variation of $L_M$ produces a doubled relative variation in $A_M$ and hence shifts the density assigned to a fixed $\nu$.

Integer filling does not identify a mechanism

At integer $\nu$, at least three conceptually different gaps are possible.

1. A band gap appears because an integer number of one-particle subbands is filled.
2. A Mott-like gap appears because changing the local integer occupation costs interaction energy even though the noninteracting active band would be partially filled.
3. A flavor-ordering gap appears when interactions polarize or coherently mix internal states and thereby reconstruct the bands.

These are limiting descriptions, not always sharply separated phases. A flavor-polarized state can also have strong local charge correlations; a lattice relaxation can produce a one-body gap that interactions enhance; and a Chern insulator can arise after flavor polarization of a topological band.

Fractional filling gives a stronger hint of translation-breaking or topological order, but it is still not decisive. A larger structural supercell, single-particle miniband splitting, disorder-induced puddles, or a fractional Chern phase can all produce nontrivial fractions. The correct denominator is the capacity of the experimentally established primitive cell, not an assumed ideal cell.

The tuning space is multidimensional

Gate voltages can vary density, displacement field, dielectric screening, and layer polarization at once. Twist angle changes the bandwidth, wave functions, topology, and relaxation. Pressure changes tunneling and structure. Magnetic field couples to spin, orbital magnetization, and Chern number. A useful phase diagram therefore reports the experimentally controlled coordinates rather than compressing them into a single nominal ratio such as $U/W$.

Mott-Like and Wigner-Like Pictures

A minimal extended lattice model

When isolated moiré bands admit sufficiently localized orbitals, a useful starting point is

$$\begin{aligned} H_{\mathrm{ext}} &= H_t+H_V+H_{\mathrm{local}}+H_{\mathrm{rest}}, \\ H_t &= -\sum_{ij}\sum_{ab} t_{ij}^{ab} c_{ia}^{\dagger}c_{jb} , \\ H_V &= \frac12 \sum_{ij} V_{ij}n_i n_j , \\ H_{\mathrm{local}} &= \sum_i H_i^{\mathrm{local}}, \\ H_{\mathrm{rest}} &= H_{\mathrm{exchange}} + H_{\mathrm{assisted}} +\cdots . \end{aligned}$$

Assume $t_{ij}^{ab}=(t_{ji}^{ba})^*$ so the hopping is Hermitian. Here $i,j$ label moiré orbitals or sites, $a,b$ label spin, valley, layer, or orbital flavors, and $n_i=\sum_a c_{ia}^{\dagger}c_{ia}$. The local term can include intra-orbital repulsion, interorbital repulsion, Hund-like exchange, and flavor anisotropy. Nonlocal exchange, density-assisted hopping, pair hopping, and coupling to remote bands may be important. In this partition $V_{ii}=0$: onsite interactions are assigned to $H_{\mathrm{local}}$, while $H_V$ contains intersite density interactions. For an unscreened long-range interaction, include the neutralizing background; equivalently use density deviations with the associated one-body terms absorbed into the potential. The Coulomb matrix $V_{ij}$ depends on Wannier extent, gates, dielectric interfaces, and dynamical screening. Wu et al. (2018) construct a generalized triangular-lattice Hubbard description for a specified class of TMD moiré bands; this is a model-building example, not an identification of every moiré insulator.

This Hamiltonian is a hypothesis whose basis and truncation must be validated. Topological or fragile-topological active bands may obstruct a symmetry-preserving, exponentially localized, few-orbital representation. Even without an obstruction, broad moiré Wannier functions make neighboring interaction matrix elements comparable enough that an onsite-only model can fail.

Mott-like localization

In the simplest site-centered picture, each effective site carries a commensurate integer occupation. Charge motion creates an energetically costly empty–double-occupation pair, while spin or valley degrees of freedom remain at lower energy. The charge density need not enlarge the moiré translation cell.

A Mott-like interpretation becomes compelling when the noninteracting active bands would be metallic, charge addition costs a finite energy, and local moments persist above the ordering temperature. These observations connect the gap to a local charge constraint. A gap that appears only when an order parameter develops may instead be largely caused by band reconstruction. The two effects can coexist, so temperature and bandwidth tuning are more informative than the label alone.

The canonical many-body charge-gap definition and its distinction from spectral, optical, and transport gaps are developed in Mott Insulators.

Generalized Wigner crystallization

A continuum Wigner crystal minimizes long-range Coulomb energy at low density by spontaneously forming a lattice. A generalized Wigner crystal in a moiré system instead is pinned to the pre-existing moiré lattice and forms a larger commensurate charge pattern. On a triangular lattice at $\nu=1/3$, for example, occupying one of three sublattices produces a $\sqrt3\times\sqrt3$ pattern with one carrier per enlarged cell.

In a classical occupancy limit,

$$E_{\mathrm{C}}[\{n_i\}] = \frac12 \sum_{i\ne j} V_{ij}n_i n_j - \mu\sum_i n_i.$$

Padhi, Chitra, and Phillips (2021) analyze Wigner crystallization in an underlying moiré potential, including the role of lattice pinning in the charge gap. In a lattice description, longer-range terms select among stripes, bubbles, triangular crystals, and other commensurate arrangements. Quantum hopping melts or dresses these patterns. Because the moiré potential already breaks continuous translation symmetry, “Wigner crystal” does not mean the same symmetry breaking as in a translationally invariant electron gas; the experimentally testable statement is enlarged-period charge order driven substantially by nonlocal repulsion.

A charge structure factor can diagnose that enlargement:

$$\begin{aligned} S_{\mathrm c}(\mathbf q) &= \frac{1}{N_M} \sum_{ij} e^{i\mathbf q\cdot(\mathbf R_i-\mathbf R_j)} C_{ij}, \\ C_{ij} &= \left\langle \delta n_i\,\delta n_j \right\rangle, \\ \delta n_i &= n_i-\bar n, \qquad \bar n=\frac{1}{N_M}\sum_i\langle n_i\rangle . \end{aligned}$$

Subtracting the spatially averaged filling retains a static charge modulation. Subtracting each local mean $\langle n_i\rangle$ instead would define a connected fluctuation correlator and remove the elastic Bragg contribution in a selected symmetry-broken state.

Top: occupy one of the triangular lattice’s three sublattices. The dashed charge cell has three times the primitive area. Bottom: for a perfect static pattern on periodic lattices, $S_{\mathrm c}(\mathbf K)=N_M/9$. Independent random occupations with probability $1/3$ instead give an ensemble mean $2/9$ at nonzero momentum. A peak that grows with sample size distinguishes coherent long-range order from local density fluctuations. These are calculated ideal limits, not measured data.

Bragg scaling of $S_{\mathrm c}(\mathbf Q)$ at a wavevector $\mathbf Q$ not equivalent to a primitive moiré reciprocal vector supports long-range charge order. A gap at fractional filling without such evidence should be called a fractional correlated insulator until its order or topology is established.

The boundary can be a crossover

“Mott” and “Wigner” emphasize different pieces of the same extended-interaction problem. At one carrier per moiré site, onsite avoidance may dominate while nonlocal repulsion reshapes the charge distribution inside each cell. At fractional filling, a charge crystal can still have strong local suppression of double occupancy. Broad orbitals may shift charge continuously between site-centered and interstitial regions.

Accordingly, labels such as Mott–Wigner state can be useful if the measured regime genuinely combines local constraints and nonlocal charge order. The name alone does not establish either contribution. The practical discriminator is whether an enlarged charge unit cell is required and observed, together with how the gap evolves when screening, bandwidth, and filling are tuned independently.

Spin and Valley Order

Charge localization leaves internal degrees of freedom that can order at scales far below the charge gap. For a spin–valley spinor $c_i$, convenient local operators are

$$\begin{aligned} \mathbf S_i &= \frac12 c_i^\dagger \left(\boldsymbol{\sigma}\otimes\tau_0\right)c_i, \\ \mathbf T_i &= \frac12 c_i^\dagger \left(\sigma_0\otimes\boldsymbol{\tau}\right)c_i. \end{aligned}$$

$\mathbf S_i$ acts on spin and $\mathbf T_i$ on valley. Valley polarization has $\langle T_i^z\rangle\ne0$. Intervalley coherence instead uses an off-diagonal order parameter, for example

$$\Phi_{\mathrm{IVC}} = \left\langle c_{i,K}^{\dagger}c_{i,K'} \right\rangle ,$$

which breaks an approximate valley $U(1)$ symmetry when intervalley scattering is weak. Layer polarization and combined spin–valley order require analogous operators. The symmetry actually available is material dependent: graphene often has approximate spin and valley flavor symmetries, while spin–orbit coupling in TMD valence bands can lock spin to valley and make the effective exchange strongly anisotropic.

For one spin-degenerate orbital per site, real nearest-neighbor hopping, one spin-1/2 fermion per site, and a stable singly occupied regime with $U-V_1\gg |t|$, the simplest virtual-hopping estimate is

$$J_1 \simeq \frac{4|t|^2}{U-V_1}.$$

Here $J_1$ multiplies $\mathbf S_i\cdot\mathbf S_j$ in an SU(2)-symmetric spin Hamiltonian. The intermediate empty–doubly occupied pair loses the original bond energy $V_1$ and gains $U$, giving the denominator; contributions from other neighbors cancel only in a uniform environment. This is not a generic moiré exchange formula, and a multi-flavor permutation Hamiltonian uses a different normalization. Triangular geometry frustrates antiferromagnetism; complex hopping phases, multiple orbitals, direct exchange, Hund coupling, and longer-range virtual processes generate anisotropic, ring-exchange, and chiral terms. If $U-V_1$ is not the dominant excitation energy, even the denominator changes.

The hierarchy of transition scales is informative. A charge gap can open at a temperature much higher than spin or valley ordering, leaving an intermediate local-moment regime. Conversely, an insulating gap that appears only with a flavor-order parameter may be primarily Slater-like. Magnetic-field dependence alone is insufficient because Zeeman coupling, orbital moments, Chern magnetization, and field-dependent screening can all shift a gap.

Experimental Signatures

Establish the moiré cell and filling first

Microscopy or diffraction determines the local moiré period and reconstruction; electrostatics determines the density reference and gate response. Landau fans and spectroscopy help determine the active-band capacity. These measurements should sample the region that carries the measured response, since local twist and strain can make different parts of a device reach a given filling at different gate voltages. A feature repeating every nominal $n_M=1/A_M$ suggests commensurability, but does not determine whether the electronic cell has enlarged.

Thermodynamic incompressibility is the central charge test

At zero temperature, a homogeneous charge-gapped state has fixed density over a finite interval of chemical potential. On a plot with density as the horizontal axis, this becomes a jump in chemical potential. Finite temperature and spatial variations round both descriptions. Local single-electron transistors, penetration-field capacitance, gate capacitance, and calibrated optical sensors can access related quantities. For this carrier-density convention, use the inverse density response

$$\chi_n^{-1} = \frac{\partial\mu}{\partial n},$$

with a singular or strongly enhanced response at a zero-temperature charge gap. Real devices add geometric capacitance, electrostatic image charges, finite temperature, inhomogeneous broadening, and negative electronic compressibility in nearby metallic regions. These contributions must be modeled before converting a measured capacitance dip or peak into an energy gap.

In TMD moiré heterostructures, excitons in the active or a nearby sensing layer can respond sharply to electronic screening. Such optical signatures have revealed integer and fractional incompressible states even where direct electrical contacts are difficult. The exciton is a transducer: its energy or oscillator strength must be calibrated against charge response, and it does not by itself identify the electron ordering pattern.

Transport locates phases but rarely names them

A thermally activated channel is often fitted as

$$\sigma_{xx}(T) \approx \sigma_\star \exp\!\left[ -\frac{\Delta_{\mathrm{tr}}}{2k_{\mathrm B}T} \right].$$

The factor of two assumes symmetric activation of mobile particles and holes; variable-range hopping, percolation, edge conduction, contact resistance, and a temperature-dependent prefactor change the interpretation. $\Delta_{\mathrm{tr}}$ is a transport scale, not automatically the many-body charge gap.

The most reliable transport claims reproduce filling locations across contacts and thermal cycles, map the response over displacement field and magnetic field, and compare it with a thermodynamic probe in the same density range. A resistance maximum that follows a high-resistance device region or disappears after changing contacts is weak bulk evidence.

Determine the broken symmetry

Charge order can be tested by scanning tunneling microscopy, local compressibility, microwave impedance microscopy, diffraction-sensitive methods, or spatially resolved optical response. The required observation is a reproducible modulation with a wavevector and phase relation not already imposed by the primitive structure.

Spin and valley order require probes coupled to those degrees of freedom: magnetization, susceptibility, circular dichroism, Kerr rotation, spin-resolved spectroscopy, field-angle dependence, or collective excitations. Hysteresis can support ferromagnetism but can also arise from domains and slow electrostatics. An order assignment should identify its conjugate field, symmetry, domain structure, and transition or crossover scale.

For a gapped two-dimensional phase, the Středa relation connects the density change at fixed chemical potential to the Hall response. With $e>0$ and signed $C$ chosen consistently with the carrier and Hall conventions,

$$\left. \frac{\partial n}{\partial B} \right|_{\mu} = C\,\frac{e}{h}.$$

Here $C$ is integer for an integer Chern phase; fractional Hall phases can have a rational Hall coefficient instead. A resistance-peak trajectory need not hold chemical potential fixed, so its slope is not automatically this derivative. Together with a robust Hall plateau and vanishing longitudinal response, an integer slope supports Chern number $C$. A sloped line in a density–field map by itself is not quantization; contact mixing, parallel channels, and incomplete bulk insulation must be excluded. Fractional Hall topology requires correspondingly stronger evidence, including fractional quantization and many-body gap systematics.

Measurements are most informative in combination. A chemical-potential jump establishes a charge gap; spatial imaging distinguishes uniform filling from charge order; magnetic and optical probes constrain flavor order; Hall measurements test topology. A model should explain these observations with one physically consistent parameter set and their uncertainties, rather than requiring unrelated adjustments for each measurement.

Relation to Hubbard Models

Moiré tunability makes Hubbard language tempting

Moiré periods enlarge electronic orbitals and reduce kinetic scales. Gates can alter dielectric screening, displacement fields can change orbital character, and twist can tune hopping. This makes the ratio of interaction to bandwidth unusually controllable. It does not make every moiré material a one-band Hubbard model.

The active orbitals must reproduce the relevant dispersion and wavefunctions. Their spatial extent sets the interaction matrix elements, while the filling convention fixes the model occupancy. Comparing a calculated optical spectrum or capacitance with a measurement also requires the corresponding observable and electrostatic environment, not only the many-body energy levels.

The hierarchy is often better described as an extended multi-flavor Hubbard family. A triangular TMD miniband may motivate a one-orbital triangular model in a restricted regime. Twisted graphene generally requires valley, sublattice, topology, remote-band, and interaction-form-factor care. At fractional filling with sizable $V_{ij}$, a lattice-gas or charge-order description may be more transparent than an onsite Hubbard reduction.

Parameters are linked to the device

With the displayed convention $H_t=-\sum t_{ij}^{ab}c_{ia}^\dagger c_{jb}$, Wannier matrix elements schematically obey

$$\begin{aligned} t_{ij}^{ab} &= -\langle w_{ia}\vert H_0\vert w_{jb}\rangle, \\ V_{ij}^{ab} &= \iint d\mathbf r\,d\mathbf r'\, \rho_{ia}(\mathbf r) W(\mathbf r,\mathbf r') \rho_{jb}(\mathbf r'). \end{aligned}$$

The minus in $t$ matches $H_t$. Here $H_t$ contains the entire projected one-body matrix, including diagonal onsite energies through $-t_{ii}^{aa}$; if an onsite one-body term is instead written separately, remove those matrix elements from $H_t$ so they are counted once. $H_{\mathrm{local}}$ in this partition contains local interactions. Here $W$ is the screened interaction appropriate to the chosen low-energy partition, and $\rho_{ia}=|w_{ia}|^2$ only in the simplest density–density estimate. Metallic gates change the long-distance tail; layer-polarized wave functions change both hopping and interactions; and screening by remote bands can be frequency dependent. Quoting $U$ from $e^2/(\epsilon L_M)$ is a scale estimate, not a complete downfolding.

A qualitative sequence of insulating fillings is a useful first test. Predicting gaps, ordering patterns and their response to screening with the same parameters is substantially more restrictive. Hubbard Physics in Materials develops this comparison between a device and its effective model.

Representative Experimental Lessons

Twisted graphene

The 2018 observation by Cao et al. of a half-filled insulating state near the first magic angle established that narrow graphene moiré bands can host interaction-driven phases. Subsequent compressibility, transport, and local-probe studies revealed cascades of flavor transitions, substantial device dependence, and close competition among correlated insulators, Chern states, superconductivity, and compressible metals. The early “Mott-like” analogy was productive, but a literal single-orbital Mott assignment is not universal: topology, flavor polarization, remote bands, strain, and screening all matter.

Semiconductor moiré systems

In aligned WSe$_2$/WS$_2$, Tang et al. (2020) probed charge and magnetic response near a Mott-like filling, while Regan et al. (2020) optically detected insulating states at one hole per cell and at one-third and two-thirds filling. Related heterostructures revealed further incompressible states through optical sensing, capacitance, and transport. Integer states can realize Mott-like local constraints; fractions such as $1/3$ and $2/3$ motivate generalized Wigner or charge-order pictures because long-range repulsion can favor enlarged cells. The real-space imaging of H. Li et al. (2021) resolves charge patterns at these fractions and at one-half filling, adding spatial evidence to earlier gap-based assignments. Later observations of stripe-like patterns, tunable charge gaps, and continuous metal–insulator transitions demonstrate both the power and the nonuniversality of a single minimal model.

TMD homobilayers add twist-tunable bandwidth, displacement-field control, and strong spin–orbit-coupled valley structure. They are especially useful for testing triangular-lattice Hubbard ideas, but the relevant orbital center, valley content, and topology can change across twist and field. “TMD moiré” is a platform family, not one Hamiltonian.

Distinguishing the main possibilities

Observation	What it establishes and what remains open
Integer-filling gap	Charge addition is costly; band completion, local constraints and flavor reconstruction remain possible.
Fractional-filling gap	The filling is special; charge order, topology or an enlarged structural cell must be distinguished.
Enlarged-period charge pattern	Translation symmetry is reduced; interaction and structural contributions still need comparison.
Activated transport	Mobile carriers face an energy scale; this need not equal the thermodynamic gap.

Several mechanisms may coexist. A charge crystal can suppress double occupancy, and a flavor-polarized state can carry a Chern number. Moiré Topology develops the Hall tests, while Moiré Superconductivity examines neighboring paired phases.

Exercises

Exercise 1: converting density to filling

A triangular moiré lattice has $L_M=10.0\,\mathrm{nm}$. Find its unit-cell area and the carrier density corresponding to one carrier per cell. Express the density in $\mathrm{cm}^{-2}$.

Solution

The area is

$$\begin{aligned} A_M &= \frac{\sqrt3}{2}(10.0\,\mathrm{nm})^2 \\ &= 86.6\,\mathrm{nm}^2 = 8.66\times10^{-13}\,\mathrm{cm}^2. \end{aligned}$$

Therefore

$$n_M = A_M^{-1} \simeq 1.15\times10^{12}\,\mathrm{cm}^{-2}.$$

If the local period were $2\%$ larger, the cell area would be about $4\%$ larger and the inferred one-per-cell density about $4\%$ smaller.

Exercise 2: commensurability versus mechanism

At $\nu=2$ a device shows a sharp resistance peak. A continuum calculation predicts a single-particle gap at the same density, and the measured gap changes only weakly when a nearby screening gate is moved. What is the strongest justified claim?

Solution

The data establish a commensurate insulating transport feature consistent with a band gap. They do not require a correlated insulator because the calibrated one-body model already provides a candidate explanation and screening produces little response. A bulk thermodynamic gap measurement would strengthen the insulating assignment, but interaction necessity would still require a discrepancy with the one-body prediction or another interaction-specific signature.

Exercise 3: a fractional charge pattern

On a triangular moiré lattice, construct the simplest $\nu=1/3$ charge pattern that avoids occupied nearest neighbors. How many primitive sites lie in its charge unit cell, and what new ordering wavevector should a structure-factor measurement find?

Solution

Color the triangular lattice with three sublattices and occupy one color. The occupied sites form a triangular lattice with spacing $\sqrt3$ times the original spacing. The charge cell contains three primitive moiré sites and one carrier.

The order is conventionally called $\sqrt3\times\sqrt3$. Its Bragg peaks occur at the inequivalent $K$ points of the primitive triangular-lattice Brillouin zone, modulo reciprocal lattice vectors. For a periodic sample with $N_M$ sites and dimensions divisible by three, the Fourier amplitude of this perfect pattern at $\mathbf K$ has magnitude $N_M/3$. Therefore $S_{\mathrm c}(\mathbf K)=N_M/9$. Subtracting each site’s local mean would instead remove the static pattern and miss this peak. A fractional gap alone does not demonstrate a crystal.

Exercise 4: exchange scale

Use the simplest one-band estimate for $t=1.0\,\mathrm{meV}$ and $U-V_1=20\,\mathrm{meV}$. Find $J_1$ in millielectronvolts and kelvin. Take $1\,\mathrm{meV}=11.604\,\mathrm K$.

Solution

The leading estimate gives

$$J_1 = \frac{4(1.0\,\mathrm{meV})^2} {20\,\mathrm{meV}} = 0.20\,\mathrm{meV}.$$

Thus $J_1/k_{\mathrm B}\simeq2.32\,\mathrm K$. This estimate is meaningful only if the assumed local charge excitation dominates and other orbitals, direct exchange, complex hopping, and higher-order processes are small.

Exercise 5: three gaps

A capacitance measurement gives addition and removal edges separated by $4.0\,\mathrm{meV}$. Transport fitted over a restricted range gives $\Delta_{\mathrm{tr}}=1.2\,\mathrm{meV}$, while an optical sensor shifts at the same filling. Are the three observations inconsistent?

Solution

No. The $4.0\,\mathrm{meV}$ separation estimates a thermodynamic charge gap after electrostatic corrections. The smaller transport scale can reflect mobility edges, disorder-assisted hopping, or in-gap conduction. The optical shift supports a change in screening or compressibility but requires its own calibration and need not equal either energy. Agreement in filling and systematic temperature dependence is valuable even when the extracted scales differ.

Exercise 6: combining charge and Hall measurements

A fractional-filling state has a chemical-potential jump, no detected charge modulation, a Hall resistivity magnitude $|\rho_{xy}|=h/e^2$, longitudinal resistivity $\rho_{xx}=0.2h/e^2$, and a density–field slope close to $e/h$. Assume an isotropic homogeneous sheet. Compute $|\sigma_{xy}|$ and explain what the observations establish and what remains open.

Solution

The chemical-potential jump establishes bulk incompressibility at the measured resolution. Inverting the resistivity tensor gives

$$|\sigma_{xy}|=\frac{|\rho_{xy}|}{\rho_{xx}^2+\rho_{xy}^2}=\frac{1}{1+0.2^2}\frac{e^2}{h}\simeq0.962\frac{e^2}{h}.$$

Thus a Hall resistivity of $h/e^2$ does not by itself imply a quantized Hall conductivity. The field slope is consistent with an integer Chern response only if it measures the fixed-chemical-potential derivative. The Hall value is suggestive. Because the longitudinal resistance is nonzero and exact Hall quantization is not established, the data do not yet prove a Chern insulator. The absence of detected charge modulation bounds order only at the probe’s resolution and wavevector coverage. Improved longitudinal insulation, metrological Hall accuracy, nonlocal or edge tests, and more complete imaging would discriminate a Chern phase from an inhomogeneous charge-ordered or parallel-conduction state.

References

• Y. Cao et al., “Correlated Insulator Behaviour at Half-Filling in Magic-Angle Graphene Superlattices,” Nature 556, 80–84 (2018), doi:10.1038/nature26154.
• H. Li et al., “Imaging Two-Dimensional Generalized Wigner Crystals,” Nature 597, 650–654 (2021), doi:10.1038/s41586-021-03874-9.
• B. Padhi, R. Chitra, and P. W. Phillips, “Generalized Wigner Crystallization in Moiré Materials,” Physical Review B 103, 125146 (2021), doi:10.1103/PhysRevB.103.125146.
• E. C. Regan et al., “Mott and Generalized Wigner Crystal States in WSe$_2$/WS$_2$ Moiré Superlattices,” Nature 579, 359–363 (2020), doi:10.1038/s41586-020-2092-4.
• Y. Tang et al., “Simulation of Hubbard Model Physics in WSe$_2$/WS$_2$ Moiré Superlattices,” Nature 579, 353–358 (2020), doi:10.1038/s41586-020-2085-3.
• F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, “Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands,” Physical Review Letters 121, 026402 (2018), doi:10.1103/PhysRevLett.121.026402.