Interlayer Excitonic Quantum Hall States

• Interlayer excitonic quantum Hall state is a phase where strong Coulomb interactions induce spontaneous exciton condensation across distinct layers.
• Experimental signatures such as quantized Hall drag, counterflow conductance, and Josephson-like tunneling provide clear diagnostics for interlayer coherence.
• The state's behavior varies with layer filling, spacing, and magnetic field, explaining distinct regimes observed in GaAs, graphene bilayers, and twisted systems.

Searching arXiv for recent and foundational papers on interlayer excitonic quantum Hall states. Searching arXiv for recent and foundational papers on interlayer excitonic quantum Hall states. An interlayer excitonic quantum Hall state is a quantum Hall phase in which strong Coulomb coupling between electrically distinct layers produces spontaneous phase coherence across the layers, so that electrons in one layer bind to holes, composite-fermion holes, or fractionally charged quasiholes in the other layer and reorganize the incompressible Hall fluid into an excitonic condensate. In the canonical bilayer setting this occurs when each layer is near half filling and direct interlayer tunneling is very small, but the same organizing principle now appears in graphene double layers, twisted bilayer and trilayer graphene, and fractional quantum Hall bilayers with fractionalized excitons. The defining experimental signatures recur across platforms: quantized Hall drag, perfect drag, finite counterflow conductance, Josephson-like interlayer tunneling, suppressed longitudinal transport at balanced half-filled configurations, and phase transitions driven by layer separation, displacement field, or magnetic breakdown (Huang et al., 2012, Liu et al., 2018, Kim et al., 2021, Kim et al., 2022, Zhang et al., 2024).

1. Definition and organizing principles

The basic control parameters are the layer filling factors

$\nu_i = \frac{n_i \phi_0}{B},$

or, in bilayer notation,

$\nu_T=\nu_1+\nu_2,$

with strong perpendicular magnetic field quenching kinetic energy and making Coulomb interactions dominant (Liu et al., 2018, Eisenstein et al., 2019). In the absence of appreciable interlayer tunneling, the separate layers retain an approximate relative $U(1)$ symmetry. Exciton condensation corresponds to spontaneous breaking of that relative symmetry, leaving only total charge conservation. In the electron-hole bilayer formulation, a nonzero interlayer coherence

$\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$

signals an excitonic insulator phase, and the phase $\phi$ is the Goldstone mode associated with exciton superfluidity (Shao et al., 2024).

In the conventional bilayer quantum Hall language, the same phase is described as a coherent BCS-like state of interlayer excitons or, equivalently, as pseudospin ferromagnetic order in which the layer index plays the role of pseudospin (Huang et al., 2012). The criterion that interlayer Coulomb interactions dominate while single-particle tunneling remains weak can be achieved in different ways. In GaAs double layers it is controlled by the ratio $d/l_B$, with the coherent $\nu_T=1$ phase appearing when the layer spacing is sufficiently small compared with the magnetic length (Huang et al., 2012). In large-angle twisted bilayer graphene, the atomic layer spacing $d \approx 0.34\ \text{nm}$ yields $d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$, while the large twist angle suppresses tunneling by momentum mismatch (Kim et al., 2022). In $2^\circ$ twisted bilayer graphene, the effective tunneling can instead be tuned by magnetic breakdown as the density approaches the van Hove singularity (Kim et al., 2021).

The state is not defined solely by a quantized Hall plateau. Several papers emphasize that the crucial feature is a neutral interlayer coherent sector coexisting with a gapped charge sector. This distinction underlies the use of counterflow transport, Hall drag, interlayer tunneling, and layer-resolved transport asymmetries as primary diagnostics (Huang et al., 2012, Liu et al., 2018, Zhang et al., 2024).

2. Canonical bilayer regime at $\nu_T=\nu_1+\nu_2,$0

The standard realization is a bilayer two-dimensional electron system with each layer near $\nu_T=\nu_1+\nu_2,$1, so that

$\nu_T=\nu_1+\nu_2,$2

In this regime the correlated ground state is simultaneously a quantum Hall state and an interlayer coherent exciton condensate. The state is often identified with the Halperin $\nu_T=\nu_1+\nu_2,$3 picture and exhibits nearly dissipationless counterflow, quantized Hall drag, and Josephson-like tunneling (Huang et al., 2012, Zhu et al., 2017, Eisenstein et al., 2019).

Corbino experiments directly isolated the neutral character of this phase. In ring-shaped bilayers at $\nu_T=\nu_1+\nu_2,$4, identical Josephson currents were observed at the two edges even though the edge lengths differ by a factor of about $\nu_T=\nu_1+\nu_2,$5, while the conductance between edges remained about $\nu_T=\nu_1+\nu_2,$6 at small voltages (Huang et al., 2012). By contrast, same-edge Josephson tunneling conductance rose to about $\nu_T=\nu_1+\nu_2,$7, a three-orders-of-magnitude enhancement. The critical Josephson current at one edge could be increased, reduced, more than doubled, or even driven to change sign by passing a second interlayer Josephson current at the other edge. Because the bulk remained effectively insulating to charge transport, the coupling was interpreted as mediated by the neutral excitonic condensate rather than by ordinary charge flow (Huang et al., 2012).

The phase boundary is also visible before full condensation. Tunneling spectroscopy near, but above, the $\nu_T=\nu_1+\nu_2,$8 transition showed a nonlinear collapse of the Coulomb pseudogap $\nu_T=\nu_1+\nu_2,$9 as $U(1)$0 approached the critical value, about

$U(1)$1

with the strongest suppression at $U(1)$2 and the collapse of $U(1)$3 occurring where Josephson-like zero-bias tunneling appears (Eisenstein et al., 2019). This established that interlayer electron-hole correlations already reshape the compressible bilayer before long-range phase coherence fully develops.

Numerically, the small-$U(1)$4 state is an exciton superfluid with spontaneous interlayer coherence, gapless pseudospin excitations, and finite exciton superfluid stiffness (Zhu et al., 2017). Exact diagonalization further indicated that the destruction of the $U(1)$5 condensate need not be a single direct collapse into two decoupled composite Fermi liquids. Instead, three regimes were found as $U(1)$6 increases: an exciton superfluid at small $U(1)$7, an intermediate phase for $U(1)$8, and a large-$U(1)$9 composite Fermi liquid (Zhu et al., 2017). The intermediate phase has finite pseudospin gap, flat Berry curvature, zero drag Hall conductance, finite exciton superfluid stiffness, and an even-odd effect in the energy cost of moving one electron between layers, which the authors associated with enhanced intralayer correlation.

3. Composite-fermion and fractional interlayer excitonic states

A major development was the extension of interlayer excitonic quantum Hall physics into the fractional regime. In a coupled graphene double layer, the fractional Hall effect was analyzed in terms of composite fermions with different intralayer and interlayer Chern-Simons gauge-field couplings. The effective flux and filling are

$\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$0

and transport is written as

$\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$1

This framework yields fully quantized states when both $\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$2 and $\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$3 are integers, semi-quantized states when only one is integer, and unquantized states when both are partially filled (Liu et al., 2018).

The distinctive fractional excitonic state in that system occurs at

$\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$4

Each layer then has one completely filled effective composite-fermion Landau level and one half-filled composite-fermion Landau level. The partially filled levels were interpreted as forming a Bose condensate of excitons, a composite-fermion analog of the Halperin $\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$5 state, with locked effective electric fields, incompressibility, an energy gap, and a Hall resistivity matrix matching the observed one (Liu et al., 2018). The paper explicitly argued that intralayer pairing or opposite-layer Cooper pairing would give the wrong Hall response. In the same analysis, the phase-stiffness form

$\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$6

and the fractional vortex charge

$\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$7

for $\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$8 make the fractionalized nature of the excitonic condensate explicit (Liu et al., 2018).

Fractional excitonic order is even broader in bilayer fractional quantum Hall systems. Corbino transport in graphene bilayers showed a fractional counterpart of the $\Delta \equiv h^{mf}_{eh}=|\Delta|e^{i\phi}$9 condensate at $\phi$0, identified with the Halperin $\phi$1 state, together with unconventional excitons whose constituents carry fractional charge and whose exchange statistics can be fermionic or anyonic rather than bosonic (Zhang et al., 2024). The transport criteria were the same structural combination seen at $\phi$2: suppressed parallel-flow conductance, finite counterflow conductance, and perfect drag,

$\phi$3

but now realized in a fractional Hall background (Zhang et al., 2024).

Balanced large-angle twisted bilayer graphene provides an experimental realization of the same fractional logic. At $\phi$4 and zero displacement field, transport and Monte Carlo simulations supported an interlayer coherent $\phi$5 state as the underlying ground state, described as the fractional analogue of the $\phi$6 exciton condensate (Kim et al., 2024). In that device the two graphene sheets are separated by $\phi$7, $\phi$8 can be as large as $\phi$9, and increasing displacement field drives a transition from the balanced interlayer coherent $d/l_B$0 state to a fully layer-polarized single-layer Laughlin $d/l_B$1 state (Kim et al., 2024).

The most explicit topological generalization is the proposal of anyonic exciton superfluidity in balanced bilayers of two Laughlin $d/l_B$2 states. There the minimal neutral interlayer excitation is

$d/l_B$3

with exchange statistics

$d/l_B$4

At finite density, these anyonic excitons were argued to form an exciton superfluid stitched to a definite bulk topological order and edge spectrum, especially near the direct transition between $d/l_B$5 and $d/l_B$6 states (Han et al., 20 Aug 2025). The paper predicted anomalous scaling near the critical point,

$d/l_B$7

and generalized the mechanism to the bilayer Jain sequence

$d/l_B$8

where the corresponding exciton superfluid carries pseudospin $d/l_B$9 (Han et al., 20 Aug 2025).

4. Graphene and moiré realizations

Graphene-based platforms changed the experimental landscape by combining atomically small layer spacing with geometrically suppressed tunneling. In $\nu_T=1$0 twisted bilayer graphene, odd-integer quantum Hall states at balanced density and suppressed tunneling were not anticipated, yet $\nu_T=1$1 and $\nu_T=1$2 were observed at $\nu_T=1$3 and interpreted as signatures of Coulomb interaction induced interlayer coherence and Bose-Einstein condensation of excitons that form at half filling of each layer (Kim et al., 2021). The same work showed reentrant behavior under displacement field: at fixed odd total filling, moving away from $\nu_T=1$4 produced resistance peaks and then a new incompressible state in which the two layers occupy different integer quantum Hall fillings. With increasing density toward the van Hove singularity, magnetic breakdown enhanced hybridization, and the interlayer coherent state and the phase transition vanished (Kim et al., 2021).

Large-angle twisted bilayer graphene realizes a distinct strong-coupling limit. Because the layers are separated only by the atomic interlayer spacing $\nu_T=1$5 and large twist suppresses tunneling, robust interlayer-coherent quantum Hall states were observed at odd total fillings

$\nu_T=1$6

in the $\nu_T=1$7 Landau level (Kim et al., 2022). The gaps of the odd-integer coherent states were of order $\nu_T=1$8 K, while even-integer quantum Hall states had gaps of about $\nu_T=1$9–$d \approx 0.34\ \text{nm}$0 K, and the paper emphasized that the coherent gaps are several orders of magnitude greater than those in GaAs (Kim et al., 2022). A phenomenological Hartree-Fock model attributed the odd states to nonzero off-diagonal coherence in layer space and the observed displacement-field evolution to transitions among canted antiferromagnetic, partially layer-polarized coherent, and fully layer-polarized phases.

Alternating twisted trilayer graphene extends the same logic to a three-layer system in which one layer can remain inert. Magnetotransport at $d \approx 0.34\ \text{nm}$1 showed a narrow suppressed-resistance region near

$d \approx 0.34\ \text{nm}$2

identified with the balanced configuration

$d \approx 0.34\ \text{nm}$3

In that interpretation the middle and bottom layers form an interlayer coherent excitonic quantum Hall state while the top layer acts as an inert integer quantum Hall background (Kim et al., 13 Sep 2025). The feature weakened systematically as the magnetic field was lowered from $d \approx 0.34\ \text{nm}$4 T to $d \approx 0.34\ \text{nm}$5 T, consistent with an interaction-driven phase whose gap depends on field-enhanced Landau-level splitting.

Across these graphene systems, odd-integer or balanced half-filled states are not treated as automatic consequences of single-particle quantization. They are interpreted as interaction-driven interlayer coherent states precisely because they appear where independent-layer or strongly hybridized descriptions do not predict them (Kim et al., 2021, Kim et al., 2022, Kim et al., 13 Sep 2025).

5. Collective modes, edge structure, and phase transitions

The interlayer excitonic quantum Hall state is not only a bulk transport phenomenon; it also has a distinctive neutral collective sector. In the helical quantum Hall exciton condensate of a band-inverted electron-hole bilayer, the bulk pseudospin polarization obeys

$d \approx 0.34\ \text{nm}$6

so the helical exciton condensate occupies the window $d \approx 0.34\ \text{nm}$7 where $d \approx 0.34\ \text{nm}$8 (Pikulin et al., 2015). The edge necessarily forms a pseudospin domain wall, and the charge density is tied to the texture by

$d \approx 0.34\ \text{nm}$9

In a narrow Hall bar, this leads to confinement: a charge on one edge necessarily induces the opposite charge on the other edge. The confined phase supports a one-dimensional effective Hamiltonian

$d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$0

with counterflow conductance

$d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$1

and the nonlocal signature $d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$2 at low bias (Pikulin et al., 2015).

Phase transitions out of excitonic order are correspondingly diverse. At $d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$3, exact diagonalization on the torus found a direct transition between a small-$d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$4 interlayer coherent exciton superfluid and a large-$d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$5 phase smoothly connected to $d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$6, with critical distance

$d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$7

The excitonic phase was characterized by easy-plane ferromagnetic spectra

$d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$8

gapless pseudospin excitations, and finite exciton superfluid stiffness extracted from

$d/l_B \sim \mathcal{O}(10^{-2}) \ll 1$9

while the large-$2^\circ$0 phase had $2^\circ$1 (Zhu et al., 2018). The authors interpreted the transition as a possible continuous topological phase transition beyond the Landau paradigm because it simultaneously changes topological order and breaks symmetry.

A different neutral regime appears in bilayer graphene near $2^\circ$2, where an even-denominator Hall plateau coexists with finite layer polarization. Theory and numerics attributed this to a topological exciton metal or topological exciton Fermi sea: the charge sector remains Hall quantized,

$2^\circ$3

but the neutral excitons are argued to be fermionic and to form a Fermi sea rather than a condensate (Zaletel et al., 2018). This is important because it shows that finite-density interlayer excitons in a quantum Hall background need not always condense into a bosonic superfluid.

6. Diagnostics, distinctions, and broader analogues

Several works emphasize that an interlayer excitonic quantum Hall state must be distinguished from ordinary single-particle hybridization, simple layer imbalance, or a conventional Hall plateau. In the electron-hole bilayer mean-field study of interlayer capacitance,

$2^\circ$4

oscillates with $2^\circ$5 and $2^\circ$6 because the relevant low-energy objects are excitons formed from electron and hole Landau levels (Shao et al., 2024). At sufficiently strong field, estimated from

$2^\circ$7

the excitonic order can be destroyed, leading to two independent quantum Hall liquids with

$2^\circ$8

The same paper contrasted this with a tunneling-only gap, for which $2^\circ$9 and the capacitance never drops to zero at $\nu_T=\nu_1+\nu_2,$00 (Shao et al., 2024). This provides a sharp criterion for separating a genuine excitonic gap from a single-particle hybridization gap.

Graphite in the quantum limit provides a layered analogue with a different Hall phenomenology. Above about $\nu_T=\nu_1+\nu_2,$01 T, the Hall resistivity approaches zero, $\nu_T=\nu_1+\nu_2,$02 becomes gapped or insulating, and differential magnetization shows a non-monotonic anomaly. The proposed explanation is an excitonic BCS-like state formed by pairing between electron-like and hole-like Landau subbands (Akiba et al., 2015). The same work explicitly noted that no clear plateau structure is observed and that the high-field state is not presented as a conventional quantum Hall plateau. This distinction is central: not every magnetic-field-induced excitonic phase in a layered material is a textbook interlayer coherent Hall condensate.

Finally, recent theory proposes a zero-field analogue in TMD bilayers separated by twisted hBN, where equilibrium interlayer phase coherence at total hole filling $\nu_T=\nu_1+\nu_2,$03 can produce a chiral $\nu_T=\nu_1+\nu_2,$04-wave exciton condensate with quantum anomalous Hall effect and counter-flow superfluidity (Zeng et al., 14 Sep 2025). Although this is not a Landau-level quantum Hall state, it clarifies which ingredients are essential and which are platform-specific: negligible tunneling, strong interlayer exchange, and a spontaneously generated coherent order parameter $\nu_T=\nu_1+\nu_2,$05.

Taken together, these results define the interlayer excitonic quantum Hall state as a family of correlated Hall phases rather than a single wavefunction. The recurring structure is a charge-gapped Hall fluid supplemented by a neutral interlayer coherent sector. Depending on the platform and filling, that sector may be built from electrons and holes, composite fermions, charge-$\nu_T=\nu_1+\nu_2,$06 quasiparticles, or anyonic excitons; it may produce a Bose condensate, a helical confined mode structure, or a finite-density anyon superfluid; and it may compete with composite Fermi liquids, Moore-Read product states, layer-polarized integer phases, or topological exciton metals (Liu et al., 2018, Zhu et al., 2018, Zhang et al., 2024, Han et al., 20 Aug 2025).