Scarola-Jain Bilayer Composite-Fermion Framework

• The framework extends Jain’s single-layer composite-fermion construction by adding an interlayer Jastrow factor to encode correlations between layers.
• It distinguishes intralayer from interlayer vortex attachment, classifying bilayer states with notations like {}^2_1CF and offering clear filling-factor relationships.
• The approach explains phase transitions driven by layer spacing and opens the possibility of spontaneous interlayer phase coherence in composite-fermion systems.

Searching arXiv for the cited review and closely related Scarola–Jain bilayer CF literature. The Scarola–Jain bilayer composite-fermion framework is a two-component extension of the Jain composite-fermion construction in which each layer carries a single-layer Jain state and interlayer correlations are encoded by an additional Jastrow factor. In the form reviewed in "Interlayer phase coherence and composite fermions" (Jolicoeur, 30 Jan 2025), it organizes bilayer fractional quantum Hall states by distinguishing intralayer vortex attachment from interlayer vortex attachment and labeling the resulting quasiparticles as ${}^{2p}_mCF$. The framework is most firmly established as a wavefunction-based description of bilayer incompressible states, especially in graphene double layers, while its more speculative frontier concerns whether the emergent composite fermions themselves can develop spontaneous interlayer phase coherence.

1. Single-layer antecedents and bilayer motivation

The framework inherits its logic from the single-layer Jain construction. In the lowest Landau level, each electron binds an even number $2p$ of vortices through a Jastrow factor, and the standard example given in the review is the rewriting of the Laughlin $\nu=1/3$ state as

$$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$

This motivates the replacement

$z^m \rightarrow z^m \prod_j (z-z_j),$

or more generally, before projection,

$$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$

Projection to the lowest Landau level is performed by moving $z^*$ to the left and replacing it by $2\partial/\partial z$. The resulting Jain mapping is expressed through

$B^*=B-2n\phi_0,$

with incompressible states when $\nu^*=p$, giving $2p$0, and also negative-effective-field states $2p$1 (Jolicoeur, 30 Jan 2025).

The bilayer setting introduces a second set of coordinates, conventionally $2p$2 and $2p$3, and a second organizing principle already familiar from the Halperin family,

$2p$4

Here $2p$5 is the intralayer exponent and $2p$6 the interlayer exponent. For balanced bilayers the total filling is stated to be

$2p$7

This two-component template is the structural precursor of the Scarola–Jain program: layer-resolved coordinates remain explicit, intralayer and interlayer correlations are separately parameterized, and bilayer states are classified by the pattern of vortex attachment rather than by a purely single-component effective filling.

The principal physical motivation is the coexistence, in bilayer quantum Hall systems, of two ideas often treated separately: composite-fermion topological order and spontaneous breakdown of the relative interlayer $2p$8 phase. At total filling $2p$9, bilayers with small separation can exhibit interlayer phase coherence at the electron level; recent graphene double-layer experiments, by contrast, reveal many fractional states unique to the bilayer case. The Scarola–Jain framework sits precisely at that junction, importing Jain’s single-layer composite-fermionization into a two-layer environment while retaining the possibility that additional symmetry-breaking physics may emerge.

2. Wavefunction construction and notation

The explicit bilayer trial state adopted in the review is

$\nu=1/3$0

Each factor $\nu=1/3$1 is a single-layer Jain composite-fermion state, with $\nu=1/3$2 intralayer vortices attached to each electron, while the interlayer Jastrow factor $\nu=1/3$3 introduces $\nu=1/3$4 interlayer vortices between electrons in opposite layers. The notation is explicit: $\nu=1/3$5 meaning $\nu=1/3$6 vortices attached within the same layer and $\nu=1/3$7 interlayer vortices (Jolicoeur, 30 Jan 2025).

This notation is more than bookkeeping. It separates two distinct correlation structures that are merged in the simpler Halperin notation. In the Halperin family the exponents appear directly in the total many-body polynomial, whereas in the Scarola–Jain construction the intralayer factors are already nontrivial Jain states that can represent filled composite-fermion $\nu=1/3$8 levels, not merely Laughlin states. The interlayer exponent $\nu=1/3$9 then supplements that intralayer composite-fermionization rather than replacing it.

The review states the balanced-layer filling relation as

$$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$0

where $$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$1 is the filling of each layer, $$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$2 is the filling of the single-layer Jain building block $$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$3, and $$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$4 is the effective composite-fermion filling before flux attachment. Since in a single layer $$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$5, the added $$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$6 is the interlayer-correlation contribution. In physical terms, the correlation hole experienced by an electron now counts both intralayer and interlayer vortices. For balanced states, the total filling may be written as $$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$7.

A later wide-quantum-well application recasts the same construction in the notation

$$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$8

with $$\prod_{i<j}(z_i-z_j)^3 = \det\!\left(z_i^{j-1}\prod_{k\neq i}(z_i-z_k)\right).$$9 single-layer Jain states. In that notation the Halperin $z^m \rightarrow z^m \prod_j (z-z_j),$0 state is identified as $z^m \rightarrow z^m \prod_j (z-z_j),$1, making explicit that the Scarola–Jain framework generalizes Halperin bilayer states by replacing Laughlin intralayer sectors with more general Jain sectors (Singh et al., 22 Jun 2026).

3. Filling-factor relations and canonical bilayer sequences

Within the review, the most important sequence is the $z^m \rightarrow z^m \prod_j (z-z_j),$2 family. Setting $z^m \rightarrow z^m \prod_j (z-z_j),$3 and $z^m \rightarrow z^m \prod_j (z-z_j),$4 gives

$z^m \rightarrow z^m \prod_j (z-z_j),$5

where $z^m \rightarrow z^m \prod_j (z-z_j),$6 is the number of filled composite-fermion pseudo-Landau levels in each layer. The positive-flux branch listed is

$z^m \rightarrow z^m \prod_j (z-z_j),$7

while the negative-flux branch is

$z^m \rightarrow z^m \prod_j (z-z_j),$8

These fillings are said to converge to a compressible state at

$z^m \rightarrow z^m \prod_j (z-z_j),$9

The review presents this as the bilayer analogue of the primary Jain sequence and as a prominent experimentally observed family in graphene double layers (Jolicoeur, 30 Jan 2025).

The same review maps several familiar Halperin states into the $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$0 language. At total filling $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$1, the $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$2 state is described as a $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$3 state, while the competing $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$4 state is interpreted as $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$5. At total filling $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$6, the $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$7 state is identified as $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$8, and the $$\phi_{N,m}(z,z^*)\rightarrow \phi_{N,m}(z,z^*)\prod_j (z-z_j).$$9 state is described as a product of two Laughlin $z^*$0 states, that is, a decoupled incompressible $z^*$1 state. These identifications are central because they translate between older bilayer trial-state nomenclatures and the Scarola–Jain vortex-attachment language.

A further general filling formula appears in the wide-quantum-well work: $z^*$2 with

$z^*$3

In the trivial case $z^*$4, the interlayer factor is absent and $z^*$5. This notation is used to identify the observed even-denominator state at $z^*$6 as

$z^*$7

for which $z^*$8 and hence $z^*$9; the $2\partial/\partial z$0 state is then taken to be the particle-hole conjugate of $2\partial/\partial z$1 (Singh et al., 22 Jun 2026).

These relations clarify a recurring source of confusion. The labels $2\partial/\partial z$2 in $2\partial/\partial z$3 characterize the intralayer Jain correlations in the trial state, whereas $2\partial/\partial z$4 denote the actual layer fillings. Thus a state such as $2\partial/\partial z$5 has intralayer $2\partial/\partial z$6-type Jain correlations but physical layer fillings $2\partial/\partial z$7, not $2\partial/\partial z$8. The framework therefore distinguishes the correlation label from the measurable layer occupancy.

4. Phase competition, compressible phases, and the role of layer spacing

A defining feature of the Scarola–Jain framework is that the interlayer separation $2\partial/\partial z$9 is treated as a control parameter that can drive transitions among bilayer states with different interlayer exponents and eventually toward decoupled or compressible phases. At total filling $B^*=B-2n\phi_0,$0, the review proposes the sequence

$B^*=B-2n\phi_0,$1

with the caveat that a Wigner crystal may intervene at large separation. At total filling $B^*=B-2n\phi_0,$2, it proposes

$B^*=B-2n\phi_0,$3

The underlying logic is that small $B^*=B-2n\phi_0,$4 favors stronger interlayer correlations, while increasing $B^*=B-2n\phi_0,$5 weakens them and can produce compressible or decoupled states (Jolicoeur, 30 Jan 2025).

This $B^*=B-2n\phi_0,$6-driven competition is not merely a feature of incompressible trial states. Experimental work on the compressible side of the $B^*=B-2n\phi_0,$7 transition shows that the incoherent phase can still be described as two parallel composite-fermion metals down to $B^*=B-2n\phi_0,$8, but not as two independent layers. In closely spaced bilayers, the tunneling peak full width at half maximum is about $B^*=B-2n\phi_0,$9 smaller than in widely separated bilayers, the fitted tunneling coefficient drops from $\nu^*=p$0 to $\nu^*=p$1, and the spin-polarization field decreases from about $\nu^*=p$2 T to about $\nu^*=p$3 T at matched density, implying an approximately $\nu^*=p$4 lower composite-fermion Fermi energy. The paper summarizes the overall interaction suppression as roughly $\nu^*=p$5 and attributes it to interlayer screening that softens the effective intralayer Coulomb interaction (Eisenstein et al., 2018).

For the Scarola–Jain framework, the significance of these measurements is constraining rather than classificatory. They support the existence of an incoherent bilayer composite-fermion metal but show that it is energetically renormalized by the nearby second layer. A simple product of two isolated $\nu^*=p$6 composite-fermion seas therefore misses an experimentally visible part of the physics near the excitonic boundary.

The framework also distinguishes states that are likely decoupled from those with substantial interlayer correlation. At $\nu^*=p$7 and $\nu^*=p$8, the review states that the observed fractions do not show drag Hall resistance and are likely decoupled fractional quantum Hall states, such as $\nu^*=p$9 at $2p$00 and its particle-hole partner at $2p$01. At $2p$02, a $2p$03 assignment remains possible if interlayer correlations are sufficiently strong, but the nearby decoupled $2p$04 states make the identification nontrivial. The recurring implication is that filling-factor matching alone does not uniquely identify the bilayer phase.

5. Interlayer phase coherence of electrons and of composite fermions

The established part of interlayer phase coherence in the review concerns electrons at total filling $2p$05. With small layer separation and negligible tunneling, the bilayer may spontaneously break the $2p$06 symmetry associated with conservation of layer population difference. The broken-symmetry wavefunction is written

$2p$07

where $2p$08 is the relative interlayer phase. The review identifies $2p$09 as the XY order parameter of pseudospin ferromagnetism. In particle-hole language in one layer, the same state becomes

$2p$10

which exhibits the state as an exciton condensate. For the coherent $2p$11-like regime the transport signatures are

$2p$12

while the counterflow Hall voltage vanishes at $2p$13 because the coherent electron-hole pairs are neutral (Jolicoeur, 30 Jan 2025).

The speculative extension is interlayer phase coherence of composite fermions rather than of electrons. The review asks whether the emergent bilayer composite-fermion quasiparticles, after flux attachment and formation of composite-fermion Landau levels or paired states, can themselves develop spontaneous coherence between layers. No explicit microscopic many-body coherent composite-fermion wavefunction is provided, and no microscopic order parameter formulated directly in composite-fermion variables is written down. The conjecture is instead motivated phenomenologically by half-integer fillings of the $2p$14 series: $2p$15 and, for negative flux,

$2p$16

The suggestion is that half-integer effective composite-fermion fillings may signal an instability of the $2p$17 fluid toward a symmetry-broken coherent phase rather than another ordinary incompressible Jain-like state.

A complementary field-theoretic route to interlayer-correlated composite-fermion physics appears in work on bilayer composite-fermion metals at $2p$18. Within an HLR/Chern–Simons description, the out-of-phase gauge mode mediates an attractive interlayer pairing interaction and the in-phase gauge mode mediates a repulsive one; in the large-$2p$19 local-approximation regime the pairing gap retains the inverse-square scaling

$2p$20

but is strongly suppressed by pair breaking (Deng et al., 2023). This is not the same construction as Scarola–Jain wavefunction theory, but it sharpens the dynamical side of the bilayer composite-fermion problem by showing that interlayer pairing and strong pair breaking emerge from the same gauge sector.

6. Experimental realizations, later extensions, and theoretical status

In the review, graphene double layers with very small $2p$21, negligible tunneling, and strong Coulomb coupling furnish the main experimental setting. The bilayer-unique sequence

$2p$22

is highlighted as a major success of the $2p$23 picture, and the $2p$24 state is explicitly identified with the Halperin $2p$25 state. Other observed fractions, such as $2p$26 and $2p$27, are suggested to be higher-generation fractional quantum Hall states of composite fermions, analogous to unconventional one-component fractions like $2p$28, but that identification is presented more tentatively (Jolicoeur, 30 Jan 2025).

A distinct experimental realization arises in a 72.5-nm-wide GaAs quantum well, where the charge distribution becomes bilayer-like and density tuning reduces $2p$29. In that regime the even-denominator states at $2p$30 and $2p$31 appear only when the system becomes two-component, and both disappear rapidly under layer asymmetry of only about $2p$32. The proposed interpretation is that $2p$33 is the balanced Scarola–Jain state $2p$34, while $2p$35 is its particle-hole conjugate. The same work treats $2p$36 as likely the particle-hole conjugate of $2p$37, placing the generalized Scarola–Jain hierarchy as a direct extension of the textbook $2p$38 logic to higher Jain intralayer sectors (Singh et al., 22 Jun 2026).

The framework also has clear limits. The review does not present a full Chern–Simons action or $2p$39-matrix formulation for bilayer Scarola–Jain states, does not derive a modified effective field theory for composite-fermion interlayer coherence, and does not present fresh numerical spectra or variational-energy tables. Its support is instead a combination of the established success of Jain wavefunctions in one-component systems, qualitative energetic reasoning for phase competition in bilayers, and the phenomenological alignment of observed graphene fractions with the $2p$40 sequence.

Several common misunderstandings are therefore excluded by the review’s own framing. The framework is not a completed theory of interlayer-coherent composite fermions; it is a successful bilayer wavefunction construction for many incompressible states together with a programmatic extension toward symmetry-broken composite-fermion phases. It does not imply that every bilayer fraction is coherent rather than decoupled, nor that simple filling-factor coincidence is sufficient for state identification. It also does not supersede alternative low-energy formulations. A particle-hole-symmetric Dirac-composite-fermion perspective, for example, reorganizes Jain-sequence bookkeeping around half-integer effective fillings and suggests that bilayer theories near exact particle-hole symmetry may require a different effective language, although that paper does not itself construct the bilayer Scarola–Jain theory (Son, 2015).

Taken together, these developments define the present status of the Scarola–Jain bilayer composite-fermion framework. Its established core is the wavefunction

$2p$41

the notation $2p$42, the balanced-layer filling relation

$2p$43

and the successful organization of prominent bilayer fractions, especially the $2p$44 sequence. Its open frontier is the possibility that composite fermions, not just electrons, can sustain spontaneous interlayer phase coherence, requiring explicit coherent trial wavefunctions and perhaps new effective theories still absent from the current formulation.