Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1412.2430 · Physical Review B

Bilayer mapping of the paired quantum Hall state: Instability toward anisotropic pairing

Jae‐Seung Jeong, Kwon Park

Published May 13, 2015• 18 citations
arXivHTMLPDF

Abstract

One of the most dominant candidates for the paired quantum Hall (QH) state at filling factor $\ensuremath{\nu}=5/2$ is the Moore-Read (MR) Pfaffian state. A salient problem, however, is that it does not occur exactly at the Coulomb interaction, but rather at a modified interaction, which favors particle-hole-symmetry breaking. In an effort to find a better state, in this work, we investigate the possible connection between the paired QH state and the antisymmetrized bilayer ground state, which is inspired by the intriguing identity that the MR Pfaffian state is entirely equivalent to the antisymmetrized projection of the bilayer QH state called the Halperin (331) state, which is valid at interlayer distance, $d$, roughly equal to the magnetic length, ${l}_{\mathrm{B}}$. Specifically, by using exact diagonalization in the torus geometry, we show that the exact $5/2$ state at a given Haldane pseudopotential variation is intimately connected with the antisymmetrized bilayer ground state at a corresponding $d/{l}_{\mathrm{B}}$ via one-to-one mapping, which we call the bilayer mapping. One of the most important discoveries in this work is that the paired QH state occurring at the Coulomb interaction is mapped onto the antisymmetrized bilayer ground state at $d\ensuremath{\gg}{l}_{\mathrm{B}}$, which is equivalent to the antisymmetrized product state of two composite fermion seas at quarter filling, not the MR Pfaffian state. While maintaining high overlap with the paired QH state, the antisymmetrized bilayer ground state at $d\ensuremath{\gg}{l}_{\mathrm{B}}$ exhibits an abrupt change under the influence of small anisotropy. This suggests that the paired QH state occurring at the Coulomb interaction might be susceptible to anisotropic instability, opening up the possibility of anisotropic ${p}_{x}$- or ${p}_{y}$-wave pairing instead of ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$-wave pairing in the MR Pfaffian/anti-Pfaffian state.

DOI
10.1103/physrevb.91.195119

Related papers

Pairing instabilities of Dirac composite fermions

Milica V. Milovanović, Marija Dimitrijević Ćirić, Vladimir Juričić

Recently, a Dirac (particle-hole symmetric) description of composite fermions in the half-filled Landau level (LL) was proposed [D. T. Son, Phys. Rev. X 5, 031027 (2015)], and we study its possible consequences on BCS (Cooper) pairing of composite fermions (CFs). One of the main consequences is the existence of anisotropic states in single-layer and bilayer systems, which was previously suggested in Jeong and Park [J. S. Jeong and K. Park, Phys. Rev. B 91, 195119 (2015)]. We argue that in the half-filled LL in the single-layer case the gapped states may sustain anisotropy, because isotropic pairings may coexist with anisotropic ones. Furthermore, anisotropic pairings with the addition of a particle-hole symmetry-breaking mass term may evolve into rotationally symmetric states, i.e., Pfaffian states of Halperin-Lee-Read (HLR) ordinary CFs. On the basis of the Dirac formalism, we argue that in the quantum Hall bilayer at total filling factor 1, with decreasing distance between the layers, weak pairing of $p$-wave paired CFs is gradually transformed from Dirac to ordinary, HLR-like, with a concomitant decrease in the CF number. Global characterization of low-energy spectra based on the Dirac CFs agrees well with previous calculations performed by exact diagonalization on a torus. Finally, we discuss features of the Dirac formalism when applied in this context.

arXiv:1606.03919•Physical review. B./Physical review. B•Sep 8, 2016•6 citations

Competing states for the fractional quantum Hall effect in the 1/3-filled second Landau level

Jae‐Seung Jeong, Hantao Lu, Ki Hoon Lee +3 more

In this work, we investigate the nature of the fractional quantum Hall state in the 1/3-filled second Landau level (SLL) at filling factor $\ensuremath{\nu}=7/3$ (and 8/3 in the presence of the particle-hole symmetry) via exact diagonalization in both torus and spherical geometries. Specifically, we compute the overlap between the exact 7/3 ground state and various competing states including (i) the Laughlin state, (ii) the fermionic Haffnian state, (iii) the antisymmetrized product state of two composite fermion seas at 1/6 filling, and (iv) the particle-hole (PH) conjugate of the ${Z}_{4}$ parafermion state. All these trial states are constructed according to a guiding principle called the bilayer mapping approach, where a trial state is obtained as the antisymmetrized projection of a bilayer quantum Hall state with interlayer distance $d$ as a variational parameter. Under the proper understanding of the ground-state degeneracy in the torus geometry, the ${Z}_{4}$ parafermion state can be obtained as the antisymmetrized projection of the Halperin (330) state. Specifically, while unclear at other momentum sectors, all degenerate copies of the ${Z}_{4}$ parafermion state can be obtained by antisymmetrizing those of the Halperin (330) state at the zero-momentum sector, where both states occur as the exact ground states of their respective model Hamiltonians with the same degeneracy. Meanwhile, in the spherical geometry, the ${Z}_{4}$ parafermion state is shown to be entirely equivalent to the antisymmetrized Halperin (330) state without any ground-state degeneracy issue. Similarly, it is proved in this work that the fermionic Haffnian state can be obtained as the antisymmetrized projection of the Halperin (551) state. The exact 7/3 ground state is obtained as a function of $\ensuremath{\delta}{V}_{1}^{(1)}$, the variation of the first-moment Haldane pseudopotential ${V}_{1}^{(1)}$ in the SLL with respect to the pure Coulomb interaction. It is shown that, while extremely accurate at sufficiently large positive $\ensuremath{\delta}{V}_{1}^{(1)}$, the Laughlin state loses its overlap with the exact 7/3 ground state significantly at $\ensuremath{\delta}{V}_{1}^{(1)}\ensuremath{\simeq}0$. At slightly negative $\ensuremath{\delta}{V}_{1}^{(1)}$, it is shown that the PH-conjugated ${Z}_{4}$ parafermion state has a substantial overlap with the exact 7/3 ground state, which is the highest among the above four trial states. Around the Coulomb point, the energy spectrum exhibits an intriguing change from the spectrum with the Laughlin-type magnetoroton structure to that with the specific quasidegeneracy of the ground state, which is characteristic to the PH-conjugated ${Z}_{4}$ parafermion state.

arXiv:1601.00403•Physical review. B./Physical review. B•Sep 25, 2017•61 citations

Tunneling-driven breakdown of the 331 state and the emergent Pfaffian and composite Fermi liquid phases

Zlatko Papić, M. O. Goerbig, Nicolas Regnault +1 more

We examine the possibility of creating the Moore-Read Pfaffian in the lowest Landau level when the multicomponent Halperin 331 state (believed to describe quantum Hall bilayers and wide quantum wells at the filling factor $\ensuremath{\nu}=1/2$) is destroyed by the increase of tunneling. Using exact diagonalization of the bilayer Hamiltonian with short-range and long-range (Coulomb) interactions in spherical and periodic rectangular geometries, we establish that tunneling is a perturbation that drives the 331 state into a compressible composite Fermi liquid, with the possibility for an intermediate critical state that is reminiscent of the Moore-Read Pfaffian. These results are interpreted in the two-component BCS model for Cauchy pairing with a tunneling constraint. We comment on the conditions to be imposed on a system with fluctuating density in order to achieve the stable Pfaffian phase.

arXiv:0912.3103•Physical Review B•Aug 3, 2010•36 citations

Phase diagram of compressible and paired states in the quarter-filled Landau level

Misha Yutushui, David F. Mross

Quantum Hall plateaus at quarter fillings occur in GaAs wide quantum wells, hole-doped GaAs, and bilayer graphene. However, the interactions favoring incompressible states over compressible composite-Fermi liquids at such fillings are not well understood. We devise a method for the computation of the trial energies for Haldane pseudopotentials via Monte Carlo sampling. Applying it to the quarter-filled lowest Landau level, we find that tuning the third and fifth pseudopotential values can stabilize the anti-Pfaffian, Moore-Read, and $f$-wave states. The smallest deviations from pure Coulomb interactions are required by anti-Pfaffian, whose presence is indicated by daughter states in recent experiments of bilayer graphene at $\ensuremath{\nu}=\frac{3}{4}$.

arXiv:2408.08354•Physical review. B./Physical review. B•Jan 3, 2025•7 citations

Josephson-coupled Moore-Read states

Gunnar Möller, Layla Hormozi, Johannes K. Slingerland +1 more

We study a quantum Hall bilayer system of bosons at total filling factor $\ensuremath{\nu}=1$, and study the phase that results from short-ranged pair tunneling combined with short-ranged interlayer interactions. We introduce two exactly solvable model Hamiltonians which both yield the coupled Moore-Read state [Phys. Rev. Lett. 108, 256809 (2012)] as a ground state, when projected onto fixed particle numbers in each layer. One of these Hamiltonians describes a gapped topological phase, while the other is gapless. However, on introduction of a pair-tunneling term, the second system becomes gapped and develops the same topological order as the gapped Hamiltonian. Supported by the exact solution of the full zero-energy quasihole spectrum and a conformal field-theory approach, we develop an intuitive picture of this system as two coupled composite fermion superconductors. In this language, pair tunneling provides a Josephson coupling of the superconducting phases of the two layers, and gaps out the Goldstone mode associated with particle transport between the layers. In particular, this implies that quasiparticles are confined between the layers. In the bulk, the resulting phase has the topological order of the Halperin 220 phase with U(1)${}_{2}\ifmmode\times\else\texttimes\fi{}\mathrm{U}$(1)${}_{2}$ topological order, but it is realized in the symmetric/antisymmetric basis of the layer index. Consequently, the edge spectrum at a fixed particle number reveals an unexpected U(1)${}_{4}\ifmmode\times\else\texttimes\fi{}\mathrm{U}$(1) structure.

arXiv:1409.6339•Physical Review B•Dec 1, 2014•15 citations

Quantum Hall phase diagram of half-filled bilayers in the lowest and the second orbital Landau levels: Abelian versus non-Abelian incompressible fractional quantum Hall states

Michael R. Peterson, S. Das Sarma

We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.

arXiv:0908.3481•Physical Review B•Apr 2, 2010•49 citations
One of the most dominant candidates for the paired quantum Hall (QH) state at filling factor \ensuremathν=5/2\ensuremath{\nu}=5/2\ensuremathν=5/2 is the Moore-Read (MR) Pfaffian state. A salient problem, however, is that it does not occur exactly at the Coulomb interaction, but rather at a modified interaction, which favors particle-hole-symmetry breaking. In an effort to find a better state, in this work, we investigate the possible connection between the paired QH state and the antisymmetrized bilayer ground state, which is inspired by the intriguing identity that the MR Pfaffian state is entirely equivalent to the antisymmetrized projection of the bilayer QH state called the Halperin (331) state, which is valid at interlayer distance, ddd, roughly equal to the magnetic length, lB{l}_{\mathrm{B}}lB​. Specifically, by using exact diagonalization in the torus geometry, we show that the exact 5/25/25/2 state at a given Haldane pseudopotential variation is intimately connected with the antisymmetrized bilayer ground state at a corresponding d/lBd/{l}_{\mathrm{B}}d/lB​ via one-to-one mapping, which we call the bilayer mapping. One of the most important discoveries in this work is that the paired QH state occurring at the Coulomb interaction is mapped onto the antisymmetrized bilayer ground state at d\ensuremath≫lBd\ensuremath{\gg}{l}_{\mathrm{B}}d\ensuremath≫lB​, which is equivalent to the antisymmetrized product state of two composite fermion seas at quarter filling, not the MR Pfaffian state. While maintaining high overlap with the paired QH state, the antisymmetrized bilayer ground state at d\ensuremath≫lBd\ensuremath{\gg}{l}_{\mathrm{B}}d\ensuremath≫lB​ exhibits an abrupt change under the influence of small anisotropy. This suggests that the paired QH state occurring at the Coulomb interaction might be susceptible to anisotropic instability, opening up the possibility of anisotropic px{p}_{x}px​- or py{p}_{y}py​-wave pairing instead of px\ifmmode±\else\textpm\fiipy{p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}px​\ifmmode±\else\textpm\fiipy​-wave pairing in the MR Pfaffian/anti-Pfaffian state.
Recently, a Dirac (particle-hole symmetric) description of composite fermions in the half-filled Landau level (LL) was proposed [D. T. Son, Phys. Rev. X 5, 031027 (2015)], and we study its possible consequences on BCS (Cooper) pairing of composite fermions (CFs). One of the main consequences is the existence of anisotropic states in single-layer and bilayer systems, which was previously suggested in Jeong and Park [J. S. Jeong and K. Park, Phys. Rev. B 91, 195119 (2015)]. We argue that in the half-filled LL in the single-layer case the gapped states may sustain anisotropy, because isotropic pairings may coexist with anisotropic ones. Furthermore, anisotropic pairings with the addition of a particle-hole symmetry-breaking mass term may evolve into rotationally symmetric states, i.e., Pfaffian states of Halperin-Lee-Read (HLR) ordinary CFs. On the basis of the Dirac formalism, we argue that in the quantum Hall bilayer at total filling factor 1, with decreasing distance between the layers, weak pairing of ppp-wave paired CFs is gradually transformed from Dirac to ordinary, HLR-like, with a concomitant decrease in the CF number. Global characterization of low-energy spectra based on the Dirac CFs agrees well with previous calculations performed by exact diagonalization on a torus. Finally, we discuss features of the Dirac formalism when applied in this context.
In this work, we investigate the nature of the fractional quantum Hall state in the 1/3-filled second Landau level (SLL) at filling factor $\ensuremath{\nu}=7/3$ (and 8/3 in the presence of the particle-hole symmetry) via exact diagonalization in both torus and spherical geometries. Specifically, we compute the overlap between the exact 7/3 ground state and various competing states including (i) the Laughlin state, (ii) the fermionic Haffnian state, (iii) the antisymmetrized product state of two composite fermion seas at 1/6 filling, and (iv) the particle-hole (PH) conjugate of the Z4{Z}_{4}Z4​ parafermion state. All these trial states are constructed according to a guiding principle called the bilayer mapping approach, where a trial state is obtained as the antisymmetrized projection of a bilayer quantum Hall state with interlayer distance ddd as a variational parameter. Under the proper understanding of the ground-state degeneracy in the torus geometry, the Z4{Z}_{4}Z4​ parafermion state can be obtained as the antisymmetrized projection of the Halperin (330) state. Specifically, while unclear at other momentum sectors, all degenerate copies of the Z4{Z}_{4}Z4​ parafermion state can be obtained by antisymmetrizing those of the Halperin (330) state at the zero-momentum sector, where both states occur as the exact ground states of their respective model Hamiltonians with the same degeneracy. Meanwhile, in the spherical geometry, the Z4{Z}_{4}Z4​ parafermion state is shown to be entirely equivalent to the antisymmetrized Halperin (330) state without any ground-state degeneracy issue. Similarly, it is proved in this work that the fermionic Haffnian state can be obtained as the antisymmetrized projection of the Halperin (551) state. The exact 7/3 ground state is obtained as a function of $\ensuremath{\delta}{V}_{1}^{(1)}$, the variation of the first-moment Haldane pseudopotential V1(1){V}_{1}^{(1)}V1(1)​ in the SLL with respect to the pure Coulomb interaction. It is shown that, while extremely accurate at sufficiently large positive $\ensuremath{\delta}{V}_{1}^{(1)}$, the Laughlin state loses its overlap with the exact 7/3 ground state significantly at $\ensuremath{\delta}{V}_{1}^{(1)}\ensuremath{\simeq}0$. At slightly negative $\ensuremath{\delta}{V}_{1}^{(1)}$, it is shown that the PH-conjugated Z4{Z}_{4}Z4​ parafermion state has a substantial overlap with the exact 7/3 ground state, which is the highest among the above four trial states. Around the Coulomb point, the energy spectrum exhibits an intriguing change from the spectrum with the Laughlin-type magnetoroton structure to that with the specific quasidegeneracy of the ground state, which is characteristic to the PH-conjugated Z4{Z}_{4}Z4​ parafermion state.
We examine the possibility of creating the Moore-Read Pfaffian in the lowest Landau level when the multicomponent Halperin 331 state (believed to describe quantum Hall bilayers and wide quantum wells at the filling factor $\ensuremath{\nu}=1/2$) is destroyed by the increase of tunneling. Using exact diagonalization of the bilayer Hamiltonian with short-range and long-range (Coulomb) interactions in spherical and periodic rectangular geometries, we establish that tunneling is a perturbation that drives the 331 state into a compressible composite Fermi liquid, with the possibility for an intermediate critical state that is reminiscent of the Moore-Read Pfaffian. These results are interpreted in the two-component BCS model for Cauchy pairing with a tunneling constraint. We comment on the conditions to be imposed on a system with fluctuating density in order to achieve the stable Pfaffian phase.
Quantum Hall plateaus at quarter fillings occur in GaAs wide quantum wells, hole-doped GaAs, and bilayer graphene. However, the interactions favoring incompressible states over compressible composite-Fermi liquids at such fillings are not well understood. We devise a method for the computation of the trial energies for Haldane pseudopotentials via Monte Carlo sampling. Applying it to the quarter-filled lowest Landau level, we find that tuning the third and fifth pseudopotential values can stabilize the anti-Pfaffian, Moore-Read, and fff-wave states. The smallest deviations from pure Coulomb interactions are required by anti-Pfaffian, whose presence is indicated by daughter states in recent experiments of bilayer graphene at $\ensuremath{\nu}=\frac{3}{4}$.
We study a quantum Hall bilayer system of bosons at total filling factor $\ensuremath{\nu}=1$, and study the phase that results from short-ranged pair tunneling combined with short-ranged interlayer interactions. We introduce two exactly solvable model Hamiltonians which both yield the coupled Moore-Read state [Phys. Rev. Lett. 108, 256809 (2012)] as a ground state, when projected onto fixed particle numbers in each layer. One of these Hamiltonians describes a gapped topological phase, while the other is gapless. However, on introduction of a pair-tunneling term, the second system becomes gapped and develops the same topological order as the gapped Hamiltonian. Supported by the exact solution of the full zero-energy quasihole spectrum and a conformal field-theory approach, we develop an intuitive picture of this system as two coupled composite fermion superconductors. In this language, pair tunneling provides a Josephson coupling of the superconducting phases of the two layers, and gaps out the Goldstone mode associated with particle transport between the layers. In particular, this implies that quasiparticles are confined between the layers. In the bulk, the resulting phase has the topological order of the Halperin 220 phase with U(1)${}_{2}\ifmmode\times\else\texttimes\fi{}\mathrm{U}$(1)2{}_{2}2​ topological order, but it is realized in the symmetric/antisymmetric basis of the layer index. Consequently, the edge spectrum at a fixed particle number reveals an unexpected U(1)${}_{4}\ifmmode\times\else\texttimes\fi{}\mathrm{U}$(1) structure.
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.