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

A Matrix Model for Bilayered Quantum Hall Systems

Ahmed Jellal, Michael Schreiber

Published Apr 24, 2003• 5 citations
OpenAlex

Abstract

We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at \\nu=q_iK_{ij}^{-1}q_j, where K_{ij} is a 2\\times 2 matrix and q_i is a vector. Our model allows us to reproduce several well-known wave functions. We show that the wave function \\Psi_{(m,m,n)} constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor {2\\over m+n} and in particular \\Psi_{(3,3,1)} at filling {1\\over 2} can be obtained from our vacuum configuration. The unpolarized Halperin wave function and especially that for the fractional quantum Hall state at filling factor {2\\over 5} can also be recovered from our approach. Generalization to more than 2 layers is straightforward.

Related papers

A matrix model for bilayered quantum Hall systems

Ahmed Jellal, Michael Schreiber

We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at ν = q i K −1 ij q j , where K ij is a 2 × 2 matrix and q i is a vector. Our model allows us to reproduce several well-known wavefunctions. We show that the wavefunction Ψ ( m , m , n ) constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor and in particular Ψ (3,3,1) at filling can be obtained from our vacuum configuration. The unpolarized Halperin wavefunction and especially that for the fractional quantum Hall state at filling factor can also be recovered from our approach. Generalization to more than two layers is straightforward.

arXiv:hep-th/0304207•Journal of Physics A Mathematical and General•Feb 18, 2004•4 citations

Trial wave functions for ν=12+12 quantum Hall bilayers

Gunnar Möller, Steven H. Simon, Edward H. Rezayi

Quantum Hall bilayer systems at filling fractions near $\ensuremath{\nu}=\frac{1}{2}+\frac{1}{2}$ undergo a transition from a compressible phase with strong intralayer correlation to an incompressible phase with strong interlayer correlations as the layer separation $d$ is reduced below some critical value. Deep in the intralayer phase (large separation) the system can be interpreted as a fluid of composite fermions (CFs), whereas deep in the interlayer phase (small separation) the system can be interpreted as a fluid of composite bosons (CBs). The focus of this paper is to understand the states that occur for intermediate layer separation by using trial variational wave functions. We consider two main classes of wave functions. In the first class, previously introduced in M\"oller et al. [Phys. Rev. Lett. 101, 176803 (2008)], we consider interlayer BCS pairing of two independent CF liquids. We find that these wave functions are exceedingly good for $d\ensuremath{\gtrsim}{\ensuremath{\ell}}_{0}$ with ${\ensuremath{\ell}}_{0}$ as the magnetic length. The second class of wave functions naturally follows the reasoning of Simon et al. [Phys. Rev. Lett. 91, 046803 (2003)] and generalizes the idea of pairing wave functions by allowing the CFs also to be replaced continuously by CBs. This generalization allows us to construct exceedingly good wave functions for interlayer spacings of $d\ensuremath{\lesssim}{\ensuremath{\ell}}_{0}$ as well. The accuracy of the wave functions discussed in this work, compared with exact diagonalization, approaches that of the celebrated Laughlin wave function.

arXiv:0811.4116•Physical Review B•Mar 6, 2009•52 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

Numeric and Analytic Investigation on Phase Diagrams and Phasetransitions of the ν = 2/3 Bilayer Fractional Quantum Hall Systems

Yangdong Zheng, T. Sorita

The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. We find some basic features essentially different from the bilayer integer QH systems at ν = 2, reflecting the special characteristics of the fractional QH systems. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The continuous transitions of the finite size systems between the spin-polarized and spin-unpolarized phases are determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy, and are almost not affected by ΔSAS.

Journal of Applied Mathematics and Physics•Jan 1, 2018•0 citations

A Matrix Model for νk1k2=(k1+k2)/k1k2Fractional Quantum Hall States

Ahmed Jellal, El Hassan Saidi, H. B. Geyer +1 more

We have developed a matrix model for FQH states at filling factor ν_{k_1k_2} going beyond the Laughlin theory. To illustrate our idea, we have considered an FQH system of a finite number N=(N_{1}+N_{2}) of electrons with filling factor ν_{k_{1}k_{2}} = ν_{p_{1}p_{2}}=\frac{p_{2}}{p_{1}p_{2}-1}; p_{1} is an odd integer and p_{2} is an even integer. The ν_{p_{1}p_{2}} series corresponds just to the level two of the Haldane hierarchy; it recovers the Laughlin series ν_{p_{1}} =\frac{1}{p_{1}} by going to the limit p_{2} large and contains several observable FQH states such as ν= 2/3, 2/5, >....

arXiv:hep-th/0303143•Journal of the Physical Society of Japan•Jan 3, 2003•6 citations

Halperin states can produce any filling fractions

En-Jui Kuo

We study bilayer Fractional Quantum Hall State also known as Halperin state. We prove that for any filling fractions (positive rational numbers), there are infinite solutions that imply that infinite topological states are corresponding to any given filling fraction. Our results can apply to any charge vectors.

arXiv:2004.09035•Apr 20, 2020•0 citations
We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at \nu=q_iK_{ij}^{-1}q_j, where K_{ij} is a 2\times 2 matrix and q_i is a vector. Our model allows us to reproduce several well-known wave functions. We show that the wave function \Psi_{(m,m,n)} constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor {2\over m+n} and in particular \Psi_{(3,3,1)} at filling {1\over 2} can be obtained from our vacuum configuration. The unpolarized Halperin wave function and especially that for the fractional quantum Hall state at filling factor {2\over 5} can also be recovered from our approach. Generalization to more than 2 layers is straightforward.
We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at ν = q i K −1 ij q j , where K ij is a 2 × 2 matrix and q i is a vector. Our model allows us to reproduce several well-known wavefunctions. We show that the wavefunction Ψ ( m , m , n ) constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor and in particular Ψ (3,3,1) at filling can be obtained from our vacuum configuration. The unpolarized Halperin wavefunction and especially that for the fractional quantum Hall state at filling factor can also be recovered from our approach. Generalization to more than two layers is straightforward.
Quantum Hall bilayer systems at filling fractions near $\ensuremath{\nu}=\frac{1}{2}+\frac{1}{2}$ undergo a transition from a compressible phase with strong intralayer correlation to an incompressible phase with strong interlayer correlations as the layer separation ddd is reduced below some critical value. Deep in the intralayer phase (large separation) the system can be interpreted as a fluid of composite fermions (CFs), whereas deep in the interlayer phase (small separation) the system can be interpreted as a fluid of composite bosons (CBs). The focus of this paper is to understand the states that occur for intermediate layer separation by using trial variational wave functions. We consider two main classes of wave functions. In the first class, previously introduced in M\"oller et al. [Phys. Rev. Lett. 101, 176803 (2008)], we consider interlayer BCS pairing of two independent CF liquids. We find that these wave functions are exceedingly good for $d\ensuremath{\gtrsim}{\ensuremath{\ell}}_{0}$ with ${\ensuremath{\ell}}_{0}$ as the magnetic length. The second class of wave functions naturally follows the reasoning of Simon et al. [Phys. Rev. Lett. 91, 046803 (2003)] and generalizes the idea of pairing wave functions by allowing the CFs also to be replaced continuously by CBs. This generalization allows us to construct exceedingly good wave functions for interlayer spacings of $d\ensuremath{\lesssim}{\ensuremath{\ell}}_{0}$ as well. The accuracy of the wave functions discussed in this work, compared with exact diagonalization, approaches that of the celebrated Laughlin wave function.
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.
The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. We find some basic features essentially different from the bilayer integer QH systems at ν = 2, reflecting the special characteristics of the fractional QH systems. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The continuous transitions of the finite size systems between the spin-polarized and spin-unpolarized phases are determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy, and are almost not affected by ΔSAS.
We have developed a matrix model for FQH states at filling factor ν_{k_1k_2} going beyond the Laughlin theory. To illustrate our idea, we have considered an FQH system of a finite number N=(N_{1}+N_{2}) of electrons with filling factor ν_{k_{1}k_{2}} = ν_{p_{1}p_{2}}=\frac{p_{2}}{p_{1}p_{2}-1}; p_{1} is an odd integer and p_{2} is an even integer. The ν_{p_{1}p_{2}} series corresponds just to the level two of the Haldane hierarchy; it recovers the Laughlin series ν_{p_{1}} =\frac{1}{p_{1}} by going to the limit p_{2} large and contains several observable FQH states such as ν= 2/3, 2/5, >....