A model of the non-Abelian fractional quantum Hall effect is obtained from the diagonalization of the matrix model proposed by Dorey, Tong, and Turner (DTT). The Hamiltonian is reminiscent of a spin Calogero-Moser model but involves higher-order symmetric representations of the non-Abelian symmetry. We derive the energy spectrum and show that the Hamiltonian has a triangular action on a certain class of wave functions with a free fermion expression. We deduce the expression of the ground states eigenfunctions and show that they solve a Knizhnik-Zamolodchikov equation. Finally, we discuss the emergence of Kac-Moody symmetries in the large $N$ limit using the level-rank duality and confirm the results obtained previously by DTT.A model of the non-Abelian fractional quantum Hall effect is obtained from the diagonalization of the matrix model proposed by Dorey, Tong, and Turner (DTT). The Hamiltonian is reminiscent of a spin Calogero-Moser model but involves higher-order symmetric representations of the non-Abelian symmetry. We derive the energy spectrum and show that the Hamiltonian has a triangular action on a certain class of wave functions with a free fermion expression. We deduce the expression of the ground states eigenfunctions and show that they solve a Knizhnik-Zamolodchikov equation. Finally, we discuss the emergence of Kac-Moody symmetries in the large $N$ limit using the level-rank duality and confirm the results obtained previously by DTT.

arXiv:2401.03087Jan 5, 20240 citations

Anatomy of Abelian and Non-Abelian Fractional Quantum Hall States

Bogdan Andrei Bernevig, Nicolas Regnault

We deduce a new set of symmetries and relations between the coefficients of the expansion of Abelian and non-Abelian fractional quantum Hall (FQH) states in free (bosonic or fermionic) many-body states. Our rules allow us to build an approximation of a FQH model state with an overlap increasing with growing system size (that may sometimes reach unity) while using a fraction of the original Hilbert space. We prove these symmetries by deriving a previously unknown recursion formula for all the coefficients of the Slater expansion of the Laughlin, Read-Rezayi, and many other states (all Jacks multiplied by Vandermonde determinants), which completely removes the current need for diagonalization procedures for these model Hamiltonians.We deduce a new set of symmetries and relations between the coefficients of the expansion of Abelian and non-Abelian fractional quantum Hall (FQH) states in free (bosonic or fermionic) many-body states. Our rules allow us to build an approximation of a FQH model state with an overlap increasing with growing system size (that may sometimes reach unity) while using a fraction of the original Hilbert space. We prove these symmetries by deriving a previously unknown recursion formula for all the coefficients of the Slater expansion of the Laughlin, Read-Rezayi, and many other states (all Jacks multiplied by Vandermonde determinants), which completely removes the current need for diagonalization procedures for these model Hamiltonians.

arXiv:0902.4320Physical Review LettersNov 10, 200984 citations

Conformal field theory construction for non-Abelian hierarchy wave functions

Yoran Tournois, Maria Hermanns

The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular non-Abelian ones. Here we analyze a class of non-Abelian fractional quantum Hall model states which are generalizations of the Abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey non-Abelian fusion rules of type $\mathrm{su}{\left(q\right)}_{k}$ . For a subset of these states we are able to derive the conformal field theory description that makes the topological properties---in particular braiding---of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular non-Abelian ones. Here we analyze a class of non-Abelian fractional quantum Hall model states which are generalizations of the Abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey non-Abelian fusion rules of type $\mathrm{su}{\left(q\right)}_{k}$. For a subset of these states we are able to derive the conformal field theory description that makes the topological properties---in particular braiding---of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.

arXiv:1708.00763Physical review. B./Physical review. BDec 5, 20172 citations

Chern-number matrix of the non-Abelian spin-singlet fractional quantum Hall effect

Tian-Sheng Zeng, Wei Kang Zhu

While the internal structure of Abelian topological order is well understood, how to characterize the non-Abelian topological order is an outstanding issue. We propose a distinctive scheme based on the many-body Chern number matrix to characterize non-Abelian multicomponent fractional quantum Hall states. As a concrete example, we study the many-body ground state of two-component bosons at the filling faction $\ensuremath{\nu}=4/3$ in topological flat band models. Utilizing density-matrix renormalization group and exact diagonalization calculations, we demonstrate the emergence of non-Abelian spin-singlet fractional quantum Hall effect under three-body interaction, whose topological nature is classified by sixfold degenerate ground states and a fractionally quantized Chern-number matrix.While the internal structure of Abelian topological order is well understood, how to characterize the non-Abelian topological order is an outstanding issue. We propose a distinctive scheme based on the many-body Chern number matrix to characterize non-Abelian multicomponent fractional quantum Hall states. As a concrete example, we study the many-body ground state of two-component bosons at the filling faction $\ensuremath{\nu}=4/3$ in topological flat band models. Utilizing density-matrix renormalization group and exact diagonalization calculations, we demonstrate the emergence of non-Abelian spin-singlet fractional quantum Hall effect under three-body interaction, whose topological nature is classified by sixfold degenerate ground states and a fractionally quantized Chern-number matrix.

arXiv:2110.07245Physical review. B./Physical review. BMar 21, 20227 citations

Matrix model for non-Abelian quantum Hall states

Nick Dorey A, David Tong, Carl Turner

We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal $\text{SU}(p)$ spin. The ground states of the matrix model include spin-singlet generalizations of the Moore-Read and Read-Rezayi states and, in general, lie in a class previously introduced by Blok and Wen. The effective action for these states is a $\text{U}(p)$ Chern-Simons theory. We show how the matrix model can be derived from quantization of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions in the boundary WZW model.We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal $\text{SU}(p)$ spin. The ground states of the matrix model include spin-singlet generalizations of the Moore-Read and Read-Rezayi states and, in general, lie in a class previously introduced by Blok and Wen. The effective action for these states is a $\text{U}(p)$ Chern-Simons theory. We show how the matrix model can be derived from quantization of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions in the boundary WZW model.

arXiv:1603.09688Physical review. B./Physical review. BAug 9, 201625 citations

Fractional quantum Hall effect and quantum symmetry

Gerhard Grensing

Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group ${U}_{q}{(su}_{2})$ with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group ${U}_{q}{(su}_{2})$ with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.

Physical review. B, Condensed matterFeb 15, 20001 citation

N

limit using the level-rank duality, and we confirm the results obtained previously by DTT.

N

limit using the level-rank duality and confirm the results obtained previously by DTT.

\mathrm{su}{\left(q\right)}_{k}

. For a subset of these states we are able to derive the conformal field theory description that makes the topological properties---in particular braiding---of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.

We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal

\text{SU}(p)

spin. The ground states of the matrix model include spin-singlet generalizations of the Moore-Read and Read-Rezayi states and, in general, lie in a class previously introduced by Blok and Wen. The effective action for these states is a

\text{U}(p)

Chern-Simons theory. We show how the matrix model can be derived from quantization of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions in the boundary WZW model.

{U}_{q}{(su}_{2})

with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.