Non-Abelian Chern-Simons bosonization approach and one dimensional t-J model

We determine and study the statistical interparticle potential of an ideal system of non-Abelian Chern-Simons (NACS) particles, comparing our results with the corresponding results of an ideal gas of Abelian anyons. In the Abelian case, the statistical potential depends on the statistical parameter and it has a "quasi-bosonic" behaviour for statistical parameter in the range (0,1/2) (non-monotonic with a minimum) and a "quasi-fermionic" behaviour for statistical parameter in the range (1/2,1) (monotonically decreasing without a minimum). In the non-Abelian case the behavior of the statistical potential depends on the Chern- Simons coupling and the isospin quantum number: as a function of these two parameters, a phase diagram with quasi-bosonic, quasi-fermionic and bosonic-like regions is obtained and investigated. Finally, using the obtained expression for the statistical potential, we compute the second virial coefficient of the NACS gas, which correctly reproduces the results available in literature.We determine and study the statistical interparticle potential of an ideal system of non-Abelian Chern-Simons (NACS) particles, comparing our results with the corresponding results of an ideal gas of Abelian anyons. In the Abelian case, the statistical potential depends on the statistical parameter and it has a "quasi-bosonic" behaviour for statistical parameter in the range (0,1/2) (non-monotonic with a minimum) and a "quasi-fermionic" behaviour for statistical parameter in the range (1/2,1) (monotonically decreasing without a minimum). In the non-Abelian case the behavior of the statistical potential depends on the Chern- Simons coupling and the isospin quantum number: as a function of these two parameters, a phase diagram with quasi-bosonic, quasi-fermionic and bosonic-like regions is obtained and investigated. Finally, using the obtained expression for the statistical potential, we compute the second virial coefficient of the NACS gas, which correctly reproduces the results available in literature.

arXiv:1207.1724Journal of Physics A Mathematical and TheoreticalJun 18, 20135 citations

Hard-core bosonization of the one-dimensional t-J model

F. Pan, M.-X. Xie, Jijie Lin +2 more

The European Physical Journal BNov 18, 20090 citations

Landau-Ginzburg theories of non-Abelian quantum Hall states from non-Abelian bosonization

Hart Goldman, Ramanjit Sohal, Eduardo Fradkin

It is an important open problem to understand the landscape of non-Abelian fractional quantum Hall phases which can be obtained starting from physically motivated theories of Abelian composite particles. We show that progress on this problem can be made using recently proposed non-Abelian bosonization dualities in $2+1$ dimensions, which morally relate $U{(N)}_{k}$ and $\mathrm{SU}{(k)}_{\ensuremath{-}N}$ Chern-Simons-matter theories. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings $\ensuremath{\nu}=k/(kM+2)$ by starting from $k$ layers of bosons at $\ensuremath{\nu}=1/2$ with $M$ Abelian fluxes attached. The Read-Rezayi states arise when $k$ clusters of the dual non-Abelian bosons condense. We extend this construction by showing that ${N}_{f}$ -component generalizations of the Halperin (2,2,1) bosonic states have dual descriptions in terms of $\mathrm{SU}{({N}_{f}+1)}_{1}$ Chern-Simons-matter theories, revealing an emergent global symmetry in the process. Clustering $k$ layers of these theories yields a non-Abelian $\mathrm{SU}({N}_{f})$ -singlet state at filling $\ensuremath{\nu}=k{N}_{f}/({N}_{f}+1+kM{N}_{f})$.It is an important open problem to understand the landscape of non-Abelian fractional quantum Hall phases which can be obtained starting from physically motivated theories of Abelian composite particles. We show that progress on this problem can be made using recently proposed non-Abelian bosonization dualities in $2+1$ dimensions, which morally relate $U{(N)}_{k}$ and $\mathrm{SU}{(k)}_{\ensuremath{-}N}$ Chern-Simons-matter theories. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings $\ensuremath{\nu}=k/(kM+2)$ by starting from $k$ layers of bosons at $\ensuremath{\nu}=1/2$ with $M$ Abelian fluxes attached. The Read-Rezayi states arise when $k$ clusters of the dual non-Abelian bosons condense. We extend this construction by showing that ${N}_{f}$-component generalizations of the Halperin (2,2,1) bosonic states have dual descriptions in terms of $\mathrm{SU}{({N}_{f}+1)}_{1}$ Chern-Simons-matter theories, revealing an emergent global symmetry in the process. Clustering $k$ layers of these theories yields a non-Abelian $\mathrm{SU}({N}_{f})$-singlet state at filling $\ensuremath{\nu}=k{N}_{f}/({N}_{f}+1+kM{N}_{f})$.

arXiv:1906.00983Physical review. B./Physical review. BSep 5, 201922 citations

Non-abelian bosonization in two dimensions

Edward Witten

Communications in Mathematical PhysicsDec 1, 19842,384 citations

2+1

dimensions, which morally relate

U{(N)}_{k}

and $\mathrm{SU}{(k)}_{\ensuremath{-}N}$ Chern-Simons-matter theories. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings $\ensuremath{\nu}=k/(kM+2)$ by starting from

k

layers of bosons at $\ensuremath{\nu}=1/2$ with

M

Abelian fluxes attached. The Read-Rezayi states arise when

k

clusters of the dual non-Abelian bosons condense. We extend this construction by showing that

{N}_{f}

-component generalizations of the Halperin (2,2,1) bosonic states have dual descriptions in terms of

\mathrm{SU}{({N}_{f}+1)}_{1}

Chern-Simons-matter theories, revealing an emergent global symmetry in the process. Clustering

k

layers of these theories yields a non-Abelian

\mathrm{SU}({N}_{f})

-singlet state at filling $\ensuremath{\nu}=k{N}_{f}/({N}_{f}+1+kM{N}_{f})$.