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arXiv:1906.00983 · Physical review. B./Physical review. B

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

Hart Goldman, Ramanjit Sohal, Eduardo Fradkin

Published Sep 5, 2019• 22 citations
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Abstract

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 $\ensuremath{\nu}=k{N}_{f}/({N}_{f}+1+kM{N}_{f})$.

DOI
10.1103/physrevb.100.115111

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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+12+12+1 dimensions, which morally relate U(N)kU{(N)}_{k}U(N)k​ and SU(k)\ensuremath−N\mathrm{SU}{(k)}_{\ensuremath{-}N}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 \ensuremathν=k/(kM+2)\ensuremath{\nu}=k/(kM+2)\ensuremathν=k/(kM+2) by starting from kkk layers of bosons at \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2 with MMM Abelian fluxes attached. The Read-Rezayi states arise when kkk clusters of the dual non-Abelian bosons condense. We extend this construction by showing that Nf{N}_{f}Nf​-component generalizations of the Halperin (2,2,1) bosonic states have dual descriptions in terms of SU(Nf+1)1\mathrm{SU}{({N}_{f}+1)}_{1}SU(Nf​+1)1​ Chern-Simons-matter theories, revealing an emergent global symmetry in the process. Clustering kkk layers of these theories yields a non-Abelian SU(Nf)\mathrm{SU}({N}_{f})SU(Nf​)-singlet state at filling \ensuremathν=kNf/(Nf+1+kMNf)\ensuremath{\nu}=k{N}_{f}/({N}_{f}+1+kM{N}_{f})\ensuremathν=kNf​/(Nf​+1+kMNf​).
The recent proposal of non-Abelian boson-fermion dualities in 2+12+12+1 dimensions, which morally relate U(k)NU{(k)}_{N}U(k)N​ to $SU{(N)}_{\ensuremath{-}k}$ Chern-Simons-matter theories, presents a new platform for exploring the landscape of non-Abelian quantum Hall states accessible from theories of Abelian composite particles. Here we focus on dualities relating theories of Abelian quantum Hall states of bosons or fermions to theories of non-Abelian ``composite fermions'' partially filling Landau levels. We show that these dualities predict special filling fractions where both Abelian and non-Abelian composite fermion theories appear capable of hosting distinct topologically ordered ground states, one Abelian and the other a non-Abelian, U(k)2U{(k)}_{2}U(k)2​ Blok-Wen state. Rather than being in conflict with the duality, we argue that these results indicate unexpected dynamics in which the infrared and lowest Landau level limits fail to commute across the duality. In such a scenario, the non-Abelian topological order can be destabilized in favor of the Abelian ground state, suggesting the presence of a phase transition between the Abelian and non-Abelian states that is likely to be first order. We also generalize these constructions to other non-Abelian fermion-fermion dualities, in the process obtaining new derivations of a variety of paired composite fermion phases using duality, including the anti-Pfaffian state. Finally, we describe how, in multilayer constructions, excitonic pairing of the composite fermions across NNN layers can also generate the family of Blok-Wen states with U(k)2U{(k)}_{2}U(k)2​ topological order.
Quantum Hall systems offer the most familiar setting where strong interparticle interactions combine with the topology of single-particle states to yield novel phenomena. Despite our mature understanding of these systems, an open challenge has been to develop a microscopic theory capturing both their universal and nonuniversal properties, when the Hamiltonian is restricted to the noncommutative space of the lowest Landau level. Here we develop such a theory for the Jain sequence of bosonic fractional quantum Hall states at fillings $\ensuremath{\nu}=\frac{p}{p+1}$. Building on a lowest Landau level description of a parent composite Fermi liquid at $\ensuremath{\nu}=1$, we describe how to dope the system to reach the Jain states. Upon doping, the composite fermions fill noncommutative generalizations of Landau levels, and the Jain states correspond to integer composite fermion filling. Using this approach, we obtain an approximate expression for the bosonic Jain sequence gaps with no reference to any long-wavelength approximation. Furthermore, we show that the universal properties, such as Hall conductivity, are encoded in an effective noncommutative Chern-Simons theory, which is obtained on integrating out the composite fermions. This theory has the same topological content as the familiar Abelian Chern-Simons theory on commutative space.
The Fibonacci topological order is the simplest platform for a universal topological quantum computer. While the $\ensuremath{\nu}=12/5$ fractional quantum Hall (QH) state has been proposed to support a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a QH system has been lacking. We use non-Abelian dualities to construct a Fibonacci state of bosons at filling $\ensuremath{\nu}=2$ starting from a trilayer of integer QH states. Our parent theory consists of bosonic composite vortices coupled to fluctuating U(2)U(2)U(2) gauge fields, which is dual to the theory of Laughlin quasiparticles. The Fibonacci state is obtained by interlayer clustering of the composite vortices, along with flux attachment. We use this framework to motivate a wave function for the Fibonacci state.
We show that a modified version of Son's Dirac composite fermion theory proposed by Seiberg et al gives a candidate unified description of the gapped and gapless fractional quantum Hall states within a single Landau level. Our main tool is the successive application of three-dimensional dualities to partially filled Landau levels of composite fermions, which imply that this theory has a complicated landscape of gapped vacua and critical points. This construction is the Lagrangian, or effective field theory, analogue of the flux attachment procedure. The critical points exist at even denominator filling and are well-described by a Fermi surface for a weakly coupled composite fermion coupled to an abelian Chern-Simons theory. The gapped states include odd-denominator filling fraction states with an abelian Chern-Simons description which we show matches the one expected for hierarchy states, as well as non-abelian states at even-denominator filling that arise from pair instabilities of the composite fermion's Fermi surface.
The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian $SU(N\ensuremath{-}1)$-singlet FQH states at filling fraction $\ensuremath{\nu}=k(N\ensuremath{-}1)/[N+k(N\ensuremath{-}1)m]$ labeled by integers kkk and mmm, which are potentially realized in multicomponent quantum Hall systems or SU(N)SU(N)SU(N) spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we show that their bulk quasiparticles and gapless edge excitations are both described by an $(N\ensuremath{-}1)$-component free-boson CFT and the $SU{(N)}_{k}/{[U(1)]}^{N\ensuremath{-}1}$ CFT known as the Gepner parafermion. Their generalization to different filling fractions is also proposed. In addition, we argue possible applications of these results to two kinds of lattice systems: bosons interacting via occupation-dependent correlated hoppings and an SU(N)SU(N)SU(N) Heisenberg model.
We study U(1)×U(1)⋊Z2 Chern-Simons theory with integral coupling constants (k,l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the U(1)×U(1)⋊Z2 Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that Z2 vortices in the U(1)×U(1)⋊Z2 Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of n pairs of Z2 vortices on a sphere. These results allow us to show that l=3 U(1)×U(1)⋊Z2 Chern-Simons theory is the low-energy effective theory for the Z4 parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction ν=2/2k−3. The U(1)×U(1)⋊Z2 theory is more useful than an alternative SU(2)4×U(1)∕U(1) Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the ν=2∕3 phase diagram.