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arXiv:1611.07968 · Physical review. B./Physical review. B

Non-Abelian S U ( N − 1 ) -singlet fractional quantum Hall states from coupled wires

Yohei Fuji, P. Lecheminant

Published Mar 24, 2017• 38 citations
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Abstract

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 $k$ and $m$, which are potentially realized in multicomponent quantum Hall systems or $SU(N)$ 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 $SU(N)$ Heisenberg model.

DOI
10.1103/physrevb.95.125130

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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)SU(N\ensuremath{-}1)SU(N\ensuremath−1)-singlet FQH states at filling fraction \ensuremathν=k(N\ensuremath−1)/[N+k(N\ensuremath−1)m]\ensuremath{\nu}=k(N\ensuremath{-}1)/[N+k(N\ensuremath{-}1)m]\ensuremathν=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)(N\ensuremath{-}1)(N\ensuremath−1)-component free-boson CFT and the SU(N)k/[U(1)]N\ensuremath−1SU{(N)}_{k}/{[U(1)]}^{N\ensuremath{-}1}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 compute the physical properties of non-Abelian fractional quantum Hall (FQH) states described by Jack polynomials at general filling nu = k/r. For r = 2, these states are the Zk Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k+1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5,3/7,4/9,.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and the non-Abelian quasihole propagator. The properties of the r > 2 Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator. The quasihole wave function, written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states.
We formulate a theory of non-Abelian fractional quantum Hall states by considering an anisotropic system consisting of coupled, interacting one dimensional wires. We show that Abelian bosonization provides a simple framework for characterizing the Moore Read state, as well as the more general Read Rezayi sequence. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. This construction leads to a simple interpretation of the coset construction of conformal field theory, which is a powerful method for describing non Abelian states. In the present context, the coset construction arises when the original chiral modes are fractionalized into coset sectors, and the different sectors acquire energy gaps due to coupling in "different directions". The coupled wire construction can also can be used to describe anisotropic lattice systems, and provides a starting point for models of fractional and non-Abelian Chern insulators. This paper also includes an extended introduction to the coupled wire construction for Abelian quantum Hall states, which was introduced earlier.
We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4{Z}_{4}Z4​ parafermion Read-Rezayi state. The Z4{Z}_{4}Z4​ orbifold quantum Hall states occur at filling factors $\ensuremath{\nu}=2/(2m\ensuremath{-}p)$ for odd integers mmm and ppp, and have a topological order with a neutral sector characterized by the orbifold conformal field theory with central charge c=1c=1c=1 at radius R=p/2R=\sqrt{p/2}R=p/2​. When p=1p=1p=1 the state is Abelian. The state with p=3p=3p=3 is the Z4{Z}_{4}Z4​ Read-Rezayi state, and the series of $p\ensuremath{\ge}3$ defines a sequence of non-Abelian states that resembles the Laughlin sequence. Our model is based on clustering of electrons in groups of four, and is formulated as a two fluid model in which each wire exhibits two phases: a weak clustered phase, where charge eee electrons coexist with charge 4e4e4e bosons and a strong clustered phase where the electrons are strongly bound in groups of four. The transition between these two phases on a wire is mapped to the critical point of the four-state clock model, which in turn is described by the orbifold conformal field theory. For an array of wires coupled in the presence of a perpendicular magnetic field, strongly clustered wires form a charge 4e4e4e bosonic Laughlin state with a chiral charge mode at the edge, but no neutral mode and a gap for single electrons. Coupled wires near the critical state form quantum Hall states with a gapless neutral mode described by the orbifold theory. The coupled wire approach allows us to employ the Abelian bosonization technique to fully analyze the physics of single wire, and then to extract most topological properties of the resulting non-Abelian quantum Hall states. These include the list of quasiparticles, their fusion rules, the correspondence between bulk quasiparticles and edge topological sectors, and most of the phases associated with quasiparticles winding one another.
The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $\ensuremath{\nu}=k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted 2‾k1k+1{\overline{2}}^{k}{1}^{k+1}2k1k+1, and show that they lie in the same universality classes as the particle-hole-conjugate RR (``anti-RR'') states. Our analytical results imply that a $[U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}]$ coset conformal field theory describes the edge excitations of the 2‾k1k+1{\overline{2}}^{k}{1}^{k+1}2k1k+1 state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence n‾2‾214\overline{n}{\overline{2}}^{2}{1}^{4}n2214, with n=1,2,3n=1,2,3n=1,2,3, to describe the FQH states observed at $\ensuremath{\nu}=2+1/2$, 2+2/52+2/52+2/5, and 2+3/82+3/82+3/8.
There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two-component FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, for integer nnn. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, including in particular the possibility of the non-Abelian Z4{Z}_{4}Z4​ parafermion state. In $\ensuremath{\nu}=2/3$ bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the Z4{Z}_{4}Z4​ state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the Z4{Z}_{4}Z4​ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\ensuremath{\nu}=8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.
We construct a coupled wire model for a sequence of non-Abelian quantum Hall states occurring at filling factors $\ensuremath{\nu}=2/(2M+q)$ with integers MMM and even (odd) integers qqq for fermionic (bosonic) states. They are termed ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ orbifold states, which have a topological order with a neutral sector described by the c=1c=1c=1 orbifold conformal field theory (CFT) at radius Rorbifold=p/2{R}_{\mathrm{orbifold}}=\sqrt{p/2}Rorbifold​=p/2​ with even integers ppp. When p=2p=2p=2, the state can be viewed as two decoupled layers of Moore-Read states, whose neutral sector is described by the Ising $\ifmmode\times\else\texttimes\fi{}$ Ising CFT and contains a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion subalgebra. We demonstrate that orbifold states with p>2p>2p>2, also containing a ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ fusion algebra, can be obtained by coupling together an array of $\mathrm{MR}\ifmmode\times\else\texttimes\fi{}\mathrm{MR}$ wires through local interactions. The corresponding charge spectrum of quasiparticles is also examined. The orbifold states constructed here are complementary to the Z4{Z}_{4}Z4​ orbifold states, whose neutral edge theory is described by orbifold CFT with odd integer ppp and contains a Z4{Z}_{4}Z4​ fusion algebra.