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.
Explicit relation between Laughlin state of the quantum Hall effect and one-dimensional(1D) model with long-ranged interaction ($1/r^2$) is discussed. By rewriting lowest Landau level wave functions in terms of 1D representation, Laughlin state can be written as a deformation of the ground state of Calogero-Sutherland model. Corresponding to Laughlin state on different geometries, different types of 1D $1/r^2$ interaction models are derived.
\noindent Using hydrodynamic collective field theory approach we show that one-particle density matrix of the $ν=1/m$ fractional quantum Hall edge state interpolates between chiral Luttinger liquid behavior $\langle ψ^{\dagger}(r) ψ(0) \rangle \sim r^{-m} $ and Calogero-Sutherland model behavior $\langle ψ^{\dagger}(r) ψ(0) \rangle \sim r^{-(m+1/m)/2} $ as the droplet width is varied continuously. Low-energy excitations are described by $c=1$ conformal field theory of a compact boson of radius $\sqrt m$. The result suggests complementary relation between the two-dimensional quantum Hall drople…
The main point of the construction of spin Calogero type classical integrable systems based on dynamical r-matrices, developed by L.-C. Li and P. Xu, is reviewed. It is shown that non-Abelian dynamical r-matrices with variables in a reductive Lie algebra ${\cal F}$ and their Abelian counterparts with variables in a Cartan subalgebra of ${\cal F}$ lead essentially to the same models.
It is known that non-commutative fluids used to model the Fractional Quantum Hall effect give Calogero--Moser systems. The group-theoretic description of these as reductions of free motion on type A Lie algebras leads directly to Laughlin wave functions. The Calogero--Moser models also parametrise the right ideals of the Weyl algebra, which can be regarded as labelling sources in the fluid.
We present a Chern-Simons matrix model describing the fractional quantum Hall effect on the two-sphere. We demonstrate the equivalence of our proposal to particular restrictions of the Calogero-Sutherland model, reproduce the quantum states and filling fraction and show the compatibility of our result with the Haldane spherical wavefunctions.
We report on the new approach to constructing superconformal extensions of the Calogero-type systems with an arbitrary number of involved particles. It is based upon the superfield gauging of non-abelian isometries of some supersymmetric matrix models. Among its applications, we focus on the new N=4 superconformal system yielding the U(2) spin Calogero model in the bosonic sector, and the one-particle case of this system, which is a new OSp(4|2) superconformal mechanics with non-dynamical U(2) spin variables. The characteristic feature of these models is that the strength of the conformal inve…
We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with non-trivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the Calogero-Sutherland Hamiltonian are characterize…
Quantum Hall matrix models are simple, solvable quantum mechanical systems which capture the physics of certain fractional quantum Hall states. Recently, it was shown that the Hall viscosity can be extracted from the matrix model for Laughlin states. Here we extend this calculation to the matrix models for a class of non-Abelian quantum Hall states. These states, which were previously introduced by Blok and Wen, arise from the conformal blocks of Wess-Zumino-Witten conformal field theory models. We show that the Hall viscosity computed from the matrix model coincides with a result of Read, in …
We show that the set of double-layer Fractional Quantum Hall (FQH) states with a given topological order form a finite Abelian group under a new product. This group structure makes it possible to construct new FQH states from known ones. We also introduce a new index which can be used to characterize the topological order of FQH states.
We propose a finite Chern-Simons matrix model on the plane as an effective description of fractional quantum Hall fluids of finite extent. The quantization of the inverse filling fraction and of the quasiparticle number is shown to arise quantum mechanically and to agree with Laughlin theory. We also point out the effective equivalence of this model, and therefore of the quantum Hall system, with the Calogero model.
Bosonization dualities relate two different Chern-Simons-matter theories, with bosonic matter on one side replaced by fermionic matter on the other. We first describe a more general class of non-Abelian bosonization dualities. We then explore the non-relativistic physics of these theories in the quantum Hall regime. The bosonic theory lies in a condensed phase and admits vortices which are known to form a non-Abelian quantum Hall state. We ask how this same physics arises in the fermionic theory. We find that a condensed boson corresponds to a fully filled Landau level of fermions, while boson…
We study the scaling behavior in the tunneling amplitude when quasiparticles tunnel along a straight path between the two edges of a fractional quantum Hall annulus. Such scaling behavior originates from the propagation and tunneling of charged quasielectrons and quasiholes in an effective field analysis. In the limit when the annulus deforms continuously into a quasi-one-dimensional ring, we conjecture the exact functional form of the tunneling amplitude for several cases, which reproduces the numerical results in finite systems exactly. The results for Abelian quasiparticle tunneling is cons…
We propose a quantum matrix oscillator as a model that provides the construction of the quantum Hall states in a direct way. A connection of this model to the regularized matrix model introduced by Polychronakos is established . By transferring the consideration to the Bargmann representation with the help of a particular similarity transformation, we show that the quantum matrix oscillator describes the quantum mechanics of electrons in the lowest Landau level with the ground state described by the Laughlin-type wave function. The equivalence with the Calogero model in one dimension is emphas…
We study the quantum anomalous Hall effect described by a class of two-component Haldane models on square lattices. We show that the latter can be transformed into a pseudospin triplet p+ip-wave paired superfluid. In the long wave length limit, the ground state wave function is described by Halperin's (1,1,-1) state of neutral fermions analogous to the double layer quantum Hall effect. The vortex excitations are charge e/2 abelian anyons which carry a neutral Dirac fermion zero mode. The superconducting proximity effect induces `tunneling' between `layers' which leads to topological phase tran…
We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. Fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called $W_{1+\infty}$. We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern-Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU($m$…
We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal SU(p) spin. The ground states of the matrix model include spin-singlet generalisations 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 U(p) Chern-Simons theory. We show how the matrix model can be derived from quantisation of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions i…
The quasiparticle propagator of Haldane-Rezayi(HR) fractional quantum Hall (FQH) state is calculated, based on a chiral fermion model (or a Weyl fermion model) equipped with a hidden spin SU(2) symmetry. The spectrum of the chiral fermion model for each total spin and total momentum is shown to be identical to that of the SU(2) c = -2 model introduced to describe the edge spectrum of the HR state.
The recent proposal of non-Abelian boson-fermion dualities in 2+1 dimensions, which morally relate $U(k)_N$ to $SU(N)_{-k}$ 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 theorie…
Partition functions of edge excitations are obtained for non-Abelian Hall states in the second Landau level, such as the anti-Read-Rezayi state, the Bonderson-Slingerland hierarchy and the Wen non-Abelian fluid, as well as for the non-Abelian spin-singlet state. The derivation is straightforward and unique starting from the non-Abelian conformal field theory data and solving the modular invariance conditions. The partition functions provide a complete account of the excitation spectrum and are used to describe experiments of Coulomb blockade and thermopower.
The (exclusion) statistics of parafermions is used to study degeneracies of quasiholes over the paired (or in general clustered) quantum Hall states. Focus is on the Z_k and su(3)_k/u(1)^2 parafermions, which are used in the description of spin-polarized and spin-singled clustered quantum Hall states.
We study excitations in edge theories for non-abelian quantum Hall states, focussing on the spin polarized states proposed by Read and Rezayi and on the spin singlet states proposed by two of the authors. By studying the exclusion statistics properties of edge-electrons and edge-quasiholes, we arrive at a novel K-matrix structure. Interestingly, the duality between the electron and quasihole sectors links the pseudoparticles that are characteristic for non-abelian statistics with composite particles that are associated to the `pairing physics' of the non-abelian quantum Hall states.
Two fundamental aspects of so-called non-abelian quantum Hall states (the q-pfaffian states and more general) are a (generalized) pairing of the participating electrons and the non-abelian statistics of the quasi-hole excitations. In this paper, we show that these two aspects are linked by a duality relation, which can be made manifest by considering the K-matrices that describe the exclusion statistics of the fundamental excitations in these systems.
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 $SU(k)_{-N}$ 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…
We propose an effective Lagrangian for the low energy theory of the Pfaffian states of the fractional quantum Hall effect in the bulk in terms of non-Abelian Chern-Simons (CS) actions. Our approach exploits the connection between the topological Chern-Simons theory and chiral conformal field theories. This construction can be used to describe a large class of non-Abelian FQH states.
We present a simple approach to calculate the degeneracy and the structure of the ground states of non-abelian quantum Hall (QH) liquids on the torus. Our approach can be applied to any QH liquids (abelian or non-abelian) obtained from the parton construction. We explain our approach by studying a series of examples of increasing complexity. When the effective theory of a non-abelian QH liquid is a non-abelian Chern-Simons (CS) theory, our approach reproduces the well known results for the ground state degeneracy of the CS theory. However, our approach also apply to non-abelian QH liquids whos…
We determine the exclusion statistics properties of the fundamental edge quasi-particles over a specific $ν=\half$ non-abelian quantum Hall state known as the pfaffian. The fundamental excitations are the edge electrons of charge $-e$ and the edge quasi-holes of charge $+{e \over 4}$. We explicitly determine thermodynamic distribution functions and establish a duality which generalizes the duality for fractional exclusion statistics in the sense of Haldane.
The U(N) Maxwell-Chern-Simons matrix gauge theory is proposed as an extension of Susskind's noncommutative approach. The theory describes D0-branes, nonrelativistic particles with matrix coordinates and gauge symmetry, that realize a matrix generalization of the quantum Hall effect. Matrix ground states obtained by suitable projections of higher Landau levels are found to be in one-to-one correspondence with the expected Laughlin and Jain hierarchical states. The Jain composite-fermion construction follows by gauge invariance via the Gauss law constraint. In the limit of commuting, ``normal'' …
Quantum Hall inter-plateaux transitions are physical exemplars of quantum phase transitions. Near each of these transitions, the measured electrical conductivity scales with the same correlation length and dynamical critical exponents, i.e., the critical points are superuniversal. In apparent contradiction to these experiments, prior theoretical studies of quantum Hall phase transitions within the framework of Abelian Chern-Simons theory coupled to matter found correlation length exponents that depend on the value of the quantum critical Hall conductivity. Here, we use non-Abelian bosonization…
The Z_k-parafermion Hall state is an incompressible fluid of k-electron clusters generalizing the Pfaffian state of paired electrons. Extending our earlier analysis of the Pfaffian, we introduce two ``parent'' abelian Hall states which reduce to the parafermion state by projecting out some neutral degrees of freedom. The first abelian state is a generalized (331) state which describes clustering of k distinguishable electrons and reproduces the parafermion state upon symmetrization over the electron coordinates. This description yields simple expressions for the quasi-particle wave functions o…
There has been a recent surge of interest in dualities relating theories of Chern-Simons gauge fields coupled to either bosons or fermions within the condensed matter community, particularly in the context of topological insulators and the half-filled Landau level. Here, we study the application of one such duality to the long-standing problem of quantum Hall inter-plateaux transitions. The key motivating experimental observations are the anomalously large value of the correlation length exponent $ν\approx 2.3$ and that $ν$ is observed to be super-universal, i.e., the same in the vicinity of d…
Symmetry Resolved Entanglement Entropy in a Non-Abelian Fractional Quantum Hall State
SISSA — International School for Advanced Studies（国际高等研究院）；INFN（意大利国家核物理研究所）；Center for Computational Quantum Physics, Flatiron Institute（平顿研究所计算量子物理中心）；Department of Physics, University of Toronto（多伦多大学物理系）；Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité（巴黎高等师范学院物理实验室，巴黎文理研究大学，法国国家科学研究中心，索邦大学，巴黎第七大学，索邦巴黎城）；Department of Physics, Princeton University（普林斯顿大学物理系）；Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE（索邦大学，法国国家科学研究中心，理论与高能物理实验室）
Many trial wavefunctions for fractional quantum Hall states in a single Landau level are given by functions called conformal blocks, taken from some conformal field theory. Also, wavefunctions for certain paired states of fermions in two dimensions, such as p_x+ip_y states, reduce to such a form at long distances. Here we investigate the adiabatic transport of such many-particle trial wavefunctions using methods from two-dimensional field theory. One context for this is to calculate the statistics of widely-separated quasiholes, which has been predicted to be non-Abelian in a variety of cases.…
We study the properties of an ultracold Fermi gas loaded in an optical square lattice and subjected to an external and classical non-Abelian gauge field. We show that this system can be exploited as an optical analogue of relativistic quantum electrodynamics, offering a remarkable route to access the exotic properties of massless Dirac fermions with cold atoms experiments. In particular we show that the underlying Minkowski space-time can also be modified, reaching anisotropic regimes where a remarkable anomalous quantum Hall effect and a squeezed Landau vacuum could be observed.
We investigate a class of non-Abelian spin-singlet (NASS) quantum Hall phases, proposed previously. The trial ground and quasihole excited states are exact eigenstates of certain k+1-body interaction Hamiltonians. The k=1 cases are the familiar Halperin Abelian spin-singlet states. We present closed-form expressions for the many-body wave functions of the ground states, which for k>1 were previously defined only in terms of correlators in specific conformal field theories. The states contain clusters of k electrons, each cluster having either all spins up, or all spins down. The ground states …
We review a recent development in the theoretical understanding of the nu=5/2 quantum Hall plateau and propose a new conformal field theory, slightly different from the Moore-Read one, to describe another universality class relevant for this plateau. The ground state is still given by the Pfaffian and is completely polarized, however, the elementary quasiholes are charge 1/2 anyons with abelian statistics theta=pi/2, which obey complete spin-charge separation. The physical hole is represented by two such quasiholes plus a free neutral Majorana fermion. We also compute the periods and amplitude…
We compute directly the entanglement entropy of spatial regions in Chern-Simons gauge theories in 2+1 dimensions using surgery. We use these results to determine the universal topological piece of the entanglement entropy for Abelian and non-Abelian quantum Hall fluids.
We present a pure Chern-Simons formulation of families of interesting Conformal Field Theories describing edge states of non-Abelian Quantum Hall states. These theories contain two Abelian Chern-Simons fields describing the electromagnetically charged and neutral sectors of these models, respectively. The charged sector is the usual Abelian Chern-Simons theory that successfully describes Laughlin-type incompressible fluids. The neutral sector is a 2+1-dimensional theory analogous to the 1+1-dimensional orbifold conformal field theories. It is based on the gauge group O(2) which contains a ${\b…
The structure of ground states of generic FQH states on a torus is studied by using both effective theory and electron wave function. The relation between the effective theory and the wave function becomes transparent when one considers the ground state structure. We find that the non-abelian Berry's phases of the abelian Hall states generated by twisting the mass matrix are identical to the modular transformation matrix for the characters of a Gaussian conformal field theory. We also show that the Haldane-Rezayi spin singlet state has a ten fold degeneracy on a torus which indicates such a st…
We consider quantum Hall droplets on complex projective spaces with a combination of abelian and nonabelian background magnetic fields. Carrying out an analysis similar to what was done for abelian backgrounds, we show that the effective action for the edge excitations is given by a chiral, gauged Wess-Zumino-Witten (WZW) theory generalized to higher dimensions.
The degeneracy of energy levels in a quantum dot of Hall fluid, leading to conductance peaks, can be readily derived from the partition functions of conformal field theory. Their complete expressions can be found for Hall states with both Abelian and non-Abelian statistics, upon adapting known results for the annulus geometry. We analyze the Abelian states with hierarchical filling fractions, ν=m/(mp \pm 1), and find a non trivial pattern of conductance peaks. In particular, each one of them occurs with a characteristic multiplicity, that is due to the extended symmetry of the m-folded edge. E…
We study charged spin-1/2 particles in two dimensions, subject to a perpendicular non-Abelian magnetic field. Specializing to a choice of vector potential that is spatially constant but non-Abelian, we investigate the Landau level spectrum in planar and spherical geometry, paying particular attention to the role of the total angular momentum J = L +S. After this we show that the adiabatic insertion of non-Abelian flux in a spin-polarized quantum Hall state leads to the formation of charged spin-textures, which in the simplest cases can be identified with quantum Hall Skyrmions.
