Benoit Estienne

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

We study the properties of the conformal blocks of the conformal eld theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate elds, the conformal blocks obey second order dierential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero-Sutherland Hamiltonian with nontrivial braiding properties. A generalized duality property relates the two types of second order degenerate elds. By studying this duality we found that the excited states of the CalogeroSutherland Hamiltonian are characterized by two partitions, or in the case of WAk 1 theories by k partitions. By extending the conformal eld theories under consideration by a u(1) eld, we nd that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero-Sutherland Hamiltonian. When the action of the Calogero-Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonisation, these integrals of motion can be expressed as a sum of two, or in generalk, bosonic Calogero-Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.

Decomposition of fractional quantum Hall model states: Product rule symmetries and approximations

We provide a detailed description of a product rule structure of the monomial (Slater) expansion coefficients of bosonic (fermionic) fractional quantum Hall (FQH) states derived recently, which we now extend to spin-singlet states. We show that the Haldane-Rezayi spin-singlet state can be obtained without exact diagonalization through a differential equation method that we conjecture to be generic to other FQH model states. The product rule symmetries allow us to build approximations of FQH states that exhibit increasing overlap with the exact state (as a function of system size) even though our approximation omits more than half of the Hilbert space. We show that the product rule is valid for any FQH state that can be written as an expectation value of parafermionic operators.

Relating Jack wavefunctions to WA(k-1) theories

The (k, r) admissible Jack polynomials, recently proposed as many-body wavefunctions for non-Abelian fractional quantum Hall systems, have been conjectured to be related to some correlation functions of the minimal model of the algebra. By studying the degenerate representations of this conformal field theory, we provide a proof for this conjecture.

Matrix product states for trial quantum Hall states

We obtain an exact matrix-product-state (MPS) representation of a large series of fractional quantum Hall (FQH) states in various geometries of genus 0. The states in question include all paired

Braiding Non-Abelian Quasiholes in Fractional Quantum Hall States

k=2

Correlation Lengths and Topological Entanglement Entropies of Unitary and Nonunitary Fractional Quantum Hall Wave Functions

Jack polynomials, such as the Moore-Read and Gaffnian states, as well as the Read-Rezayi

D-algebra structure of topological insulators

k=3

Clustering properties, Jack polynomials and unitary conformal field theories

state. We also outline the procedures through which the MPSs of other model FQH states can be obtained, provided their wave function can be written as a correlator in a

Correlation functions with fusion-channel multiplicity in

1+1

{\mathcal{W}}_3

conformal field theory (CFT). The auxiliary Hilbert space of the MPS, which gives the counting of the entanglement spectrum, is then simply the Hilbert space of the underlying CFT. This formalism enlightens the link between entanglement spectrum and edge modes. Properties of model wave functions such as the thin-torus root partitions and squeezing are recast in the MPS form, and numerical benchmarks for the accuracy of the new MPS prescription in various geometries are provided.