Correlation Lengths and Topological Entanglement Entropies of Unitary and Non-Unitary Fractional Quantum Hall Wavefunctions

Using the newly developed Matrix Product State (MPS) formalism for non-abelian Fractional Quantum Hall (FQH) states, we address the question of whether a FQH trial wave function written as a correlation function in a non-unitary Conformal Field Theory (CFT) can describe the bulk of a gapped FQH phase. We show that the non-unitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has infinite correlation length in (at least) the non-Abelian sector, and is therefore gapless. We also compute the topological entanglement entropy of several non-abelian states with and without quasiholes. For the first time in FQH the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the non-unitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the Composite Fermion Jain state at equal filling.Comment: 5 pages, 5 figures, and supplementary material. Published versio

D-Algebra Structure of Topological Insulators

In the quantum Hall effect, the density operators at different wave-vectors generally do not commute and give rise to the Girvin MacDonald Plazmann (GMP) algebra with important consequences such as ground-state center of mass degeneracy at fractional filling fraction, and W_{1 + \infty} symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher dimensional topological insulators involves the concept of a D-algebra formed by using the fully anti-symmetric tensor in D-dimensions. For insulators in even dimensional space, the D-algebra is isotropic and closes for the case of constant non-Abelian F(k) ^ F(k) ... ^ F(k) connection (D-Berry curvature), and its structure factors are proportional to the D/2-Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one less dimension) and does not contain the Chern-Simons \theta form (F ^ A - 2/3 A ^ A ^ A in 3 dimensions). The Chern-Simons form appears in a certain combination of the parallel transport and simple translation operator which is not an algebra. The possible relation to D-dimensional volume preserving diffeomorphisms and parallel transport of extended objects is also discussed.Comment: 5 page

Matrix Product State Description and Gaplessness of the Haldane-Rezayi State

We derive an exact matrix product state representation of the Haldane-Rezayi state on both the cylinder and torus geometry. Our derivation is based on the description of the Haldane-Rezayi state as a correlator in a non-unitary logarithmic conformal field theory. This construction faithfully captures the ten degenerate ground states of this model state on the torus. Using the cylinder geometry, we probe the gapless nature of the phase by extracting the correlation length, which diverges in the thermodynamic limit. The numerically extracted topological entanglement entropies seem to only probe the Abelian part of the theory, which is reminiscent of the Gaffnian state, another model state deriving from a non-unitary conformal field theory.Comment: Corrected labels in Fig.

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 k=2 Jack polynomials, such as the Moore-Read and Gaffnian states, as well as the Read-Rezayi k=3 state. We also outline the procedures through which the MPS of other model FQH states can be obtained, provided their wavefunction can be written as a correlator in a 1+1 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 wavefunctions 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.Comment: 5 pages, 1 figure, published versio

Braiding non-Abelian quasiholes in fractional quantum Hall states

Quasiholes in certain fractional quantum Hall states are promising candidates for the experimental realization of non-Abelian anyons. They are assumed to be localized excitations, and to display non-Abelian statistics when sufficiently separated, but these properties have not been explicitly demonstrated except for the Moore-Read state. In this work, we apply the newly developed matrix product state technique to examine these exotic excitations. For the Moore-Read and the $\mathbb{Z}_3$ Read-Rezayi states, we estimate the quasihole radii, and determine the correlation lengths associated with the exponential convergence of the braiding statistics. We provide the first microscopic verification for the Fibonacci nature of the $\mathbb{Z}_3$ Read-Rezayi quasiholes. We also present evidence for the failure of plasma screening in the non-unitary Gaffnian wave function.Comment: 9 pages, 9 figures; published versio

Matrix product state representation of non-Abelian quasiholes

We provide a detailed explanation of the formalism necessary to construct matrix product states for non-Abelian quasiholes in fractional quantum Hall model states. Our construction yields an efficient representation of the wave functions with conformal-block normalization and monodromy, and complements the matrix product state representation of fractional quantum Hall ground states.Comment: 14 pages, 2 figures; published versio

Spin-Singlet Quantum Hall States and Jack Polynomials with a Prescribed Symmetry

We show that a large class of bosonic spin-singlet Fractional Quantum Hall model wave-functions and their quasi-hole excitations can be written in terms of Jack polynomials with a prescribed symmetry. Our approach describes new spin-singlet quantum Hall states at filling fraction nu = 2k/(2r-1) and generalizes the (k,r) spin-polarized Jack polynomial states. The NASS and Halperin spin singlet states emerge as specific cases of our construction. The polynomials express many-body states which contain configurations obtained from a root partition through a generalized squeezing procedure involving spin and orbital degrees of freedom. The corresponding generalized Pauli principle for root partitions is obtained, allowing for counting of the quasihole states. We also extract the central charge and quasihole scaling dimension, and propose a conjecture for the underlying CFT of the (k, r) spin-singlet Jack states.Comment: 17 pages, 1 figur