Fractional Quantum Hall Wave Functions Research Articles

Spin-statistics relation and Abelian braiding phase for anyons in the fractional quantum Hall effect

Quasihole excitations in fractional quantum Hall (FQH) systems exhibit fractional statistics and fractional spin, but how the spin-statistics relation emerges from many-body physics remains poorly understood. Here we prove a spin-statistics relation using only FQH wave functions, on both the sphere and disk geometry. In particular, the proof on the disk generalizes to all quasiholes in realistic systems, which have a finite size and could be deformed into arbitrary shapes. Different components of the quasihole spins are linked to different conformal Hilbert spaces (CHS), which are nullspaces of model Hamiltonians that host the respective FQH ground states and quasihole states. Understanding how the intrinsic spin of the quasiholes is linked to different CHS is crucial for the generalized spin-statistics relation that takes into account the effect of metric deformation. In terms of the experimental relevance, this enables us to study the effect of deformation and disorder that introduces an additional source of Berry curvature, an aspect of anyon braiding that has been largely neglected in previous literature.

Construction of a series of new ν=2/5 fractional quantum Hall wave functions by conformal field theory

In this paper, a series of $\nu=2/5$ fractional quantum Hall wave functions are constructed from conformal field theory(CFT). They share the same topological properties with states constructed by Jain's composite fermion approach. Upon exact lowest Landau level(LLL) projection, some of Jain composite fermion states would not survive if constraints on Landau level indices given in the appendices of this paper were not satisfied. By contrast, states constructed from CFT always stay in LLL. These states are characterized by different topological shifts and multibody relative angular momenta. As a by-product, in the appendices we prove the necessary conditions for general $ \nu=p/(2p+1) $ composite fermion states to have nonvanishing LLL projection.

Geometric Construction of Quantum Hall Clustering Hamiltonians

Many fractional quantum Hall wave functions are known to be unique and highest-density zero modes of certain "pseudopotential" Hamiltonians. Examples include the Read-Rezayi series (in particular, the Laughlin, Moore-Read and Read-Rezayi Z_3 states), and more exotic non-unitary (Haldane-Rezayi, Gaffnian states) or irrational states (Haffnian state). While a systematic method to construct such Hamiltonians is available for the infinite plane or sphere geometry, its generalization to manifolds such as the cylinder or torus, where relative angular momentum is not an exact quantum number, has remained an open problem. Here we develop a geometric approach for constructing pseudopotential Hamiltonians in a universal manner that naturally applies to all geometries. Our method generalizes to the multicomponent SU(n) cases with a combination of spin or pseudospin (layer, subband, valley) degrees of freedom. We demonstrate the utility of the approach through several examples, including certain non-Abelian multicomponent states whose parent Hamiltonians were previously unknown, and verify the method by numerically computing their entanglement properties.

Projective construction of theZkRead-Rezayi fractional quantum Hall states and their excitations on the torus geometry

Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the ${\mathbb{Z}}_{k}$ Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of ${\mathbb{Z}}_{k}$ ground states on the torus. We develop an alternative projective construction of ${\mathbb{Z}}_{k}$ Read-Rezayi states that complements the existing one. On the multilayer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend our symmetrization scheme to the plane and sphere geometries.

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

Using the newly developed matrix product state 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 nonunitary conformal field theory (CFT) can describe the bulk of a gapped FQH phase. We show that the nonunitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of a Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has an 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 the FQH effect the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the nonunitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the composite fermion Jain state at equal filling.

Crystal-symmetry preserving Wannier states for fractional Chern insulators

Recently, many numerical evidences of fractional Chern insulator, i.e., the fractional quantum Hall states on lattices, are proposed when a Chern band is partially filled. Some trial wave functions of fractional Chern insulators can be obtained by mapping the fractional quantum Hall wave functions defined in the continuum onto the lattice through the Wannier state representation [Phys. Rev. Lett. 107, 126803 (2011)] in which the single particle Landau orbits in the Landau levels are identified with the one-dimensional Wannier states of the Chern bands with Chern number $C=1$. However, this mapping generically breaks the lattice point group symmetry. In this paper we discuss a general approach of modifying the mapping to accommodate the lattice rotational symmetry. The wave functions constructed through this modified mapping should serve as better trial wave functions in the thermodynamical limit and on the rotationally invariant finite lattice. Also these wave functions will form a good basis for the construction of lattice symmetry preserving pseudopotential formalism for fractional Chern insulators. The focus of this paper shall be mainly on the ${C}_{4}$ rotational symmetry of square lattices. Similar analysis can be straightforwardly generalized to triangular or hexagonal lattices with ${C}_{6}$ symmetry. We also generalize the discussion to the lattice symmetry of fractional Chern insulators with high Chern number bands.

Extracting Excitations from Model State Entanglement

We extend the concept of an entanglement spectrum from the geometrical to the particle bipartite partition. We apply this to several fractional quantum Hall wave functions on both sphere and torus geometries to show that this new type of entanglement spectra completely reveals the physics of bulk quasihole excitations. While this is easily understood when a local Hamiltonian for the model state exists, we show that the quasihole wave functions are encoded within the model state even when such a Hamiltonian is not known. As a nontrivial example, we look at Jain's composite fermion states and obtain their quasiholes directly from the model state wave function. We reach similar conclusions for wave functions described by Jack polynomials.

Composite-fermion wave functions as correlators in conformal field theory

It is known that a subset of fractional quantum Hall wave functions has been expressed as conformal field theory (CFT) correlators, notably the Laughlin wave function at filling factor $\nu=1/m$ ($m$ odd) and its quasiholes, and the Pfaffian wave function at $\nu=1/2$ and its quasiholes. We develop a general scheme for constructing composite-fermion (CF) wave functions from conformal field theory. Quasiparticles at $\nu=1/m$ are created by inserting anyonic vertex operators, $P_{\frac{1}{m}}(z)$, that replace a subset of the electron operators in the correlator. The one-quasiparticle wave function is identical to the corresponding CF wave function, and the two-quasiparticle wave function has correct fractional charge and statistics and is numerically almost identical to the corresponding CF wave function. We further show how to exactly represent the CF wavefunctions in the Jain series $\nu = s/(2sp+1)$ as the CFT correlators of a new type of fermionic vertex operators, $V_{p,n}(z)$, constructed from $n$ free compactified bosons; these operators provide the CFT representation of composite fermions carrying $2p$ flux quanta in the $n^{\rm th}$ CF Landau level. We also construct the corresponding quasiparticle- and quasihole operators and argue that they have the expected fractional charge and statistics. For filling fractions 2/5 and 3/7 we show that the chiral CFTs that describe the bulk wave functions are identical to those given by Wen's general classification of quantum Hall states in terms of $K$-matrices and $l$- and $t$-vectors, and we propose that to be generally true. Our results suggest a general procedure for constructing quasiparticle wave functions for other fractional Hall states, as well as for constructing ground states at filling fractions not contained in the principal Jain series.

Chiral operator product algebra hidden in certain fractional quantum Hall wave functions

In this paper we study the conditions under which an N-electron wave function for a fractional quantum Hall (FQH) state can be viewed as an N-point correlation function in a conformal field theory (CFT). Several concrete examples are presented to illustrate, when these conditions are satisfied, how to “derive” or “uncover” a relevant operator algebra in the associated CFT from the FQH wave functions. Besides the known pfaffian state, the states studied here include three d-wave paired states, one for spinless electrons and two for spin- 1 2 electrons (one of them is the Haldane-Rezayi state). It is suggested that the non-abelian topological order hidden in these states can be characterized by their associated chiral operator product algebra, from which one may infer the quantum numbers of quasiparticles and calculate their wave functions.