We give a complete definition of the entanglement gap separating low-energy, topological levels, from high-energy, generic ones, in the "entanglement spectrum" of Fractional Quantum Hall (FQH) states. By removing the magnetic length inherent in the FQH problem -a procedure which we call taking the "conformal limit", we find that the entanglement spectrum of an incompressible ground-state of a generic (i.e. Coulomb) lowest Landau Level Hamiltonian re-arranges into a low-(entanglement) energy part separated by a full gap from the high energy entanglement levels. As previously observed [1], the counting of these levels starts off as the counting of modes of the edge theory of the FQH state, but quickly develops finite-size effects which we show can also serve as a fingerprint of the FQH state. As the sphere manifold where the FQH resides grows, the level spacing of the states at the same angular momentum goes to zero, suggestive of the presence of relativistic gapless edge-states. By using the adiabatic continuity of the low entanglement energy levels, we investigate whether two states are topologically connected.PACS numbers: 03.67. Mn, 05.30.Pr, 73.43.f Topological phases of matter generally lack local order parameters that can distinguish them from trivial ones. Moreover, extracting the topological order directly from the ground-state wavefunction is a nontrivial task. For incompressible states, several non-local indicators of the topological nature of the phase, such as ground-state degeneracy on compact high genus manifolds, the structure of edge modes and their scaling exponents, as well as quantum dimension analysis exist, but still do not fully describe the topological phase. The measure of choice has so far been the entanglement entropy (EE), especially its topological part [2,3]. For a given state |Ψ 0 and according density matrix ρ = |Ψ 0 Ψ 0 |, let the Hilbert space be decomposed as a direct product H = H A ⊗ H B . Defining ρ A ≡ Tr B [ρ], the EE with respect to the partitioning (A, B) is defined by S A = −Tr A [ρ A lnρ A ]. For twodimensional quantum systems, except in special cases where analytical solutions can be found [4], extracting the topological part of the EE becomes a highly nontrivial (and almost impossible) task.While the EE is just one number, it was recently proposed and numerically substantiated [1] that the entanglement spectrum (ES), i.e. the full set of eigenvalues of ρ A , understood as a geometric partition of the quantum Hall sphere [5], is a better indicator of topological order in the ground state of FQH systems. Writing the eigenvalues as the spectrum of a fictitious Hamiltonian, ρ A = exp(−H), where one can think of the H eigenvalues ξ as a quasi-energy (or entanglement energy), Li and Haldane [1] showed that the low quasi-energy spectrum for generic gapped ν = 5/2 states exhibits a universal structure, related to conformal field theory. A few of the eigenvalues displaying this CFT counting are separated from a non-universal high energy spectrum by an entanglement gap which ...

We extend the concept of entanglement spectrum from the geometrical to the particle bipartite partition. We apply this to several Fractional Quantum Hall (FQH) wavefunctions 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 quasiholes wavefunctions 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 wavefunction. We reach similar conclusions for wavefunctions described by Jack polynomials.PACS numbers: 03.67. Mn, 05.30.Pr, Topological phases are highly nontrivial states of matter whose complete characterization has, despite intense effort, remained elusive. The parade example of a topological ordered phase, and the only one so far realized in experiments, is the Fractional Quantum Hall (FQH) effect. Active ongoing efforts to understand the physics of these systems focus on several issues: an important longstanding research direction in topological phases focuses on finding the best numerical techniques to identify topological order in realistic systems. Due to the absence of a local order parameter, this is a highly nontrivial task. A related recent research direction has focused on extracting as much information as possible about a topological phase -including information about its excitationspurely from its ground state wavefunction. The deep conceptual question is whether the ground state of a generic Hamiltonian encodes the complete information about the universality class of a topologically ordered system, and if so, what is the best way to extract it.Towards this end, it has been recently proposed and numerically substantiated that the physical properties of the FQH edge can be obtained from the ground state using the entanglement spectrum (ES) [1]. For a single nondegenerated ground state |Ψ , ES can be defined through the Schmidt decomposition of |Ψ in two regions A, B (not necessarily spatial):where Ψ can be regarded as the eigenvalues and eigenstates of a fictitious hamiltonian H. The ES is the spectrum associated to H.It has been shown numerically on a case by case basis [1,2] that the counting of the low energy part of the ξ i 's in the ES matches the counting of the edge modes of the respective FQH state. In the case of a realistic system whose ground state is close to a model FQH wavefunction, such as the Coulomb ground state in the lowest Landau level (LLL) at filling factor ν = 1/3 (close to the Laughlin state), an entanglement gap can be defined separating a low energy part matching the model state ξ's and a non-universal high energy spectrum [3]. The entanglement gap is numerically conjectured to remain finite in the thermodynamic (TD) limit if the realistic system is in the same universality class as the model wavefunction. The ES has also been ap...

We report the observation of a new series of Abelian and non-Abelian topological states in fractional Chern insulators (FCI). The states appear at bosonic filling ν = k/(C + 1) (k, C integers) in several lattice models, in fractionally filled bands of Chern numbers C ≥ 1 subject to on-site Hubbard interactions. We show strong evidence that the k = 1 series is Abelian while the k > 1 series is non-Abelian. The energy spectrum at both groundstate filling and upon the addition of quasiholes shows a low-lying manifold of states whose total degeneracy and counting matches, at the appropriate size, that of the Fractional Quantum Hall (FQH) SU (C) (color) singlet k-clustered states (including Halperin, non-Abelian spin singlet(NASS) states and their generalizations). The groundstate momenta are correctly predicted by the FQH to FCI lattice folding. However, the counting of FCI states also matches that of a spinless FQH series, preventing a clear identification just from the energy spectrum. The entanglement spectrum lends support to the identification of our states as SU (C) color-singlets but offers new anomalies in the counting for C > 1, possibly related to dislocations that call for the development of new counting rules of these topological states.