We study the "entanglement spectrum" (a presentation of the Schmidt decomposition analogous to a set of "energy levels") of a many-body state, and compare the Moore-Read model wave function for the nu=5/2 fractional quantum Hall state with a generic 5/2 state obtained by finite-size diagonalization of the second-Landau-level-projected Coulomb interactions. Their spectra share a common "gapless" structure, related to conformal field theory. In the model state, these are the only levels, while in the "generic" case, they are separated from the rest of the spectrum by a clear "entanglement gap", which appears to remain finite in the thermodynamic limit. We propose that the low-lying entanglement spectrum can be used as a "fingerprint" to identify topological order.
Determining correlated ground states directly in the thermodynamic limit remains a central challenge in quantum many-body physics. Here we develop a quantum many-body bootstrap that yields a rigorous and tight lower bound on the thermodynamic ground-state energy of a fractional quantum Hall system together with an optimized reduced density matrix
ρW⋆ on a spatial window. For sufficiently tight bounds, we find that
ρW⋆ accurately reproduces the ground-state reduced density matrix, as directly tested by its fidelity with known model states. For exactly solvable Laughlin and Moore--Read parent Hamiltonians, the bootstrap also reproduces pair correlations and orbital entanglement spectra (OES). For Landau-level-projected Coulomb interactions across multiple fillings, the energy bounds closely track established thermodynamic estimates, while the window density matrix recovers the familiar
ν=1/3 Laughlin and
ν=5/2 Moore--Read states and the composite Fermi liquid at
ν=1/2. More importantly, the same framework also resolves unsettled regimes. At
ν=1/4, the OES and real-space correlations identify a composite Fermi liquid, while at
ν=12/5 the combined diagnostics suggest
RR3 as the ground-state order. Our bootstrap therefore provides direct access to both ground-state energies and many-body structure in the thermodynamic limit.
Entanglement in topological phases of matter has so far been investigated through the perspective of their ground-state wave functions. In contrast, we demonstrate that the excitations of fractional quantum Hall (FQH) systems also contain information to identify the system's topological order. Entanglement spectrum of the FQH quasihole (QH) excitations is shown to differentiate between the conformal field theory (CFT) sectors, based on the relative position of the QH with respect to the entanglement cut. For Read-Rezayi model states, as well as Coulomb interaction eigenstates, the counting of the QH entanglement levels in the thermodynamic limit matches exactly the CFT counting, and sector changes occur as non-Abelian quasiholes successively cross the entanglement cut.
We conjecture that the counting of the levels in the orbital entanglement spectra (OES) of finite-size Laughlin fractional quantum Hall (FQH) droplets at filling $\ensuremath{\nu}=1/m$ is described by the Haldane statistics of particles in a box of finite size. This principle explains the observed deviations of the OES counting from the edge-mode conformal field theory counting and directly provides us with a topological number of FQH states inaccessible in the thermodynamic limit---the boson compactification radius. It also suggests that the entanglement gap in the Coulomb spectrum in the conformal limit protects a universal quantity---the statistics of the state. We support our conjecture with ample numerical checks.
The edge of the electronic fractional quantum Hall (FQH) system obeys the law of the chiral Luttinger liquid theory due to its intrinsic topological properties and the relation of bulk-edge correspondence. However, in a realistic experimental system, such as the usual Hall bar setup, the softening of the background confinement potential can induce the reconstruction of the edge spectrum, which breaks the chirality and universality of the FQH edge. The entanglement spectrum (ES) of the FQH ground state has the same counting structure as that of the energy spectrum indicating the topological characters of the quantum state. In this work, we report that the ES can also have an edge reconstruction while sweeping the area of the subsystem in real space cut. Moreover, we found that the critical area of the subsystem matches accurately with the intrinsic building block of the FQH liquids, namely the correlation hole of the FQH liquids. The above results seem to be universal after our study of a series of typical FQH states, such as two Laughlin states at $\ensuremath{\nu}=1/3$ and
1/5, and the Moore-Read state for $\ensuremath{\nu}=5/2$.
The topological entanglement entropy is a robust measurement of the quantum many-body state with topological order. In fractional quantum Hall (FQH) state, it has a connection to the quantum dimension of the state itself and its quasihole excitations from the conformal field theory (CFT) description. We systematically study the entanglement entropy in the Laughlin, Moore-Read, and Read-Rezayi FQH states. The Abelian nature of the quasihole manifests in the invariant of the entanglement entropy and spectrum while it is created in the subsystem. Whereas the non-Abelian quasihole excitation induces an extra correction of the topological entanglement entropy and the changing of the structure of the entanglement spectrum. The quantum dimension of the quasihole can be obtained from the entropy difference before and after inserting the quasihole. On the other hand, the entanglement entropy behaviors similarly to the density profile and has less oscillations far away from the center of quasihole. It gives us a better definition of the quasihole boundary and the measurement of the size of the quasihole.