Numerical studies of quantum entanglement in fractional quantum Hall effect systems

Fractional quantum Hall effects have attracted broad interest since the phenomena were discovered. In two-dimensions, electron systems subject to perpendicular magnetic fields behave very strangely, appearing to contain fractionally charged particles that obey fractional statistics.;Recently, there has been increasing interest in using quantum entanglement as a probe to detect topological properties of many-body quantum states, in particular the states exhibiting the fractional quantum Hall effect. Among various measures of quantum entanglement, the entanglement entropy has by far been the favorite. It has been shown that the topological entanglement entropy of fractional quantum Hall states is closely related to its topological order.;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 model wavefunction for both the various fractional quantum Hall state with generic states at appropriate filling (nu = 1/3, 5/2 etc) obtained by finite-size diagonalization of the Landau-level-projected Coulomb interactions. Their spectra share a common "gapless" structure, related to conformal field theory. In the model states, 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.;Assuming that the gap does remain finite in the thermodynamic limit, characterization of the entanglement spectrum is a reliable way to identify a topologically ordered state (the low-lying entanglement spectrum can be used as a "fingerprint"). While finite-size numerical studies often show impressive overlaps between model wave-functions and "realistic" states at intermediate system sizes, this cannot persist in the thermodynamic limit. Furthermore, the entanglement spectrum is a property of the ground state wave-function itself, as oppose to the physical excitations of a system with boundaries, so allows direct comparison between model states and physical ones.