Entanglement spectrum
of Fractional Quantum Hall states and Quantum spin chains
R. Thomale
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 and quantum spin chains. 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, 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. For the spin chains, the entanglement spectrum from a cut in momentum space allows to study the dimerization transition, bulk excitation state counting, and the manifestation of logarithmic CFT correction purely from the ground state wave function. It provides a new formulation of non-local order in quantum spin chains.