Spectrum of Quantum Entanglement in Fractional Quantum Hall States

We present numerical studies of the bipartite entanglement in fractional quantum Hall (FQH) states. We partitioned the (spherical geometry) Landau-level orbitals into two hemispheres: the entanglement spectrum derives from the Schmidt decomposition $|\psi\rangle = {\sum}_{i}\exp(-{\beta_{i}}/{2}) |\psi_{A}^{i}\rangle\otimes|\psi_{B}^{i}\rangle$, where $|\psi_{A}^{i}\rangle$ (or $|\psi_{B}^{i}\rangle$) are orthonormal. The $\beta_{i}$ are ``energy levels'' of a system with thermodynamic entropy at ``temperature'' $k_{B}T =1$ equivalent to the entanglement entropy. The \textit{entanglement spectrum}, \textit{i.e.}, the relation between the $\beta_i$ and the quantum numbers that classify $|\psi_{A}^{i}\rangle$ (or $|\psi_{B}^{i}\rangle$), serves as a ``fingerprint'' of the topological phase of the FQH state, and reveals much more information than just the entanglement entropy, a single number. The spectrum of, \textit{e.g.}, the $1/3$ Laughlin state has far fewer levels than expected for a generic wavefunction, and its low-energy spectrum corresponds to that of a conformal field theory (CFT). We studied the wavefunctions that interpolate between the Laughlin state and the ground state of a realistic Coulomb interaction potential at $\nu = 1/3$: the generic number of levels is restored, but the low-lying CFT structure remains essentially unchanged. We also describe the interpolation between the Moore-Read state and the Coulomb interaction ground state at $\nu = 5/2$.