We consider the trial wave functions for the fractional quantum Hall effect that are given by conformal blocks, and construct their associated edge excited states in full generality. The inner products between these edge states are computed in the thermodynamic limit, assuming generalized screening (i.e., short-range correlations only) inside the quantum Hall droplet and using the language of boundary conformal field theory (boundary CFT). These inner products take universal values in this limit: they are equal to the corresponding inner products in the bulk two-dimensional chiral CFT which underlies the trial wave function. This is a bulk/edge correspondence; it shows the equality between equal-time correlators along the edge and the correlators of the bulk CFT up to a Wick rotation. This approach is then used to analyze the entanglement spectrum of the ground state obtained with a bipartition $A\ensuremath{\cup}B$ in real space. Starting from our universal result for inner products in the thermodynamic limit, we tackle corrections to scaling using standard field-theoretic and renormalization-group arguments. We prove that generalized screening implies that the entanglement Hamiltonian ${H}_{E}=\ensuremath{-}\mathrm{ln}{\ensuremath{\rho}}_{A}$ is isospectral to an operator that is local along the cut between $A$ and $B$ . We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wave functions given by conformal blocks and tensor product states, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ paired superfluids are treated in a similar fashion in the Appendixes.We consider the trial wave functions for the fractional quantum Hall effect that are given by conformal blocks, and construct their associated edge excited states in full generality. The inner products between these edge states are computed in the thermodynamic limit, assuming generalized screening (i.e., short-range correlations only) inside the quantum Hall droplet and using the language of boundary conformal field theory (boundary CFT). These inner products take universal values in this limit: they are equal to the corresponding inner products in the bulk two-dimensional chiral CFT which underlies the trial wave function. This is a bulk/edge correspondence; it shows the equality between equal-time correlators along the edge and the correlators of the bulk CFT up to a Wick rotation. This approach is then used to analyze the entanglement spectrum of the ground state obtained with a bipartition $A\ensuremath{\cup}B$ in real space. Starting from our universal result for inner products in the thermodynamic limit, we tackle corrections to scaling using standard field-theoretic and renormalization-group arguments. We prove that generalized screening implies that the entanglement Hamiltonian ${H}_{E}=\ensuremath{-}\mathrm{ln}{\ensuremath{\rho}}_{A}$ is isospectral to an operator that is local along the cut between $A$ and $B$. We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wave functions given by conformal blocks and tensor product states, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ paired superfluids are treated in a similar fashion in the Appendixes.

arXiv:1207.7119Physical Review BDec 13, 2012150 citations

Band structure in bulk entanglement spectrum of quantum Hall state

Chi-Ken Lu, Dah-Wei Chiou, Fei Lin

Bulletin of the American Physical SocietyMar 14, 20170 citations

Particle entanglement spectra for quantum Hall states on lattices

Antoine Sterdyniak, Nicolas Regnault, Gunnar Möller

We use particle entanglement spectra to characterize bosonic quantum Hall states on lattices, motivated by recent studies of bosonic atoms on optical lattices. Unlike for the related problem of fractional Chern insulators, very good trial wave functions are known for fractional quantum Hall states on lattices. We focus on the entanglement spectra for the Laughlin state at $\ensuremath{\nu}=1/2$ for the non-Abelian Moore-Read state at $\ensuremath{\nu}=1$. We undertake a comparative study of these trial states to the corresponding ground states of repulsive two-body or three-body contact interactions on the lattice. The magnitude of the entanglement gap is studied as a function of the interaction strength on the lattice, giving insight into the nature of Landau-level mixing. In addition, we compare the performance of the entanglement gap and overlaps with trial wave functions as possible indicators for the topological order in the system. We discuss how the entanglement spectra allow to detect competing phases such as a Bose-Einstein condensate.We use particle entanglement spectra to characterize bosonic quantum Hall states on lattices, motivated by recent studies of bosonic atoms on optical lattices. Unlike for the related problem of fractional Chern insulators, very good trial wave functions are known for fractional quantum Hall states on lattices. We focus on the entanglement spectra for the Laughlin state at $\ensuremath{\nu}=1/2$ for the non-Abelian Moore-Read state at $\ensuremath{\nu}=1$. We undertake a comparative study of these trial states to the corresponding ground states of repulsive two-body or three-body contact interactions on the lattice. The magnitude of the entanglement gap is studied as a function of the interaction strength on the lattice, giving insight into the nature of Landau-level mixing. In addition, we compare the performance of the entanglement gap and overlaps with trial wave functions as possible indicators for the topological order in the system. We discuss how the entanglement spectra allow to detect competing phases such as a Bose-Einstein condensate.

arXiv:1207.6065Physical Review BOct 12, 201246 citations

Quantum Hall plateau transitions in the bulk entanglement spectrum

Xin Wan, Qiong Zhu, Guang-Ming Zhang

Bulletin of the American Physical SocietyMar 3, 20150 citations

Bulk-edge correspondence in entanglement spectra

Anushya Chandran, Maria Hermanns, Nicolas Regnault +1 more

Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground states of time-reversal-breaking topological phases [fractional quantum Hall (FQH) states] contains information about the counting of their edge modes when the ground state is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one-to-one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum (PES), whose counting of eigenvalues for each good quantum number is identical to the counting of bulk quasiholes (up to accidental zero eigenvalues of the reduced density matrix), and the orbital entanglement spectrum (OES), considered by Li and Haldane. By using a set of clustering operators that have their origin in conformal-field-theory (CFT) operator expansions, we show that the counting of the OES eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk. The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Our results can be interpreted as a bulk-edge correspondence in entanglement spectra. Moreover, we show that the counting of the PES and OES is identical even for CFT states that are likely bulk gapless, such as the Gaffnian wave function.Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground states of time-reversal-breaking topological phases [fractional quantum Hall (FQH) states] contains information about the counting of their edge modes when the ground state is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one-to-one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum (PES), whose counting of eigenvalues for each good quantum number is identical to the counting of bulk quasiholes (up to accidental zero eigenvalues of the reduced density matrix), and the orbital entanglement spectrum (OES), considered by Li and Haldane. By using a set of clustering operators that have their origin in conformal-field-theory (CFT) operator expansions, we show that the counting of the OES eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk. The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Our results can be interpreted as a bulk-edge correspondence in entanglement spectra. Moreover, we show that the counting of the PES and OES is identical even for CFT states that are likely bulk gapless, such as the Gaffnian wave function.

arXiv:1102.2218Physical Review BNov 22, 2011169 citations

A

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. We also show that a similar analysis can be carried out for particle partition. We discuss the close analogy between the formalism of trial wave functions given by conformal blocks and tensor product states, for which results analogous to ours have appeared recently. Finally, the edge theory and entanglement spectrum of ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ paired superfluids are treated in a similar fashion in the Appendixes.