We present analytic and numerical calculations on the bipartite entanglement entropy in fractional quantum Hall states of the fermionic Laughlin sequence. The partitioning of the system is done both by dividing Landau-level orbitals and by grouping the fermions themselves. For the case of orbital partitioning, our results can be related to spatial partitioning, enabling us to extract a topological quantity (the "total quantum dimension") characterizing the Laughlin states. For particle partitioning we prove a very close upper bound for the entanglement entropy of a subset of the particles with the rest, and provide an interpretation in terms of exclusion statistics.
We present analytic and numerical calculations on the bipartite entanglement entropy in fractional quantum Hall states of the fermionic Laughlin sequence. The partitioning of the system is done both by dividing Landau-level orbitals and by grouping the fermions themselves. For the case of orbital partitioning, our results can be related to spatial partitioning, enabling us to extract a topological quantity (the "total quantum dimension") characterizing the Laughlin states. For particle partitioning we prove a very close upper bound for the entanglement entropy of a subset of the particles with the rest, and provide an interpretation in terms of exclusion statistics.
We present a detailed analysis of bipartite entanglement entropies in fractional quantum Hall (FQH) states, considering both Abelian (Laughlin) and non-Abelian (Moore-Read) states. We derive upper bounds for the entanglement between two subsets of the particles making up the state. We also consider the entanglement between spatial regions supporting a FQH state. Using the latter, we show how the so-called topological entanglement entropy $\ensuremath{\gamma}$ of a FQH state can be extracted from wave functions for a limited number of particles.
Through exact numerical diagonalization, the von Neumann entropy is calculated for the Laughlin and Pfaffian quantum Hall states in rotating interacting Bose gases at zero temperature in the lowest Landau level limit. The particles comprising the states are indistinguishable, so the required spatial bipartitioning is effected by tracing over a subset of single-particle orbitals. The topological entropy is then extracted through a finite-size scaling analysis. The results for the Laughlin and the Pfaffian states are consistent with the expected values of ln2 and ln4, respectively.
We theoretically examine entanglement in fractional quantum hall states, explicitly taking into account and emphasizing the quasi-two-dimensional nature of experimental quantum Hall systems. In particular, we study the entanglement entropy and the entanglement spectrum as a function of the finite layer thickness, d, of the quasi-two-dimensional system for a number of filling fractions $\ensuremath{\nu}$ in the lowest and the second Landau levels: $\ensuremath{\nu}$ = 1/3, 7/3, 1/2, and 5/2. We observe that the entanglement measures are dependent on which Landau level the electrons fractionally occupy, and find that filling fractions 1/3 and 7/3, which are considered to be Laughlin states, weaken with d in the lowest Landau level ($\ensuremath{\nu}=1/3$) and strengthen with d in the second Landau level ($\ensuremath{\nu}=7/3$). For the enigmatic even-denominator $\ensuremath{\nu}=5/2$ state, we find that entanglement in the ground state is consistent with that of the non-Abelian Moore-Read Pfaffian state at an optimal thickness, d. We also find that the single-layer $\ensuremath{\nu}=1/2$ system is not a fractional quantum Hall state consistent with the experimental observation. In general, our theoretical findings based on entanglement considerations are completely consistent with the results based on wave-function overlap calculations.
We investigate the entanglement spectra arising from sharp real-space partitions of the system for quantum Hall states. These partitions differ from the previously utilized orbital and particle partitions and reveal complementary aspects of the physics of these topologically ordered systems. We show, by constructing one to one maps to the particle partition entanglement spectra, that the counting of the real-space entanglement spectra levels for different particle number sectors versus their angular momentum along the spatial partition boundary is equal to the counting of states for the system with a number of (unpinned) bulk quasiholes excitations corresponding to the same particle and flux numbers. This proves that, for an ideal model state described by a conformal field theory, the real-space entanglement spectra level counting is bounded by the counting of the conformal field theory edge modes. This bound is known to be saturated in the thermodynamic limit (and at finite sizes for certain states). Numerically analyzing several ideal model states, we find that the real-space entanglement spectra indeed display the edge modes dispersion relations expected from their corresponding conformal field theories. We also numerically find that the real-space entanglement spectra of Coulomb interaction ground states exhibit a series of branches, which we relate to the model state and (above an entanglement gap) to its quasiparticle-quasihole excitations. We also numerically compute the entanglement entropy for the $\ensuremath{\nu}=1$ integer quantum Hall state with real-space partitions and compare against the analytic prediction. We find that the entanglement entropy indeed scales linearly with the boundary length for large enough systems, but that the attainable system sizes are still too small to provide a reliable extraction of the subleading topological entanglement entropy term.
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with a fixed number parity yet not necessarily a fixed particle number. The mode entanglement between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space H is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of H is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined by J. Schliemann et al. [Phys. Rev. A 64, 022303 (2001)], implying that particle entanglement may be seen as minimum mode entanglement. It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that in the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy is evaluated analytically, extending the results in previous reference to general states.
We calculate the typical bipartite entanglement entropy ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{N}$ in systems containing indistinguishable particles of any kind as a function of the total particle number N, the volume V, and the subsystem fraction f=VA/V, where VA is the volume of the subsystem. We expand our result as a power series ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{N}=afV+b\sqrt{V}+c+o(1)$, and find that c is universal (i.e., independent of the system type), while a and b can be obtained from a generating function characterizing the local Hilbert space dimension. We illustrate the generality of our findings by studying a wide range of different systems, e.g., bosons, fermions, spins, and mixtures thereof. We provide evidence that our analytical results describe the entanglement entropy of highly excited eigenstates of quantum-chaotic spin and boson systems, which is distinct from that of integrable counterparts.