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arXiv:1008.5289 · Physical Review B

Entanglement spectrum and entanglement thermodynamics of quantum Hall bilayers at ν=1

John Schliemann

Published Mar 16, 2011• 64 citations
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Abstract

We study the entanglement spectra of bilayer quantum Hall systems at total filling factor $\ensuremath{\nu}=1$. In the interlayer-coherent phase at layer separations smaller than a critical value, the entanglement spectra show a striking similarity to the energy spectra of the corresponding monolayer systems around half-filling. The transition to the incoherent phase can be followed in terms of low-lying entanglement levels. Finally, we describe the connection between those two types of spectra in terms of an effective temperature leading to relations for the entanglement entropy which are in full analogy to canonical thermodynamics.

DOI
10.1103/physrevb.83.115322

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We study the entanglement spectra of bilayer quantum Hall systems at total filling factor \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1. In the interlayer-coherent phase at layer separations smaller than a critical value, the entanglement spectra show a striking similarity to the energy spectra of the corresponding monolayer systems around half-filling. The transition to the incoherent phase can be followed in terms of low-lying entanglement levels. Finally, we describe the connection between those two types of spectra in terms of an effective temperature leading to relations for the entanglement entropy which are in full analogy to canonical thermodynamics.
We study Hofstadter bilayers, i.e. coupled hopping models on two-dimensional square lattices in a perpendicular magnetic field. Upon tracing out one of the layers, we find an explicit expression for the resulting entanglement spectrum in terms of the energy eigenvalues of the underlying monolayer system. For strongly coupled layers, the entanglement Hamiltonian is proportional to the energetic Hamiltonian of the monolayer system. The proportionality factor, however, cannot be interpreted as the inverse thermodynamic temperature, but represents a phenomenological temperature scale. We derive an explicit relationship between both temperature scales which is in close analogy to a standard result of classic thermodynamics. In the limit of vanishing temperature, thermodynamic quantities such as entropy and inner energy approach their ground-state values, but show a fractal structure as a function of magnetic flux.
We investigate further the relationship between the entanglement spectrum of a composite many-body system and the energy spectrum of a subsystem making use of concepts of canonical thermodynamics. In many important cases the entanglement Hamiltonian is, in the limit of strong coupling between subsystems, proportional to the energy Hamiltonian of the subsystem. The proportionality factor is an appropriately defined coupling parameter, suggesting to interpret the latter as a inverse temperature. We identify a condition on the entanglement Hamiltonian which rigorously guarantees this interpretation to hold and removes any ambiguity in the definition of the entanglement Hamiltonian regarding contributions proportional to the unit operator. Illustrations of our findings are provided by spin ladders of arbitrary spin length, and by bilayer quantum Hall systems at total filling factor ν = 2. Within mean-field description, the latter system realizes an entanglement spectrum of free fermions with just two levels of equal modulus where the analogies to canonical thermodynamics are particularly close.
We study the phase diagram of quantum Hall bilayer systems with total filing ν_{T}=1/2+1/2 of the lowest Landau level as a function of layer distances d. Based on numerical exact diagonalization calculations, we obtain three distinct phases, including an exciton superfluid phase with spontaneous interlayer coherence at small d, a composite Fermi liquid at large d, and an intermediate phase for 1.1<d/l_{B}<1.8 (l_{B} is the magnetic length). The transition from the exciton superfluid to the intermediate phase is identified by (i) a dramatic change in the Berry curvature of the ground state under twisted boundary conditions on the two layers and (ii) an energy level crossing of the first excited state. The transition from the intermediate phase to the composite Fermi liquid is identified by the vanishing of the exciton superfluid stiffness. Furthermore, from our finite-size study, the energy cost of transferring one electron between the layers shows an even-odd effect and possibly extrapolates to a finite value in the thermodynamic limit, indicating the enhanced intralayer correlation. Our identification of an intermediate phase and its distinctive features shed new light on the theoretical understanding of the quantum Hall bilayer system at total filling ν_{T}=1.
We study coherence and entanglement properties of the state space of a composite bi-fermion (two electrons pierced by λ magnetic flux lines) at one Landau site of a bilayer quantum Hall system. In particular, interlayer imbalance and entanglement (and its fluctuations) are analyzed for a set of U(4) coherent (quasiclassical) states generalizing the standard pseudospin U(2) coherent states for the spin-frozen case. The interplay between spin and pseudospin degrees of freedom opens new possibilities with regard to the spin-frozen case. Actually, spin degrees of freedom make interlayer entanglement more effective and robust under perturbations than in the spin-frozen situation, mainly for a large number of flux quanta λ. Interlayer entanglement of an equilibrium thermal state and its dependence with temperature and bias voltage is also studied for a pseudo-Zeeman interaction.
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 Moore-Read model wave function for the nu=5/2 fractional quantum Hall state with a generic 5/2 state obtained by finite-size diagonalization of the second-Landau-level-projected Coulomb interactions. Their spectra share a common "gapless" structure, related to conformal field theory. In the model state, 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. We propose that the low-lying entanglement spectrum can be used as a "fingerprint" to identify topological order.
At small layer separations, the ground state of a nu = 1 bilayer quantum Hall system exhibits spontaneous interlayer phase coherence. The evolution of this state with increasing layer separation d has been a matter of controversy. We report on small system exact diagonalization calculations which suggest that a single-phase transition, likely of first order, separates incompressible states with strong interlayer correlations from compressible states with weak interlayer correlations. We find a dependence of the phase boundary on d and interlayer tunneling amplitude that is in very good agreement with recent experiments.