Husimi function and phase-space analysis of bilayer quantum Hall systems at ν = 2/λ

Abstract

We propose localization measures in phase space of the ground state of bilayer quantum Hall (BLQH) systems at fractional filling factors ν = 2/λ, to characterize the three quantum phases (shortly denoted by spin, canted and ppin) for arbitrary U (4)-isospin λ. We use a coherent state (Bargmann) representation of quantum states, as holomorphic functions in the 8-dimensional Grassmannian phase-space G 4 2 = U (4)/[U (2)×U (2)] (a higher-dimensional generalization of the Haldane's 2-dimensional sphere S 2 = U (2)/[U (1) × U (1)]). We quantify the localization (inverse volume) of the ground state wave function in phase-space throughout the phase diagram (i.e., as a function of Zeeman, tunneling, layer distance, etc, control parameters) with the Husimi function second moment, a kind of inverse participation ratio that behaves as an order parameter. Then we visualize the different ground state structure in phase space of the three quantum phases, the canted phase displaying a much higher delocalization (a Schrödinger cat structure) than the spin and ppin phases, where the ground state is highly coherent. We find a good agreement between analytic (variational) and numeric diagonalization results.

Role of mixed permutation symmetry sectors in the thermodynamic limit of critical three-level Lipkin-Meshkov-Glick atom models

“…Intermediate, fractionary, parastatistics also play a fundamental role in the quasiparticle zoo [30], that provides a deep understanding of complex phenomena in many-body and condensed matter physics. Recently, a proposal to describe composite fermions (in multicomponent fractional quantum Hall systems) in terms rectangular Young tableaux has been put forward [31,32] and showed to describe the quantum phases of bilayer quantum Hall systems with U (4) dynamical symmetry [33,34]. This is also an excellent area to explore the role of permutation symmetry.…”

“…The eigenvector composition and degeneracy of higher excited states is a bit more involved. Note that all GT vectors |m in (19) are eigenvectors of the free Hamiltonian density H (0) and their eigenvalues can be easily calculated as 35) and (34), respectively. This degeneracy is partially lifted when the two-body interaction [with coupling constant λ, like in (9)] is introduced.…”

Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models

“…The parity symmetry is spontaneously broken in the thermodynamic limit N → ∞ and degenerated ground states arise. Parity-adapted coherent states are then good variational states, reproducing the energy of the ground state of these quantum critical models in the thermodynamic limit N → ∞, namely in matter-field interactions (Dicke model) of two-level [27,28] and three-level [29,30] atoms, BEC [31], U(3) vibron models of molecules [32,33], bilayer quantum Hall systems [34] and (LMG) models for two-level atoms [35][36][37]. Quantum information (fidelity, entropy, fluctuation, entanglement, etc.)…”

Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin-Meshkov-Glick D-level atom models

“…The parity symmetry is spontaneously broken in the thermodynamic limit N → ∞ and degenerated ground states arise. Parity adapted coherent states are then good variational states, reproducing the energy of the ground state of these quantum critical models in the thermodynamic limit N → ∞, namely in matter-field interactions (Dicke model) of two-level [27,28] and three-level [29,30] atoms, BEC [31], U(3) vibron models of molecules [32,33], bilayer quantum Hall systems [34] and (LMG) models for two-level atoms [35][36][37]. Quantum information (fidelity, entropy, fluctuation, entanglement, etc) measures have proved to be useful in the analysis of the highly correlated ground state structure of these many-body systems and the identification of critical points across the phase diagram.…”