Hilbert space and ground-state structure of bilayer quantum Hall systems at ν=2/λ

Abstract

We analyze the Hilbert space and ground state structure of bilayer quantum Hall (BLQH) systems at fractional filling factors ν = 2/λ (λ odd) and we also study the large SU (4) isospin-λ limit. The model Hamiltonian is an adaptation of the ν = 2 case [Z.F. Ezawa et al., Phys. Rev. B71 (2005) 125318] to the many-body situation (arbitrary λ flux quanta per electron). The semiclassical regime and quantum phase diagram (in terms of layer distance, Zeeeman, tunneling, etc, control parameters) is obtained by using previously introduced Grassmannian G 4 2 = U (4)/[U (2) × U (2)] coherent states as variational states. The existence of three quantum phases (spin, canted and ppin) is common to any λ, but the phase transition points depend on λ, and the instance λ = 1 is recovered as a particular case. We also analyze the quantum case through a numerical diagonalization of the Hamiltonian and compare with the mean-field results, which give a good approximation in the spin and ppin phases but not in the canted phase, where we detect exactly λ energy level crossings between the ground and first excited state for given values of the tunneling gap. An energy band structure at low and high interlayer tunneling (spin and ppin phases, respectively) also appears depending on angular momentum and layer population imbalance quantum numbers.

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

“…For λ = 1 it coincides with the variational ground state condition provided in [6]. For BLQH systems at ν = 2/λ we have seen in [22] that the spin and ppin phases are characterized by maximum values of S 2 = λ 2 and P 2 = λ 2 , respectively. The Husimi function Q ψ (Z) of a given state |ψ is the CS expectation value Q ψ (Z) = Z|ρ|Z of the corresponding density matrix ρ = |ψ ψ| (this definition can be directly extended to mixed states).…”

“…The orthonormal basis vectors | j,m qa,q b are now indexed by four (half-)integer numbers subject to constraints. We shall provide here a brief summary with the basic expressions, in order to make the article more self-contained (more information can be found in references [18][19][20][21][22]).…”

“…we discard therms proportional to ε D and ∆ b ). Using the parametrization (26) of Z, we found in [22] the common relations (39) in all phases. This leaves only two free parameters, for instance, ϑ + and θ b .…”

“…For ∆ t < ∆ sc t (λ) the BLQH system at ν = 2/λ is in the spin phase, for ∆ sc t (λ) ≤ ∆ t ≤ ∆ cp t (λ) it is in the canted phase and for ∆ t > ∆ cp t (λ) it is in the ppin phase [see [22] for more details]. Note that we have two different solutions of (ϑ c + , θ c b ) in the canted phase, given by the signs (+, −) and (−, +) in equation (41), leading to the same minimum energy Z c (θ a,b , φ a,b , ϑ ± , β ± )| c ± the corresponding stationary point in the Grassmannian G 4 2 for any of the two solutions (+) = (+, −) and (−) = (−, +) together with the common restrictions (39).…”

Analogies between the topological insulator phase of 2D Dirac materials and the superradiant phase of atom‐field systems

“…We shall show that an information‐theoretical analysis in the phase space of topological and standard quantum phases reveals a similar structure around the critical points, especially between edge and ground states. Actually, localization, entropy, and entanglement measures of Hamiltonian eigenstates have proven to be good markers of the QPT for the Dike model of matter‐radiation interaction, [ 17–20 ] vibron model of molecules, [ 21–23 ] the ubiquitous Lipkin‐Meshkov‐Glick, [ 24–27 ] Bose‐Einstein condensates, [ 28 ] bilayer quantum Hall effect, [ 29–31 ] etc. As shown in, [ 32 ] these entropic measures are even capable of identifying the order of the corresponding QPT.…”