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arXiv:1706.04793 · Journal of Statistical Mechanics Theory and Experiment

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

Manuel Calixto, Carlos Peón-Nieto

Published May 1, 2018• 7 citations
arXivHTMLPDF

Abstract

Abstract We propose localization measures in phase space of the ground state of bilayer quantum Hall systems at fractional filling factors , to characterize the three quantum phases (shortly denoted by spin, canted and ppin) for arbitrary -isospin λ . We use a coherent state (Bargmann) representation of quantum states, as holomorphic functions in the 8-dimensional Grassmannian phase-space (a higher-dimensional generalization of the Haldane’s 2-dimensional sphere ). 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.

DOI
10.1088/1742-5468/aabfcb

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Abstract We propose localization measures in phase space of the ground state of bilayer quantum Hall systems at fractional filling factors , to characterize the three quantum phases (shortly denoted by spin, canted and ppin) for arbitrary -isospin λ . We use a coherent state (Bargmann) representation of quantum states, as holomorphic functions in the 8-dimensional Grassmannian phase-space (a higher-dimensional generalization of the Haldane’s 2-dimensional sphere ). 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.
Bilayer quantum Hall (BLQH) systems, which underlie a U (4) symmetry, display unique quantum coherence effects. We study coherent states (CS) on the complex Grassmannian , orthonormal basis, U (4) generators and their matrix elements in the reproducing kernel Hilbert space of analytic square-integrable holomorphic functions on , which carries a unitary irreducible representation of U (4) with index . A many-body representation of the previous construction is introduced through an oscillator realization of the U (4) Lie algebra generators in terms of eight boson operators. This particle picture allows us to make a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index λ is related to the number of flux quanta bound to a bi-fermion in the composite fermion picture of Jain for fractions of the filling factor ν = 2. The simpler, and better known, case of spin- s CS on the Riemann–Bloch sphere is also treated in parallel, of which Grassmannian -CS can be regarded as a generalized (matrix) version.
We present an exact diagonalization study of bilayer quantum Hall systems at filling factor nu = 2 in the spherical geometry. We find the high-Zeeman-coupling phase boundary of the broken symmetry canted antiferromagnet is given exactly by previous Hartree-Fock mean-field theories, but that the state's stability at weak Zeeman coupling has been qualitatively overestimated. In the absence of interlayer tunneling, degeneracies occur between total spin multiplets due to the Hamiltonian's invariance under independent spin rotations in top and bottom two-dimensional electron layers.
The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. We find some basic features essentially different from the bilayer integer QH systems at ν = 2, reflecting the special characteristics of the fractional QH systems. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The continuous transitions of the finite size systems between the spin-polarized and spin-unpolarized phases are determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy, and are almost not affected by ΔSAS.
We study phase space properties of critical, parity symmetric, N-qudit systems undergoing a quantum phase transition (QPT) in the thermodynamic N→∞ limit. The D=3 level (qutrit) Lipkin-Meshkov-Glick model is eventually examined as a particular example. For this purpose, we consider U(D)-spin coherent states (DSCS), generalizing the standard D=2 atomic coherent states, to define the coherent state representation Q_{ψ} (Husimi function) of a symmetric N-qudit state |ψ〉 in the phase space CP^{D-1} (complex projective manifold). DSCS are good variational approximations to the ground state of an N-qudit system, especially in the N→∞ limit, where the discrete parity symmetry Z_{2}^{D-1} is spontaneously broken. For finite N, parity can be restored by projecting DSCS onto 2^{D-1} different parity invariant subspaces, which define generalized "Schrödinger cat states" reproducing quite faithfully low-lying Hamiltonian eigenstates obtained by numerical diagonalization. Precursors of the QPT are then visualized for finite N by plotting the Husimi function of these parity projected DSCS in phase space, together with their Husimi moments and Wehrl entropy, in the neighborhood of the critical points. These are good localization measures and markers of the QPT.
Quantum Hall systems have a one-body energy spectrum consisting of dispersion-less Landau levels. Electron-electron interactions thus dominate in partially filled Landau levels, which exhibit a myriad of strongly correlated phases such as quantum hall ferromagnets and fractional quantum Hall phases. We study two examples of these phenomena. In the first project, we explore the ground state of a system with an interface between two semi-infinite regions with fillings ν= 4 and ν= 3 respectively. The width of the interface can be controlled by varying the background potential, which provides an additional tuning parameter. For a certain range of interaction strengths, the ν= 4 bulk is unpolarized whereas the ν= 3 bulk is fully polarized. In the parameter space spanned by the interaction strength and width of the interface, we find two phases at the interface. Phase A has spin as a good quantum number, and the long-wavelength spin edge excitations are gapped. In phase B, spin rotation symmetry is spontaneously broken at the mean-field level. Using symmetry arguments we find the effective theory near the interface of phase B. This effective theory is known to have gapless long-wavelength spin excitations. In the second project, we study the ground state of a tunnel-decoupled double-layer graphene system when both layers are undoped. We find a simple Hamiltonian in the continuum limit from symmetries of the system. Using the Hartree-Fock approximation we find a state with inter-layer coherence with broken layer U(1) symmetry. This phase becomes magnetized in presence of a non-zero Zeeman field. A first-order phase transition can be driven from the ferromagnetic phase to the magnetized inter-layer coherent phase by increasing the Zeeman field. We predict the number of gapless modes in the bulk.
There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two-component FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, for integer nnn. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, including in particular the possibility of the non-Abelian Z4{Z}_{4}Z4​ parafermion state. In $\ensuremath{\nu}=2/3$ bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the Z4{Z}_{4}Z4​ state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the Z4{Z}_{4}Z4​ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\ensuremath{\nu}=8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.