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arXiv:1512.04914 · Physical Review Letters

Optical Emission Spectroscopy Study of Competing Phases of Electrons in the Second Landau Level

Antonio Levy, Ursula Wurstbauer, Y. Y. Kuznetsova, Aron Pinczuk, Loren N. Pfeiffer, K. W. West, Michael J. Manfra, Geoffrey C. Gardner, John Watson

Published Jan 8, 2016• 13 citations
arXivHTMLPDF

Abstract

Quantum phases of electrons in the filling factor range 2≤ν≤3 are probed by the weak optical emission from the partially populated second Landau level and spin wave measurements. Observations of optical emission include a multiplet of sharp peaks that exhibit a strong filling factor dependence. Spin wave measurements by resonant inelastic light scattering probe breaking of spin rotational invariance and are used to link this optical emission with collective phases of electrons. A remarkably rapid interplay between emission peak intensities manifests phase competition in the second Landau level.

DOI
10.1103/physrevlett.116.016801

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Quantum phases of electrons in the filling factor range 2≤ν≤3 are probed by the weak optical emission from the partially populated second Landau level and spin wave measurements. Observations of optical emission include a multiplet of sharp peaks that exhibit a strong filling factor dependence. Spin wave measurements by resonant inelastic light scattering probe breaking of spin rotational invariance and are used to link this optical emission with collective phases of electrons. A remarkably rapid interplay between emission peak intensities manifests phase competition in the second Landau level.
At a very low-temperature of 9 mK, electrons in the second Landau level of an extremely high-mobility two-dimensional electron system exhibit a very complex electronic behavior. With a varying filling factor, quantum liquids of different origins compete with several insulating phases leading to an irregular pattern in the transport parameters. We observe a fully developed nu=2+2/5 state separated from the even-denominator nu=2+1/2 state by an insulating phase and a nu=2+2/7 and nu=2+1/5 state surrounded by such phases. A developing plateau at nu=2+3/8 points to the existence of other even-denominator states.
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.
Collective spin excitations were utilized to probe many-particle energies of strongly correlated quantum Hall ferromagnets (FMs) at $\frac{1}{3}\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}1$. For this purpose, experiments on inelastic light scattering and polarized photoluminescence (PL) were performed on a MgZnO/ZnO heterostructure, containing a high-mobility two-dimensional electron system with the Wigner-Seitz parameter ${r}_{s}\ensuremath{\approx}11$. The dispersion of the Larmor spin exciton was utilized to extract the spin stiffness, which was found to have maxima at $\ensuremath{\nu}=\frac{1}{3}, \ensuremath{\nu}=1$, and to be strongly suppressed at intermediate filling factors. Measurements of circularly polarized PL helped to confirm a FM configuration of the system. In the close vicinity of $\ensuremath{\nu}=\frac{1}{3}$, a hybrid spin-density excitation was discovered, the energy of which enables us to probe the correlation energy of the Laughlin liquid. The measured correlation and exchange energies at $\ensuremath{\nu}=\frac{1}{3}$ and ${r}_{s}\ensuremath{\gg}1$ turned out to be much less suppressed by the Landau level mixing than the exchange energy at integer filling factors. The same answer follows from the calculations performed using the exact diagonalization method.
We study two interacting electrons in a two-dimensional system under a strong magnetic field and show that their numerically exact solutions organize into a set of {\em sub-Landau levels} characterized by relative angular momentum quantum number mmm. These sub-levels define correlation-resolved subspaces of the Landau-level Hilbert space, while retaining the full degeneracy associated with center-of-mass motion. Within this structure, the accessible states in each correlation channel are effectively reduced, leading to a natural organization of guiding-center states consistent with a fractional occupancy. We further analyze the role of electron correlation, Zeeman splitting, and disorder in stabilizing spin-polarized electron-pair states. Building on the two-electron states, we construct a class of many-electron trial wavefunctions based on correlated electron pairs with fixed mmm, which encode short-range correlations through the vanishing of the pair wavefunction at small separation. Our results establish a direct connection between exact two-body physics and the organization of correlated many-electron states in the lowest Landau level, providing a microscopic perspective on how relative angular momentum structures can underpin the emergence of correlated phases in quantum Hall systems.