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International Journal of Modern Physics B

QUANTUM HALL EFFECT: WHAT IS REALLY GOING ON AT HALF-FILLING?

Erio Tosatti

Published Mar 1, 1992• 1 citation
Publisher

Abstract

This paper reviews the authors' theoretical understanding for the quantum Hall anomalies recently observed at fractional fillings with even denominators, more particularly at filling 1/2, where the electron spins are still fully polarized. After a brief reminder of the basic experimental evidence dip of {rho}{sub xx}, lack of plateau in {rho}{sub xx}, temperature behavior, the author discusses the important role of electron-hole (e-h) symmetry at half-filling. In particular the two alternative possibilities of spontaneous e-h symmetry breaking or non-breaking are discussed along with their physical implications. Hartree-Fock charge density waves are presented as an explicit example of spontaneous e-h symmetry breaking. Next, several different scenarios are discussed for the ground state at half-filling. One, largely discussed in the experimental papers is that of Halperin, involving phase separation. Another is that of Fano et al. where the half-filling ground state is supposed to be incompressible and homogeneous, but deformable. Pursuing further the second scenario, numerical evidence from the recent work of Chen is presented, indicating incompressibility and deformability. This work also demonstrates a rather abrupt switching of short-range electron correlations. Just below half-filling, an electron is typically surrounded by six close-by electrons (like a triangular lattice).

DOI
10.1142/s0217979292000475

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This paper reviews the authors' theoretical understanding for the quantum Hall anomalies recently observed at fractional fillings with even denominators, more particularly at filling 1/2, where the electron spins are still fully polarized. After a brief reminder of the basic experimental evidence dip of {rho}{sub xx}, lack of plateau in {rho}{sub xx}, temperature behavior, the author discusses the important role of electron-hole (e-h) symmetry at half-filling. In particular the two alternative possibilities of spontaneous e-h symmetry breaking or non-breaking are discussed along with their physical implications. Hartree-Fock charge density waves are presented as an explicit example of spontaneous e-h symmetry breaking. Next, several different scenarios are discussed for the ground state at half-filling. One, largely discussed in the experimental papers is that of Halperin, involving phase separation. Another is that of Fano et al. where the half-filling ground state is supposed to be incompressible and homogeneous, but deformable. Pursuing further the second scenario, numerical evidence from the recent work of Chen is presented, indicating incompressibility and deformability. This work also demonstrates a rather abrupt switching of short-range electron correlations. Just below half-filling, an electron is typically surrounded by six close-by electrons (like a triangular lattice).
We compare quantum Hall systems at filling factors $\ensuremath{\nu}=2$ to $\ensuremath{\nu}=\frac{2}{3}$ and 25\frac{2}{5}52​, corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P=1P=1P=1) and paramagnetic (P=0)(P=0)(P=0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At $\ensuremath{\nu}=2$, we find spin domains at half-polarization (P=12)(P=\frac{1}{2})(P=21​), while antiferromagnetic order seems most favorable in the CF systems. The transition between P=0P=0P=0 and 1, as seen when, e.g., the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompressible half-polarized state is found in some models.
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.
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.
A quantum Hall system is investigated by exact numerical diagonalization of the Hamiltonian. The energy, off-diagonal long-range order, crystal long-range order, and the mean occupation number \ensuremath{\rho}(j)=〈${\mathit{c}}_{\mathit{j}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$cj{\mathit{c}}_{\mathit{j}}cj​〉 of the jth Landau orbital for \ensuremath{\nu}=1/2 ground state are calculated. Our results indicate no sign of either electron pairing or phase separation, the two possible features previously suggested for the \ensuremath{\nu}=1/2 ground state. A possible quasiparticle pairing mechanism is proposed in trying to account for the numerical results and experimental observations.
Two-dimensional interacting electron systems become strongly correlated if the electrons are subject to a perpendicular high magnetic field. After introducing the physics of the quantum Hall regime the incompressible many-particle ground state and its excitations are studied in detail at fractional filling factors for spin-polarized electrons. The spin degree of freedom whose importance was shown in recent experiments is considered by studying the thermodynamics at filling factor one and near one.
Two-dimensional interacting electron systems become strongly correlated if the electrons are subject to a perpendicular high magnetic field. After introducing the physics of the quantum Hall regime the incompressible many-particle ground state and its excitations are studied in detail at fractional filling factors for spin-polarized electrons. The spin degree of freedom whose importance was shown in recent experiments is considered by studying the thermodynamics at filling factor one and near one.