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arXiv:cond-mat/9809024 · Physical Review Letters

Transition from Quantum Hall to Compressible States in the Second Landau Level: New Light on the ν=5/2 Enigma

R. Morf

Published Feb 16, 1998• 406 citations
arXivHTMLPDF

Abstract

Quantum Hall states at filling fraction $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5/2$ are examined by numerical diagonalization. Spin-polarized and spin-unpolarized states of systems with $N\ensuremath{\le}18$ electrons are studied, neglecting effects of Landau level mixing. We find that the ground state is spin polarized. It is incompressible and has a large overlap with paired states like the Pfaffian. For a given sample, the energy gap is about 11 times smaller than at $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3$. Evidence is presented of phase transitions to compressible states, driven by the interaction strength at short distance. A reinterpretation of experiments is suggested.

DOI
10.1103/physrevlett.80.1505

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•Aug 10, 2021•0 citations
Quantum Hall states at filling fraction \ensuremathν=5/2\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5/2\ensuremathν=5/2 are examined by numerical diagonalization. Spin-polarized and spin-unpolarized states of systems with N\ensuremath≤18N\ensuremath{\le}18N\ensuremath≤18 electrons are studied, neglecting effects of Landau level mixing. We find that the ground state is spin polarized. It is incompressible and has a large overlap with paired states like the Pfaffian. For a given sample, the energy gap is about 11 times smaller than at \ensuremathν=1/3\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3\ensuremathν=1/3. Evidence is presented of phase transitions to compressible states, driven by the interaction strength at short distance. A reinterpretation of experiments is suggested.
We numerically study a 52\frac{5}{2}25​ fractional quantum Hall system with even number of electrons using the exact diagonalization where both the strong Landau-level (LL) mixing and a finite width of the quantum well have been considered and adapted into a screened Coulomb interaction. With the principal component analysis, we are able to recognize a compressible-incompressible phase transition in the parameter space made of the magnetic field and the quantum well width by the competition between the first two leading components of the ground-state wave functions, which is consistent with the low-lying spectral feature and previous works in the odd-electron system. In addition, the presence of the subdominant third component suggests an incompressible transition occurring as the LL mixing strength grows into a certain parameter region associated with the ZnO experiments. We further investigate the strongly LL-mixed phase in this emerging region with the Hall viscosity, wave function overlaps, and the entanglement spectra. Results show it can be well described as a particle-hole symmetrized Pfaffian state with the dual topological properties of the Pfaffian and the anti-Pfaffian states.
The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with k+1k+1k+1-body interactions, for all integers k> 1.k>~1.k> 1. The remarkably simple wave functions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The k=2k=2k=2 case is the Pfaffian. For k> 2,k>~2,k> 2, the quasiparticle excitations of these systems are expected to possess non-Abelian statistics, like those of the Pfaffian. For k=3,k=3,k=3, these ground states have large overlaps with the ground states of the (two-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors $\ensuremath{\nu}=2+3/5,2+2/5.$
The $\ensuremath{\nu}=5/2$ fractional quantum Hall effect is a system of intense experimental and theoretical interest as its ground state may host non-Abelian excitations, but the exact nature of the ground state is still undetermined. We present the results of an exact diagonalization study of an electron system in the disk configuration, including the effects of Landau level (LL) mixing and the finite thickness of the quantum well confining the electrons. The degeneracy between the two leading candidates for the ground state, the Pfaffian and anti-Pfaffian, is broken by interactions with a neutralizing background, in addition to the inclusion of two- and three-body interactions via LL mixing. As a result of the neutralizing background in the disk configuration, there is a phase transition from the anti-Pfaffian to the Pfaffian as LL mixing is turned on, in stark contrast to what is observed in spherical geometry. This behavior is in agreement with existing experiments, showing the appearance of the Pfaffian state at strong LL mixing before the system enters a compressible phase. The inclusion of LL mixing leads to an increased charge e/4e/4e/4 quasihole size. LL mixing interactions are also shown to overcome the effects of edge reconstruction. Due to finite thickness effects, these properties are enhanced dramatically. We also find that only the Pfaffian and anti-Pfaffian states continue to possess energy gaps at finite width, while gaps for compressible stripe states close, which is in agreement with available experimental data.
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.
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.
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.