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arXiv:1310.2263 · Physical Review B

Quasiholes of 13 and 73 quantum Hall states: Size estimates via exact diagonalization and density-matrix renormalization group

Sonika Johri, Zlatko Papić, R. N. Bhatt, Peter Schmitteckert

Published Mar 24, 2014• 34 citations
arXivHTMLPDF

Abstract

We determine the size of the elementary quasihole in $\ensuremath{\nu}=1/3$ and $\ensuremath{\nu}=7/3$ quantum Hall states via exact-diagonalization and density-matrix renormalization group calculations on the sphere and cylinder, using a variety of short- and long-range pinning potentials. The size of the quasihole at filling factor $\ensuremath{\nu}=1/3$ is estimated to be $\ensuremath{\approx}4{\ensuremath{\ell}}_{B}$, and that of $\ensuremath{\nu}=7/3$ is $\ensuremath{\approx}7{\ensuremath{\ell}}_{B}$, where ${\ensuremath{\ell}}_{B}$ is the magnetic length. In contrast, the size of the Laughlin quasihole, expected to capture the basic physics in these two states, is around $\ensuremath{\approx}2.5{\ensuremath{\ell}}_{B}$. Our work supports the earlier findings that the quasihole in the first excited Landau level is significantly larger than in the lowest Landau level.

DOI
10.1103/physrevb.89.115124

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We determine the size of the elementary quasihole in \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 and \ensuremathν=7/3\ensuremath{\nu}=7/3\ensuremathν=7/3 quantum Hall states via exact-diagonalization and density-matrix renormalization group calculations on the sphere and cylinder, using a variety of short- and long-range pinning potentials. The size of the quasihole at filling factor \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 is estimated to be \ensuremath≈4\ensuremathℓB\ensuremath{\approx}4{\ensuremath{\ell}}_{B}\ensuremath≈4\ensuremathℓB​, and that of \ensuremathν=7/3\ensuremath{\nu}=7/3\ensuremathν=7/3 is \ensuremath≈7\ensuremathℓB\ensuremath{\approx}7{\ensuremath{\ell}}_{B}\ensuremath≈7\ensuremathℓB​, where \ensuremathℓB{\ensuremath{\ell}}_{B}\ensuremathℓB​ is the magnetic length. In contrast, the size of the Laughlin quasihole, expected to capture the basic physics in these two states, is around \ensuremath≈2.5\ensuremathℓB\ensuremath{\approx}2.5{\ensuremath{\ell}}_{B}\ensuremath≈2.5\ensuremathℓB​. Our work supports the earlier findings that the quasihole in the first excited Landau level is significantly larger than in the lowest Landau level.
We use the dynamical structure factors of the quantum Hall states at $\ensuremath{\nu}=1/3$ and 1/21/21/2 in the lowest Landau level to study their excitation spectrum. Using the density matrix renormalization group in combination with the time-dependent variational principle on an infinite cylinder geometry, we extract the low-energy properties. At $\ensuremath{\nu}=1/3$, a sharp magnetoroton mode and the two-roton continuum are present and the finite-size effects can be understood using the fractional charge of the quasiparticle. At $\ensuremath{\nu}=1/2$, we find low-energy modes with linear dispersion and the static structure factor $\overline{s}(q)\ensuremath{\sim}{(q\ensuremath{\ell})}^{3}$ in the limit $q\ensuremath{\ell}\ensuremath{\rightarrow}0$. The properties of these modes agree quantitatively with the predictions of the composite-fermion theory placed on the infinite cylinder.
We develop the density-matrix renormalization group (DMRG) technique for numerically studying incompressible fractional quantum Hall (FQH) states on the sphere. We calculate accurate estimates for ground-state energies and excitation gaps at FQH filling fractions nu=1/3 and nu=5/2 for systems that are considerably larger than the largest ever studied by exact diagonalization. We establish, by carefully comparing with existing numerical results on smaller systems, that DMRG is a highly effective numerical tool for studying incompressible FQH states.
The single-particle spectral function for an incompressible fractional quantum Hall state in the presence of a scalar short-ranged attractive impurity potential is calculated via exact diagonalization within the spherical geometry. In contrast to the noninteracting case, where only a single bound state below the lowest Landau level forms, electron–electron interactions strongly renormalize the impurity potential, effectively giving it a finite range, which can support many quasi-bound states (long-lived resonances). Averaging the spectral weights of the quasi-bound states and extrapolating to the thermodynamic limit, for filling factor ν = 1/3 we find evidence consistent with localized fractionally charged e /3 quasi-particles. For ν = 2/5, the results are slightly more ambiguous, due to finite size effects and possible bunching of Laughlin-quasi-particles.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
Using exact diagonalization in the spherical geometry, we investigate systems of localized quasiholes at $\ensuremath{\nu}=5/2$ for interactions interpolating between the pure Coulomb and the three-body interaction for which the Moore-Read state is the exact ground state. We show that the charge e/4e/4e/4 quasihole can be easily localized by means of a $\ensuremath{\delta}$-function pinning potential. Using a tuned smooth pinning potential, the quasihole radius can be limited to approximately three magnetic length units. For systems of two quasiholes, adiabatic continuity between the Moore-Read and the Coulomb limit holds for the ground state, while for four quasiholes, the lowest two energy states exhibit adiabatic continuity. This implies the existence of a Majorana fermion for pure Coulomb interaction. We also present preliminary results in the Coulomb limit for braiding in systems containing four quasiholes, with up to 14 electrons, diagonalizing in the full spin-polarized sector of the second Landau-level Hilbert space.
The nature of the state at low Landau-level filling factors has been a long-standing puzzle in the field of the fractional quantum Hall effect (FQHE). While theoretical calculations suggest that a crystal is favored at filling factors $\ensuremath{\nu}\ensuremath{\lesssim}\frac{1}{6}$, experiments show, at somewhat elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect at $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{11}, \frac{2}{13}, \frac{3}{17}, \frac{3}{19}, \frac{1}{9}, \frac{2}{15}$, and 217\frac{2}{17}172​, which belong to the standard sequences $\ensuremath{\nu}=n/(6n\ifmmode\pm\else\textpm\fi{}1)$ and $n/(8n\ifmmode\pm\else\textpm\fi{}1)$. To address this paradox, we investigate the nature of some of the low-$\ensuremath{\nu}$ states, specifically $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{13}$, and 19\frac{1}{9}91​, by variational Monte Carlo, density matrix renormalization group, and exact diagonalization methods. We conclude that in the thermodynamic limit, these are likely to be incompressible fractional quantum Hall liquids, albeit with strong short-range crystalline correlations. This suggests a natural explanation for the experimentally observed behavior and a rich phase diagram that admits, in the low-disorder limit, a multitude of crystal-FQHE liquid transitions as the filling factor is reduced.