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arXiv:2112.02113 · Physical review. B./Physical review. B

Neutral excitations of quantum Hall states: A density matrix renormalization group study

Prashant Kumar, F. D. M. Haldane

Published Aug 8, 2022• 23 citations
arXivHTMLPDF

Abstract

We use the dynamical structure factors of the quantum Hall states at $\ensuremath{\nu}=1/3$ and $1/2$ 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.

DOI
10.1103/physrevb.106.075116

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We investigate the ground state properties of a two-dimensional electron gas in the lowest Landau level using the Density Matrix Renormalization Group. The electron gas is confined to a cylinder with a strong magnetic field perpendicular to the surface. For a thin cylinder, the ground state is for generic filling factors a charge density wave state, however, at ν=1/2ν=1/2ν=1/2 a homogeneous and apparently gapless ground state is found and particle and hole excitations are studied.
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.
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We investigate how cavity-mediated attractive electron-electron interactions modify the excitation gaps of fractional quantum Hall states within the composite-fermion framework. We compute both the neutral magnetoroton excitation spectrum and the charged excitation gap relevant to transport experiments for the Laughlin ν=1/3ν=1/3ν=1/3 and ν=1/5ν=1/5ν=1/5 states. We consider a spin-polarized lowest-Landau-level model in which the interaction is mediated by a cavity mode with a spatially uniform vacuum-field gradient and a finite interaction range controlled by a long-distance cutoff. Finite-size scaling reveals that the transport gap is consistently enhanced by the cavity-induced interaction, with the gap enhancement scaling quadratically with the electron number and with the fourth power of the vacuum-field gradient. By contrast, the magnetoroton spectrum exhibits a richer dependence on the interaction range. The high-kkk magnetoroton gap is enhanced for all interaction ranges considered, consistent with its close connection to the charged excitation gap, even with the long-range character of the interaction.
We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where L1{L}_{1}L1​ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd qqq, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd qqq, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L1{L}_{1}L1​ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1{L}_{1}L1​; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.