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arXiv:cond-mat/0001036 · Journal of Physics Condensed Matter

Roton instability of the spin-wave excitation in the fully polarized quantum Hall state and the phase diagram at nu = 2

K Park, Jainendra K. Jain

Published Apr 4, 2000• 6 citations
arXivHTMLPDF

Abstract

We consider the effect of interaction on electrons confined to two dimensions at Landau level filling = 2, with the specific aim of determining the range of parameters where the fully polarized state is stable. We calculate the charge- and the spin-density collective modes in the random-phase approximation (RPA) including vertex corrections (also known as the time-dependent Hartree-Fock approximation), and treating the Landau level mixing accurately within the subspace of a single particle-hole pair. It is found that the spin-wave excitation mode of the fully polarized state has a roton minimum which deepens as a result of the interaction-induced Landau level mixing, and the energy of the roton vanishes at a critical Zeeman energy, signalling an instability of the fully polarized state at still lower Zeeman energies. The feasibility of the experimental observation of the roton minimum in the spin-wave mode and its softening will be discussed. The spin- and charge-density collective modes of the unpolarized state are also considered, and a phase diagram for the = 2 state as a function of r S and the Zeeman energy is obtained.

DOI
10.1088/0953-8984/12/16/304

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We consider the effect of interaction on electrons confined to two dimensions at Landau level filling = 2, with the specific aim of determining the range of parameters where the fully polarized state is stable. We calculate the charge- and the spin-density collective modes in the random-phase approximation (RPA) including vertex corrections (also known as the time-dependent Hartree-Fock approximation), and treating the Landau level mixing accurately within the subspace of a single particle-hole pair. It is found that the spin-wave excitation mode of the fully polarized state has a roton minimum which deepens as a result of the interaction-induced Landau level mixing, and the energy of the roton vanishes at a critical Zeeman energy, signalling an instability of the fully polarized state at still lower Zeeman energies. The feasibility of the experimental observation of the roton minimum in the spin-wave mode and its softening will be discussed. The spin- and charge-density collective modes of the unpolarized state are also considered, and a phase diagram for the = 2 state as a function of r S and the Zeeman energy is obtained.
We report inelastic light scattering measurements of dispersive spin and charge density excitations in dilute 2D electron systems reaching densities less than ${10}^{10}{\mathrm{cm}}^{\ensuremath{-}2}$. In the quantum Hall state at $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$, roton critical points in the spin inter-Landau level mode show a pronounced softening as rs{r}_{s}rs​ is increased. Instead of a soft mode instability predicted by Hartree-Fock calculations for ${r}_{s}\ensuremath{\sim}3.3$, we find evidence of multiple rotons in the dispersion of the softening spin excitations.
The spin transitions in the fractional quantum Hall effect provide a direct measure of the tiny energy differences between differently spin-polarized states and thereby serve as an extremely sensitive test of the quantitative accuracy of the theory of the fractional quantum Hall effect, and, in particular, of the role of Landau-level mixing in lifting the particle-hole symmetry. We report on an accurate quantitative study of this physics, evaluating the effect of Landau-level mixing in a nonperturbative manner using a fixed-phase diffusion Monte Carlo method. We find excellent agreement between our calculated critical Zeeman energies and the experimentally measured values. In particular, we find, as also do experiments, that the critical Zeeman energies for fractional quantum Hall states at filling factors ν=2-n/(2n±1) are significantly higher than those for ν=n/(2n±1), a quantitative signature of the lifting of particle-hole symmetry due to Landau-level mixing.
We present the magnetic phase diagram of the $\ensuremath{\nu}=2$ quantum Hall system on the whole (rs,EZ)({r}_{s},{E}_{Z})(rs​,EZ​) plane. We fix the phase boundaries of the paramagnetic and ferromagnetic states by looking for a softening of spin-density excitations in the time-dependent Hartree-Fock theory. A nontrivial phase is obtained in the self-consistent Hartree-Fock theory for ${r}_{s}\ensuremath{\sim}2$ and ${E}_{Z}\ensuremath{\lesssim}0.06\ensuremath{\hbar}{\ensuremath{\omega}}_{c}$, where both the paramagnetic and ferromagnetic states show spin instability. We show that the obtained phase is the spin-density wave (SDW) state, and explain the mechanism how the SDW stabilizes.
We present the magnetic phase diagram of the ν = 2 quantum Hall system on the whole (rs, EZ ) plane.We fix the phase boundaries of the paramagnetic and ferromagnetic states by looking for a softening of spin density excitations (SDE) in the time-dependent Hartree-Fock theory.A new phase is obtained in the self-consistent Hartree-Fock theory for rs ∼ 2 and EZ 0.06 ωc, where both the paramagnetic and ferromagnetic states show spin instability.We show that the new phase is the spin density wave (SDW) state, and explain the mechanism how the SDW stabilizes.
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In this dissertation, we investigate the edge spin excitations and reconstructions of quantum Hall (QH) liquids. Edge spin reconstructions reflect the interplay of confining potential, electron-electron interaction, and Zeeman splitting at the edges of QH liquids. The main method used in this dissertation is the exact diagonalization calculation for finite size systems. The QH liquids studied in this dissertation are classified into three kinds, based on their topological properties and statistics of quasiparticles (quasiholes). The first kind of QH liquids is spin-polarized QH liquids. The QH liquids with filling factors ν = 1 and 1/3 are studied in this kind. We find the low-energy excitations of the ν = 1 and 1/3 are bosonic edge spin waves. Instabilities of these ferromagnetic state with altering confinement strength result from the softening of these spin waves and formation of edge spin textures. A microscopic trial wave function is used to study the edge instabilities of larger systems with particle number up to 40, and the obtained results are consistent to the ones calculated by exact diagonalization. The second kind of QH liquids is spin-unpolarized QH liquids. The liquids with ν = 2, 2/5 and 2/3 are studied in this kind. When ν = 2, exact diagonalization on edge electron systems indicates that compact Hartree-Fock states with different total spin always become ground states in some regions of parameter space, and the ground states appearing between two compact states are their edge spin waves. The initial ν = 2 instability is toward the compact state with total spin 1. At ν = 2/5, the edge instabilities is also triggered by softening of the edge spin waves with smoother confinement. The situation of ν = 2/3 unpolarized QH liquids is a little different, because it has one backward-moving edge spin wave mode and one forward-moving magnetoplasmon mode. There is no mystery about what kind of instability would occur with stronger or smoother confinement. The exact diagonalization calculation of finite systems can reproduce the instability induced by edge spin wave at the stronger confinement side and gives an estimate about the critical value of confining potential. But instability due to edge magnetoplasmon mode can not be realized in small size systems because of the high velocity of this mode. The last kind is the non-Abelian QH liquid (at ν = 5/2). Its edge instability triggered by softening of edge magnetoplasmon excitations with smoother confinement. Three different confinements are compared in the study of integer QH liquids in finite size systems. And some quantitative predictions on the edge instabilities for a certain type of confining potential are reached in the thermodynamic limit. The effect of Zeeman splitting to the edge instabilities is also considered in this dissertation.