Landau-Level Mixing and Particle-Hole Symmetry Breaking for Spin Transitions in the Fractional Quantum Hall Effect

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 $\nu =2-n/(2n\pm 1)$ are significantly higher than those for $\nu =n/(2n\pm 1)$, a quantitative signature of the lifting of particle-hole symmetry due to Landau-level mixing.

Figure 1

The theoretical critical Zeeman energies ${\alpha }_Z^{\mathrm{crit}}=E_Z^{\mathrm{crit}}/(e^2/\epsilon \ell )$ for $\nu =2/5$ (left), $3/7$ (middle), $4/9$ (right) are shown with empty symbols with error bars, calculated from both DMC (solid line) and VMC (dashed line) methods, for an ideal 2D system ($w=0$), quantum wells with widths $w=30$ and 50 nm, and HJ. The symbol $\rho$ denotes the electron density. The solid lines for quantum wells and heterojunction display a “hill” shape, where, roughly speaking, LL mixing correction dominates on the left of the hill (at small $\rho$) and finite width correction on the right (at large $\rho$).

Figure 2

Comparison between experimental values (stars) and theoretical DMC values (empty circles) of ${\alpha }_Z^{\mathrm{crit}}=E_Z^{\mathrm{crit}}/(e^2/\epsilon \ell )$ for a $w=65\mathrm{nm}$ quantum well (blue) from Liu et al. [40], and heterojunctions (red) from Engel et al. [30] and Kang et al. [32]. (For the experiment of Kang et al., we estimate the value of the Landé factor $g_0$ by assuming that it changes linearly and passes through zero at a pressure of roughly 18 Kbar [62].) The filling factors $\nu =n/(2n+1)$ are shown on top and $1/n$ at the bottom. The black circles show the results obtained from ED without including any LL mixing or finite-width corrections [41] (these do not involve the DMC calculation). The results for the 65 nm quantum well are shifted down by 0.005 for ease of depiction. The dashed lines are a guide to the eye. For the heterojunction, some other experimental values (theoretical predictions) of ${\alpha }_Z^{\mathrm{crit}}$ are 0.0109 [0.0076(4)] [29] and 0.0078 [0.0065(4)] [30] at $\nu =2/3$, and 0.0081[0.0080(20)] [30] at $\nu =3/5$; these are not shown on the figure to avoid clutter.

Figure 3

Theoretical critical Zeeman energies for the $w=0$ model as a function of the LL mixing parameter $\kappa$ obtained from the DMC method for $\nu =4/3$ (green circle), $2/3$ (blue square), $4/9$ (magenta downward triangle), $3/7$ (black upward triangle), and $2/5$ (red diamond). For the fractions $n/(2n+1)$, the wave functions of Eq. (2) are used to fix the phase. For $\nu =2/3$ and $\nu =4/3$, the exact Coulomb state in the LLL is used to fix the phase of the wave function. The solid lines are an approximate guide to the eye. The filled symbols indicate the experimental data from heterojunction samples at $\nu =2/3$ (light blue) and $4/3$ (green) taken from Eisenstein et al. [29] (circle), Engel et al. [30] (diamond), and Du et al. [31] (rightward triangle).