Anomalously Low Magnetoroton Energies of the Unconventional Fractional Quantum Hall States of Composite Fermions

We show a generic formation of the primary magnetorotons in the collective modes of the observed “unconventional” fractional quantum Hall effect states of the composite fermions at the filling factors $4/11$, $4/13$, $5/13$, $5/17$, and $3/8$ at very low wave vectors with anomalously low energies which do not have any analog to the conventional fractional quantum Hall states. Rather slow decay of the oscillations of the pair-correlation functions in these states is responsible for the low-energy magnetorotons. This is a manifestation of the distinct topology predicted previously for these fractional quantum Hall effect states. Experimental consequences of our theory are also discussed.

Figure 1

Calculated pair-correlation functions $g(r)$ for finite systems and their thermodynamic extrapolation in the damped-oscillatory form $g(r)=1+A(r/l{)}^{-\alpha }\sin (\beta r/l-\gamma )$ used earlier [28], with $A,\alpha$, $\beta$, and $\gamma$ being numerical constants, for (a) $\nu =4/11$, (b) $\nu =5/13$, (c) $\nu =3/8$, and (d) $\nu =4/13$ and $5/17$. All of these are zoomed for showing oscillations in $g(r)$. Inset (c) shows $g(r)$ for $\nu =3/8$ without zooming. Insets of (a), (b), and (d) show $g(r)$ for $\nu =1/3$, $2/5$, and $2/7$, respectively.

Figure 2

LLL-projected static structure factor $\overline{S}(k)$ for (a) $\nu =4/11$, (b) $\nu =5/13$, (c) $\nu =3/8$, and (d) $\nu =4/13$ and $5/17$. $\overline{S}(k)$ for neighboring Jain states at $\nu =1/3$, $2/5$, and $2/7$ are shown in the insets of (a), (b), and (d), respectively.

Figure 3

Dispersion of the GMP mode $\mathrm{\Delta }(k)$ in the unit of $e^2/\epsilon l$ for (a) $\nu =4/11$, (b) $\nu =5/13$, (c) $\nu =3/8$, and (d) $\nu =4/13$ and $5/17$. $\mathrm{\Delta }(k)$ for neighboring Jain states at $\nu =1/3$, $2/5$, and $2/7$ are shown as the insets of (a), (b), and (d), respectively. While there is only one magnetoroton mode at $\nu =1/3$ and two magnetoroton modes at $\nu =2/5$ and $2/7$, all the other states in the range $2/5>\nu >2/7$ have two prominent magnetorotons and some not so prominent magnetorotons.