Weiss oscillations and particle-hole symmetry at the half-filled Landau level

Particle-hole symmetry in the lowest Landau level of the two-dimensional electron gas requires the electrical Hall conductivity to equal $\pm e^2/2h$ at half filling. We study the consequences of weakly broken particle-hole symmetry for magnetoresistance oscillations about half filling in the presence of an applied periodic one-dimensional electrostatic potential using the Dirac composite fermion theory proposed by Son [Son, Phys. Rev. X 5, 031027 (2015)]. At fixed electron density, the oscillation minima are asymmetrically biased towards higher magnetic fields, while at fixed magnetic field the oscillations occur symmetrically as the electron density is varied about half filling. We find an approximate “sum rule” obeyed for all pairs of oscillation minima that can be tested in experiment. The locations of the magnetoresistance oscillation minima for the composite fermion theory of Halperin, Lee, and Read (HLR) and its particle-hole conjugate agree exactly. Within the current experimental resolution, the locations of the oscillation minima produced by the Dirac composite fermion coincide with those of HLR. These results may indicate that all three composite fermion theories describe the same long-wavelength physics.

Figure 1

${\sigma }_{yy}^{\psi }$ for the Dirac composite fermions as numerically calculated using the general formula of Eq. (2.24). The $x$ axis is $B_{1/2}/b$ where $b$ is the effective magnetic field $b=B_{1/2}-B$ with fixed electron density $n_e=B_{1/2}/2{\mathrm{\Phi }}_0$. $B_d/B_{1/2}=0.001$ where $B_d=\hslash /e{d}^2$ is the magnetic field associated with the modulation period $d$. Two temperatures are shown: $k_{B}T=0.0008\sqrt{2B_{1/2}}$ (low temperature) and $k_{B}T=0.02\sqrt{2B_{1/2}}$ (high temperature). At low temperatures, two types of oscillations are apparent. The longer period oscillations are the Weiss oscillations while at high effective magnetic fields (small $1/b$) the Weiss oscillations are superimposed with shorter period Shubnikov–de Haas-type oscillations due to Landau levels crossing the chemical potential. At higher temperatures, the Weiss oscillations remain while the Shubnikov–de Haas-type oscillations are strongly suppressed. The inset shows the small $B_{1/2}/b$ region more clearly. For reference, the first four SdH-type peaks are identified with the integer filling fractions of the Dirac composite fermion Landau levels.

Figure 2

Oscillations in ${\rho }_{xx}$ for electrons near half filling as predicted by the Dirac composite fermion theory when the electron density $n_e$ is fixed and the external magnetic field is varied. Associating a magnetic field $B_d=\hslash /e{d}^2$ with the modulation period $d$, the result above corresponds to the choice: $B_d/B_{1/2}=0.001$, $v_F=1$, and $k_{B}T=0.06\sqrt{2B_{1/2}}$. The vertical dashed bars correspond to the positions of the minima found in recent experiments [20, 21].

Figure 3

Oscillations in ${\rho }_{xx}$ for electrons near half filling as predicted by the Dirac composite fermion theory when the external magnetic field $B$ is fixed and the electron density $n_e$ is varied about its half-filling value of $n_{e,1/2}=B_{1/2}/4\pi$. The result above corresponds to the choice: $B_d/B_{1/2}=0.001$, $v_F=1$, and $k_{B}T=0.06\sqrt{2B_{1/2}}$. The vertical dashed bars correspond to the positions of the minima found in recent experiments [20, 21].