Abelian Chern-Simons theory for the fractional quantum Hall effect in graphene

We develop a theory for the pseudorelativistic fractional quantum Hall effect in graphene, which is based on a multicomponent Abelian Chern-Simons theory in the fermionic functional integral approach. Calculations are performed in the Keldysh formalism, directly giving access to real-time correlation functions at finite temperature. We obtain an exact effective action for the Chern-Simons gauge fields, which is expanded to second order in the gauge field fluctuations around the mean-field solution. The one-loop fermionic polarization tensor as well as the electromagnetic response tensor in random phase approximation are derived, from which we obtain the Hall conductivities for various FQH states, lying symmetrically around charge neutrality.

Figure 1

Schwinger-Keldysh closed time contour in the complex time plane with forward (${\mathcal{C}}_{+}$) and backward time branch (${\mathcal{C}}_{-}$). Here, $t_0$ is a reference time, where an initial density matrix enters the theory. Since we are only interested in the system's linear response properties close to thermal equilibrium, we send the reference time to the remote past ($t_0=-\infty$) outright.

Figure 2

(a) Charge carrier density at constant chemical potential as a function of the effective magnetic field $B_{\mathrm{eff}}^{\alpha }$ at $T=10$ K (blue). The straight dotted lines (orange) indicate the first few Landau levels. At vanishing magnetic field, the charge carrier density scales quadratically with the chemical potential [5, 10]. (Without loss of generality the sign of the chemical is assumed to be positive). Upon increasing (the absolute value of) the effective magnetic field $B_{\mathrm{eff}}^{\alpha }$—while keeping the chemical potential fixed—the carrier density shows oscillations in the regime ${\omega }_{\mathrm{c}}^{\alpha }<{\mu }_{\alpha }$, whereas for ${\omega }_{\mathrm{c}}^{\alpha }\gg {\mu }_{\alpha }$ it grows linearly as a function of the magnetic field. This behavior is readily explained by the formation of Landau levels, and the dependence of their degeneracy and relative energetic separation on the magnetic field. (b) Filling fraction per flavor ${\nu }_{\alpha }$ at constant effective magnetic field $B_{\mathrm{eff}}^{\alpha }=15\mathrm{T}$ and temperature $T=10\mathrm{K}$ as a function of the chemical potential. The plateaus occur at half-integer filling fractions ${\nu }_{\alpha }=\pm (n_{\alpha }+1/2)$. The transitions between the plateaus are smeared out due to temperature. (c) Derivative of ${\nu }_{\alpha }$ with respect to the chemical potential as a measure for temperature-induced Landau level broadening at $B_{\mathrm{eff}}^{\alpha }=15\mathrm{T}$ for the temperatures $T=5,25,50,\text{and}75\mathrm{K}$. Increasing the temperature clearly leads to a broadening of the discrete energy levels. Since the relativistic Landau levels are not equidistant in energy space, the level broadening causes the Landau levels to overlap significantly away from the charge neutrality point. The Landau level located directly at the charge neutrality point, however, remains well-defined up to rather large temperatures.

Figure 3

Total filling fraction ${\nu }_G=\frac{4\nu }{2k·4\nu +1}$ as a function of the chemical potential $\mu$ at $B_{\mathrm{eff}}=15\mathrm{T}$ and $T=10\mathrm{K}$ for $k=+1$ (blue) and $k=-1$ (orange). The finite temperature smears out the transitions from one Hall plateau to another, similar to the noninteracting case, cf. Fig. 2. Note that the plateaus occur in pairs, lying symmetrically around the charge neutrality point ${\nu }_G=0$. In this simple case, one can construct a manifestly particle-hole symmetric filling fraction by considering the regimes $\mu <0$ and $\mu >0$ and flip the sign of $k$ at $\mu =0$, see Fig. 4, which yields the two branches $|{\nu }_G|<|1/2k|$ and $|{\nu }_G|>|1/2k|$.

Figure 4

Particle-hole symmetric total filling fractions for ${\nu }_G=\frac{({\nu }_{\uparrow }+{\nu }_{\downarrow })}{2k·({\nu }_{\uparrow }+{\nu }_{\downarrow })+1}$, with ${\nu }_{\uparrow }={\nu }_1+{\nu }_2,{\nu }_{\downarrow }={\nu }_3+{\nu }_4$, as a function of the chemical potential $\mu$ at $B_{\mathrm{eff}}=15\mathrm{T}$ and $T=10\mathrm{K}$. (a) Zero Zeeman splitting, implying ${\nu }_{\uparrow }={\nu }_{\downarrow }=2\nu$, for $\left|k\right|=1,...,4$. (b) Finite spin Zeeman term with Zeeman energies $E_{\mathrm{Z}}=0.0,0.1,...,0.4\times \hslash {\omega }_{\mathrm{c}}^{\mathrm{eff}}$ for $\left|k\right|=1$. The finite Zeeman term leads to the formation of new plateaus, with the ${\nu }_G=0$ plateau being the most dominant one.