More realistic Hamiltonians for the fractional quantum Hall regime in GaAs and graphene

We construct an effective Hamiltonian for electrons in the fractional quantum Hall regime for GaAs and graphene that takes into account Landau level mixing (for both GaAs and graphene) and subband mixing (for GaAs, due to the nonzero width of the quantum well). This mixing has the important qualitative effect of breaking particle-hole symmetry as well as renormalizing the strength of the interparticle interactions. Both effects could have important consequences for the prospect that the fractional quantum Hall effect at $\nu =\frac{5}{2}$ is described by states that support non-Abelian excitations such as the Moore-Read Pfaffian or anti-Pfaffian states. For GaAs, Landau level and subband mixing break particle-hole symmetry in all Landau levels and subband mixing, due to finite thickness, causes additional short-distance softening of the Coulomb interaction, further renormalizing the Hamiltonian; additionally, the Landau level and subband energy spacings are comparable so it is crucial to consider both effects simultaneously. We find that in graphene, Landau level mixing only breaks particle-hole symmetry outside of the lowest Landau level ($N\ne 0$). Landau level mixing is likely to be especially important in graphene since the Landau level mixing parameter is independent of the external magnetic field and is of order one. Our realistic Hamiltonians will serve as starting points for future numerical studies.

Figure 1

The bare interaction Feynman diagram. The Coulomb interaction (augmented or not) can not switch the spin so the first term on the right-hand side has ${\delta }^{\alpha {\alpha }^{\prime }}{\delta }^{\beta \beta }$ and the second term has ${\delta }^{\alpha {\beta }^{\prime }}{\delta }^{\beta {\alpha }^{\prime }}$. The labels 1, 2, 3, and 4 correspond to $1=\{s_1,n_1,m_1\}$, etc.

Figure 2

(a) The ZS Feynman diagram: $V_{x3,x^{\prime }1}^{\gamma {\alpha }^{\prime },{\gamma }^{\prime }\alpha }{V}_{4x^{\prime },2x}^{{\beta }^{\prime }{\gamma }^{\prime },\beta \gamma }$. (b) The ZS${}^{\prime }$ Feynman diagram: $V_{x4,x^{\prime }1}^{\gamma {\beta }^{\prime },{\gamma }^{\prime }\alpha }{V}_{3x^{\prime },2x}^{{\alpha }^{\prime }{\gamma }^{\prime },\beta \gamma }$. (c) The BCS Feynman diagram: $V_{43,x{x}^{\prime }}^{{\beta }^{\prime }{\alpha }^{\prime },\gamma {\gamma }^{\prime }}{V}_{x{x}^{\prime },21}^{\gamma {\gamma }^{\prime },\beta \alpha }$.

Figure 3

One of the nine three-body diagrams.

Figure 4

The Landau level mixing induced corrections to the two-body pseudopotentials for $M=1$–6 in the lowest ($N=0$, left figures) and second ($N=1$, right figures) Landau levels. The top left plot in both panels shows all $\delta V_M^{\left(2\right)}(N,d/{\ell }_0)$ versus $d/{\ell }_0$ on the same plot for an easier comparison. Note that for the lowest Landau level ($N=0)$ the $M=0$ plot is on a different vertical scale than all the others since its value is so much larger. For the $N=0$ Landau level, the corrections are less than 10$\%$ (for all $d/{\ell }_0$) of the bare pseudopotential value, discounting $M=0$ which is nearly 30$\%$ (for $d/{\ell }_0=0$) the bare pseudopotential. For $N=1$, the corrections are much larger. For $M=0$, the correction is nearly 60$\%$ and remains near 50$\%$ for $M<3$. Of course, for both Landau levels the corrections are mitigated by finite width (increasing $d/{\ell }_0$). All energies are given in units of $e^2/\epsilon {\ell }_0$.

Figure 5

$\delta V_M^{\left(2\right)}(N,d/{\ell }_0)$ as a function of $M$ for $d/{\ell }_0=0$, 1, 2, 3 and 4. The left panel corresponds to the lowest Landau level ($N=0$) while the right panel corresponds to the first excited Landau level ($N=1$). As in the case of the bare pseudopotentials, the corrections to the two-body pseudopotentials decrease with increasing $M$. All energies are given in units of $e^2/\epsilon {\ell }_0$.

Figure 6

The three-body pseudopotentials $V_M^{\left(3\right)}(N,d/{\ell }_0)$ as a function of $d/{\ell }_0$ for the lowest Landau level (left panel) and the first excited Landau level (right panel). Note that these are numerically small compared to the bare two-body pseudopotentials. All energies are given in units of $e^2/\epsilon {\ell }_0$.

Figure 7

$V_M^{\left(3\right)}(N,d/{\ell }_0)$ as a function of $M$ for $d/{\ell }_0=0$, 0.5, 1–6 for the lowest Landau level (left panel) and the first excited Landau level (right panel). The magnitude decreases with $M$ except for the nontrivial oscillatory behavior for $M\ge 5$. All energies are given in units of $e^2/\epsilon {\ell }_0$.

Figure 8

The two-body and three-body pseudopotentials $V_M^{\left(2\right)}(N,d/{\ell }_0)$ and $V_M^{\left(3\right)}(N,d/{\ell }_0)$ as functions of $M$ for $\kappa =0$–3 for $N=0$ (top panels), $N=1$ (bottom panels) for $d/{\ell }_0=0$ (left), 2 (middle), and 3 (right). Obviously, $V_M^{\left(2\right)}(N,d/{\ell }_0)$ is a sum of the bare two-body term plus $\kappa$ times the two two-body corrections. The two-body pseudopotentials are color coded such that $\kappa =0$ corresponds to red and $\kappa =3$ corresponds to yellow, i.e., the color coding is “hot.” The three-body terms are color coded “cold,” i.e., $\kappa =0$ is green and the color changes continuously to blue for $\kappa =3$. Generally, increasing $d/{\ell }_0$ and $\kappa$ reduces the two-body pseudopotentials compared to the bare values ($\kappa =0$). The three-body pseudopotentials are directly proportional to $\kappa$ so increase in magnitude for increasing $\kappa$, however, increasing $d/{\ell }_0$ mitigates this effect by decreasing the coefficients themselves, as expected. Please note that any two-body corrections for $M\ge 10$ are expected to be small and unimportant. From inspection of this figure, one would expect that Landau level mixing is likely to be unimportant for the FQHE in the $N=0$ Landau level for spin-polarized electrons. For the $N=1$ Landau level, however, it is not clear what effect these changes to the pseudopotentials will have on the physics and the answer will have to await future exact diagonalization studies. Of course, even in the $N=0$ Landau level, the effect of the three-body terms cold produce nontrivial effects. All energies are given in units of $e^2/\epsilon {\ell }_0$.

Figure 9

${\tilde{V}}_M^{\left(3\right)}\left(N\right)$ as a function of $M$ for $N=0$–6. Note that for $N=0$ all three-body pseudopotentials exactly vanish. All energies are given in units of $e^2/\epsilon {\ell }_0$.

Figure 10

The two-body pseudopotential corrections $\delta {\tilde{V}}_M^{\left(2\right)}\left(N\right)$ for $N=0$, 1, and 2 for graphene as a function of $M$. It is clear from this figure that the pseudopotential corrections increase for increasing Landau level $N$, reflecting the conjecture that the effective Landau level mixing parameter is actually $\propto \tilde{\kappa }\sqrt{N}$ and not simply $\tilde{\kappa }$. All energies are given in units of $e^2/\epsilon {\ell }_0$.

Figure 11

Both the two- and three-body pseudopotentials as functions of $M$ for $\tilde{\kappa }=0–3$ for graphene, in the $N=0$, 1, and 2 Landau levels. For the lowest Landau level ($N=0$) there is no three-body term (it exactly vanishes) and the two-body corrections are modest. Hence, one would expect, from merely studying the pseudopotentials, that the FQHE in $N=0$ Landau level for graphene would be nearly identical to that of GaAs. However, for $N\ne 0$, we observe significant corrections to the two-body pseudopotentials and significant three-body terms. It is not at all clear what these effects will have on the FQHE. Naively, one is tempted to suggest that the FQHE will not be observed in $N\ne 0$ Landau levels in graphene, or if it is observed, it will be of an exotic variety. All energies are given in units of $e^2/\epsilon {\ell }_0$.