Effective low-energy Hamiltonians for interacting nanostructures

We present a functional renormalization group (fRG) treatment of trigonal graphene nanodisks and composites thereof, modeled by finite-size Hubbard-like Hamiltonians with honeycomb lattice structure. At half filling, the noninteracting spectrum of these structures contains a certain number of half-filled states at the Fermi level. For the case of trigonal nanodisks, including interactions between these degenerate states was argued to lead to a large ground state spin with potential spintronics applications [M. Ezawa, Eur. Phys. J. B 67, 543 (2009)]. Here we perform a systematic fRG flow where the excited single-particle states are integrated out with a decreasing energy cutoff, yielding a renormalized low-energy Hamiltonian for the zero-energy states that includes effects of the excited levels. The numerical implementation corroborates the results obtained with a simpler Hartree-Fock treatment of the interaction effects within the zero-energy states only. In particular, for trigonal nanodisks the degeneracy of the one-particle-states with zero energy turns out to be protected against influences of the higher levels. As an explanation, we give a general argument that within this fRG scheme the zero-energy degeneracy remains unsplit under quite general conditions and for any size of the trigonal nanodisk. We also discuss a second class of nanostructures, bow-tie-shaped systems, where the zero-energy states are not protected.

Figure 1

Two different kinds of graphene-nanodisks. (a) Trigonal zigzag-nanodisks that can be characterized by the size parameter $N$. Here the zero-energy-degeneracy $\eta$ is equal to the sublattice imbalance $L_B-L_A=N$ and (b) bow-tie-shaped nanostructure with zero sublattice mismatch and $\eta =2$.

Figure 2

(Color online) Noninteracting energy spectra of the trigonal nanodisk for $N=1,2,\dots ,5$ and for the bow-tie-shaped nanodisk. The degeneracy of energy levels is marked by different colors.

Figure 3

(Color online) Flow of the coupling functions in the zero-energy-sector of trigonal nanodisks with $N=2,3,4$ down to a small scale around the zero-energy states. For the parameters of the Hamiltonian (1) we set $U=3t$ and $V_1=2t$. The kinks of the flowing coupling functions result from integrating out the discrete energy levels of ${\widehat{H}}_0$. In the end of the flow the coupling functions can be interpretated as matrix elements of an effective Hamiltonian for the zero-energy states.

Figure 4

(Color online) Flow of the one-particle energies (upper plot) and various coupling functions (lower plot) for the bow-tie-shaped nanostructure with $\eta =2$ for $U=3t$ and $V_1=2t$.