We investigate the low-lying excitation spectrum and ground-state properties of narrow graphene nanoribbons with zigzag edge configurations. Nanoribbons of comparable widths have been synthesized very recently [P. Ruffieux et al., Nature (London) 531, 489 (2016)], and their descriptions require more sophisticated methods since in this regime conventional methods, like mean-field or density-functional theory with local-density approximation, fail to capture the enhanced quantum fluctuations. Using the unbiased density-matrix renormalization-group algorithm, we calculate the charge gaps with high accuracy for different widths and interaction strengths and compare them with mean-field results. It turns out that the gaps are much smaller in the former case due to the proper treatment of quantum fluctuations. Applying the elements of quantum information theory, we also reveal the entanglement structure inside a ribbon and examine the spectrum of subsystem density matrices to understand the origin of entanglement. We examine the possibility of magnetic ordering and the effect of magnetic field. Our findings are relevant for understanding the gap values in different recent experiments and the deviations between them.
Using the density-matrix renormalization group method and quantum Monte Carlo simulation, we studied numerically the energy spectrum and thermodynamics of the quantum Heisenberg spin model for narrow graphene nanoribbons and their derivatives with periodically embedded heteroatoms. For several nanoribbon structures we found macroscopic ground state spin, gapless lowest excitation spectra and intermediate magnetization plateaus at low temperatures. We also studied the lowest energy states of frustrated systems formed by triangular graphitic clusters connected by bridged ions of transition metals. On the base of many-body perturbation theory and the exact diagonalization method, we showed the possibility of spin switching for this model due to the change the corresponding coupling parameters.
A $\ensuremath{\pi}$-electron network in nanographite systems with zigzag edges exhibits strongly localized edge states, which are expected to have peculiar properties. We study effects of electron-electron interactions on ground-state properties of zigzag nanographite ribbons and open-ended zigzag nanotubes by means of the weak-coupling renormalization group and the density-matrix renormalization-group method. It is shown that the ground state is a spin-singlet Mott insulator with finite charge and spin gaps. We also find that the edge states are robust against the electronic correlations, resulting in edge-effective spins that can flip almost freely. The schematic picture for the low-energy physics of the systems is discussed.
We examine the magnetic correlations in quantum spin models that were derived recently as effective low-energy theories for electronic correlation effects on the edge states of graphene nanoribbons. For this purpose, we employ quantum Monte Carlo simulations to access the large-distance properties, accounting for quantum fluctuations beyond mean-field-theory approaches to edge magnetism. For certain chiral nanoribbons, antiferromagnetic interedge couplings were previously found to induce a gapped quantum disordered ground state of the effective spin model. We find that the extended nature of the intraedge couplings in the effective spin model for zigzag nanoribbons leads to a quantum phase transition at a large, finite value of the interedge coupling. This quantum critical point separates the quantum disordered region from a gapless phase of stable edge magnetism at weak intraedge coupling, which includes the ground states of spin-ladder models for wide zigzag nanoribbons. To study the quantum critical behavior, the effective spin model can be related to a model of two antiferromagnetically coupled Haldane-Shastry spin-half chains with long-ranged ferromagnetic intrachain couplings. The results for the critical exponents are compared also to several recent renormalization-group calculations for related long-ranged interacting quantum systems.
We present a completely unbiased and controlled numerical method to solve quantum impurity problems in
d-dimensional lattices. This approach is based on a canonical transformation, of the Lanczos form, where the complete lattice Hamiltonian is exactly mapped onto an equivalent one-dimensional system, in the same spirit as Wilson's numerical renormalization, and Haydock's recursion method. We introduce many-body interactions in the form of a Kondo or Anderson impurity and we solve the low-dimensional problem using the density matrix renormalization group. The technique is particularly suited to study systems that are inhomogeneous, and/or have a boundary. The resulting dimensional reduction translates into a reduction of the scaling of the entanglement entropy by a factor ${L}^{d\ensuremath{-}1}$, where
L is the linear dimension of the original
d-dimensional lattice. This allows one to calculate the ground state of a magnetic impurity attached to an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice and an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ cubic lattice with
L up to 140 sites. We also study the localized edge states in graphene nanoribbons by attaching a magnetic impurity to the edge or the center of the system. For armchair metallic nanoribbons we find a slow decay of the spin correlations as a consequence of the delocalized metallic states. In the case of zigzag ribbons, the decay of the spin correlations depends on the position of the impurity. If the impurity is situated in the bulk of the ribbon, the decay is slow as in the metallic case. On the other hand, if the adatom is attached to the edge, the decay is fast, within few sites of the impurity, as a consequence of the localized edge states, and the short correlation length. The mapping can be combined with ab initio band structure calculations to model the system, and to understand correlation effects in quantum impurity problems starting from first principles.
Based on the effective spin-ladder model, we compute the negativity and geometric quantum discord (GQD) between two edge spins in zigzag graphene nanoribbons (ZGNRs) thermalized with a reservoir at temperature T (canonical ensemble). The results show that the entanglement only exists in antiferromagnetically coupled spin pairs, and completely vanishes between any two spins at the same ribbon edge. Intriguingly, the dominant entanglement always occurs in the nearest inter-edge coupled spin pairs, which is very robust even at room temperature. By contrast, almost all edge spin pairs can hold non-zero GQD, which strongly depends on the ribbon width. To get further insight into the properties of entanglement, the multiparticle entanglement (ME) shared between three spins is studied. The results show that ME always exists among spins located at different edges, and it is more robust than the corresponding two-spin entanglement in against the temperature. These remarkable properties make ZGNRs very promising for possible applications in quantum information processing.