Magnetism of zigzag graphene edges was predicted by M. Fujita (1996) et al. long before graphene was isolated. Since that time about 2000 theoretical papers were published on this topic.

The number of experimental works is 100 times less. They confirm the existence of peculiar erge states (Wang 2008, Tao 2011). The direct evidence for edge magnetism is still absent. The claims are based on indirect evidence (STM spectroscopy Magda 2014; transport Baringhaus 2014,) or on artifacts, e.g. measurements on unzipped carbon nanotubes with ferromagnetic catalyst. For a review see Lado, SyMet 210, 56 (2016).

There are two main reasons for the absence of experimental evidence of zigzag edge magnetism. First, zigzag edge is very reactive and immediately becomes contaminated; moreover, zigzag edges are energetically unstable and they undergo reconstruction via the so-called Stone-Wales transformations. Second, there is lack of understanding what shall be measured, and the puirsuit of ferromagnetic loops generates misleading publications.

From the very beginning, the discoverers of peculiar edge states created the correct magnetic model of graphene zigzag nanoribbon: spin ladder model. Among the theoretical papers, there are a few treatments beyond the mean-field approximation, in which quantum fluctuation is fully taken into account. These works are cited below.

## Ground state of graphite

**Spin Wave Mode of Edge-Localized Magnetic States in Nanographite Zigzag Ribbons:** Based on this spin structure, the effective spin ladder model is proposed and the existence of the spin gap is claimed. (K. Wakabayashi, M. Sigrist and M. Fujita: Spin Wave Mode of Edge-Localized Magnetic States in Nanographite Zigzag Ribbons J. Phys. Soc. Jpn. 67 2089-2093 (1998).

**Electron-Electron Interaction in Nanographite Ribbons** (K. Nakada, M. Igami, M. Fujita; J. of the Phys. Soc. of Japan, 67, 2388 (1998)

**Spin- and charge-polarized states in nanographene ribbons with zigzag edges**. Atsushi Yamashiro, Yukihiro Shimoi, Kikuo Harigaya, Katsunori Wakabayashi, Physical Review B, 2003.

**Spin Excitation in Nano-Graphite Ribbons with Zigzag Edges**. This work condiders one of funsamental problems of two-leg and three legladders. It concluds that spin excitation has a gap when the number of the zigzag lines is even (Hideo Yoshioka, Spin Excitation in Nano-Graphite Ribbons with Zigzag Edges. J. Phys. Soc. Jpn 72 (9), 2145 – 2148 (2003).

**Ground-state properties of nanographite systems with zigzag edges. **Hikihara ** **investigated the ground state and the excitation around it of the zigzag NGR; the ground state is a spin-singlet Mott insulator with finite charge and spin gap (Ground-state properties of nanographite systems with zigzag edge,s T. Hikihara, X. Hu, H.H. Lin, C.Y. Mou, PRB 68 035432 (2003).

**Ground state of graphite ribbons with zigzag** **edges.** There are two possible phases: the singlet superconducting (SS) phase and the excitonic insulator (EI). All excitations in the EI phase, especially the spin excitations, are gapped. For the EI and spin liquid, Temperature dependence of the uniform magnetic susceptibility, at low temperature, it will exhibit the activated behavior in the SL due to the spin gap, whereas the Curie-like behavior is expected for the gapless edge states. (Y.-L. Lee and Y.-W. Lee, Phys. Rev. B 66, 245402 (2002)

**Parity law of the singlet-triplet gap in graphitic ribbons. **Hajj transferred to graphene the parity law of the singlet-triplet gap established for square ladders, gapped for even number of legs, gapless for odd number of legs. (M. Al Hajj, F. Alet, S. Capponi, M.B. Lepetit, J.-P. Malrieu, and S. Todo: Eur. Phys. J. B 51 (2006) 51).

## Realization of zigzag edge magnetism on zigzag interfaces

Zigzag nanoridge is a chain of sp3 hybridized atoms running in zigzag direction. To evaluate the magnetic properties of a nanoridge while taking quantum and thermal fluctuations into consideration, we used a spin-ladder model; see Scheme. The sites of the ladder are occupied by spin-½ units, with N_{s} spins on each of the two legs. The number of spins is a parameter of the chain, and, as a rough estimate in agreement with the DFT calculations, is a fraction of the number of fluorine atoms in the chain. In the spin-ladder model, the intra-leg spin interaction J provides the ferromagnetic ordering on the edges. The inter-leg interaction J_{AF} is antiferromagnetic.

We solve the model in the limit of a strong intra-leg ferromagnetic coupling J and temperatures well below J. In this limit, spin-wave excitations within the short ferromagnetic legs can be ignored, such that the spins remain aligned within each of the legs. In this limit, the only relevant degree of freedom is the orientation of the collective moments.

The ground state corresponds to the singlet state S =0 which is separated by the spin gap D = J_{AF} /N_{s} from the first excited state S = 1. The spin gap vanishes in the limit N_{s} à ¥ so that the gap is an attribute of a nanoscale system consisting of only a few spins.

At low temperatures, the nanoridge exists predominantly in the singlet ground state *S = 0*, which is magnetically inert. The magnetically active states, , require thermal activation across the spin gap, and, for this reason, the magnetic susceptibility exhibits activated exponential behaviour, as observed in the experiment using fluorinated graphene planes.