2 FIG. 1: (a) Armchair graphene nanoribbon of width N (where N is the number of dimer lines) with two H atoms (shown in green) adsorbed in the middle of the ribbon at a distance d from each other. The inset shows a detail of the adsorption ge- ometry. (b) Pictorical representation of the one-orbital tight- binding model where the presence of H atoms is simulated by removing sites in a head-to-head configuration. (c) Same as in (b), but for the opposite ordering (tail-to-tail). (d) Same as in (b), but for a finite semiconducting AGNR connected to metallic nanoribbons at the edges. FIG. 2: Pictorical view of the antiferromagnetic state (a) and the magnetic-field driven ferromagnetic state (b) where the magnetic moments localized around two hydrogen atoms are depicted by red arrows (the orientation of the arrows with respect to the graphene plane is arbitrary since spin-orbit coupling is neglected here). states can be substantial and translate into a magnetore- sistive response as large as 100% for distances between the H atoms of the order of few nanometers. Practical implications of these results are discussed in Sec. IV and a brief summary is presented in Sec. V ˆ n$_{iη}$ = ˆ c + $_{iη}$ˆ c$_{iη}$ are the number operators associated to each π orbital with spin η . These two different lev- els of approximation to the electronic structure have been shown to yield similar results for nanoribbons 24 and nanographenes 13 where a full passivation by H of the σ bonds is assumed. The bulk adsorption geometry of a H atom has been obtained by relaxing the C atom bonded to the H and the nearest C atoms until the characteristic sp 3 hybridization is obtained [see inset in Fig. 1(a)]. The bonding between a H atom and a C atom results in the effective removal of the π orbital from the low energy sector, so that the H adsorption is simulated by simply removing a site in the one-orbital mean-field Hubbard model [see Fig. 1(b- d)]. Our results, as shown below, provide further support to the use of the Hubbard model in graphene systems, as an alternative to the computationally more demand- ing LSDA and extend the range of applicability of Lieb’s theorem 25 to a wider set of situations. For the calculation of the electronic structure of hydro- genated AGNR’s we use both ab initio techniques within the local spin density approximation (LSDA), aided by the CRYSTAL03 package $^{23}$, and a one-orbital ( π ) first- neighbor tight-binding model where the electronic repul- sion is treated by means of a Hubbard-like interaction U in the mean field approximation: The first term represents the kinetic energy with first neighbors hopping t between π orbitals. The remain- ing terms account for the electronic interactions where To calculate the transport properties we use the standard Green’s function partitioning method as im- plemented, e.g., in the quantum transport package ALACANT $^{26}$. To this purpose, the infinite system is ˆ H = ∑ i,j t ˆ c † $_{i}$ˆ c$_{j}$ + U ∑ i (ˆ n$_{i}$$_{↑}$ 〈 ˆ n$_{i}$$_{↓}$ 〉 +ˆ n$_{i}$$_{↓}$ 〈 ˆ n$_{i}$$_{↑}$ 〉 ) − U ∑ i 〈 ˆ n$_{i}$$_{↓}$ 〉〈 ˆ n$_{i}$$_{↑}$ 〉 . (1) II. COMPUTATIONAL DETAILS