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considerations are left aside, the AB (or BA) configura- tion represents the simplest case of a random ensemble of H atoms which, in average, will equally populate the two graphene sublattices. 

In Fig. 3(a) we plot the LSDA total energy for H atoms in the middle of the AGNR as a function of d . The S = 0 state is always the ground state. This implies antiferromagnetic coupling, except at short distances for which the local magnetization, quantified through Σ = √ ∑ $_{i}$〈 m$_{i}$ 〉 $^{2}$, vanishes altoghether [see Fig. 4(c)]. The quenching of the magnetization is easily understood in terms of the formation of a spin singlet $^{27}$. At the minimum distance for which Σ ̸ = 0, the energy differ- ence presents a maximum and decays exponentially for larger d [see inset in Fig. 3(a)]. As d → ∞ the spin clouds do not interact anymore and both F and AF so- lutions tend to have the same energy. In summary, for any distance between H atoms the ground state presents S = 0, following Lieb’s theorem $^{25}$, but the overall spin texture strongly depends on their mutual distance. We note that it also depends on the ordering (AB or BA) of the H atoms. Whereas in bulk the spin cloud associ- ated to the H atom would be invariant under rotations of 120 degrees, in a nanoribbon there is a preferential direction along the ribbon axis which is different for H atoms located on A and B sublattices [see Fig. 1(b-c)]. We refer to the preferential direction as the head and the tail to the opposite one. Thus, a head-to-head coupling (AB) is expected to be much stronger than a tail-to-tail coupling (BA). The magnetization clouds are shown for an AB (or head-to-head) case in Fig. 4(a-b). It is easy to appreciate the strong directionality just alluded to. As a consequence, when reversing the ordering of the H atoms to a BA (or tail-to-tail) configuration [see Fig. 1(c)], these do not couple magnetically at any distance, except when in very close proximity for which Σ = 0. All these results are similar to the ones reported using the one-orbital mean-field Hubbard model $^{27}$. 

In Fig. 3(b) we present the LSDA total energy of the F and AF states when the H atoms are placed near one of the edges. Again the ground state is the AF state for any distance. The proximity to the edge decreases the localization length of the spin texture, increasing Σ [see Fig. 4(d-e)], and thereby, decreasing the critical distance below which the magnetization disappears [see Fig. 4(f)]. The perturbation of the edge modifies the spin texture around each atom and, contrary to the previous case, when reversing the ordering to a tail-to-tail configura- tion, the H atoms couple magnetically in a finite range of distances (see below). All energies, including those de- picted in Fig. 3(a), are referred to E$_{0}$ , the lowest energy solution for H atoms close to the edge, corresponding to the smallest distance. Note that for the same distance between H atoms the total energy is always smaller when these are closer to the edge, which reflects that the bind- ing energy is higher there by approximately 1 eV $^{28}$. 

Finally we present in Fig. 3(c) the case where the H atoms are placed on opposite edges for an N = 13 

<img src='content_image/429398.jpg'>

semiconducting AGNR. Placing the atoms on different edges allows us now to explore the energetics of differ- ent magnetic coupling orientations (tail-to-tail for d > 0 and head-to-head for d < 0) at a reasonable computa- tional cost ( d is now the longitudinal distance between H atoms). As expected, due to the strong anisotropy of the spin texture, the magnetic coupling strongly de- pends on the ordering of the atoms and not only on their mutual distance. While for d < 0 the energy difference between the AF and F states is large, for d > 0 this dif- ference decreases substantially and does not depend too much on d for the values considered. This asymmetry is easily understood since for d > 0( d < 0) the spin tex- tures approximately couple in a tail-to-tail(head-to-head) manner. As shown below, this may have important ex- perimental consequences. 

We now turn our discussion to the implications these results may have on the electrical transport. Under the influence of a magnetic field, the hydrogenated AGNR behaves like a diluted paramagnetic semiconductor 29 for small concentrations of H. At large concentrations, when the spin density is zero everywhere, the influence of the field can only give rise to a minor diamagnetic response. At intermediate concentrations, where the magnetization clouds induced by the H atoms interact with each other, one can switch from the AF to the F state by apply- ing a sufficiently strong magnetic field. In analogy with the H$_{2}$ molecule, where switching from the singlet to the triplet state modifies the orbital part of the wavefunc- 

## B. Magnetoresistance 

