10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 0.0 0.2 0.4 0.6 0.8  B=0.0020545  B=0.00206  B=0.00207  B=0.0021  B=0  B=0.002  B=0.00204  B=0.00205 -2 -1 0 1 2 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 -2 -1 0 1 2 0.0 0.1 0.2 0.3 b a A P ↑ /A 0 A P ↓ /A 0 ω/T$_{K}$ ω/T K ω/T K J = 0 0.0 0.1 0.2 0.3 0.4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 0.0 0.2 0.4 0.6 0.8  B=0.001928  B=0.001929  B=0.001931  B=0.00194  B=0  B=0.00192  B=0.001925  B=0.001927 -0.4 -0.2 0.0 0.2 0.4 0.0 0.4 0.8 -0.4 -0.2 0.0 0.2 0.4 0.0 0.2 0.4 d c PSfrag replacem ents A P $_{↑}$/ A 0 A P $_{↓}$/ A 0 ω/T$_{K}$ J = 0 . 17 ω/T K ω/T K 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 0.0 0.1 0.2 0.3 0.4  B=0.00191621  B=0.001918  B=0.00193  B=0.002 0.00 0.05 0.10 0.15 0.20  B=0  B=0.0019  B=0.001914 -0.2 -0.1 0.0 0.1 0.2 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2 0.0 0.1 0.2 0.3 0.4 f e PSfrag replacem ents A P $_{↑}$/ A 0 A P $_{↓}$/ A 0 ω/T$_{K}$ J = 0 . 17875 ω/T K ω/T K 7 FIG. 7: (color online) The spectral function of the d -level operator in the parallel magnetic configuration for the asymmetric Anderson model in the presence of external magnetic field B applied along the z th direction for different values of exchange coupling constant J = 0 (a,b), J = 0 . 17 (c,d) and J = 0 . 17875 (e,f). The other parameters are the same as in Fig. 5. in the spectral density can be restored for both spin orien- tations by properly tuned external magnetic field, which happens for B = B$_{c}$ = 0 . 0020545, where B$_{c}$ (in the units of D) denotes the compensating field. This is in agree- ment with the result obtained earlier. 6,7 Similar behavior also appears for larger positive J , e.g. for J = 0 . 17 shown in Fig. 7(c) and (d). Now, the Kondo resonance becomes restored when the compensating field is B$_{c}$ = 0 . 001928. Note that the magnitude of magnetic field necessary for full restoration of the Kondo effect slightly decreases as J increases. The question which arises now is whether such a restoration by magnetic field is also possible for larger values of J . By fine-tuning in the parameter space of J and B , we have found that this is the case for J below a certain critical value J < ˜ J P c = 0 . 17875. Here ˜ J P c de- notes the critical value of J in the parallel configuration and in the presence of the compensating magnetic field. From numerical results (not shown here), follows that for J > ˜ J P c , the magnetic field can only partially restore the Kondo effect, leading to small side peaks in the spectral function, while the full Kondo peak at ω = 0 cannot be restored. One may now expect that for J = ˜ J P c , the mag- netic field should also restore the quantum critical state. Indeed, by fine-tuning in the parameter space we have found that the quantum critical state can be recovered for B$_{c}$ = 0 . 00191621. This situation is shown explicitly in Fig. 7(e) and (f). Thus, we have shown that in the parallel configuration the properly-tuned magnetic field can restore both the full Kondo effect for J < ˜ J P c as well as the quantum critical state for J = ˜ J P c . can note that the compensating field B$_{c}$ decreases with increasing the exchange coupling J . This is explicitly shown in Fig. 8, where we have calculated the depen- dence of B$_{c}$ on the exchange coupling J . For J < ˜ J P c , the Kondo resonance can be fully restored by applying compensating field B$_{c}$ . On the other hand, when J > ˜ J P c , the magnetic field cannot compensate the exchange field, so the notion of compensating field becomes meaningless. By comparing numerical curves presented in Fig. 7, one FIG. 8: The dependence of the compensating magnetic field B c on the exchange coupling constant J in the case of the parallel configuration and asymmetric Anderson model. The other parameters are the same as in Fig. 5. J and B c are in units of D = 1. 0.00 0.04 0.08 0.12 0.16 0.00190 0.00192 0.00194 0.00196 0.00198 0.00200 0.00202 0.00204 0.00206 B c J