where~s is the spin operator for the electrons at each site. Also in this model there is an antiferromagnetic phase at half-filling, which eventually becomes unstable towards a spin-density wave [157] (and references within). Figure 5.8 shows an example for a stable spin density wave forJ/W = 0.5,µ/W =−0.225, T ≈ 0, and periodicity N = 14. One should notice that the occupation and magnetization seems to oscillate with a period of N ≈ 3, which however did not stabilize the system.
The example given shows that such order is possible. Nevertheless, the method used for stabilizing such states cannot be recommended, as the numerical effort is very high. Additionally, such solutions seem to be rather unstable towards numerical errors, making it very hard to find them.
5.3 Comparison of DMRG and NRG
In principle the DMRG is an alternative to the NRG as impurity solver for the DMFT [115–118]. For the Bethe lattice it is possible to formulate the DMFT self-consistency equation as [70]
G−1(z) =t2G(z),
in which t is the hopping parameter in the Hubbard model, G(z) the impu-rity Green’s function andG the effective medium of the DMFT. Hence, for the Bethe lattice it is not necessary to calculate the self-energy for determining the effective medium. As it is simpler within the DMRG to calculate the impu-rity Green’s function than calculating the self-energy, I used this formulation of the self-consistency. In the DMRG one usually calculates G(ω+iη). Even though the self-consistency can be formulated for arbitraryη, for determining the hopping parameters from the effective medium, which are needed in the next DMRG calculation, one needs G−1(ω +i0). Therefore a deconvolution must be performed after every impurity calculation. The PMIT was analyzed within the DMFT and DMRG as impurity solver in several works [158–160].
They showed that it is possible to use DMRG as impurity solver for DMFT and discussed spectral features close to the Mott transition. I will here concen-trate on magnetic phases. Because the antiferromagnetic phase at half-filling is the most pronounced magnetic phase in the one-orbital Hubbard model, I will concentrate on this state.
Figure 5.9 shows the antiferromagnetic DMFT results for U/W = {0.5; 0.8;
1.0; 1.5} and half-filling. The solutions were calculated separately with the DMRG and the NRG. The DMRG was performed for a bandwidth ofW = 0.2 and a broadening η = 0.05. The shown DMRG results are already deconvo-luted. For the NRG calculations, I used the same bandwidth and the usual frequency dependent broadening with b = 0.8, as described in chapter 3.3.3.
ForU/W = 0.05 (upper left panel) the antiferromagnetic gap is too small to be
-1.5 -1 -0.5 0 0.5 1 1.5
Figure 5.9: Antiferromagnetic DMFT calculations performed with DMRG and NRG as impurity solver for different interaction strengthsU/W. The DMRG results are already deconvoluted. ForU/W = 0.5 the gap is to small to be resolved by the DMRG. Thus the DMRG failed to stabilize the antiferromagnetic phase.
resolved by the DMRG. The gap-width for this interaction strength, as calcu-lated by the NRG, is ∆AF ≈0.015. With the mentioned Lorentzian broadening this cannot be resolved and is too small even after deconvolution. Thus the solu-tion of the DMFT with DMRG for this interacsolu-tion is a half-filled paramagnetic metal. In contrast, the NRG can resolve this gap resulting in an antiferromag-netic insulator with strong van-Hove singularities at the gap edges. For stronger interactions DMFT with DMRG can stabilize the correct solution, meaning an antiferromagnetic state. If one looks carefully, one will notice that even if there is an antiferromagnetic solution for U/W = 0.8 and U/W = 1.0, DMRG fails to really open the gap. The DMRG results would actually predict an antiferro-magnetic metallic state for these intermediate interactions. The reason for this is again the broadening. The gap-width in both system has the same order of magnitude as the broadeningη. Only for stronger interactionsU/W >1.5 one can clearly see a gaped antiferromagnetic state in the DMRG calculations. The interaction strengths, at which the DMRG is able to open the gap and form an antiferromagnetic solution, depend on the ratio η/W. Using smaller broaden-ing will make it possible to better resolve the low frequency parts includbroaden-ing the antiferromagnetic gap. But using a smaller broadening η will, of course, result in a heavier numerical task.
Another big difference between the spectral functions of the NRG and the DMRG is the structure of the Hubbard bands. As the broadening in the NRG is large for large frequencies, the Hubbard bands do not show additional structures for U/W ≥ 0.8. In the NRG one can only see for U/W = 0.5 van-Hove
5.3 Comparison of DMRG and NRG 79
Figure 5.10: DMRG and NRG doped spectral function forU/W = 1 andT ≈0. The left panel shows the paramagnetic solution for µ/W = 0.1. The right panel shows antiferromagnetic solutions forµ/W= 0.3.
singularities and a shoulder in the Hubbard bands. The DMRG results include always two structures in each Hubbard band. There are theoretical results on additional structures in the antiferromagnetic state [153, 161–164]. These additional structures are created by virtual movement of electrons or holes in long paths. If the structures found here are related to this must be further investigated. It is possible that they are only artifacts of the interplay between broadening and deconvolution. All shown results in figure 5.9 were calculated using the same broadeningη= 0.05. One should do the same calculations using different η. If the structures remain at their positions, they are most likely of physical origin.
Besides the antiferromagnetic behavior for different interaction strengths, one should also compare the doping behavior. Figure 5.10 shows the spectral func-tions of antiferromagnetic calculafunc-tions for two different chemical potentials µ/W = 0.1 and µ/W = 0.3. For U/W = 1 the system jumps directly from a paramagnetic metal away from half-filling into an antiferromagnetic insulator at µ/W ≈ 0.27, as I have discussed above. Notice that although the NRG solution in the left panel of figure 5.10 is half-filled, it is asymmetric. The DMFT/DMRG calculations show a change in their behavior at approximately the same value of the chemical potential. In contrast to the NRG results, DMRG stabilizes a metallic antiferromagnetic state away half-filling. The filling of the system ishni ≈ 0.9 and lies therefore exactly in the phase separated region in the DMFT/NRG calculations. Although one cannot be completely sure, there is strong evidence for phase separation between the paramagnetic metal and the antiferromagnetic insulator in this parameter regime [142–144]. The reason for
the stable antiferromagnetic metal is most likely again the interplay between the broadening and deconvolution.