Comparison with CTQMC Results

all calculations with µ=−3 we observe convergence after six iterations, which results in very fast runtimes ranging from one hour to one day. Of course, this behaviour is strongly influenced by the chemical potential, which not only determines the filling of the lattice and the different patches but is also setting the complexity of the hybridisation, the con-vergence properties, and whether the system exhibits metallic or insulating behaviour.

We will discuss this in more detail in the next sections.

While for single-site and two-site calculations no convergence issues are observed for any choice of µ, for four-site DCA it is strongly recommended to use momentum quantum numbers to achieve convergence over a wide range of values for the chemical potential.

Only with the momentum quantum numbers we were able to determine global ground states in each iteration reliably. This is in strong contrast to the single-site and two-site case where global ground states can be found easily and repeatedly even without momen-tum quanmomen-tum numbers. Since for higher number of patches the Hamiltonian is getting more complicated and the Hilbert space more fractured into different symmetry sectors, it is comprehensible that DMRG is strongly dependent on the randomly generated starting state if not all symmetries are implemented.

−4 −3 −2 −1 0 1 2 3 µ/t

0.6 0.8 1.0 1.2 1.4

n(µ)

1s2s 4s4s^∗

Figure 6.4: Total electron densityndependent on the chemical potentialµfor single-site DMFT and different multi-site DCA calculations. Our DMRG results (dots) are in agreement with the CTQMC results (lines) from Gull et al.^[53]. The temperature difference between the two methods explains why DMRG predicts a slightly larger gap in the Mott insulating regime. However, all patch choices show the same behaviour for high electron- and hole-doping and differ close to half-filling dependent on the number of patches.

agreement with previous studies^[9].

Our DMRG results show very good agreement with all CTQMC results. We only see small deviations next to the Mott plateau where DMRG predicts a slightly larger gap.

This is reasonable since DMRG produces results at T = 0 while the finite temperature in CTQMC causes the onset of metallic behaviour for slightly smaller chemical potentials due to thermal excitations.

Fig. 6.5 and Fig. 6.6 show the partial occupation of the patches nk as a function of the chemical potential µ and as a function of the total filling n. Our DMRG+DMFT results are, again, in full agreement with the CTQMC results by Gull et al.^[53] and show slightly larger gaps in correspondence with the total filling of the lattice. The behaviour of nk(n) in Fig. 6.6 shows that with the alternative four-site patching the (π,0) sector is incompressible while the other sectors can still be doped. This feature is reproduced for higher cluster DCA calculations with CTQMC and thus can be considered as robust.

This physical behaviour is a signature of a sector-selective regime^[167], and it is interesting to see that the conventional patching does not exhibit this feature. Gull et al.^[53] explain this by the missing possibility to distinguish the nodal and anti-nodal excitations in the conventional patching. The inset of Fig. 6.6 shows clearly that the sector-selective regime is only located on the hole-doped site, which is in agreement with higher cluster CTQMC results.

−4 −3 −2 −1 0 1 2 3 µ/t

0.0 0.5 1.0 1.5 2.0

nk(µ)

(0,0) (π, π)

0.6 0.8 1.0 1.2 1.4

n 0.0

0.5 1.0 1.5 2.0

nk(n)

(0,0) (π, π)

Figure 6.5: Occupation in the different patches n_k as a function of the chemical potential µ (left) and the total electron density n(right) for two-site DCA. We observe very good agreement between CTQMC (lines) and DMRG results (dots). Despite the fact that the bands have different fillings, they behave qualitatively similar. The slightly larger gap for DMRG compared to CTQMC observed in Fig. 6.4 can also be seen in the partial occupations.

While the sector-selective regime is not present in the conventional four-site patching, the region where the (π,0) patch is half-filled is found for both patchings. Gull et al.

suggest that the conventional (π,0) patch has to represent the pseudogap behaviour of the (π,0) patch as well as the physics of the gapless nodal quasiparticles located at the Fermi surface. The alternative patching avoids this problem by locating the nodal portion of the Fermi surface in the (0,0) patch.

Our DMRG+DMFT results are again in excellent agreement with the CTQMC+DMFT results while being computed extraordinarily fast. For the single-site and two-site case we observe convergence in less than ten iterations and runtimes for each iteration of around 15 minutes in the case of single-site DMFT and 40 minutes in the case of two-site DCA while using six cores. Unfortunately, in the four-site calculations the runtime per iteration increases from five to 15 hours. However, convergence is, in general, still observed in less than ten iterations, which results in total runtimes for DMFT of around a day or two with ten cores.

The runtime for the four-site cases can probably be reduced by not using momentum quantum numbers. For the ground state search we observe similar runtimes when using or not using momentum quantum numbers. But for the time evolutions using these additional quantum numbers slows down the computation by roughly a factor of two.

However, using these quantum numbers is vital for finding the global ground state reliably, which means they cannot simply be abandoned. Rather, we suggest to determine the global ground state with the help of the momentum quantum numbers. Afterwards, the ground state has to be computed a second time without the quantum numbers in the pre-determined symmetry sector. Unfortunately, this second step can require several

−4 −3 −2 −1 0 1 2 3 µ/t

0.0 0.2 0.4 0.6 0.8 1.0

nk(µ) (0,0)

(π,0) (π, π)

−4 −3 −2 −1 0 1 2 3

µ/t 0.0

0.2 0.4 0.6 0.8 1.0

nk(µ) (0,0)

(π,0) (π, π)

0.6 0.8 1.0 1.2 1.4

n 0.0

0.2 0.4 0.6 0.8 1.0

nk(n) (0,0)

(π,0) (π, π)

0.6 0.8 1.0 1.2 1.4

n 0.0

0.2 0.4 0.6 0.8 1.0

nk(n) 0.92 0.96 1.00 1.04 0.48 0.52

Figure 6.6: Occupation in the different patches nk as a function of the chemical potential µ (upper panels) and the total electron density n (lower panels) for four-site DCA with the con-ventional (left) and the alternative patching proposed by Gull et al.^[53] (right). We observe very good agreement between CTQMC and DMRG results. Most notably, a range of chemical poten-tials exist where the(π,0)patch is half-filled. The partial occupationn_k as a function of the total density n reveals qualitative differences between the conventional and alternative patching. In the latter, the(π,0)remains at half-filling for a range of densities while the other patches accom-modate the electrons. The inset shows the zoomed-in area of the plateau. Using the conventional patching this behaviour is missing.

attempts due to the previously described convergence problems of the DMRG calculations in multi-site DCA calculations. However, for four-site DCA this should still be a net-win in computation times since a single DMRG run only takes a minute.

We want to mention here that we observed convergence issues for the four-site calculation with standard patching and µ = −2. The issues were neither related to overfitting nor was DMFT oscillating between two solutions. Up to now, we were not able to deduce

what exactly causes the seemingly random behaviour.