The Two-Dimensional Hubbard Model
6.2 Bath Size
2 4 6 8 10 12 Lb
10−8 10−6 10−4 10−2 100
χ(Lb)
1s2s 4s4s∗
2 4 6 8 10 12
Lb
0.66 0.68 0.70 0.72 0.74 0.76
n(Lb)
1s2s 4s4s∗
Figure 6.1: Left panel: Cost function value χ of the (0,0) patch for different numbers of bath sites Lb. We display results for single-site DMFT (1s), two-site DCA (2s) and results obtained with two different four-site patchings (4s and 4s∗). χ decreases exponentially with the number of bath sites. Higher DCA approximations lead to better fit results for the same number of bath sites. In all cases a region of overfitting is reached when the cost function drops to approximately 10−8. This can not be seen for four-site DCA since we only considered Lb < 8 bath sites due to the very long runtimes for bigger bath sizes. Right panel: The converged total filling of the latticen. Small even-odd oscillations can be seen around the converged filling from Lb = 4 bath sites on for all cluster sizes. The value ofn depends on the number of patches but is located in the same region for all DCA calculations.
Lb in all cases. Since the Hamiltonian parameters, obtained with the hybridisation fit, vary from iteration to iteration by 10−3, it is reasonable that occupation values vary on the same scale. This is also the case for calculations with different bath sizes. An even or odd number of bath sites can bias the fit for certain solutions and occupation numbers.
However, it is obvious that the filling of the lattice is, in general, already converged for relatively small bath sizes of around four bath sites. This corresponds to cost function values of the order 10−3−10−2, which indicates relatively bad fits of the hybridisation.
Furthermore, the filling of the lattice is dependent on the number of patches. We will discuss this in more detail when we compare our results with CTQMC.
In Fig. 6.2 the cost function values of all hybridisations in the first and last iteration of single-site DMFT and different multi-site DCA calculations are displayed. In general, the fit results of the later iterations are significantly better. It seems that the non-interacting hybridisation, which is used to initialise DMFT, is a comparable bad choice.
Interestingly, the hybridisations of the distinct patches behave significantly different. Es-pecially the patches located near the (π, π) patch are, in general, fitted much better than all others when using the same number of bath sites. Since these patches are connected to energetically higher states in the lattice, they are only occupied by a small amount of electrons. This seems to make the interaction with the environment much simpler, which
2 4 6 8 10 12 Lb
10−8 10−6 10−4 10−2 100
χ(Lb)
1stlast
2 4 6 8 10 12
Lb
10−8 10−6 10−4 10−2 100
χ(Lb)
(0,0),1st (π, π),1st (0,0),last (π, π),last
2 4 6 8 10 12
Lb 10−9
10−7 10−5 10−3 10−1
χ(Lb)
(0,0),1st (π,0),1st (π, π),1st (0,0),last (π,0),last (π, π),last
2 4 6 8 10 12
Lb 10−9
10−7 10−5 10−3 10−1
χ(Lb)
(0,0),1st (π,0),1st (π, π),1st (0,0),last (π,0),last (π, π),last
Figure 6.2: Cost function values of the fits of the hybridisation for all patches in the first (circles) and last iteration (squares) of a single-site DMFT calculation (upper left), two-site DCA calculation (upper right), and four-site DCA calculation with conventional (lower left) and alternative patching (lower right). The qualitative behaviour in all cases is similar. The fit results can differ by up to one order of magnitude between the first and last iteration. In general, the hybridisation of the (π, π) patch is fitted much better than the other patches for the same number of bath sites.
is reflected in the small number of bath sites needed for a good description of ∆(iωn). If only a certain quality of the hybridisation fit is needed, this fact can be used to reduce computation times by attributing different bath sizes to the impurity sites in order to reduce the total system size.
This explanation is supported by the behaviour of the four-site DCA system with alter-native patching. The different way of dividing the Brillouin zone changes the filling of the impurity sites and thus the physics the hybridisations have to describe. This clearly changes the behaviour of χ(Lb), especially for the patch around the (π, π) point. And indeed, the occupation of the (π, π) patch is lower for the alternative patching compared
0 2 4 6 8 10 12 14 iωn
10−10 10−8 10−6 10−4 10−2 100
|<(∆(iωn)−∆discr(iωn)|
Lb= 4 Lb= 6 Lb= 8 Lb= 9 Lb= 10 Lb= 12
0 2 4 6 8 10 12 14 iωn
10−10 10−8 10−6 10−4 10−2 100
|=(∆(iωn)−∆discr(iωn))|
Lb= 4 Lb= 6 Lb= 8 Lb= 9 Lb= 10 Lb= 12
Figure 6.3: Difference of the real (left) and imaginary (right) part of the hybridisation ∆(iωn) and its discretisation∆discr(iωn) for different numbers of bath sitesLb in the case of single-site DMFT. The hybridisation is fitted better for small frequencies sinceα= 1in the cost functionχ.
Because of the limited number of parameters, the fit oscillates around the hybridisation, which can be seen by the kinks in the curves. As already shown in Fig. 6.2, the results for Lb = 10 and higher numbers of baths sites are qualitatively similar. This can be seen explicitly for the imaginary part where we either observe no improvements at all or only small ones. Nevertheless, the parameter sets obtained from the fit can differ significantly.
to the usual one, as can be seen in the following section.
We based our analysis so far only on the cost function valuesχ. For the sake of complete-ness, we now want to show the actual differences between a typical hybridisation ∆(iωn) and its discretised version ∆discr(iωn) in Fig. 6.3. Obviously, since we chooseα = 1 in the cost function (see Eq. (5.3.11)), the fits are in better agreement with the hybridisation for small than for large frequencies. Due to the finite number of parameters and the finite superposition of summands in Eq. (5.3.11), the fits oscillate around ∆(iωn), which is indicated by the kinks in the plot. These kinks correspond to intersections between the fit and the hybridisation. We see that the real and imaginary parts of the hybridisation are fitted equally well and that the differences between ∆(iωn) and thew fit are typically of the order √χ.
In agreement to the behaviour of χ, the absolute value of the differences between the hybridisation and its discretised version are staying nearly constant for a high number of bath sites. However, the parameter sets can differ strongly, which can be seen especially well for the results withLb = 10 andLb = 12 where the real parts are remarkably different for the very small difference in the cost function value χ.
To avoid the regime of overfitting and the possible corresponding convergence problems, we choose to use Lb = 9 bath sites for single-site DMFT, Lb = 8 bath sites for two-site DCA, and Lb = 5 for both four-site DCA calculations. With this choice we observe very good fitting results for the first as well as the last iterations of DMFT and DCA. For
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.