Other Network Topologies

Instead of using an one-dimensional chain as our system topology, together with C.Hubig we developed the idea to use binary tree tensors (BBT) depicted in Fig. 3.8. In this sense, we introduce four-dimensional auxiliary tensors (grey sites in the figure) each connect-ing two impurity tensors. If there are more than two impurities, the auxiliary tensors are connected by another auxiliary tensor and so on. This reduces the distance between strongly interacting sites, e.g. the impurity sites with their bath sites and the impurity sites with themselves. For a large number of impurity sites it also avoids the entangle-ment information between two impurities being transported through impurities located in between. Instead the information about the entanglement of two impurities is only transported through auxiliary tensors. The drawback is that the auxiliary sites are four-dimensional tensors which have much worse scaling properties than the three-four-dimensional ones of normal MPS.

However, DMRG and time evolution methods are all easily generalised to BTTs. In Fig. 3.8 a) we also display a typical sweep through a BTT which produces, as is the case of standard MPS on an one-dimensional chain, a unique left or right canonical state.

This structure looks advantageous compared to a normal one-dimensional MPS chain.

Especially if we compare which sites are interacting with each other (Fig. 3.8 b) and Fig. 3.4), the BTT structure seems to be more local. Unfortunately, entanglement is not only generated by direct interactions between sites. As can be seen in the data of the mutual information, the bath sites of different impurities are also entangled with each other. Especially the degenerate sites are good examples for this. BTTs can increase the distance of these entangled bath sites compared to reordered MPS systems, which makes them unfavourable for small systems. Since the average distance between two sites in an one-dimensional chain increases much faster than in our BTT structure, we expect better performance of BTTs for large systems. This was confirmed with our first test calcula-tions where we observed that BTTs perform better than MPS only for systems which are sufficiently large enough, i.e. at least four impurity sites and each impurity site coupled to at least six bath sites. Unfortunately, we do not deal with such big systems in our DMFT calculations.

A more common approach used in quantum chemistry[21,44,90,97] and since 2017 also in the many-body community^[40]are so-called tree tensor networks (TTN) or fork tensor-product states. The fork tensor-product states introduced by Bauernfeind et al. are comparable to our BTTs without the auxiliary tensors. Therefore, the impurity sites itself are repre-sented by four-dimensional tensors. As we already discussed, this leads to tensors with very high bond dimensions, limiting the usage of these structures. The model discussed by Bauernfeind et al.^[40]was a very simple, completely degenerate three-band model. To our knowledge any step towards more realistic models was accompanied with an unfeasible increase of bond dimensions and computation times.

The TTN are more general since they allow the dimension of tensors to vary from site to site depending on the entanglement structure of the system. Instead of fixing the topol-ogy based on intuition and on possible interactions as done with our BTTs and the fork tensor-product stated of Bauernfeind et al., for TTN the mutual information of a system

a) b)

Figure 3.9: a) An example of an approximated minimum spanning tree (MST). Each dot rep-resents a site described by a tensor connected via the black lines to different number of other tensors. For the sake of clarity we resigned from drawing the lines for the physical indices. The maximum dimension of the tensors is therefore four. The structure is given completely by the entanglement structure of a foregoing calculations or by the existing interactions of the model. b) An example of a minimum entangled tree (MET). Again, we neglected the lines for the physical indices which have to be added to each site. The shape of the tree for a given maximal dimension (in this case 4) is given by the minimisation of the number of renormalised states needed for an exact solution. After the shape of the tree is determined, the sites of the lattice have to be mapped to it via an optimisation of the overall mutual information, e.g. a genetic algorithm.

is calculated and used to build up a general network system. In general, one can dis-tinguish minimum-spanning trees (MST) and minimum-entangled trees (MET)^[44]. The MST (Fig. 3.9a) ) are obtained by taking the entanglement spectrum and ordering the sites according to the strongest entanglement interactions. The dimension of the tensors can vary depending on the number of strongly entangled sites. E.g. if a site is strongly entangled with three other sites, it will be connected directly to all three of them, forming a node. A site only strongly entangled with a single other site will only be connected to that site, forming the end of a branch. Thus, an MST is an irregular formed network with probably the lowest bond dimension needed to obtain a good ground state. This does not necessarily mean that this structure is the optimal topology for DMRG and time evolu-tion methods. Convergence and sweep times can be slower because the irregular structure can lead to very slow information transport through the system or a lot of unnecessary optimisations are performed if a branch is already converged but must be transferred. A different approach is to optimise the convergence properties by defining the tree such that the number of renormalised states required to achieve an exact calculation is minimised.

After obtaining such a so-called MET (Fig. 3.9b) ), a genetic algorithm^[98] is used to place the system sites optimally in this tree. Nakatani et al.^[44] showed that MST

con-verge slower but produce better energies as METs which confirms our argumentations.

Using entanglement information to reorder not only the order of the tensor sites but the network topology itself seems to be a promising ansatz to improve computations even further. The entanglement is not only determined by the present interactions between sites but also by their filling, by the physical parameters of the system and by the state we are calculating. Thus, it is recommended to use generic algorithms to determine the optimal topology for each ground state search and each time evolution performed in the context of DMFT. For further improvements we definitely recommend to consider MST and MET as an alternative route to normal one-dimensional MPS or BTTs.