Tree Tensor Networks (TTN)

Tree Tensor Network (TTN) is a loop-free hierarchical tensor network (see belew).

Tree Tensor Network for a system of size 16

Due to enhanced connectivity in the structure of TTN, it can support long-range quantum correlations comapred to MPS. Moreover, simulations for systems with periodic boundary conditions (PBC) are almost as efficient as the same with open boundary conditions (OBC). TTNs have found their place for simulating 2D or pseudo-2D ladder/cylinder systems with PBC where the performance of MPS is not adequate.

TenNetLib.jl implements the unconstrained optimization for obtaining ground as well as excited states as described in Phys. Rev. B 90, 125154 (2014) and SciPost Phys. Lect. Notes 8 (2019).

Each tensor in the TTN are indexed by a pair (Tuple) of Ints = (ll, nn).

const Int2 = Tuple{Int, Int}

In the default scenario, ll denotes the layer index and nn denotes the tensor index at each layer (see the image above). The counting for ll, in the default case, starts at the bottom (towards to top level), while nn counts from left. The structure of the network is defined by a Graph{Int2} object. The link / bond between two neighboring nodes is denoted by a LinkTypeTTN object, an unordered pair of Int2s.

struct LinkTypeTTN
    first::Int2
    second::Int2
end

TTN for system-size 12	TTN for system-size 14
Sites 1, 4, 7, 11 are directly connected to ll=2 tensors	Sites 5, 12 are directly connected to ll=2 tensors
Nodes (1,1), (1,3), (1,5), (1,7) are removed	Nodes (1,3), (1,7) are removed