Causal Cone

The causal cone of the MERA is a powerful feature that emerges and is intimately linked to the fact that it is efficient to contract the tensor network to compute operators and

correlation functions. The causal cone of a site (or set of sites) is all the tensors that can be reached from the site(s) of support for the operator(s) if we only move along the edges and up the graph in the renormalisation direction. [26, 19]

The importance of the causal cone comes from the fact that the tensors in the causal cone of some operator(s) are the only ones that effect the outcome of the expectation value for those operators. When computing the expectation value of an operator, said operator is sandwiched between the MERA and its dual. The dual of the MERA is the complex conjugate transpose, in the MERA the transpose component of the MERA corresponds to the special direction imposed by the renormalisation direction of the network. The transpose of this tensor networks then looks as if the tensor network has been flipped around the x-axis, the complex conjugate can be applied separately to each tensor in the network. The two tensor networks are then contracted by joining corresponding bonds at all sites where the operator(s) doesn’t have support, with the operators sandwiched between at all other points. A large portion of the tensor network is then annihilated by the isometric properties of the tensors giving rise to the causal cone.

Three examples of computing operators and the resulting casual cones are given in figure 4.10. The first two examples are examples of single site operators with the causal cones indicated by the orange background, the third example shows the causal cone for two separate operators. The causal cone in this final examples starts off as two separate re-gions at the shortest length scales before joining into a single region at the longer length scales. With a little bit of analysis of the networks in figure 4.10 it can be seen that all tensors outside the causal cones are contracted with their dual tensors which is the identity as in figure 4.5b, in these diagrams the duals are indicated by an asterisk.

This layered construction means that while the MERA is designed to describe the ground state of a given microscopic model we can naturally transition to any length scale. If we take the first n layers to renormalise the operators to some longer length scale, the re-maining layers are interpreted as describing the quantum state at that new length scale.

So by taking all the renormalisation layers in the MERA besides the bottom one we now have a tensor network description of the quantum state at a length scale larger then the microscopic one by a factor of three. This interpretation is simply a mathematical slight of hand so that the expectation value of operators at any length scale are exactly the same, just as we would hope for from renormalisation.

a) b) c)

Figure 4.10: Examples of the causal cone for several different operator locations in the ternary MERA. In all diagrams the dual tensors are indicated by an asterisk and the causal cone is indicated by an orange background. All tensors outside the causal cone cancel out with their dual as per the definitions in figure 4.5b. a) The simplest causal cone, a single tensor placed at the bottom of a spine, a special site where there are only isometries directly above it. b) Another example of the causal cone for a different choice of initial operator site, in this case the renormalised operator saturates to a two site operator. c) An example of a two site operator, the casual cone starts in two separate locations before merging higher up in the MERA renormalisation scale. Analysis of the network indicates that after a certain number of layers the casual cone of any sized network will reduce to an operator on two sites in the majority of cases, in the worst case this is a three site operator.

Renormalisation of operators is also important in the optimisation of the MERA where the renormalised Hamiltonians are constructed at each layer and used to optimise the tensors of the corresponding layer. The fact that this computation remains tractable at higher layers is again connected to the casual cone of the MERA. Any local Hamiltonian is renormalised to another local Hamiltonian, the number of sites which the Hamiltonian has support on growing no larger then a fixed number. In 1D this means the MERA for a n-local Hamiltonian (where the Hamiltonian is comprised of interactions of no more then n neighbouring sites) will be renormalised to an n-local effective Hamiltonian for some n. This n is dependent on the specific MERA scheme chosen, and since this thesis focuses on the ternary MERA this means n = 2. However this is different for different MERA where the binary MERA (the first one proposed) is n = 3 and the modified binary MERA is again n = 2, these alternative MERA are given in figure 4.6.

This locality restriction for renormalisation means that any operator with support ini-tially on no more then n sites will be renormalised to have support on no more then n

sites after any number of renormalisation steps. Furthermore if the support is on no more then mn sites for integer m, such as for products of adjacent operators, then by the same argument the renormalised operator will be acting on no more then mn sites. However in all 1D MERA this actually results in a renormalised Hamiltonian with a support on much fewer then mn sites. For the ternary MERA this maximal support for the renor-malisation of operators turns out to be 3 in the least optimal location, the three sites centred under the spine of isometries (which form the causal cone in figure 4.10c). For the binary MERA this worst case turns out to be 4 and for the modified binary MERA it is again 3. This upper bound holds regardless of the initial size and location of the operator(s) at the shortest length scale.