Tensor Network Analogues

Both block renormalisation, as discussed in section 4.1, and entanglement renormali-sation, as discussed in section 4.2, can be written in the language of tensor networks [19, 69, 73]. The block renormalisation procedure discussed is mapped onto a tensor net-work called the tree tensor netnet-work (TTN) and constructed from isometries W , which, in general, takes b sites into a single effective site. In this case, as in the example above, I will demonstrate it with three sites combined into one effective site in each layer. This tensor network is given in figure 4.4a where the renormalisation direction of the network is pointed out, and figure 4.4b describes the isometric property of W , the asterisk indi-cating the complex conjugate transpose. Comparing with figure 4.1 it is clear that this tensor network represents block renormalisation and W corresponds to a change of basis combining three sites into one effective site.

As discussed in section 4.1 there is an issue with block renormalisation for critical models which is demonstrated by figure 4.4c. This can be seen by considering the entropy of a region A. A number of examples of A are given in figure 4.4c by solid lines under the tensor network. Because all the tensors are isometries and thus a change of basis, remov-ing a coarse grainremov-ing tensor which acts only within region A can’t change the resultremov-ing entropy. Therefore in 4.4c green dotted lines and blue dashed lines indicate the minimal number of effective sites that are in the region.

The fact that block renormalisation (and by extension the TTN) will not work for a critical spin system can be seen from a simple analysis of this diagram. The green lines indicate that for any size L there is always a region of that size or larger which cannot have more entropy then the entropy of a single effective site. If the size of that single

Renormalisation Steps

a) b) c)

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Figure 4.4: a) The tensor network which corresponds to block renormalisation, the circles indicate the sites where the blocks were in figure 4.1 and it can be seen that the tensors therefore correspond to the block renormalisation procedure. b) The isometric property of the component tensors in the network, the asterisks indicates that we applied the complex conjugate transpose to the tensor, partially seen by it being flipped in the vertical direction. c) Examples of the minimal number of sites for a selection of regions A, the corresponding region is indicated by solid lines below the network and the minimal number of sites is indicated by the number of bonds cut by either a dotted line or a dashed line.

A special family of regions are indicated by the green lines where the entropy is bounded by the entropy of one effective site, indicating a size independent entropy bound. This constant entropy bound is the reason that block renormalisation fundamentally cannot work for critical spin systems. The blue regions are another family of regions where this approach suggests the entropy bounds grow logarithmically. Therefore we must take care that the entropy does not appear to grow at a higher rate then it truly does when using this procedure.

site (which corresponds to the bond dimension of the bond we cut) is bounded then this means there is a saturation bound to the amount of entropy in a region regardless of the size of A and so the TTN can only described gapped systems. In addition to the family of green regions in figure 4.4c there is also a family of blue regions where this approach suggests that the entropy grows logarithmically with size (assuming all bonds are equal in bond dimension). Therefore we must be careful when trying to work out how the entropy grows using this procedure and consider only the lower bounds on the maximum amount of entanglement the tensor network graph allows.

This approach to upper bounding the entropy is discussed in subsection 4.4.2 but the simple rules are as follows. For a 1D connected region A the maximum entropy can be worked out by finding the line connecting one boundary of A and the other one which cuts the fewest bonds (while going into the tensor network and not passing through

ten-sors). To associate a maximum entropy to each line we define a function S_max which takes a Hilbert space (e.g. H_χ for a space of dimension χ) and returns the maximum entropy possible for that space (e.g. log(χ)). Then the maximum entropy is the sum of the maximum entropy associated to each bond that is cut, so if it cuts n bonds, each of which have a bond dimension of χ, then the maximum entropy is nS_max(H_χ). This will be used again when analysing the entanglement renormalisation tensor network analogue.

If we make the modifications proposed by entanglement renormalisation, i.e. removing short ranged entanglement before coarse graining with the isometries via a distentangler, then we get the MERA as shown in figure 4.5a. The distinguishing feature in the network is the fact that the existance of disentanglers introduces loops into the network rather then it just being a tree diagram. Just like in the tree tensor network we have to addi-tionally impose that all the tensors are isometric so that W^†W = I and U^†U = I ⊗ I which is shown in figure 4.5b. In figure 4.5c the analogous region to that in figure 4.4a is considered. When doing the minimum effective sites analysis the number of bonds cut continues to grow (logarithmically) with the length of the region considered. As I will discuss in section 4.6 this actually predicts that the maximum entropy will grow loga-rithmically with the region size, just as would be predicted for a conformal field theory or critical spin system from chapter 2, however before that it is worthwhile to introduce some important features of the MERA.[22, 23, 75, 26, 76]

Because the MERA is constructed only from isometric tensors then it turns out to be possible to efficiently compute the expectation values of local operators, as well as for the correlations between local operators. This property is known as the causal cone structure of the MERA, and discussed in subsection 4.4.3. Efficient computation of local operators (and correlations between them) also holds true for the tree tensor network as it can be viewed as a subset of possible MERA networks where all disentanglers are the identity tensor.

The MERA algorithm also incorporates two other types of tensors which don’t play a part in the tensor network, the local Hamiltonians at each length scale, and the reduced density matrices at each length scale. The local Hamiltonians are local nearest or next-nearest interactions that make up the full Hamiltonian when summed together. As the MERA is a representation of renormalisation we can obtain local Hamiltonians at each length scale, which are a key component for the update steps (see section 4.5.1 for de-tails). As part of the input for the algorithm the local Hamiltonians for the shortest

Renormalisation Steps

a) b) c)

*

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Figure 4.5: a) The tensor network which corresponds to entanglement renormalisation, the circles indicate the sites where the blocks were in figure 4.2 the additional modifica-tion of removing short ranged entanglement makes its appearance in disentanglers (blue squares) introduced in this figure. b) The isometric properties arising in the MERA net-work, just as for the tree tensor network in figure 4.4b, the complex conjugate transpose is indicated by an asterisks. To be precise, the blue squares, called disentanglers, are uni-tary as opposed to isometric and therefore we could reverse the order of tensors and still get the identity, but that fact is irrelevant for MERA calculations. c) The corresponding minimal bonds for the regions which only cut a single bond in the tree tensor network (see figure 4.4), as can be seen the number of minimal bonds scale with the logarithm of the area (as it is proportional to the number of layers we moved through which goes with the logarithm of the region size).

length scales are given, with all others constructed from these. The second type of tensor is the reduced density matrix, as part of the updating algorithm there must be a reduced density matrix for each local Hamiltonian (at the same length scale and with the same support).

The idea behind the causal cone is that we exploit the local renormalisation structure of the MERA to renormalise the operators from the shortest length scale upto some rel-evant length scale. For single site operators this renormalisation is not required as we have computed a series of local reduced density matrices at all scales. However for op-erators over multiple sites (including correlation functions between single site opop-erators) this renormalisation procedure is applied until the number of sites is reduced to no more then 2 neighbouring sites.

These procedures can also be used to study infinite systems so we can work with states in the thermodynamic limit. This is done by having a number of transitional

renormali-sation layers, beyond which all renormalirenormali-sation layers are the same and referred to as the scale invariant layer. The idea of the scale invariant layers is that the renormalisation flow generated by the transitional layers takes us close enough to a critical spin system fixed point that the scale invariance of critical spin system can be assumed.

The MERA network shown in figure 4.5 is a particular version of the 1D MERA and called the ternary MERA. There are three different standard 1D MERA which are called the binary, ternary and modified binary and they are given in figure 4.6.[26]

a) b) c)

Figure 4.6: The three standard 1D MERA schemes: a) the binary MERA, b) the ternary MERA, and c) the modified Binary MERA. There are similarities and differences between all of these schemes. For example in both binary models (a and c) two sites are taken to one effective site in each layer while in the ternary case (b) three sites are taken to one effective site. There is also a difference in how local the corresponding Hamiltonians and reduced density matrices are in the two models, in (a) these tensors have support on three adjacent sites, while in (b and c) they have support on only two sites. (c) Is also slightly different from the other two for the translationally invariant case, normally this translational invariance would extend to the Hamiltonians and density matrices, however for (c) this becomes translationally invariant at two step intervals. This is because there are two types of coarse graining operations which must be considered separately (one coloured dark green and one coloured light green). When we are not assuming translational invariance this point is not important as we will treat all isometric tensors independently in the first place.