Will the Tensor network ansatz work in two dimensions?

In German ansatz means "educated guess".
For quantum many-body physics two dimensions is very different to one.

In the last two days I have heard talks from graduate students of my UQ colleague Guifre Vidal [who is moving to Perimeter Institute] about using tensor network states to describe quantum many-body states. A nice statement of the problem and the approach is in a Physics Viewpoint by Subir Sachdev.

It is first important to appreciate that Tensor Network states are essentially a convenient way to write a variational wave function for the ground state of a quantum many-body system. Like any such wave function they will only be useful/accurate/reliable if this choice is specific enough to capture the essential physics and/or if it is general enough to describe any state. Writing down a good variational wave function is an art worthy of a Nobel Prize (BCS, Laughlin, Anderson,...).

The one-dimensional version of a Tensor Network is a matrix product state (MPS). They work extremely well in one dimension because they can capture all the quantum entanglement. Essentially what the DMRG (Density Matrix Renormalisation Group)  does is find MPS states. It seems that MPS can be used to represent the ground state of any physically reasonable Hamiltonian with short range interactions.

But how about in two dimensions? As Sachdev says,

There is an alphabet soup of proposals [5], including MPS, projected entangled-pair states (PEPS), multiscale renormalization ansatz (MERA) [6], tensor renormalization group (TRG) [7], and now the tensor entanglement-filtering renormalization (TEFR) of Gu and Wen.  These methods are connected to each other, and differ mainly in the numerical algorithm used to explore the possible states. So far no previously unsolved model H has been moved into the solved column, although recent results from Evenbly and Vidal [8] show fairly conclusive evidence for VBS order on the kagome lattice, and there is promising progress on frustrated square lattice antiferromagnets [9].

 However, since Sachdev wrote that in 2009 it turns out that these results for the Kagome model turned out not to get the correct ground state. So it is still not clear that these are appropriate and useful variational wave functions for two dimensions.

Should we be optimistic? We need to keep in mind that two dimensions is a lot richer and  more complicated than one dimension. There are many unique things about one dimension which greatly restrict the possible type of quantum many-body states one can have. One can only scatter particles forward and backwards. One cannot have any continuous broken symmetries. Many quantum lattice models are integrable. Conformal field theory provides a means to classify all possible critical theories.... Nevertheless, it is surprising to me that MPS can capture all states.

In contrast, experiment shows that two dimensions produces a plethora of strange ground states in two dimensions, e.g., high-Tc superconductors, strange metals, non-Fermi liquids, topological order, spin liquids, fractional quantum Hall effect with quasi-particles with anyonic statistics. Hence, I will be surprised (and delighted) if one can really capture all these states in terms of just one class of wavefunctions such as PEPS.