I'll start with an intro to quantum critical phenomena, points at which physical observables have power-law correlations in space and time governed by a universal set of scaling exponents. The ground state at a critical point is entangled at all length scales, leading to log divergent entanglement in 1D, making them difficult to describe using finitely entangled ansatz like matrix product states.

 

The hierarchical structure of entanglement motivated G. Vidal to propose a new tensor network ansatz, the "multiscale entanglement renormalization ansatz (MERA)." MERA elegantly encodes the data of a critical point  - scaling exponents, operator product expansions, and the central charge - and can even be carried out numerically given sufficient determination.