Renormalization group constructions of topological quantum liquids and beyond

We give a detailed physical argument for the area law for entanglement entropy in gapped phases of matter arising from local Hamiltonians. Our approach is based on renormalization group (RG) ideas and takes a resource oriented perspective. We report four main results. First, we argue for the ``weak area law'': any gapped phase with a unique ground state on every closed manifold obeys the area law. Second, we introduce an RG based classification scheme and give a detailed argument that all phases within the classification scheme obey the area law. Third, we define a special subclass of gapped phases, topological quantum liquids, which captures all examples of current physical relevance, and we rigorously show that topological quantum liquids obey an area law. Fourth, we show that all topological quantum liquids have MERA representations which achieve unit overlap with the ground state in the thermodynamic limit and which have a bond dimension scaling with system size $L$ as ${e}^{c{log}^{d(1+\ensuremath{\delta})}(L)}$ for all $\ensuremath{\delta}g0$. For example, we show that chiral phases in $d=2$ dimensions have an approximate MERA with bond dimension ${e}^{c{log}^{2(1+\ensuremath{\delta})}(L)}$. We discuss extensively a number of subsidiary ideas and results necessary to make the main arguments, including field theory constructions. While our argument for the general area law rests on physically motivated assumptions (which we make explicit) and is therefore not rigorous, we may conclude that ``conventional'' gapped phases obey the area law and that any gapped phase which violates the area law must be a dragon.