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 $e^{c{log}^{d(1+\delta )}\left(L\right)}$ for all $\delta >0$. For example, we show that chiral phases in $d=2$ dimensions have an approximate MERA with bond dimension $e^{c{log}^{2(1+\delta )}\left(L\right)}$. 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.

Figure 1

A $d=1$ version of the RG transformation.

Figure 2

Coupling $H$ (blue disk) to its orientation reversed partner $H_{\text{rev}}$ (red disk) along their common boundary, we can produce the gapped sphere Hamiltonian.

Figure 3

An array of wormholes in $d=2$.

Figure 4

The transformation to a trivial state using an expanding wormhole array. The white spaces denote product states or just empty space. We have suppressed the wormholes and are effectively viewing the whole system as a composite of $H$ and $H_{\text{rec}}$ on a system with boundary.

Figure 5

The red dot is local term in the Hamiltonian which is smeared into a quasilocal constraint. The dashed circle is a cutoff where we truncate the quasilocal constraint to a strictly local constraint. Gappability of an edge suggests the constraint can be chosen to live strictly within $A$.

Figure 6

The staggered circuit composed of blocks of size $\widehat{\ell }$ which approximates the action of the quasilocal unitary mapping $|{\psi }_L\rangle$ to $|{\psi }_{L/2}{\rangle |0\rangle }^{L/2}$ in $d=1$ for an $s=1$ fixed point. The colors of the circuit elements are coordinated with the colors of the terms in the equation in the figure.

Figure 7

The three layers of a $d=2$ circuit approximation of the quasilocal unitary transformation. In layer 1 we apply $K$ in the red boxes to leave a quasi-one-dimensional network which is dealt with in layers 2 and 3 using the blue and purple unitaries similar to Fig. 6. We have $|{\psi }_L\rangle \approx U_3{U}_2{U}_1|{\psi }_{L/2}{\rangle |0\rangle }^{3L^2/4}$. The colors of circuit elements in the figure are coordinated with the colors of terms in the previous equation.

Figure 8

The blocking scheme in $d=3$. First, we deal with the red blocks. Then, we deal with the blue faces. Finally, we are left with a quasi-one-dimensional network where the blue faces intersect.

Figure 9

$A,B,C$ label regions used in the definition of the topological entanglement entropy. The ancillas which are unentangled before the action of the quasilocal unitary are not pictured. The gray disks represent regions of linear size $\widehat{\ell }$, on which a single layer of the staggered circuit representations of the quasilocal unitary has support, as in Fig. 6.