Real-Space Renormalization Yields Finite Correlations

Real-space renormalization approaches for quantum lattice systems generate certain hierarchical classes of states that are subsumed by the multiscale entanglement renormalization Ansatz (MERA). It is shown that, with the exception of one spatial dimension, MERA states are actually states with finite correlations, i.e., projected entangled pair states (PEPS) with a bond dimension independent of the system size. Hence, real-space renormalization generates states which can be encoded with local effective degrees of freedom, and MERA states form an efficiently contractible class of PEPS that obey the area law for the entanglement entropy. It is further pointed out that there exist other efficiently contractible schemes violating the area law.

Figure 1

A 1D MERA with linear branching ratio $b=2$. Circles, squares, and the triangle denote tensors, the lines denote contractions of those tensors. The squares are isometries that map two local subsystems ${\mathcal{H}}_i^{\tau }$ and ${\mathcal{H}}_{i+1}^{\tau }$ into one ${\mathcal{H}}_{i/2}^{\tau +1}$ as in Kadanoff’s block spin transformation. The circles denote unitary operators, disentanglers, reducing the entanglement between ${\mathcal{H}}_i^{\tau }\otimes {\mathcal{H}}_{i+1}^{\tau }$ and the rest of the system before the action of the isometry. Tensor positions are chosen according to Eq. (6) such that stacking of tensors is avoided.

Figure 2

(a) Procedure for mapping a TNS (left) to a 2D PEPS (right), by assigning tensors to lattice sites and contraction lines to paths on the lattice. (b) The elements of the PEPS tensors are determined by the elements of the tensors composing the TNS.

Figure 3

(a) Unit cell of a specific 2D MERA state. With each layer, corresponding to a single RG step, unitary disentanglers are applied that reduce the entanglement between blocks of $2\times 2$ sites with the rest of the system. Then, an isometry maps from those $2\times 2$ sites (dots) into one (crosses). (b) Mapping of this MERA state to a PEPS. The diagram shows the assignment of two layers of the MERA, composed of disentanglers $\widehat{u}$ and isometries $\widehat{w}$, to the physical lattice.

Figure 4

Disjoint sublattices ${\mathcal{V}}_{\tau }\subset {\mathcal{V}}_{\mathrm{phys}}$ to which MERA tensors are assigned and disjoint subsets of edges ${\mathcal{E}}_{\tau }\subset {\mathcal{E}}_{\mathrm{phys}}$ to which MERA contraction lines are assigned for $b=2$. In our construction, the paths assigned to contraction lines from tensors of layer 2 to tensors in layer 3 are, e.g., restricted to edges from ${\mathcal{E}}_2\cup {\mathcal{E}}_3$.