MERA Overview

The multiscale entanglement renormalization ansatz (MERA) represents a state by layers of unitary disentanglers and isometries. Its computational advantage comes from cancellation outside the causal cone of a local observable. Its extra scale structure is useful for critical states, but choosing a scale-invariant ansatz is a physical restriction that must be tested.

Required background. Tensor-network contractions supplies the computational notation; entanglement and criticality supplies the physical motivation.

Isometries and unitary disentanglers

For a binary one-dimensional construction, let $w:\mathbb C^\chi\to\mathbb C^\chi\otimes\mathbb C^\chi$ embed a coarse site into two finer sites, and let $u$ act on neighboring fine sites. The constraints are

$$w^\dagger w=I,\qquad u^\dagger u=uu^\dagger=I.$$

The state is built downward from a small top tensor. A disentangler acts across a prospective block boundary before the next coarse-graining step. It can remove correlations that otherwise burden a local block truncation. The isometry then selects a smaller effective space. Unlike an arbitrary tensor network, these constraints produce exact cancellations in a bra-ket contraction.

The precise tensor pattern—binary, ternary, modified binary—changes causal-cone width and cost. Quote that pattern when reporting a complexity exponent; a universal cost formula in $\chi$ would be misleading.

Ascending operators and descending states

To evaluate a local expectation, move the operator upward through the few tensors in its causal cone. All tensors outside it cancel against their conjugates. Alternatively descend a reduced density matrix toward the physical layer. The two routes should agree:

$$\operatorname{tr}[\rho\,\mathcal A(O)] =\operatorname{tr}[\mathcal D(\rho)\,O],$$

where ascending and descending maps are adjoints under the trace pairing. This equality is an excellent contraction test with randomly generated complex isometries.

A finite chain has $O(\log N)$ layers. Bounded causal width prevents the local observable from expanding over the entire chain, even though it can retain contributions from many length scales.

Scale-invariant layers and scaling dimensions

If an optimized scale-invariant region rescales lengths by $b$, its scaling superoperator obeys $\mathcal S(\phi_\alpha)=\lambda_\alpha\phi_\alpha$. For a scaling operator with a positive eigenvalue in the applicable normalization,

$$\Delta_\alpha=-\frac{\log\lambda_\alpha}{\log b}.$$

For example, with $b=2$ and $\lambda=1/2$, the extracted scaling dimension is one. A corresponding two-point function scales as distance to the power $-2\Delta$ when the state and operator satisfy the scaling assumptions. Oscillatory, complex, or symmetry-resolved eigenmodes require additional interpretation; fitting every eigenvalue to a real scalar exponent is unsafe.

The identity eigenoperator should have eigenvalue one. Failure of that check can reveal violated tensor constraints or an improperly normalized ascending map.

Optimization and convergence

Optimize each tensor using its environment while preserving isometric or unitary constraints, for example through a polar/SVD step or constrained manifold optimization. A Euclidean unconstrained gradient step generally destroys the cancellation identities, even when it lowers a local objective.

Check constraint residuals, energy density, ascending/descending consistency, and relevant scaling eigenvalues. Increase $\chi$, vary the number of transitional layers before scale invariance is imposed, and compare different starts. A scale-invariant ansatz can force apparent algebraic behavior on a system that should flow away from criticality.

For a finite-system benchmark, contract a small MERA into a full state, verify its norm, and compare direct local expectations with causal-cone contractions. This tests implementation before extracting any critical data.

Exercises

If $w^\dagger w=I$, is $ww^\dagger=I$ on the fine Hilbert space?

Solution

No. The coarse space has dimension $\chi$ and the fine space has dimension $\chi^2$. The operator $ww^\dagger$ is a rank-$\chi$ orthogonal projector onto the retained subspace. Replacing it by the fine-space identity would erase the truncation and give incorrect ascending maps.

References

• Evenbly, Glen, and Guifré Vidal, “Algorithms for entanglement renormalization,” Physical Review B 79, 144108 (2009), doi:10.1103/PhysRevB.79.144108 — causal-cone contractions and constrained optimization.
• Vidal, Guifré, “Entanglement renormalization,” Physical Review Letters 99, 220405 (2007), arXiv:cond-mat/0512165 — disentanglers and scale-dependent coarse graining.