The ERG formalism was further developed in [57, 58, 59, 60]. See [32, 61] for a review and list of applications. For some more modern applications, see [51, 62, 63].

See [64, 65] for attempts to generalize Gaussian cMERA in perturbation theory.

The erasure of the large momentum correlations in the ERG can be made precise using the language of quantum error correction codes [30, 31]. See also Sec. 7.

These are the standard free field creation and annihilation operators when the regulating function is set to 1. Note that this definition has an infrared divergence, and the resulting operator $a(\mathbf{p},\mathrm{\Lambda })$ should be interpreted in the usual distributional sense.

The cutoff $\mathrm{\Lambda }$ after the rescaling is now a fixed parameter.

When the regulating function $K$ is taken to be a sharp momentum space cutoff, it generates a coarse-graining local in momentum space, whereas for a more smooth momentum space cutoff, it is more natural to view ${\widehat{K}}_s$ as a disentangler in position space.

As a concrete example, one can consider $K_{\beta }(s)=\frac{1}{2}+\frac{1}{2}\tanh (\beta (1-s))$ and take the limit $\beta \to \infty$.

It is straightforward to check that no other terms in ${\delta }_{S_I}\rho$ can contribute matrix elements with odd particle number. Moreover, the operator $\mathcal{S}$ in (92) cannot change the particle number of a given term in the perturbative expansion, and so $\mathcal{S}(|{\mathrm{\Omega }}_{\lambda }⟩⟨{\mathrm{\Omega }}_{\lambda }|)$ never has odd particle matrix elements, simply by the parity symmetry of the action.

These factors are always $\mathrm{\Lambda }$ independent and so do not affect any of our manipulations in this section.

Matching propagators and momentum-conserving $\delta$ functions is straightforward, but the energy factors of $E_1$ and $E_2$ require some more care to understand. See Appendix pp5 for details.

A quantum channel is a completely positive trace-preserving map, and its dual is a completely positive map that preserves the identity operator (a unital map) [30].

It is natural to interpret this as a quantum detailed balance condition, even though the thermal density matrix is not invariant under the flow.

The behavior of the result in arbitrary dimensions is similar and can be expressed generally in terms of hypergeometric functions.

The infinite temperature limit of bosonic states does not technically exist, but the relative entropy remains finite in the limit.