Entanglement renormalization, scale invariance, and quantum criticality

Abstract

The use of entanglement renormalization in the presence of scale invariance is investigated. We explain how to compute an accurate approximation of the critical ground state of a lattice model, and how to evaluate local observables, correlators and critical exponents. Our results unveil a precise connection between the multi-scale entanglement renormalization ansatz (MERA) and conformal field theory (CFT). Given a critical Hamiltonian on the lattice, this connection can be exploited to extract most of the conformal data of the CFT that describes the model in the continuum limit.

Figures (3)

• Figure 1: (Color online) ($i$) Two lowest rows of disentanglers $u$ and isometries $w$ of the ternary MERA. They map the original infinite lattice $\mathcal{L}_0\equiv \mathcal{L}$ into increasingly coarse-grained lattices $\mathcal{L}_{1}$ and $\mathcal{L}_2$. Notice that three sites of $\mathcal{L}_{\tau-1}$ become one site of $\mathcal{L}_{\tau}$, hence the use of $\log_3$ throughout the paper. ($ii$)-($iv$) Under the coarse-graining transformation defined by the MERA, two-site operators supported on three different pairs of sites of $\mathcal{L}_{\tau-1}$ become supported on the same pair of sites of $\mathcal{L}_{\tau}$. ($v$) Accordingly, the scaling superoperator $\mathcal{S}$ is the average of three contributions, each of which (and thus also their average) is unital and contractive thanks to the isometric character of $u$ and $w$MERA.
• Figure 2: (Color online) ($i$) One-site operators on special sites are coarse-grained into one-site operators. ($ii$) Scaling superoperator for one-site operators. ($iii$) In computing correlators on specific sites $x$ and $y$ (or $x$, $y$ and $z$), one-site operators are coarse-grained individually according to $\mathcal{S}^{(1)}$ until they become nearest neighbors (which in this case occurs at lattice $\mathcal{L}_{2}$, $q=2$).
• Figure 3: (Color online) Scaling dimensions $\Delta_{\alpha}$ obtained from the spectrum of the scaling superoperator $\mathcal{S}$. Circles indicate primary fields. Left: For the Ising model we can identify the scaling dimensions of the three primary fields, the so-called identity $\mathbb{I}$, spin $\sigma$ and energy $\epsilon$, together with several of their descendants. Right: The spectrum of $\mathcal{S}$ for the 3-level Potts model shows some of its primary fields, including its primary fields with multiplicity two, namely the spins $\sigma_1$ and $\sigma_2$ and the pair $Z_1$ and $Z_2$Francesco.