Holographic Geometry of Entanglement Renormalization in Quantum Field Theories

TL;DR

The authors propose a concrete link between AdS/CFT holography and real-space entanglement renormalization by defining a continuum MERA (cMERA) and an emergent bulk metric purely from quantum-field-theoretic data. They compute this metric for free scalar and free fermion theories, showing AdS-like geometries for CFTs, capping by mass gaps, and time-dependent growth after quantum quenches. The work also analyzes how holographic notions such as entanglement entropy, bulk diffeomorphisms, and extremal surfaces emerge from tensor-network data, suggesting a broad framework to interpret quantum phases and gravity-like structures from entanglement. These results offer a pathway to studying holography beyond AdS and motivate further exploration of bulk geometry in non-conformal or non-AdS settings, including flat-space holography and finite-temperature cases.

Abstract

We study a conjectured connection between the AdS/CFT and a real-space quantum renormalization group scheme, the multi-scale entanglement renormalization ansatz (MERA). By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction, which is formulated purely in terms of quantum field theoretical data. Using the continuum version of the MERA (cMERA), we calculate this emergent holographic metric explicitly for free scalar boson and free fermions theories, and check that the metric so computed has the properties expected from AdS/CFT. We also discuss the cMERA in a time-dependent background induced by quantum quench and estimate its corresponding metric.

Figures (9)

• Figure 1: The matrix product state (MPS) for a quantum spin chain. The upper left diagram represents a tensor network for the MPS, which consists of "tripods" shown in the upper left diagram. The corresponding quantum state is shown below.
• Figure 2: The tensor network diagrams and the estimation of the entanglement entropy $S_A$ for the matrix product state (MPS) (left) and for the tree tensor network (TTN) (right).
• Figure 3: (Left) The tensor network for the MERA and the estimation of the entanglement entropy. The brown trees describes the coarse-graining of the original spin chain. The red horizontal bonds describe the disentanglers, which is an unitary transformation acting on each pair of two spins. It is clear from this picture that we can estimate the entanglement entropy as $\hbox{Min}_{\gamma_A}\left[{\#}\hbox{Bonds}(\gamma_A)\right]\sim \log L$. (Right) The MERA tensor network consists of disentanglers ("tetrapods") and isometries implementing coarse-graining ("tripod").
• Figure 4: The time-evolution of $\tilde{G}_{12}=g(u)$ after a quantum quench, computed for $m=0.1$ and $m_0=1$ using the same approximation in (\ref{['gfu']}). We fixed the phase ambiguity as $\theta_0(t)=0$. The horizontal and vertical coordinate correspond to $1/|k|=\epsilon e^{-u}$ and $t$, respectively. The former, $1/|k|$, can be interpreted as the $z$ coordinate in the AdS as we explain in section 4. With this interpretation, we can observe, qualitatively, that the excitations induced by the quantum quench are within the light-cone: $z<t$ in the AdS.
• Figure 5: The parallelism between the calculations of the entanglement entropy $S_A$ in the MERA (left) and AdS/CFT (right). The green surface represents $\gamma_A$ in both pictures. The red bonds in the left denote the disentanglers.