Loop quantum gravity, exact holographic mapping, and holographic entanglement entropy

The relation between loop quantum gravity (LQG) and tensor networks is explored from the perspectives of the bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $\mathrm{\Sigma }$ with boundary $\partial \mathrm{\Sigma }$ is an exact holographic mapping similar to the proposal in [X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282]. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse-graining procedure of spin networks. Furthermore, when a region $A$ and its complement $\overline{A}$ are specified on the boundary $\partial \mathrm{\Sigma }$, we show that the boundary entanglement entropy $S(A)$ of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e., $S(A)$ is proportional to the minimal area of the bulk surface attached to the boundary of $A$ in $\partial \mathrm{\Sigma }$.

Figure 1

The entanglement entropy of a region is bounded by a minimal curve (red curve) cutting through the MERA tensor network. (Figure adapted and modified from Ref. [13].)

Figure 2

Various examples of tensor networks: MPS on the left and PEP on the right. (Figure adapted and modified from Ref. [16].)

Figure 3

The spatial region $\mathrm{\Sigma }$ with boundary $\partial \mathrm{\Sigma }$ and its semiclassical geometry are built by a large number of polyhedra $\mathfrak{p}$ (tetrahedra shown in figure) with semiclassical geometry. The semiclassical geometry of $\mathfrak{p}$ is fundamentally made by the spin networks $|{\mathrm{\Gamma }}_{\mathfrak{p}},\{j_e,j_l\},\{I_v\},\{m_l\}⟩$ with a large number of edges and vertices in the graph ${\mathrm{\Gamma }}_{\mathfrak{p}}$, shown in panel (c). Each edge carries a spin $j$ as the quanta of area at the Planck scale, while each vertex carries a intertwiner $I_v$ as the quanta of volume. The spin networks with a large number of degree of freedom can be coarse grained to the picture shown in panel (b). Each of the four legs in panel (b) represents the Hilbert space ${\mathcal{H}}_{\partial }(f)$, whose basis is labeled by $|{\mu }_f⟩$. ${\mathcal{H}}_{\partial }(f)$ includes all microstates $|j_l,m_l⟩$ (the boundary microstates of $\mathfrak{p}$) carried by the dangling edges in the spin-network graph ${\mathrm{\Gamma }}_{\mathfrak{p}}$. The middle ball in panel (b) represents the Hilbert space ${\mathcal{H}}_b(\mathfrak{p})$, whose basis is labeled by $|{\xi }_{\mathfrak{p}}^{\overrightarrow{\mu }}⟩$. ${\mathcal{H}}_b(\mathfrak{p})$ includes all the spins on the internal edges of ${\mathrm{\Gamma }}_{\mathfrak{p}}$ and intertwiners on all vertices (the bulk microstate inside $\mathfrak{p}$). The coarse-grained spin networks give the exact holographic mapping, and lead to the tensor network representing the ground state of the boundary CFT. Panels (a), (b), and (c) are the physical pictures at three different scales: (a) is at the macroscopic scale, where the typical length scale $L$ is the mean curvature radius of the semiclassical geometry; (b) is at the microscopic scale, where the tensor network lives, and the typical (squared) length scale is the mean face area ${\mathbf{Ar}}_f$ of a polyhedron; (c) is at the Planck scale, where the spin network lives, and the typical length scale is the Planck length ${\ell }_P$. The semiclassical regime of LQG is given by ${\ell }_P^2\ll A_f\ll L^2$. In this regime, we reproduce correctly the Ryu-Takayanagi formula of the holographic entanglement entropy.

Figure 4

An Ising configuration with a domain wall separating two domains with opposite spins.

Figure 5

(a) The space $\mathrm{\Sigma }$ with boundary $\partial \mathrm{\Sigma }$ divided into regions $A$ and $\overline{A}$. $\mathrm{\Sigma }$ contains the domain walls $(1),(2),\dots ,(8)$, which divide the bulk of $\mathrm{\Sigma }$ into regions I, II, $\dots$, VI. The domain wall $(1)\cup (2)$ is the unique surface with minimal area. Each bulk region associates a permutation $g_{I,II,\dots ,VI}$, with $g_I=I$ and $g_{II}=({\mathcal{C}}^{(n)}{)}^{-1}$. Each domain wall associates a number of cycles $\chi (g^{-1}{g}^{\prime })=\chi ({g^{\prime }}^{-1}g)$, with $g$, $g^{\prime }$ on two sides of the domain wall. (b) The domain walls using flow lines: each domain wall carries the Cayley weight $n-\chi (g^{-1}{g}^{\prime })$ of a permutation $g^{-1}{g}^{\prime }$. Since the flow lines only present the value of the Cayley weight, there is no need to include any crossing of flow lines. The flow line picture is always planar.

Figure 6

A graphical presentation of the triangle inequality (a2).