Emergent bulk gauge field in random tensor networks

Abstract

Random tensor network states are toy models for holographic duality, which have entanglement properties determined by graph geometry. In this paper, we propose a generalization of the random tensor network states which describe an ensemble of states preserving a given global symmetry. We show that Renyi entropy for this family of states can be described by a quantum extremal surface formula, with corrections to the area law term determined by a bulk gauge theory wavefunction. This provides a toy model of the correspondence between boundary global symmetry and bulk gauge symmetry in holographic duality. We discuss the boundary physical consequences of the bulk deconfined and confined phases.

Figures (3)

• Figure 1: (a) Illustration of a tensor network defined on a trivalent graph. (b) Each tensor is a three-qudit state with the Hilbert space of each qudit decomposed into irreducible representations of the symmetry group. (b) Illustration of the decomposition of the symmetric random tensor network. The vertex tensor is defined to be a product of Clebsch-Gordan coefficients $C_{\mu\nu\sigma}^{jkl}$, a normalized random tensor $T_{abc}^{jkl}$ that is independent for different representations and different vertices, and a weight $\lambda^{jkl}$. The tensor network state can be expressed as the vertex projection by random tensors $T_{abc}^{jkl}$ on a state $\left|\tilde{\Psi}_P\right\rangle=\left|\Psi_L\right\rangle\otimes\left|\Psi_g\right\rangle$, with $\left|\Psi_g\right\rangle$ defining the bulk gauge field wavefunction in the basis of representations. The yellow ball and the yellow cone both refers to the projection to the random state $\left|T_x\right\rangle$ defined by tensor $T_{abc}^{jkl}$.
• Figure 2: The wormhole geometry corresponding to two entangled boundaries in a TFD state. (a) and (b) illustrate the two topological sectors in the case of $Z_2$ gauge theory. The blue circle is the black hole horizon, which is also the RT surface of a single boundary.
• Figure 3: (a) A small subsystem $A$ (red arc) with a singly connected entanglement wedge $\Sigma_A$ (orange shaded region). (b) A large subsystem $A$ (red arc) with a doubly connected entanglement wedge $\Sigma_A$. The RT surface $\gamma_A=\gamma_{\overline{A}}\cup A_{BH}$ with $A_{BH}$ the black hole horizon.