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Tensor Network and Black Hole

Hiroaki Matsueda, Masafumi Ishihara, Yoichiro Hashizume

TL;DR

The paper develops a dual tensor-network framework based on thermofield dynamics to represent finite-temperature quantum states, identifying the interface between the original and tilde Hilbert spaces as an event horizon with entropy scaling as $S \sim \ln \chi$. By decomposing the thermal vector into a truncated MERA network, an emergent AdS black-hole geometry is obtained, with the maximal entanglement entropy $S_{EE}^{max}=\frac{L}{z_H}\ln m$ and temperature scaling $T\sim z_H^{-1}$, consistent with holographic expectations. A MERA–tilde MERA combined network is introduced to model finite-temperature effects, linking horizon structure, entropy, and boundary temperature within a discrete AdS/CFT setting. Overall, the work provides a concrete, graphical realization of finite-temperature AdS/CFT via MERA and offers insights for numerical RG approaches in condensed matter at finite temperature.

Abstract

A tensor network formalism of thermofield dynamics is introduced. The formalism relates the original Hilbert space with its tilde space by a product of two copies of a tensor network. Then, their interface becomes an event horizon, and the logarithm of the tensor rank corresponds to the black hole entropy. Eventually, multiscale entanglement renormalization anzats (MERA) reproduces an AdS black hole at finite temperature. Our finding shows rich functionalities of MERA as efficient graphical representation of AdS/CFT correspondence.

Tensor Network and Black Hole

TL;DR

The paper develops a dual tensor-network framework based on thermofield dynamics to represent finite-temperature quantum states, identifying the interface between the original and tilde Hilbert spaces as an event horizon with entropy scaling as . By decomposing the thermal vector into a truncated MERA network, an emergent AdS black-hole geometry is obtained, with the maximal entanglement entropy and temperature scaling , consistent with holographic expectations. A MERA–tilde MERA combined network is introduced to model finite-temperature effects, linking horizon structure, entropy, and boundary temperature within a discrete AdS/CFT setting. Overall, the work provides a concrete, graphical realization of finite-temperature AdS/CFT via MERA and offers insights for numerical RG approaches in condensed matter at finite temperature.

Abstract

A tensor network formalism of thermofield dynamics is introduced. The formalism relates the original Hilbert space with its tilde space by a product of two copies of a tensor network. Then, their interface becomes an event horizon, and the logarithm of the tensor rank corresponds to the black hole entropy. Eventually, multiscale entanglement renormalization anzats (MERA) reproduces an AdS black hole at finite temperature. Our finding shows rich functionalities of MERA as efficient graphical representation of AdS/CFT correspondence.

Paper Structure

This paper contains 8 sections, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Tensor Network Representation of Thermal State
  3. Entanglement between Dual Hilbert Spaces and Thermal Entropy
  4. Vector Product Form of Thermal State
  5. Decomposition of Vector into Truncated MERA Network and Emergence of AdS Black Hole
  6. MERA - Tilde MERA Combined Network
  7. Discussion
  8. Summary

Figures (4)

  • Figure 1: 2D binary MERA network. Filled dots, diamonds, and filled triangles are the original sites in a quantum 1D critical system, disentangler tensors, and isometries, respectivery. The vertical direction denotes renormalization flow. A red line represents the causal cone enclosing a partial system. The surface area of the causal cone is given by the sum of the number of the boundary points in each layer.
  • Figure 2: Decomposition of the vector $A$ into a set of tensors.
  • Figure 3: Combination of MERA (lower half) and tilde MERA (upper half) networks. A red wavy line represents an event horizon. Two pairs of isometries across the horizon are entangled.
  • Figure 4: Higher temperature case of MERA - tilde MERA combined network. When we fix system size $L$, larger $\chi$ leads to the smaller layer number.