# nLab tree tensor network state

Contents

### Context

#### Monoidal categories

monoidal categories

# Contents

## Idea

What are called tree tensor network states (TTN states) in quantum physics (specifically in solid state physics and in AdS/CFT) are those tensor network states of the form of tree, often considered of fixed valence.

For example, given a metric Lie algebra $\mathfrak{g}$ (with string diagram-notation as discussed there) its Lie bracket-tensor $f \coloneqq [-,-] \in \mathfrak{g}^\ast \otimes \mathfrak{g}^\ast \otimes \mathfrak{g}$ gives rise to tree tensor network of the following form:

graphics from Sati-Schreiber 19c

The tree tensor network states in the form of Bruhat-Tits trees play a special role in the AdS/CFT correspondence, either as

1. a kind of lattice QFT-approximation to an actual boundary CFT quantum state,

3. as a reflection of actual crystal-site quantum states in AdS/CFT in solid state physics:

graphics from Sati-Schreiber 19c

(As in HMSS 16, HLM 19. But maybe one wants the Poincaré-dual networks, instead, as in HMPS 18?)

## References

### General

For more see the references at tensor network state.

• Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Luca Tagliacozzo, Gang Su, Maciej Lewenstein, Section 2.3.3 of: Tensor Network Contractions, Lecture Notes in Physics, Springer (2020) (arXiv:1708.09213, doi:10.1007/978-3-030-34489-4)

### Application to entanglement entropy

Application to entanglement entropy;

### Application in solid state physics

Application in solid state physics:

• Valentin Murg, Örs Legeza, Reinhard M. Noack, Frank Verstraete, Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks, Phys. Rev. B 82, 205105 (2010) (arXiv:1006.3095)

### Application in quantum chemistry

For application in quantum chemistry:

• Naoki Nakatani, Garnet Kin-Lic Chan, Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm, J. Chem. Phys. 138, 134113 (2013) (arXiv:1302.2298)

• Klaas Gunst, Frank Verstraete, Sebastian Wouters, Örs Legeza, Dimitri Van Neck, T3NS: three-legged tree tensor network states, Chem. Theory Comput. 2018, 14, 4, 2026-2033 (arXiv:1801.09998)

• Henrik R. Larsson, Computing vibrational eigenstates with tree tensor network states (TTNS), J. Chem. Phys. 151, 204102 (2019) (arXiv:1909.13831)

### Application to $p$-adic AdS/CFT correspondence:

Discussion of Bruhat-Tits tree-tensor networks in the context of the p-adic AdS/CFT correspondence:

• Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, Tensor network and ($p$-adic) AdS/CFT, JHEP 1801 (2018) 139 (arXiv:1703.05445)

• Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, $p$-adic CFT is a holographic tensor network (arXiv:1902.01411)

### Application in machine learning

Application in machine learning:

• Ding Liu, Shi-Ju Ran, Peter Wittek, Cheng Peng, Raul Blázquez García, Gang Su, Maciej Lewenstein, Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure, New Journal of Physics, 21, 073059 (2019) (arXiv:1710.04833)

• Song Cheng, Lei Wang, Tao Xiang, Pan Zhang, Tree Tensor Networks for Generative Modeling, Phys. Rev. B 99, 155131 (2019) (arXiv:1901.02217)

Last revised on February 9, 2020 at 05:42:38. See the history of this page for a list of all contributions to it.