nLab
tree tensor network state

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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 𝔤\mathfrak{g} (with string diagram-notation as discussed there) its Lie bracket-tensor f[,]𝔤 *𝔤 *𝔤f \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,

  2. as the p-adic geometric incarnation of anti de Sitter spacetime in the p-adic AdS/CFT correspondence,

  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?)

quantum probability theoryobservables and states

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 pp-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 (pp-adic) AdS/CFT, JHEP 1801 (2018) 139 (arXiv:1703.05445)

  • Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, pp-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.