nLab string-net model

Idea

In condensed matter theory, string-net models were introduced as lattice models for 2-dimensional topological phases of matter exhibiting topological order. The name essentially derives from the fact that the main degrees of freedom are located on the lattice edges.

Related concept

• fusion category, anyons

• topological order

• string diagrams

References

Original articles:

• Michael Levin, Xiao-Gang Wen, String-net condensation: A physical mechanism for topological phases, Phys.Rev. B 71 (2005) 045110 $[$arXiv:cond-mat/0404617$]$

  (see also at Levin-Wen model)

• Michael Levin, Xiao-Gang Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett., 96, 110405 (2006) (arXiv:cond-mat/0510613)

  (introducing topological entanglement entropy with regards to string-net models)

Textbook account:

• Xiao-Gang Wen: String-condensation – a unification of light and fermions, chapter 10 in: Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons, Oxford Academic (2007) [doi:10.1093/acprof:oso/9780199227259.001.0001, pdf]

In terms of tensor networks:

• O. Buerschaper, M. Aguado, G. Vidal, Explicit tensor network representation for the ground states of string-net models, Phys. Rev. B 79 (2009) 085119 $[$arXiv:0809.2393, doi:10.1103/PhysRevB.95.195110$]$

• Zheng-Cheng Gu, Michael Levin, Brian Swingle, Xiao-Gang Wen, Tensor-product representations for string-net condensed states, Phys. Rev. B 79 (2009) 085118 $[$doi:10.1103/PhysRevB.79.085118$]$

Further development:

• Ling-Yan Hung, Yidun Wan, String-Net Models with $\mathbb{Z}_N$ Fusion Algebra, Phys. Rev. B 86 (2012) 235132 $[$arXiv:1207.6169$]$

• Chien-Hung Lin, Multi-flavor string-net models, Phys. Rev. B 95 195110 (2017) $[$arXiv:1611.08288, doi:10.1103/PhysRevB.95.195110$]$

Further review and exposition:

• Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen:

  Sections 6.4-6.9 in: Quantum Information Meets Quantum Matter – From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (QST), Springer (2019) [arXiv:1508.02595, doi:10.1007/978-1-4939-9084-9]

• Chien-Hung Lin, Michael Levin, Fiona J. Burnell, Generalized string-net models: A thorough exposition, Phys. Rev. B 103 (2021) 195155 [arXiv:2012.14424, doi:10.1103/PhysRevB.103.195155]

Constructing correlators in rational conformal field theory via string net models:

• Jürgen Fuchs, Christoph Schweigert, Yang Yang, String-Net Construction of RCFT Correlators, Springer Briefs in Mathematical Physics 45, Springer (2023) [doi:10.1007/978-3-031-14682-4, arXiv:2112.12708]

Constructing modular functors from pivotal bicategories using string net models:

• Jürgen Fuchs, Christoph Schweigert, Yang Yang, String-net models for pivotal bicategories [arXiv:2302.01468]

Exposition and review:

• Christoph Schweigert, String-net methods for CFT correlators, talk at QFT and Cobordism, CQTS (Mar 2023) [web]

• Jürgen Fuchs, Christoph Schweigert, Simon Wood, Yang Yang, pp. 18 of: Algebraic structures in two-dimensional conformal field theory, Encyclopedia of Mathematical Physics [arXiv:2305.02773]

On the membrane analog related to the Yetter model for strict 2-groups:

• Alex Bullivant, Clement Delcamp. Excitations in strict 2-group higher gauge models of topological phases (2019). (arXiv:1909.07937).

• Clement Delcamp. Tensor network approach to electromagnetic duality in (3+1)d topological gauge models (2021). (arXiv:2112.08324).