nLab 3d toric code

Idea

What has come to be called the “3d toric code” is a 4d TQFT (i.e. with 3 spatial dimensions, whence the name) which is the higher-dimensional analog of the 3d TQFT known as the toric code.

References

The model is due to:

• Alioscia Hamma, Paolo Zanardi, Xiao-Gang Wen, String and Membrane condensation on 3D lattices, Phys. Rev. B72:035307, 2005 (arXiv:cond-mat/0411752, doi:10.1103/PhysRevB.72.035307)

Further discussion of defects in the model (such as defect strings):

• Liang Kong, Yin Tian, Zhi-Hao Zhang, Defects in the 3-dimensional toric code model form a braided fusion 2-category, J. High Energ. Phys. 2020, 78 (2020) (arXiv:2009.06564, doi:10.1007/JHEP12(2020)078)

Relation of extended 3d TQFT to lattice models for topological phases of matter (like the 3d toric code):

• Daniel S. Freed, Michael J. Hopkins, Constantin Teleman: Discrete quantum systems from topological field theory [arXiv:2506.05131]