# String and membrane condensation on three-dimensional lattices

@article{Hamma2005StringAM, title={String and membrane condensation on three-dimensional lattices}, author={Alioscia Hamma and Paolo Zanardi and Xiao-Gang Wen}, journal={Physical Review B}, year={2005}, volume={72}, pages={035307} }

In this paper, we investigate the general properties of latt ice spin models that have string and/or membrane condensed ground states. We discuss the properties needed to define a string or membrane operator. We study three 3D spin models which lead to Z2 gauge theory at low energies. All the three models are exactly soluble and produce topologically ordered ground states. The first mode l contains both closed-string and closed-membrane condensations. The second model contains closed-string…

## 45 Citations

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## References

Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons

- Physics
- 2004

1. Introduction 2. Path integral formulation of quantum mechanics 3. Interacting boson systems 4. Free fermion systems 5. Interacting fermion systems 6. Quantum gauge theories 7. Theory of quantum…