Incondensed matter physics,astring-netis an extended object whose collective behavior has been proposed as a physical mechanism fortopological orderbyMichael A. LevinandXiao-Gang Wen.A particular string-net model may involve only closed loops; or networks of oriented, labeled strings obeying branching rules given by somegauge group;or still more general networks.[1]

Overview

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The string-net model is claimed to show the derivation of photons, electrons, and U(1) gauge charge, small (relative to thePlanck mass) but nonzero masses, and suggestions that theleptons,quarks,andgluonscan be modeled in the same way. In other words, string-net condensation provides a unified origin forphotonsandelectrons(orgauge bosonsandfermions). It can be viewed as an origin oflightandelectron(orgauge interactionsandFermi statistics). However, their model does not account for thechiralcoupling between thefermionsand the SU(2)gauge bosonsin thestandard model.

For strings labeled by the positive integers, string-nets are thespin networksstudied inloop quantum gravity.This has led to the proposal by Levin and Wen,[2]and Smolin, Markopoulou and Konopka[3]that loop quantum gravity's spin networks can give rise to thestandard modelofparticle physicsthrough this mechanism, along withfermi statisticsandgauge interactions.To date, a rigorous derivation from LQG's spin networks to Levin and Wen's spin lattice has yet to be done, but the project to do so is calledquantum graphity,and in a more recent paper, Tomasz Konopka,Fotini Markopoulou,Simone Severiniargued that there are some similarities to spin networks (but not necessarily an exact equivalence) that gives rise to U(1) gauge charge and electrons in the string net mechanism.[4]

Herbertsmithitemay be an example of string-net matter.[5][6]

Examples

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Z2 spin liquid

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Z2 spin liquidobtained usingslave-particle approachmay be the first theoretical example of string-net liquid.[7][8]

The toric code

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Thetoric codeis a two-dimensional spin-lattice that acts as a quantum error-correcting code. It is defined on a two-dimensional lattice withtoricboundary conditionswith a spin-1/2 on each link. It can be shown that the ground-state of the standard toric code Hamiltonian is anequal-weight superpositionof closed-string states.[9]Such a ground-state is an example of a string-net condensate[10]which has the sametopological order as the Z2 spin liquid above.

References

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  1. ^Levin, Michael A. & Xiao-Gang Wen (12 January 2005). "String-net condensation: A physical mechanism for topological phases".Physical Review B.71(45110): 21.arXiv:cond-mat/0404617.Bibcode:2005PhRvB..71d5110L.doi:10.1103/PhysRevB.71.045110.S2CID51962817.
  2. ^Levin, Michael; Wen, Xiao-Gang (2005). "Photons and electrons as emergent phenomena".Rev. Mod. Phys.77(3): 871–879 [878].arXiv:cond-mat/0407140.Bibcode:2005RvMP...77..871L.doi:10.1103/RevModPhys.77.871.S2CID117563047.loop quantum gravity appears to be a string net condensation...
  3. ^Konopka, Tomasz; Markopoulou, Fotini; Smolin, Lee (2006). "Quantum Graphity".arXiv:hep-th/0611197.We argue (but do not prove) that under certain conditions the spins in the system can arrange themselves in regular, lattice-like patterns at low temperatures.
  4. ^Konopka, Tomasz; Markopoulou, Fotini; Severini, Simone (May 2008). "Quantum graphity: A model of emergent locality".Phys. Rev. D.77(10): 19.arXiv:0801.0861.Bibcode:2008PhRvD..77j4029K.doi:10.1103/PhysRevD.77.104029.S2CID6959359.The characterization of the string-condensed ground state is difficult but its excitations are expected to be that of a U(1) gauge theory,... The two main differences between this model and the original string-net condensation model proposed by Levin and Wen are that in the present case the background lattice is dynamical and has hexagonal rather than square plaquettes.
  5. ^Bowles, Claire."Have researchers found a new state of matter?".Eureka Alert.Retrieved29 January2012.
  6. ^Merali, Zeeya (2007-03-17)."The universe is a string-net liquid".New Scientist.193(2595): 8–9.doi:10.1016/s0262-4079(07)60640-x.Retrieved29 January2012.
  7. ^Read, N.; Sachdev, Subir (1 March 1991). "Large-Nexpansion for frustrated quantum antiferromagnets".Physical Review Letters.66(13). American Physical Society (APS): 1773–1776.Bibcode:1991PhRvL..66.1773R.doi:10.1103/physrevlett.66.1773.ISSN0031-9007.PMID10043303.
  8. ^Xiao-Gang Wen,Mean Field Theory of Spin Liquid States with Finite Energy Gaps and Topological Orders,Phys. Rev. B44, 2664 (1991).
  9. ^Kitaev, Alexei, Y.; Chris Laumann (2009). "Topological phases and quantum computation".arXiv:0904.2771[cond-mat.mes-hall].{{cite arXiv}}:CS1 maint: multiple names: authors list (link)
  10. ^Morimae, Tomoyuki (2012). "Quantum computational tensor network on string-net condensate".Physical Review A.85(6): 062328.arXiv:1012.1000.Bibcode:2012PhRvA..85f2328M.doi:10.1103/PhysRevA.85.062328.S2CID118522495.