String-net liquid: Difference between revisions

Revision as of 10:59, 29 March 2013

Revision as of 20:02, 29 May 2013

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{{Beyond the Standard Model|cTopic=[[Quantum gravity]]}}

In [[condensed matter physics]], a '''string-net''' is an extended object whose collective behavior has been proposed as a physical mechanism for [[topological order]] by [[Michael A. Levin]] and [[Xiao-Gang Wen]]. A particular string-net model may involve only closed loops; or networks of oriented, labeled strings obeying branching rules given by some [[gauge group]]; or still more general networks. <ref>{{cite journal |author=Levin, Michael A. and Xiao-Gang Wen |title=String-net condensation: A physical mechanism for topological phases |journal=Physical Review B: Condensed Matter and Materials Physics |volume=71 |issue=045110 |pages=21 |date=12 January 2005|arxiv=cond-mat/0404617 |doi=10.1103/PhysRevB.71.045110 |bibcode=2005PhRvB..71d5110L }}</ref>

Their model purports to show the derivation of photons, electrons, and U(1) gauge charge, small (relative to the [[planck mass]]) but nonzero masses, and suggestions that the [[leptons]], [[quarks]], and [[gluons]], can be modeled in the same way. In other words, string-net condensation provides an unification of [[photon]] and [[electron]] (or [[gauge bosons]] and [[fermions]]). It can be viewed as an origin of [[light]] and [[electron]] (or [[gauge bosons|gauge interactions]] and [[Fermi statistics]]).

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For strings labeled by the positive integers, string-nets are the [[spin network]]s studied in [[loop quantum gravity]]. This has led to the proposal by Levin and Wen,<ref>Photons and electrons as emergent phenomena Michael Levin, Xiao-Gang Wen http://arxiv.org/abs/cond-mat/0407140 page 8 "loop quantum gravity appears to be a string net condensation..."</ref> and Smolin, Markopoulou and Konopka<ref>{{cite arxiv |eprint=hep-th/0611197|quote=we argue, but do not prove, that loop quantum gravity's spin networks can reproduce Levin and Wen's string net condensation in quantum gravity... |author1=Tomasz Konopka |author2=Fotini Markopoulou |author3=Lee Smolin |title=Quantum Graphity |class=hep-th |year=2006}}</ref> that loop quantum gravity's spin networks can give rise to the [[standard model]] of [[particle physics]] through this mechanism, along with [[fermi statistics]] and [[gauge bosons|gauge 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 called "[[quantum graphity]]", and in a more recent paper, Tomasz Konopka, [[Fotini Markopoulou]], [[Simone Severini]] argued 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.<ref>Quantum Graphity: a model of emergent locality http://arxiv.org/abs/0801.0861 page #19 "the excitation of the ground state...is expected to give rise to U(1) gauge charge, ...main difference between this model and the original model of Levin and Wen is that in the present case the background is dynamical, and has hexagonal rather than square plaquettes</ref>

[[Herbertsmithite]] may be an example of string-net matter.<ref name=eureka>{{cite web|last=Bowles|first=Claire|title=Have researchers found a new state of matter?|url=http://www.eurekalert.org/pub_releases/2007-03/ns-hrf031407.php|publisher=Eureka Alert|accessdate=29 January 2012}}</ref><ref name=newscientist>{{cite journal|last=Merali|first=Zeeya|title=The universe is a string-net liquid.|journal=New Scientist|date=3/2007-03-17/2007|year=2007|volume=193|issue=2595|pages=8–9|url=http://www.newscientist.com/article/mg19325954.200|accessdate=29 January 2012}}</ref>

==Examples==

===Z2 spin liquid===

[[Z2 spin liquid]] obtained using [[slave-particle approach]] may be the first theoretical example of string-net liquid.<ref>N. Read and Subir Sachdev, Large-N expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66 1773 (1991)</ref><ref>[[Xiao-Gang Wen]], Mean Field Theory of Spin Liquid States with Finite Energy Gaps and Topological Orders, [http://link.aps.org/doi/10.1103/PhysRevB.44.2664 Phys. Rev. B44, 2664 (1991)]. </ref>

===The toric code===

The [[toric code]] is a two dimensional spin lattice, that acts as a quantum error-correcting code. It is defined on a two dimensional lattice with [[torus|toric]] [[boundary conditions]] with a spin-1/2 on each link. It can be shown that the ground state of the standard toric code Hamiltonian is an ''equal weight superposition'' of closed string states.<ref name=kitaevlaumann>{{cite journal|last=Kitaev|first=Alexei, Y.|coauthors=Chris Laumann|title=Topological phases and quantum computation|journal=ArXiv|year=2009|url=http://arxiv.org/abs/0904.2771}}</ref> Such a ground state is an example of a string-net condensate,<ref>{{cite journal|last=Morimae|first=Tomoyuki|title=Quantum computational tensor network on string-net condensate|journal=ArXiv|year=2011|url=http://arxiv.org/pdf/1012.1000v2.pdf}}</ref> which has the same [[topological order]]

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==References==

{{Reflist|2}}

{{State of matter}}

{{DEFAULTSORT:String-Net}}

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[[Category:Chemical engineering]]

[[Category:Phases of matter| ]]

{{State of matter}}

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