String-net liquid: Difference between revisions

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{{Short description|Condensed matter physics model involving only closed loops}}

{{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. |author2=Xiao-Gang Wen |lastname-authorlist-ampstyle=yesamp |title=String-net condensation: A physical mechanism for topological phases |journal=Physical Review B |volume=71 |issue=45110 |pages=21 |date=12 January 2005|arxiv=cond-mat/0404617 |doi=10.1103/PhysRevB.71.045110 |bibcode=2005PhRvB..71d5110L |s2cid=51962817 }}</ref>

==Overview==

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However, their model does not account for the [[chiral]] coupling between the [[fermions]] and the SU(2) [[gauge bosons]] in the [[standard model]].

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>{{cite Journaljournal |title=Photons and electrons as emergent phenomena |first1=Michael |last1=Levin |first2=Xiao-Gang |last2=Wen |journal=Rev. Mod. Phys. |volume=77 |pages=871–879 [878] |date=2005 |issue=3 |quote=loop quantum gravity appears to be a string net condensation …... |doi=10.1103/RevModPhys.77.871 |arxiv=cond-mat/0407140 |bibcode=2005RvMP...77..871L |s2cid=117563047 }}</ref> and Smolin, Markopoulou and Konopka<ref>{{cite arXiv |eprint=hep-th/0611197 |quote=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. |first1=Tomasz |last1=Konopka |first2=Fotini |last2=Markopoulou |first3=Lee |last3=Smolin |title=Quantum Graphity |date=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 [[Event symmetry#Quantum_graphity_and_other_random_graph_models|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>{{cite journal |title=Quantum graphity: A model of emergent locality |first1=Tomasz |last1=Konopka |first2=Fotini |last2=Markopoulou |first3=Simone |last3=Severini |journal=Phys. Rev. D |volume=77 |issue=10 |page=19 |date=May 2008 |doi=10.1103/PhysRevD.77.104029 |arxiv=0801.0861 |bibcode=2008PhRvD..77j4029K |s2cid=6959359 |quote=The characterization of the string-condensed ground state is difficultdifficult but its excitations are expected to be that of a U(1) gauge theory, …... The two main differencesdifferences 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.}}</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=2007-03-17|volume=193|issue=2595|pages=8–9|url=https://www.newscientist.com/article/mg19325954.200|accessdate=29 January 2012|doi=10.1016/s0262-4079(07)60640-x|url-access=subscription}}</ref>

==Examples==

===Z2 spin liquid===

[[Z2 spin liquid]] obtained using [[slave-particle approach]] may be the first theoretical example of string-net liquid.<ref name="Read Sachdev pp. 1773–1776">{{cite journal | lastlast1=Read | firstfirst1=N. | last2=Sachdev | first2=Subir | title=Large-Nexpansion for frustrated quantum antiferromagnets | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=66 | issue=13 | date=1 March 1991 | issn=0031-9007 | doi=10.1103/physrevlett.66.1773 | pages=1773–1776 | pmid=10043303| bibcode=1991PhRvL..66.1773R }}</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 toricToric 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 arXiv|last=Kitaev|first=Alexei, Y.|author2=Chris Laumann |title=Topological phases and quantum computation|date=2009|eprint=0904.2771|class=cond-mat.mes-hall}}</ref> Such a ground -state is an example of a string-net condensate<ref>{{Cite journal|journal=Physical Review A|volume=85|issue=6|page=062328|doi=10.1103/PhysRevA.85.062328|last=Morimae|first=Tomoyuki|title=Quantum computational tensor network on string-net condensate|date=2012|arxiv=1012.1000|bibcode = 2012PhRvA..85f2328M |s2cid=118522495}}</ref> which has the same [[topological order]]

as the [[Z2 spin liquid]] above.

==References==