String-net liquid: Difference between revisions

Revision as of 18:03, 5 June 2007

Latest revision as of 15:29, 30 August 2025

(62 intermediate revisions by 50 users not shown)

Line 1:

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

{{expert}}

{{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 |name-list-style=amp |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>

'''String-net liquid''' is the phrase used for a hypothetical [[state of matter]] in which the atoms do not line up in opposing "spins", but in a more erratic order, as if they had partial spins or charges. [[Herbertsmithite]], a crystalline material occurring in nature, may have such qualities. 2+2=@#$%^^ Chino= andresorue , Chino dejate de babosadas vea algo exagerado:

888_8:::888888:::::::::::::::::::::::::88888888888___888 ___________88:::::88888888::::::m::::::::::::::88888888888____8 _________888888888888888888:M::::::::::::::::8888888888888 ________88888888888888888888:::::::::::::::::M88888888888888 ________8888888888888888888888:::::::::::::M8888888888888888 _________8888888888888888888888::::::::::M888888888888888888 ________8888888888888888::88888:::::::::M88888888888888888888 ______88888888888888888:::88888::::::::M888888888888888___8888 _____88888888888888888:::88888::::::M:::::;o*M*o;888888888____88 ____88888888888888888:::8888:::::::M::::::::::::::::::88888888____8 ___88888888888888888::::88:::::::::M:;:::::::::::::::::::888888888 __8888888888888888888:::8:::::::::M::::aAa::::::::::::M8888888888_______8 __88___8888888888::88::::8::::::::M::::::::::::::::::::888888888888888_8888 _88__88888888888:::8::::::::::::::M:::::::::::;::::::::88:88888888888888888 _8__8888888888888::::::::::::::::M::::@@@@::::::::8w8888888888888888 __88888888888:888:::::::::::::::M:::::::@a@:::::::M8i888888888888888 _8888888888::::88:::::::::::::::M888:::::::::::::::::M88z88888888888888888 8888888888:::::8::::::::::::::::M88888::::::::::::MM888!888888888888888888 888888888:::::8::::::::::::::::M8888888MAmAMVMM88*88888888___88888888 888888_M::::::::::::::::::::::M888888888:::::::::MM8888888888888___8888888 8888___M:::::::::::::::::::::M88888888888:::::::MM88888888888888____88888 _888___M::::::::::::::::::::M8888888888888M:::::mM888888888888____888 __888__M:::::::::::::::::::M8888:8888888888888:::m::Mm8888_8888___888 ___88__M:::::::::::::::::::8888:8888888888888888:::::::::Mm8___8888___888 ___88__M::::::::::::::::8888M::88888::888888888888::::::::::Mm8888____88 ___8___MM:::::::::::::8888M::::8888:::::888888888888::::::::::::Mm8_____4 _______8M::::::::::::8888M:::::::888::::::::88:::8888888::::::::::::::Mm_____2 ______88MM:::::::::8888M::::::::::88:::::::::8:::::888888:::::::::M::::::M _____8888M:::::::::888MM::::::::::::8::::::::::::M::::8888:::::::::::M::::M ____88888M:::::::::88:M:::::::::::::::8:::::::::::::M:::8888::::::::::::M::M ___88_888MM::::::888:M:::::::::::::::::::::::::::::::M:8888::::::::::::::M: ___8_88888M::::::88::M:::::::::::::::::::::::::::::::::MM:88:::::::::::::::::M _____88888M::::::88::M:::::::::::::::::*88*::::::::::::M:88::::::::::::::::::::M ____888888M::::::88::M:::::::::::::::88@@88::::::::::M::88:::::::::::::::::::::M ____888888MM::::88::MM:::::::::::::88@@88:::::::::::M::::8:::::::::::::::::::::*8 ____88888__M::::::8::MM:::::::::::::::*88*:::::::::::::M::::::::::::::::::::::::::::88@ ____8888___MM:::::::::MM::::::::::::::::::::::::::::::MM:::::::::::::::::::::::::::88@@ _____888____M:::::::::::MM:::::::::::::::::::::::::::MM::M::::::::::::::::::::::::::888/ _____888____MM:::::::::::MMM:::::::::::::::::::::MM::::MM:::::::::::::::::::::::::MM ______88_____M:::::::::::::MMMM::::::::::::MMMM::::::::MM::::::::::::::::::::MMM _______88____MM:::::::::::::::MMMMMMMMMMM:::::::::::::MMM:::::::MMMMMM ________88____MM:::::::::::::::::::MMMMM::::::::::::::::::::MMMMMMMMMMII _________88___8MM::::::::::::::::::::::::::::::::::::::::::::::::::MMMMMMMM __________8___88MM:::::::::::::::::::::::::::::::::M:::M:::::::::::MMM ______________888MM:::::::::::::::::::::::::::MM::::::::MM::::::MM _____________88888MM::::::::::::::::::::::MMM:::::::::mM:::::MM _____________888888MM:::::::::::::::::::MMM:::::::::::::MMM::M ____________88888888MM::::::::::::::::MMM::::::::::::::::MM:::M ___________88_8888888M:::::::::::::MMM:::::::::::::::::::::M::::M ___________8__888888_M::::::::::MM::::::::::::::::::::::::::M::::M: ______________888888_M::::::::M:::::::::::::::::::::::::::::M:::::MM _____________888888__M:::::::M::::::::::::::::::::::::::::::::::::::::M _____________888888__M:::::::M::::::::::::::@.:::::::::::::::::::::::M _____________88888___M:::::::::::::::::::::@@:::::::::::::::::::::::M ____________88888___M::::::::::::::::::::@@@:::::::::::::::::::::::M ___________88888___M:::::::::::::::::::::@@::::::::::::::::::::::::::::M __________88888___M::::::m::::::::::::::@:::::::::::::::Mm::::::::::::M __________8888___M::::::M:::::::::::::::::::::::::::::::::::MM::::::::::::M _________8888___M::::::M::::::::::::::::::::::::::::::::::MMM:::::::::::::M ________888____M::::::Mm:::::::::::::::::::::::::::::::MMM::::::::::::::::M ______8888____M:::::::Mm::::::::::::::::::::::::::::MMMM:::::::::::::::::M _____888______M:::::::M:::::::::::::::::::::::::::MMM:::::::::::::::::::::::M: __8888_______M:::::::::::::::::::::::::::::::::::MM:::::::::::::::::::::::::::

See also [[string-net]].

== External linksOverview==

The string-net model is claimed 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 a unified origin for [[photons]] and [[electrons]] (or [[gauge bosons]] and [[fermions]]). It can be viewed as an origin of [[light]] and [[electron]] (or [[gauge bosons|gauge interactions]] and [[Fermi statistics]]).

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 journal |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 [[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 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.}}</ref>

* [http://www.newscientist.com/article.ns?id=mg19325954.200&feedId=online-news_rss20 The universe is a string-net liquid] - "New Scientist" (15 March 2007).

[[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==

{{physics-stub}}

===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 | last1=Read | first1=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>

===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 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==

{{Reflist}}

{{State of matter}}

{{DEFAULTSORT:String-Net}}

[[Category:Quantum phases]]

[[Category:Condensed matter physics]]

[[Category:Chemical engineering]]

[[Category:Phases of matter]]