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.[1]
For strings labeled by the positive integers, string-nets are the spin networks studied in loop 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 the standard model of particle physics through this mechanism, along with fermi statistics and 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.[4]
Z2 spin liquid obtained using slave-particle approach may be the first theoretical example of string-net liquid.[7][8]
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 toricboundary 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.[9] Such a ground-state is an example of a string-net condensate[10] which has the same topological order as the Z2 spin liquid above.
References
^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. S2CID 51962817.
^Levin, Michael; Wen, Xiao-Gang (2005). "Photons and electrons as emergent phenomena". Rev. Mod. Phys. 77: 871–879 [878]. arXiv:cond-mat/0407140. Bibcode:2005RvMP...77..871L. doi:10.1103/RevModPhys.77.871. S2CID 117563047. loop quantum gravity appears to be a string net condensation ...
^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.
^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. S2CID 6959359. 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.
^Bowles, Claire. "Have researchers found a new state of matter?". Eureka Alert. Retrieved 29 January 2012.
^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. Retrieved 29 January 2012.
^Read, N.; Sachdev, Subir (1 March 1991). "Large-Nexpansion for frustrated quantum antiferromagnets". Physical Review Letters. American Physical Society (APS). 66 (13): 1773–1776. Bibcode:1991PhRvL..66.1773R. doi:10.1103/physrevlett.66.1773. ISSN 0031-9007. PMID 10043303.
^Xiao-Gang Wen, Mean Field Theory of Spin Liquid States with Finite Energy Gaps and Topological Orders, Phys. Rev. B44, 2664 (1991).
^Kitaev, Alexei, Y.; Chris Laumann (2009). "Topological phases and quantum computation". arXiv:0904.2771 [cond-mat.mes-hall].
string, liquid, condensed, matter, physics, string, extended, object, whose, collective, behavior, been, proposed, physical, mechanism, topological, order, michael, levin, xiao, gang, particular, string, model, involve, only, closed, loops, networks, oriented,. 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 1 Contents 1 Overview 2 Examples 2 1 Z2 spin liquid 2 2 The toric code 3 ReferencesOverview EditThe 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 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 networks studied in loop 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 the standard model of particle physics through this mechanism along with fermi statistics and 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 4 Herbertsmithite may be an example of string net matter 5 6 Examples EditZ2 spin liquid Edit Z2 spin liquid obtained using slave particle approach may be the first theoretical example of string net liquid 7 8 The toric code Edit 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 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 9 Such a ground state is an example of a string net condensate 10 which has the same topological order as the Z2 spin liquid above References Edit Levin Michael A amp 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 S2CID 51962817 Levin Michael Wen Xiao Gang 2005 Photons and electrons as emergent phenomena Rev Mod Phys 77 871 879 878 arXiv cond mat 0407140 Bibcode 2005RvMP 77 871L doi 10 1103 RevModPhys 77 871 S2CID 117563047 loop quantum gravity appears to be a string net condensation 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 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 S2CID 6959359 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 Bowles Claire Have researchers found a new state of matter Eureka Alert Retrieved 29 January 2012 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 Retrieved 29 January 2012 Read N Sachdev Subir 1 March 1991 Large Nexpansion for frustrated quantum antiferromagnets Physical Review Letters American Physical Society APS 66 13 1773 1776 Bibcode 1991PhRvL 66 1773R doi 10 1103 physrevlett 66 1773 ISSN 0031 9007 PMID 10043303 Xiao Gang Wen Mean Field Theory of Spin Liquid States with Finite Energy Gaps and Topological Orders Phys Rev B44 2664 1991 Kitaev Alexei Y Chris Laumann 2009 Topological phases and quantum computation arXiv 0904 2771 cond mat mes hall 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 S2CID 118522495 Retrieved from https en wikipedia org w index php title String net liquid amp oldid 1114317623, wikipedia, wiki, book, books, library,
