In condensed matter physics, a string-net is an extended object whose collective behavior has been proposed as a physical mechanism for

topological order In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian ...

by Michael A. Levin and

Xiao-Gang Wen Xiao-Gang Wen (; born November 26, 1961) is a Chinese-American physicist. He is a Cecil and Ida Green Professor of Physics at the Massachusetts Institute of Technology and Distinguished Visiting Research Chair at the Perimeter Institute for Theor ...

. A particular string-net model may involve only closed loops; or networks of oriented, labeled strings obeying branching rules given by some

gauge group In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...

; or still more general networks.

Overview

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 In particle physics, a lepton is an elementary particle of half-integer spin (spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutr ...

quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...

, and

gluons A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...

can be modeled in the same way. In other words, string-net condensation provides a unified origin for photons and

electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...

(or

gauge bosons In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge ...

and

fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...

). It can be viewed as an origin of

light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...

and

electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...

(or gauge interactions and Fermi statistics). However, their model does not account for the

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...

coupling between the

and the SU(2)

in the standard model. For strings labeled by the positive integers, string-nets are the

spin network In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear f ...

s studied in loop quantum gravity. This has led to the proposal by Levin and Wen, and Smolin, Markopoulou and Konopka that loop quantum gravity's spin networks can give rise to the standard model of

particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...

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 Simone Severini is an Italian-born British computer scientist. He is currently Professor of Physics of Information at University College London, and Director of Quantum Computing at Amazon Web Services. Work Severini worked in quantum informati ...

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.

Herbertsmithite Herbertsmithite is a mineral with chemical structure Zn Cu3( OH)6 Cl2. It is named after the mineralogist Herbert Smith (1872–1953) and was first found in 1972 in Chile. It is polymorphous with kapellasite and closely related to paratacamit ...

may be an example of string-net matter.

Examples

Z2 spin liquid

Z2 spin liquid Z (or z) is the 26th and last letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its usual names in English are ''zed'' () and ''zee'' (), with an occasi ...

obtained using slave-particle approach may be the first theoretical example of string-net liquid.

The toric code

The

toric code The toric code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice. It is the simplest and most well studied of the quantum double models. It is also the simplest example o ...

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. Such a ground-state is an example of a string-net condensate which has the same

as the Z2 spin liquid above.

References

{{DEFAULTSORT:String-Net Quantum phases Condensed matter physics Chemical engineering Phases of matter