condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...

, 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. 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 neutra ...

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 common ...

, 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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...

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 n ...

(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 tera ...

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 kn ...

(or gauge interactions and

Fermi statistics Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...

). 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 ...

coupling between the

and the SU(2)

in the

standard model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...

. 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 Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...

. 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

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 Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...

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 Induced gravity (or emergent gravity) is an idea in quantum gravity that spacetime curvature and its dynamics emerge as a mean field approximation of underlying microscopic degrees of freedom, similar to the fluid mechanics approximation of Bose ...

, and in a more recent paper, Tomasz Konopka,

Fotini Markopoulou Fotini G. Markopoulou-Kalamara ( el, Φωτεινή Μαρκοπούλου-Καλαμαρά; born April 3, 1971) is a Greek theoretical physicist interested in quantum gravity, foundational mathematics, quantum mechanics and a design engineer wo ...

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 In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...

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