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 sub ...

, a string-net is an extended object whose collective behavior has been proposed as a physical mechanism for topological order by

Michael A. Levin Michael may refer to: People * Michael (given name), a given name * Michael (surname), including a list of people with the surname Michael Given name "Michael" * Michael (archangel), ''first'' of God's archangels in the Jewish, Christian and ...

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.

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, quarks, and gluons 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, so they alway ...

and electrons (or gauge bosons and fermions). It can be viewed as an origin of light and electron (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 coupling between the fermions and the SU(2) gauge bosons 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 multiline ...

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

of particle physics 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 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 may be an example of string-net matter.

Examples

Z2 spin liquid

Z2 spin liquid obtained using slave-particle approach may be the first theoretical example of string-net liquid.

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 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 topological order as the Z2 spin liquid above.

References

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