nLab internal symmetry

Idea

In physics by an “internal symmetry” one broadly means a symmetry other than spacetime symmetry, i.e. a symmetry not due to nor reflecting movement in space and time, but due to “internal degrees of freedom”.

In high energy physics

Notably gauge symmetry is counted as internal symmetry.

Or a spinor field, in the absence of any magnetic field or spin-orbit interaction or any other interaction involving the spin will have a dynamics which is invariant under transformations that just flips the spin, and this is then an internal symmetry.

In solid state physics

In condensed matter theory, internal symmetries are also referred to as “on-site” symmetries, this referring to the sites of the atomic nuclei in the crystal lattice.

For example, the Ising model has an on-site “spin flip”-$\mathbb{Z}/2$symmetry $\vert \uparrow \rangle \xleftrightarrow{S} \vert \downarrow \rangle$, while the Heisenberg model has the full Spin(3)=SU(2)-group acting as an on-site symmetry on the electron spins (cf. CGW11, p. 5).

(table from SS 22)

Related concepts

• symmetry group

• Coleman-Mandula theorem

• symmetry protected topological phase

• crystallographic group

References

Explicit definition of internal symmetry condensed matter theory:

• Charles Zhaoxi Xiong, p. 13 of: Classification and Construction of Topological Phases of Quantum Matter $[$arXiv:1906.02892$]$

Beware that old historical articles (eg. Barlow 1883) used “internal symmetry” to refer to crystallographic symmetry which now is referred to as “spatial symmetry” and regarded as non-internal.

See also:

• Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen, Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B 83 035107 (2011) $[$doi:10.1103/PhysRevB.83.035107$]$