Quantum phase transition

2013-07-15T16:29:31Z

209.119.70.1: The text (and a relevant link) has been updated to reflect the fact that the "order" of a phase transition refers to the derivative of free energy which is discontinuous at the transition rather than to an order of approximation.

{{Group theory sidebar |Topological}}

In [[mathematics]], '''SO(8)''' is the [[special orthogonal group]] acting on eight-dimensional [[Euclidean space]]. It could be either a real or complex [[simple Lie group]] of rank 4 and dimension 28.

==Spin(8)==
Like all special orthogonal groups of <math>n > 2</math>, SO(8) is not [[simply connected]], having a [[fundamental group]] [[isomorphic]] to [[cyclic group|'''Z'''2]]. The [[universal cover]] of SO(8) is the [[spin group]] '''Spin(8)'''.

==Center==
The [[center of a group|center]] of SO(8) is '''Z'''2, the diagonal matrices {±I} (as for all SO(2''n'') for 2''n'' > 2), while the center of Spin(8) is '''Z'''2×'''Z'''2 (as for all Spin(4''n''), 4''n'' > 0).

==Triality==
{{main|Triality}}
[[Image:Dynkin diagram D4.png|thumb|[[Dynkin diagram]] of SO(8), (D4)]]

SO(8) is unique among the [[simple Lie group]]s in that its [[Dynkin diagram]] (shown right) (D4 under the Dynkin classification) possesses a three-fold [[symmetry]]. This gives rise to peculiar feature of Spin(8) known as [[triality]]. Related to this is the fact that the two [[spinor]] [[group representation|representations]], as well as the [[fundamental representation|fundamental]] vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). The triality [[automorphism]] of Spin(8) lives in the [[outer automorphism group]] of Spin(8) which is isomorphic to the [[symmetric group]] S3 that permutes these three representations. The automorphism group acts on the center '''Z'''2 x '''Z'''2 (which also has automorphism group isomorphic to ''S''3 which may also be considered as the [[general linear group]] over the finite field with two elements, ''S''3 ≅GL(2,2)). When one quotients Spin(8) by one central '''Z'''2, breaking this symmetry and obtaining SO(8), the remaining [[outer automorphism group]] is only '''Z'''2. The triality symmetry acts again on the further quotient SO(8)/'''Z'''2.

Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a [[semidirect product]]: Aut(Spin(8)) ≅ Spin (8) ⋊ ''S''3.

==[[Root system]]==
<math>(\pm 1,\pm 1,0,0)</math>

<math>(\pm 1,0,\pm 1,0)</math>

<math>(\pm 1,0,0,\pm 1)</math>

<math>(0,\pm 1,\pm 1,0)</math>

<math>(0,\pm 1,0,\pm 1)</math>

<math>(0,0,\pm 1,\pm 1)</math>

==[[Weyl group]]==
Its [[Weyl group|Weyl]]/[[Coxeter group]] has 4!×8=192 elements.

==[[Cartan matrix]]==
<math>
\begin{pmatrix}
2 & -1 & -1 & -1\\
-1 & 2 & 0 & 0\\
-1 & 0 & 2 & 0\\
-1 & 0 & 0 & 2
\end{pmatrix}
</math>

==See also==
* [[Octonions]]
* [[Clifford algebra]]
* [[G2 (mathematics)|''G''2]]

==References==
*{{citation|last=Adams|first=J.F.|authorlink=Frank Adams|title=Lectures on exceptional Lie groups|series=Chicago Lectures in Mathematics|publisher= [[University of Chicago Press]]|year= 1996|isbn= 0-226-00526-7}}
*{{citation|last=Chevalley|first=Claude|authorlink=Claude Chevalley|title=The algebraic theory of spinors and Clifford algebras|series=Collected works|volume=2|publisher=Springer-Verlag|year=1997|isbn= 3-540-57063-2}} (originally published in 1954 by [[Columbia University Press]])
*{{citation|last=Porteous|first= Ian R.|authorlink=Ian R. Porteous|title=Clifford algebras and the classical groups|series=
Cambridge Studies in Advanced Mathematics|volume= 50|publisher= [[Cambridge University Press]]|year= 1995|isbn= 0-521-55177-3}}

[[Category:Lie groups]]

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Quantum phase transition