Tuesday, 18 October 2016

condensed matter - Fractionalization and the structure of spin rotation group?


As we know, the phenomena of fractionalizations in condensed matter physics is fantastic, like fractional spin, fractional charge , fractional statistics, .... And one key point is that the quasiparticals must be created or annihilated by pair.


On the other hand, consider the groups SU(2) and SO(3), they are the rotation groups for half-integer and integer spins, respectively. And we know that SU(2)/Z2=SO(3), which means that each element in SO(3) can be viewed as one pair (U,U), where USU(2) (otherwise put: the coset {U,U}SU(2) in the quotient group SU(2)/Z2 is our element in SO(3)).



So I wonder that whether is there any underlying connection between the pair nature of quasiparticals in topological phase in physics side and the pair structure relating SU(2) and SO(3) in mathematics side?


Thank you very much.



Answer



The group of rotations of an N-dimensional space is SO(N). Being a symmetry of nature, classical systems transform according to representations of SO(N).


Quantum mechanics, on the other hand, allows systems which transform according to the universal covering groups of classical symmetries. This is the reason why we get in three dimensional quantum theory representations of SU(2) which are not true representations of SO(3), (the half integer spin representations). More generally, we have, in quantum theory, representations of Spin(N)=SO(N)Z2.


However in the case of a two spatial dimensions, SO(2)U(1), and the universal covering of U(1) is not Spin(2) but rather R.


In contrast to SO(2) or U(1) which allow discrete values of the two dimensional spin: u=einθ nZ, 0θ<2π, the universal covering R allows a continuum of spin values.


This is the basic reason of the fractionalization of spin in two dimensions.


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