# Can a Hamiltonian have spin(N) or pin(N) structure if the Hamiltonian is time-reversal invariant?

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I am trying to understand when a model Hamiltonian could correspond to the Lie group Spin(N), the double cover of SO(N), or Pin(N), the double cover of O(N). More specifically, I mean the ground state manifold of the Hamiltonian is Spin(N) or Pin(N) or close to this. Is this structure enforced by time reversal invariance of the model? The time reversal operator $\cal{T}$ corresponds to Pin groups in some fashion, but it is not clear to me if the discussion I see in HEP literature implies what I am interested in. What I have seen from HEP, I think, is that fermions with spinor representation and some corresponding choice for the time-reversal operator are elements of the Pin$_{-}$ group. But I don't see how this implies a time-reversal invariant Hamiltonian has any pin group or spin group structure itself. Any help or thoughts would be appreciated.

asked Nov 19, 2020 in cond-mat
edited Nov 20, 2020

''What I have seen from HEP'' - Could you please add a reference?

Hi Arnold,

This was the paper I was looking at.

Reviews in Mathematical PhysicsVol. 13, No. 08, pp. 953-1034 (2001)
THE PIN GROUPS IN PHYSICS: C, P AND T
MARCUS BERG, CÉCILE DeWITT-MORETTE, SHANGJR GWO and ERIC KRAMER

https://arxiv.org/abs/math-ph/0012006

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