Symmetry, gauge, and projective symmetry group (PSG)?
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As discussed by Prof.Wen in the context of the quantum orders of spin liquids, PSG is defined as all the transformations that leave the mean-field ansatz invariant, IGG is the so-called invariant gauge group formed by all the gauge-transformations that leave the mean-field ansatz invariant, and SG denotes the usual symmetry group (e.g., lattice space symmetry, time-reversal symmetry, etc), and these groups are related as follows SG=PSG/IGG, where SGcan be viewed as the quotient group.
However, in math, the name of projective group is usually referred to the quotient group, like the so-called projective special unitary group PSU(2)=SU(2)/Z2PSU(2)=SU(2)/Z2, and here PSU(2)PSU(2) is in fact the group SO(3)SO(3).
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However, in math, the name of projective group is usually referred to the quotient group, like the so-called projective special unitary group PSU(2)=SU(2)/Z2PSU(2)=SU(2)/Z2, and here PSU(2)PSU(2) is in fact the group SO(3)SO(3).
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More specifically, it seems generally impossible to write UR−1U−1UR−1U−1 as an gauge operator generated by J=12ψ†τψJ=12ψ†τψ.
However, if we generalize the definition of SU(2)SU(2) gauge operators RR to those satisfying 3 properties A: Unitary; B: RψiR−1=Wiψi,Wi∈SU(2)RψiR−1=Wiψi,Wi∈SU(2) matrices, which implies that physical spins should be gauge invariant (e.g., RSiR−1=SiRSiR−1=Si); and C: RP=PR=P with projection operator PP, which implies that physical spin-space should be gauge invariant.
Then one can show that UR−1U−1UR−1U−1 indeed fulfills the above 3 properties A,B,C where UU is time reversal, SU(2)SU(2) spin rotation, or lattice symmetries.
(Furthermore, RR respects A,B,C ⇒R−1⇒R−1 respects A,B,C; R1,R2R1,R2 both respect A,B,C ⇒R1R2⇒R1R2 respects A,B,C.) Therefore, the expression RGUUR−1U−1RGUUR−1U−1 in (1) is an SU(2)SU(2) gauge operator in the sense A,B,C.
If UU represents some symmetry (e.g., time reversal, SU(2)SU(2) spin rotation, or lattice symmetries), then does URUR or RURU still represent the same physical symmetry? Where RR is an SU(2)SU(2) gauge operator (in the sense A,B,C mentioned above).
One can show that U′=URU′=UR or RURU represents the same physical symmetry as UU in the following sense: U′SiU′−1=USiU−1U′SiU′−1=USiU−1 and U′ϕ=UϕU′ϕ=Uϕ, where ϕ=Pϕ∈ϕ=Pϕ∈ physical spin space.
(Note that U′ϕ=UϕU′ϕ=Uϕ is still a physical spin state due to [P,U]=[P,U′]=0[P,U]=[P,U′]=0.) Therefore, the U′U′ and U′′U″ in expressions (2) and (3) indeed represent the same physical symmetry as UU.
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More specifically, it seems generally impossible to write UR−1U−1UR−1U−1 as an gauge operator generated by J=12ψ†τψJ=12ψ†τψ.
However, if we generalize the definition of SU(2)SU(2) gauge operators RR to those satisfying 3 properties A: Unitary; B: RψiR−1=Wiψi,Wi∈SU(2)RψiR−1=Wiψi,Wi∈SU(2) matrices, which implies that physical spins should be gauge invariant (e.g., RSiR−1=SiRSiR−1=Si); and C: RP=PR=P with projection operator PP, which implies that physical spin-space should be gauge invariant.
Then one can show that UR−1U−1UR−1U−1 indeed fulfills the above 3 properties A,B,C where UU is time reversal, SU(2)SU(2) spin rotation, or lattice symmetries.
(Furthermore, RR respects A,B,C ⇒R−1⇒R−1 respects A,B,C; R1,R2R1,R2 both respect A,B,C ⇒R1R2⇒R1R2 respects A,B,C.) Therefore, the expression RGUUR−1U−1RGUUR−1U−1 in (1) is an SU(2)SU(2) gauge operator in the sense A,B,C.
If UU represents some symmetry (e.g., time reversal, SU(2)SU(2) spin rotation, or lattice symmetries), then does URUR or RURU still represent the same physical symmetry? Where RR is an SU(2)SU(2) gauge operator (in the sense A,B,C mentioned above).
One can show that U′=URU′=UR or RURU represents the same physical symmetry as UU in the following sense: U′SiU′−1=USiU−1U′SiU′−1=USiU−1 and U′ϕ=UϕU′ϕ=Uϕ, where ϕ=Pϕ∈ϕ=Pϕ∈ physical spin space.
(Note that U′ϕ=UϕU′ϕ=Uϕ is still a physical spin state due to [P,U]=[P,U′]=0[P,U]=[P,U′]=0.) Therefore, the U′U′ and U′′U″ in expressions (2) and (3) indeed represent the same physical symmetry as UU.
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