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Proof. [T1, MH†] =- t 2N i,j,k,σ, σ′ [ni σ̄ c iσ † cjσ hj σ̄, σ σ′] (-1) k, [T1, MH†] = - t 2N i,j,k,σ, σ′ ni σ̄ { [c iσ †, nk, σ′] cjσ + c iσ † [cjσ, nk, σ′] } × hj σ̄ σ σ, σ′ z (-1) k, hence, [T1, MH†] =- t 2N i,j,σ ni σ̄ c iσ † cjσ hj σ̄ [- (-1) i + (- 1j)] σ σ,σ z. Since (-1) i =- (-1) j for nearest neighbors, we get the final result and can generalize it for T-1 and T0.
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Proof. [T1, MH†] =- t 2N i,j,k,σ, σ′ [ni σ̄ c iσ † cjσ hj σ̄, σ σ, σ′ z nk, σ′] (-1) k, [T1, MH†] = - t 2N i,j,k,σ, σ′ ni σ̄ { [c iσ †, nk, σ′] cjσ + c iσ † [cjσ, nk, σ′] } × hj σ̄ σ σ, σ′ z (-1) k, hence, [T1, MH†] =- t 2N i,j,σ ni σ̄ c iσ † cjσ hj σ̄ [- (-1) i + (- 1j)] σ σ,σ z. Since (-1) i =- (-1) j for nearest neighbors, we get the final result and can generalize it for T-1 and T0.
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Proof. Same calculation as in Eqs. 48 49 50, but for second or third nearest neighbors, (-1) i = (-1) j.
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Proof. Same calculation as in Eqs. 48 49 50, but for second or third nearest neighbors, (-1) i = (-1) j.
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