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P0 H0 P0 contributes with a constant, and thus will be set to zero as it does not change the eigenstates.
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We rewrite Hab as H (ab) =- - + dE E k>0 ψ0 | U Pk U | ψ0 δ (E- Ek + E0) and use the integral representation of the Dirac delta function δ (E) = (2π) -1 - + dt eiEt, and the fact that ψ0 | ei E0 t U ei Ek t Pk U | ψ0 = ψ0 | U (t) Pk U | ψ0, where U (t) = ei H0 t U e-i H0 t is the probe-spin-chain coupling in the Heisenberg representation for the spin chain. Finally we can set U 0 =0 to our advantage, by including the term k=0 in the sum over k and, since 1=k Pk, we obtain the stated result.
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We rewrite Hab as H (ab) =- - + dE E k>0 ψ0 | U Pk U | ψ0 δ (E- Ek + E0) and use the integral representation of the Dirac delta function δ (E) = (2π) -1 - + dt eiEt, and the fact that ψ0 | ei E0 t U ei Ek t Pk U | ψ0 = ψ0 | U (t) Pk U | ψ0, where U (t) = ei H0 t U e-i H0 t is the probe-spin-chain coupling in the Heisenberg representation for the spin chain. Finally we can set U 0 =0 to our advantage, by including the term k=0 in the sum over k and, since 1=k Pk, we obtain the stated result.
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23
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40849088684
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The local terms in H (ab) read HL (ab) = α,β=1 p (γαa aβa Cmα,mβ Aα Aβ 1b + γαb γβb Cnα,nβ 1a Bα Bβ) and HL a (b) = α=1 p γα a (b) A (B) α Omα. These terms are either constants or zero in the cases we consider.
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The local terms in H (ab) read HL (ab) = α,β=1 p (γαa aβa Cmα,mβ Aα Aβ 1b + γαb γβb Cnα,nβ 1a Bα Bβ) and HL a (b) = α=1 p γα a (b) A (B) α Omα. These terms are either constants or zero in the cases we consider.
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24
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40849087055
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More precisely, due to the perturbative nature of our formalism with expansion parameter Jp /Δ, the negativity or any other entanglement monotone will deviate from the value of maximal entanglement (the singlet state) with corrections of the order of (Jp /Δ) 2, which are negligible for weakly interacting probes, i.e., E (ρab) =1-O (Jp2 / Δ2).
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More precisely, due to the perturbative nature of our formalism with expansion parameter Jp /Δ, the negativity or any other entanglement monotone will deviate from the value of maximal entanglement (the singlet state) with corrections of the order of (Jp /Δ) 2, which are negligible for weakly interacting probes, i.e., E (ρab) =1-O (Jp2 / Δ2).
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27
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40849087057
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In a conformally invariant two-dimensional system with scaling dimension η the two-point correlation function has the transformation law G g′ (w1, w2) = | dw (z1) /d z1 dw (z2) /d z2 | -(η/2) Gg (z1, z2) generated by any analytic function w (z). The subscripts g and g′ refer to the boundary geometry where the theory is defined.
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In a conformally invariant two-dimensional system with scaling dimension η the two-point correlation function has the transformation law G g′ (w1, w2) = | dw (z1) /d z1 dw (z2) /d z2 | -(η/2) Gg (z1, z2) generated by any analytic function w (z). The subscripts g and g′ refer to the boundary geometry where the theory is defined.
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