In this appendix, we offer an alternative route to the RG scaling equations, equation (19), at the 2CKfixed point via bosonization. The Hamiltonian at 2CK equation (5) can be expressed more rigorously by including the Klein factors [34], which proves to be important in deriving the RG scaling equation:
δ π
where F Fs, f,Fsfare Klein factors satisfying the following relations [34]:
=
Following the approach in appendixA, wefirst focus on the termJ2xyδJ1z, which will contribute to the renormalization of j2
xy. One of its contributions is given by:
∫ ∫
Following similar steps to those shown in equation (A.17), we may rewrite equation (D.3) as:
∫ ∫
contribution in the bracket ⋯[ ]of equation (D.4) is given by:π θ τ ϕ τ
With the help of equation (A.20), carrying out the integral over the fast modes and performing rescaling, we arrive at:
∫
2 is used. Therefore, we have reproduced the RG scaling equation for j2
xyat the 2CKfixed point in equation (19) with the proper definitions for the renormalized Kondo couplings:
≡ρ μ −
Next, we compute the renormalization ofδj1zcontributed from the J(2xy)2term:
∫ ∫
Carrying out averaging over the fast modes, we have:
∫ ∫
With the help of the identity equation (A.20), equation (D.12) can be re-written as:
∫
τ∫
τ′ τ ⎡⎣θ τ −θ τ′ ⎤⎦ ≈∫
τ τ ⎡ ′τ ∂ ϕ τ + ′∂ θ τThefirst term in equation (D.13) can be simplified in the limit of ≈ →x a 0 as: function and therefore its charge density vanishes at x = 0). Therefore, the remaining part in equation (D.14) becomes
It is straightforward to see that˜ is a highly irrelevant operator with a scaling dimension[ ˜ ] =2+ > 1
K 1
2 for
anyK>0; we therefore ignore it here.
The second term in equation (D.13) (proportional to θ∂x s(0, )τ ) will contribute to the renormalization of the j˜1zterm. Combining everything from equation (D.7) to equation (D.16), the one-loop RG scaling equation forδj1zbecomes:
With the proper rescaling of J2xy
: →
, wefinally reproduce the RG scaling equation for j1z
in equation (19).
In fact, the above results can be understood alternatively in terms of non-vanishing correlator
τ τ δ τ
〈 〉 ≡ 〈ˆ jˆ (0, ) ˆ (0, ) ˆ ( , ˜)2xy j2xy ′ j1z x 〉, which measures the cross-correlations between theδj1zand j2xy
terms at the 2CKfixed point under one-loop RG. Here, δjˆ1z, jˆ2xyrefer to the bosonic operators associated with theδj1zand j2xy
terms, respectively. A typical term inˆ reads:
θ τ
With the above relations, equation (D.18) becomes:
θ τ
where we have dropped the term τ θ∂τ τ = ∂ϕ τ vanishes. It is clear from equation (D.21) that the correlator gets afinite expectation value as
θ τ θ τ our previous derivations for the RG scaling equations equation (19) via re-fermionization.
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