Dataset (analytical calculations) for the paper "Spin and thermal current scaling at a Y -junction of XX spin chains" by D. Giuliano and F. Buccheri

Dataset (analytical calculations) for the paper “Spin and thermal current scaling at a Y-junction of XX spin chains” Domenico Giuliano Dipartimento di Fisica, Universit`a della Calabria Arcavacata di Rende I-87036, Cosenza, Italy and I.N.F.N., Gruppo collegato di Cosenza Arcavacata di Rende I-87036, Cosenza, Italy Francesco Buccheri Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy and INFN Sezione di Torino, Via P. Giuria 1, I-10125, Torino, Italy (Dated: August 14, 2025) We provide computation details about the derivation of our results shown and discussed in the quoted paper. Specifically, we discuss in detail the long derivation of the renormalization group equations (15,16) of our paper and of the residual boundary Hamiltonian HI1, equation (42). 2 I. MATHEMATICAL DERIVATION OF THE RENORMALIZATION GROUP EQUATIONS FOR THE RUNNING KONDO PARAMETERS In this Section we derive the renormalization group equations for the running boundary couplings in the Kondo interaction Hamiltonian HK,f . Within imaginary time formalism, the Euclidean action takes the form S=S(0) +SK, with S(0) = 3 X λ=1 S(0) λ S(0) λ=ˆβ 0 dτ    L X j=1 c† j,λ(τ)[∂τ+H]cj,λ(τ)−J L X j=1 [c† j,λ(τ)cj+1,λ(τ) + c† j+1,λ(τ)cj,λ(τ)]    SK=J1+J2 23 X λ=1 ˆβ 0 dτ [−iηλ(τ)ηλ+1(τ)] [−i(c† 1,λ(τ)c1,λ+1(τ) + c1,λ(τ)c† 1,λ+1(τ))] +J1−J2 23 X λ=1 ˆβ 0 dτ [−iηλ(τ)ηλ+1(τ)] [−i(c1,λ(τ)c1,λ+1(τ) + c† 1,λ(τ)c† 1,λ+1(τ))] .(1) To resort to the renormalization group approach to the running boundary (Kondo) interaction strengths, we infer the equations for the corresponding running coupling strengths by computing the average values of Oλ 1(τ) and of Oλ 2(τ), respectively defined as Oλ 1(τ)=[−iηλ(τ)ηλ+1 (τ)] [−ic† 1,λ(τ)c1,λ+1(τ)] Oλ 2(τ)=[−iηλ(τ)ηλ+1(τ)] [−ic1,λ(τ)c1,λ+1(τ)] .(2) To first-order in SK, we obtain ⟨Oλ 1(τ)⟩1=−J1+J2 23 X λ1=1 ˆβ 0 dτ1⟨Tτ[−iηλ(τ)ηλ+1 (τ)][−iηλ1(τ1)ηλ1+1 (τ1)] ⟩× ⟨Tτ[−ic† 1,λ(τ)c1,λ+1(τ)[−i(c† λ1,1(τ1)c1,λ+1(τ1)+c1,λ1(τ1)c† 1,λ1+1(τ1))] ⟩ =J1+J2 2ˆβ 0 dτ1gλ(τ1−τ)gλ+1 (τ−τ1),(3) with Tτbeing the imaginary time ordering operator. In Eq.(3) we have used ⟨Tτηλ(τ)ηλ′(τ′)⟩=δλ,λ′ϵ(τ−τ′),(4) with ϵ(τ) being the sign function, and with gλ(τ−τ′)=⟨Tτc1,λ(τ)c† 1,λ(τ′)⟩.(5) (See below for a detailed discussion of the explicit form of gλ). Similarly, we obtain ⟨Oλ 2(τ)⟩1=−J1−J2 23 X λ1=1 ˆβ 0 dτ1⟨Tτ[−iηλ(τ)ηλ+1 (τ)][−iηλ1(τ1)ηλ1+1 (τ1)] ⟩× ⟨Tτ[−ic1,λ(τ)c1,λ+1(τ)[−i(cλ1,1(τ1)c1,λ+1(τ1)+c† 1,λ1(τ1)c† 1,λ1+1(τ1))] ⟩ =−J1−J2 2ˆβ 0 dτ1gλ(τ−τ1)gλ+1 (τ−τ1),(6) Moving to Fourier-Matsubara space (see below for a definition of the various quantities in frequency space), we obtain: ⟨Oλ 1⟩1=J1+J2 21 βX iω gλ(iω)gλ+1(iω) ⟨Oλ 2⟩1=−J1−J2 21 βX iω gλ(iω)gλ+1(−iω).(7) 3 Let us, now, look for the second-order corrections: these are given by ⟨Oλ 1(τ)⟩2=1 2J1+J2 223 X λ1,λ2=1 ˆβ 0 dτ1dτ2× ⟨Tτ[−iηλ(τ)ηλ+1(τ)] [−iηλ1(τ1)ηλ1+1(τ1)] [−iηλ2(τ2)ηλ2+1 (τ2)] ⟩× ⟨Tτ[−ic† 1,λ(τ)c1,λ+1(τ)] [−i(c† 1,λ1(τ1)c1,λ1+1 (τ1)+c1,λ1(τ1)c† 1,λ1+1(τ1))] × [−i(c† 1,λ2(τ2)c1,λ2+1 (τ2)+c1,λ2(τ2)c† 1,λ2+1(τ2))]⟩ +1 2J1−J2 223 X λ1,λ2=1 ˆβ 0 dτ1dτ2× ⟨Tτ[−iηλ(τ)ηλ+1(τ)] [−iηλ1(τ1)ηλ1+1(τ1)] [−iηλ2(τ2)ηλ2+1 (τ2)] ⟩× ⟨Tτ[−ic† 1,λ(τ)c1,λ+1(τ)] [−i(c1,λ1(τ1)c1,λ1+1 (τ1)+c† 1,λ1(τ1)c† 1,λ1+1(τ1))] × [−i(c1,λ2(τ2)c1,λ2+1 (τ2)+c† 1,λ2(τ2)c† 1,λ2+1(τ2))]⟩,(8) as well as ⟨Oλ 2(τ)⟩2=1 2J1+J2 2J1−J2 23 X λ1,λ2=1 ˆβ 0 dτ1dτ2× ⟨Tτ[−iηλ(τ)ηλ+1(τ)] [−iηλ1(τ1)ηλ1+1(τ1)] [−iηλ2(τ2)ηλ2+1 (τ2)] ⟩× ⟨Tτ[−ic1,λ(τ)c1,λ+1(τ)] [−i(c† 1,λ1(τ1)c1,λ1+1 (τ1)+c1,λ1(τ1)c† 1,λ1+1(τ1))] × [−i(c1,λ2(τ2)c1,λ2+1 (τ2)+c† 1,λ2(τ2)c† 1,λ2+1(τ2))]⟩ +1 2J1+J2 2J1−J2 23 X λ1,λ2=1 ˆβ 0 dτ1dτ2× ⟨Tτ[−iηλ(τ)ηλ+1(τ)] [−iηλ1(τ1)ηλ1+1(τ1)] [−iηλ2(τ2)ηλ2+1 (τ2)] ⟩× ⟨Tτ[−ic1,λ(τ)c1,λ+1(τ)] [−i(c1,λ1(τ1)c1,λ1+1 (τ1)+c† 1,λ1(τ1)c† 1,λ1+1(τ1))] × [−i(c† 1,λ2(τ2)c1,λ2+1 (τ2)+c1,λ2(τ2)c† 1,λ2+1(τ2))]⟩,(9) The key quantity we need, now, is the tensor ⟨Tτ[−iηλ(τ)ηλ+1(τ)][−iηλ1(τ1)ηλ1+1 (τ1)] [−iηλ2(τ2)ηλ2+1(τ2)] ⟩= iδλ+1,λ1δλ1+1,λ2δλ2+1,λ ϵ(τ−τ1)ϵ(τ1−τ2)ϵ(τ2−τ) + iδλ+1,λ2δλ2+1,λ1δλ1+1,λ ϵ(τ−τ2)ϵ(τ2−τ1)ϵ(τ1−τ).(10) From Eq.(10), we obtain ⟨Oλ 1(τ)⟩2=−J1+J2 22ˆβ 0 dτ1dτ2ϵ(τ−τ1)ϵ(τ1−τ2)ϵ(τ2−τ)× gλ(τ2−τ)gλ+1 (τ−τ1)gλ+2 (τ1−τ2) +J1−J2 22ˆβ 0 dτ1dτ2ϵ(τ−τ1)ϵ(τ1−τ2)ϵ(τ2−τ)× gλ(τ2−τ)gλ+1 (τ−τ1)gλ+2 (τ2−τ1),(11) and by ⟨Oλ 2(τ)⟩2=J1+J2 2J1−J2 2ˆβ 0 dτ1dτ2ϵ(τ−τ1)ϵ(τ1−τ2)ϵ(τ2−τ)× gλ(τ−τ2)gλ+1 (τ−τ1)gλ+2(τ1−τ2) −J1+J2 2J1−J2 2ˆβ 0 dτ1dτ2ϵ(τ−τ1)ϵ(τ1−τ2)ϵ(τ2−τ)× gλ(τ−τ2)gλ+1 (τ−τ1)gλ+2(τ2−τ1).(12) 4 To recover the explicit form of the single-fermion Green’s functions we start from the mode expansion of the lattice fermionic field operator in the imaginary-time interaction representation, given by cj,λ(τ) = r2 ℓ+ 1 X k ck,λ sin(kj)e−ϵkτ,(13) with ϵk=−2Jcos(k) + H, which implies gλ(τ−τ′)=⟨Tτc1,λ(τ)c† 1,λ(τ′)⟩=2 ℓ+ 1 X k sin2(k){[1−fλ(ϵk)] e−ϵk(τ−τ′)θ(τ−τ′)−fλ(ϵk)e−ϵk(τ−τ′)θ(τ′−τ)},(14) with fλbeing Fermi distribution function for chain-λ. In Fourier-Matsubara space, we obtain gλ(iω) = ˆβ 0 dτ eiωτ gλ(τ)=−2 ℓ+ 1 X ksin2(k) iω −ϵk.(15) Next, we introduce the tilded Green’s functions ˜gλ(τ), that are defined as ˜gλ(τ)=gλ(τ)ϵ(τ).(16) Multiplying the Green’s function by ϵ(τ) trades it from fermionic to bosonic. Indeed, taking into account the periodicity over β, we obtain that the Fourier transform of ϵ(τ) is given by ϵ(iω) = 1 2ˆβ −β dτ eiωτ ϵ(τ)=−2 iω .(17) We therefore obtain that, for a bosonic Matsubara frequency Ω ˜gλ(iΩ) = ˆβ 0 dτ eiΩτ1 β2X iω1X iω2 e−i(ω1+ω2)τgλ(iω1)ϵ(iω2) = 1 βX iω gλ(iω)ϵ(iΩ−iω).(18) Next, we use the identity 1 βX iω 2 i(Ω −ω) 1 iω −ϵk=− tanh βϵk 2 iΩ−ϵk ,(19) to eventually get ˜gλ(iΩ) = 2 ℓ+ 1 X k  sin2(k) tanh βϵk 2 iΩ−ϵk .(20) Having introduced the tilded Green’s functions, we may rewrite the right-hand side of Eq.(15) as ⟨Oλ 1(τ)⟩2=−J1+J2 221 βX iΩ ˜gλ(iΩ)˜gλ+1(iΩ)˜gλ+2 (iΩ) −J1−J2 221 βX iΩ ˜gλ(iΩ)˜gλ+1 (iΩ)˜gλ+2(−iΩ) .(21) Similarly, Eq.(13) can be rewritten as ⟨Oλ 2(τ)⟩2=−J1+J2 2J1−J2 21 βX iΩ ˜gλ(iΩ)˜gλ+1 (−iΩ)˜gλ+2(−iΩ) −J1+J2 2J1−J2 21 βX iΩ ˜gλ(iΩ)˜gλ+1 (−iΩ)˜gλ+2(iΩ) (22) 5 To recover the RG equations, let us rewrite Eqs.(21,22) as: ⟨Oλ 1(τ)⟩2=−J1+J2 221 β3X iω1,iω2 gλ(iω1)gλ+1(iω2)X iΩϵ(iΩ−iω1)ϵ(iΩ−iω2)˜gλ+2 (iΩ) −J1−J2 221 β3X iω1,iω2 gλ(iω1)gλ+1(iω2)X iΩϵ(iΩ−iω1)ϵ(iΩ−iω2)˜gλ+2 (−iΩ),(23) and as ⟨Oλ 2(τ)⟩2=−J2 1−J2 2 4 1 β3X iω1,iω2 gλ(iω1)gλ+1 (−iω2)X iΩϵ(iΩ−iω1)ϵ(−iΩ+iω2)˜gλ+2(−iΩ) −J2 1−J2 2 4 1 β3X iω1,iω2 gλ(iω1)gλ+1 (−iω2)X iΩϵ(iΩ−iω1)ϵ(−iΩ+iω2)˜gλ+2(iΩ).(24) The sums over iΩ can be performed in the standard way. Specifically, we obtain: •First sum: X iΩ ϵ(iΩ−iω1)ϵ(iΩ−iω2)˜gλ+2(iΩ) = −2X iω3δω1,ω2 iω1−iω3 +δω1,ω3 iω1−iω2−δω2,ω3 iω1−iω2gλ+2 (iω3).(25) •Second sum: 1 βX iΩ ϵ(iΩ−iω1)ϵ(iΩ−iω2)˜gλ+2(−iΩ) = 2X iω3δω1,ω2 iω1−iω3 +δω1,ω3 iω1−iω2−δω2,ω3 iω1−iω2gλ+2 (−iω3).(26) •Third sum: 1 βX iΩ ϵ(iΩ−iω1)ϵ(−iΩ+iω2)˜gλ+2(−iΩ) = −2X iω3δω1,ω2 iω1−iω3 +δω1,ω3 iω1−iω2−δω2,ω3 iω1−iω2gλ+2 (−iω3).(27) •Fourth sum: 1 βX iΩ ϵ(iΩ−iω1)ϵ(−iΩ+iω2)˜gλ+2(iΩ) = 2X iω3δω1,ω2 iω1−iω3 +δω1,ω3 iω1−iω2−δω2,ω3 iω1−iω2gλ+2 (iω3).(28) Thus, we obtain (assuming that the RG equations are derived with the three leads at equilibrium, which implies that we can drop the apex λfrom the Jordan-Wigner fermion Green’s functions): ⟨Oλ 1(τ)⟩2=2 β2J1+J2 22 3X iω1,iω2[g(iω1)]2g(iω2) iω1−iω2 +2 β2J1−J2 22 X iω1,iω2[g(iω1)]2−g(−iω2) iω1−iω2 + 2 g(−iω1) g(iω1)g(iω2) iω1−iω2 ,(29) 6 and ⟨Oλ 2(τ)⟩2=−2 β2 J2 1−J2 2 4X iω1,iω2g(iω1)g(−iω1)−g(iω1) g(−iω1)g(−iω2) iω1−iω2 + 2 g(iω2) iω1−iω2 −2 β2 J2 1−J2 2 43X iω1,iω2g(iω1)g(−iω1)g(iω1) g(−iω1)g(iω2) iω1−iω2.(30) An important result is the explicit expression for the single-fermion Green’s function gλ(iω) in the large-ℓlimit (which we take in the following). Specifically, we obtain: gλ(iω) = 2 ℓ+ 1 X ksin2(k) ϵk−iω →L→∞ 2 πˆπ 0 dk sin2(k) ϵk−iω  =1 πJ ˆ2J −2J dϵ  q1−ϵ 2J2 ϵ+H−iω  =i Js1−iω −H 2J2 .(31) Now, following the recipe of Ref.[2], we recover the RG equations for the running coupling associated to the running couplings Jg=J1+J2 2and Ju=J1−J2 2by first of all trading the sums over iω2for integrals and by introducing a pertinent cutoff, according to 1 βX iω2−→ ˆD −D dω2 2π.(32) Next, we rescale the cutoff D→D−δD and renormalize the running coupling accordingly: Jg,u −→ Jg,u +δJg,u(D). Finally, we divide by δD the parameter renormalization. As a result, in the limit in which we assume that the relevant frequencies are ≪D, we obtain dJg dD =−6 πJD φ(D, H){J2 g+J2 u} dJu dD =−6 πJD φ(D, H) 2JgJu,(33) with φ(D, H) = ℜes1 + D+iH 2J2 =(1 + D2−H2 2J2+D2+H2 4J22)1 4 cos 1 2atan 2HD 4J2+D2−H2 .(34) Eqs.(33) decouple, once written in terms of the original running couplings J1,2(D). In particular, one gets dJ1,2 dD =−6 πJDφ(D, H)J2 1,2,(35) which implies J1,2(D) = J1,2(D0) 1 + 6J1,2(D0) πJ ´D0 Dhφ(u,H) uidu .(36) Eq.(36) implies, for the Kondo temperatures associated to the running couplings, the expressions Tk,(1,2), defined as πJ 6J1,2(D0)=ˆD0 kBTk,(1,2) φ(u, H) udu . (37) A numerical integration of Eqs.(35) is straightforward and would yield, as a byproduct, the full dependence of TK,(1,2) on H. 7 II. CONSTRUCTION OF THE BOUNDARY HAMILTONIAN HI1 To construct HI1, we temporarily resort to a lattice description of our system, in which the bulk Hamiltonian is given by HLat =−2iJ 3 X λ=1 ℓ−1 X j=−ℓ{¯ ξj,λ ¯ ξj+1,λ +¯ ζj,λ ¯ ζj+1,λ}.(38) At the same time, HK,f,1takes the form HK,f,1=J1 3 X λ=1 [−i(¯ ξ1,λ +¯ ξ−1,λ)(¯ ξ1,λ+1 +¯ ξ−1,λ+1)][−iηληλ+1].(39) To minimize HK,f,1in the large-J1limit, we introduce the Dirac fermionic operators dλ=1 √2¯ ξ1,λ +¯ ξ−1,λ √2+iηλ,(40) so that we may rewrite HK,f,1as HK,f,1=−2J1(3 X λ=1 d† λdλ−1 2)2 +3¯ J1 2.(41) From Eq.(41) we see that the groundstate of HK,f,1is twofold degenerate. Indeed, denoting with |n1, n2, n3⟩the state with occupation number of mode dλequal to nλ(λ= 1,2,3), both |0,0,0⟩and |1,1,1⟩have energy −3J1. At variance, the higher energy eigenvalue J1is sixfold degenerate, as the corresponding eigenspace is spanned by all the states with one of the nλ’s equal to 0 and the other two equal to 1, or vice versa. We now build the leading boundary perturbation at the (so far) “putative” 2CK fixed point, in which we have set ¯ J2= 0 and ¯ J1very large. To do so, we connect the central triangle to the leads by singling out of the lattice bulk Hamiltonian the term containing the ¯ ξ1,λ and the ¯ ξ−1,λ operators. This is given by Ht=−iτ 2 3 X λ=1{¯ ξ1,λ ¯ ξ2,λ +¯ ξ−2,λ ¯ ξ−1,λ},(42) with τ=J. Let us, for the time being, assume that τis a parameter unrelated to J. The relevant Hamiltonian operators are H′=iτ 3 X λ=1 ξ2,λ (dλ+d† λ) H′′ =¯ J2 3 X λ=1{ζ1,λζ1,λ+1[dλdλ+1 +d† λd† λ+1 −d† λdλ+1 −dλ.d† λ+1]}.(43) When acting on the twofold degenerate groundstate, the total Hamiltonian H′+H′′ yields [H′+H′′ ]|0,0,0⟩=iτ{ξ2,1|1,0,0⟩+ξ2,2|0,1,0⟩+ξ2,3|0,0,1⟩} +¯ J2{ζ1,1ζ1,2|1,1,0⟩+ζ1,2ζ1,3|0,1,1⟩+ζ1,3ζ1,1|1,0,1⟩} [H′+H′′ ]|1,1,1⟩=iτ{ξ2,1|0,1,1⟩+ξ2,2|1,0,1⟩+ξ2,3|1,1,0⟩} +¯ J2{ζ1,1ζ1,2|0,0,1⟩+ζ1,2ζ1,3|1,0,0⟩+ζ1,3ζ1,1|0,1,0⟩} .(44) The projector PGover the groundstate manifold is given by PG=|0,0,0⟩⟨0,0,0|+|1,1,1⟩⟨1,1,1|.(45) The effective boundary interaction arising from the Schrieffer-Wolff procedure has necessarily to exhibit a matrix structure with respect to the degenerate groundstate in the strongly coupled limit. To implement it, we first of all 8 project the time-independent Schr¨odinger equation over the lowest energy subspace Gspanned by |0,0,0⟩and by |1,1,1⟩and over the orthogonal subaspace. We therefore obtain EPG|Ψ⟩=E0PG|Ψ⟩+PG(H′+H′′ ){I−PG}|Ψ⟩ E{I−PG}|Ψ⟩=EX{I−PG}|Ψ⟩+{I−PG}(H′+H′′ ){I−PG}|Ψ⟩+{I−PG}(H′+H′′ )PG|Ψ⟩,(46) with E0=−3¯ J1, EX=¯ J1. Formally, within the Schrieffer-Wolff procedure we find a first boundary interaction that shows up to second order in H′+H′′ . This is given by δ(2) H=PG(H′+H′′ )(I−PG)(H′+H′′ )PG −4¯ J1 =−3(τ2+¯ J2 2) 4¯ J1{|0,0,0⟩⟨0,0,0|+|1,1,1⟩⟨1,1,1|} ,(47) which, being proportional to the identity operator over G, just provides a trivial additive renormalization to the total energy. As a preliminary step, in order to compute the third-order contribution in the interaction strengths, we have to make H′and H′′ act on the excited states. Specifically, we obtain •First contribution (H′+H′′ )|1,0,0⟩=iτ{ξ2,1|0,0,0⟩+ξ2,2|1,1,0⟩+ξ2,3|1,0,1⟩} +¯ J2{ζ1,2ζ1,3|1,1,1⟩−ζ1,1ζ1,2|0,1,0⟩−ζ3,1ζ1,1|0,0,1⟩} ,(48) •Second contribution (H′+H′′ )|0,1,0⟩=iτ{ξ2,1|1,1,0⟩+ξ2,2|0,0,0⟩+ξ2,3|1,0,1⟩} +¯ J2{ζ1,3ζ1,1|1,1,1⟩−ζ1,1ζ1,2|1,0,0⟩−ζ2,2ζ2,3|0,0,1⟩,(49) •Third contribution (H′+H′′ )|0,0,1⟩=iτ{ξ2,1|1,0,1⟩+ξ2,2|0,1,1⟩+ξ2,3|0,0,0⟩} +¯ J2{ζ1,1ζ1,2|1,1,1⟩−ζ1,2ζ1,3|0,1,0⟩−ζ1,3ζ1,1|1,0,0⟩} ,(50) •Fourth contribution (H′+H′′ )|1,1,0⟩=iτ{ξ2,1|0,1,0⟩+ξ2,2|0,1,0⟩+ξ2,3|1,1,1⟩} +¯ J2{ζ1,1ζ1,2|0,0,0⟩−ζ1,2ζ1,3|1,0,1⟩−ζ1,3ζ1,1|0,1,1⟩} ,(51) •Fifth contribution (H′+H′′ )|1,0,1⟩=iτ{ξ2,1|0,0,1⟩+ξ2,2|1,1,1⟩+ξ2,3|1,0,0⟩} +¯ J2{ζ1,3ζ1,1|0,0,0⟩−ζ1,1ζ1,2|0,1,1⟩−ζ1,2ζ1,3|1,1,0⟩} ,(52) •Sixth contribution (H′+H′′ )|0,1,1⟩=iτ{ξ2,1|1,1,1⟩+ξ2,2|0,0,1⟩+ξ2,3|0,1,0⟩} +¯ J2{ζ1,2ζ1,3|0,0,0⟩−ζ1,1ζ1,2|1,0,1⟩−ζ1,3ζ1,1|1,1,0⟩} .(53) Next, we obtain (H′)2=−2τ2{3+ξ2,1ξ2,2(d1+d† 1)(d2+d† 2)+ξ2,2ξ2,3(d2+d† 2)(d3+d† 3)+ξ2,3ξ2,1(d3+d† 3)(d1+d† 1)} (H′′ )2= 2( ¯ J2)2{3+ζ1,1ζ1,2(d1−d† 1)(d2−d† 2)+ζ1,2ζ1,3(d2−d† 2)(d3−d† 3)+ζ3,1ζ1,1(d3−d† 3)(d1−d† 1)} [H′, H′′ ]=0 .(54) 9 Accordingly, we obtain [{I−PG}H′]2|0,0,0⟩=−2τ2{ξ2,1ξ2,2|1,1,0⟩+ξ2,2ξ2,3|0,1,1⟩+ξ2,3ξ2,1|1,0,1⟩} [{I−PG}H′]2|1,1,1⟩=−2τ2{ξ2,1ξ2,2|0,0,1⟩+ξ2,2ξ2,3|1,0,0⟩+ξ2,3ξ2,1|0,1,0⟩} [{I−PG}H′′ ]2|0,0,0⟩= 2( ¯ J2)2{ζ1,1ζ1,2|1,1,0⟩+ζ1,2ζ1,3|0,1,1⟩+ζ1,3ζ1,1|1,0,1⟩} [{I−PG}H′′ ]2|1,1,1⟩= 2( ¯ J2)2{ζ1,1ζ1,2|0,0,1⟩+ζ1,2ζ1,3|1,0,0⟩+ζ1,3ζ1,1|0,1,0⟩} .(55) Also, we get ⟨0,0,0|H′[{I−PG}H′]2|0,0,0⟩= 0 ⟨1,1,1|H′′ [{I−PG}H′′ ]2|0,0,0⟩= 0 ⟨0,0,0|H′[{I−PG}H′]2|1,1,1⟩= 3iτ3ξ2,1ξ2,2ξ2,3 ⟨0,0,0|H′′ [{I−PG}H′′ ]2|0,0,0⟩=⟨1,1,1|H′′ [{I−PG}H′′ ]2|1,1,1⟩= 6( ¯ J2)3 ⟨0,0,0|H′[{I−PG}H′′]2|0,0,0⟩= 0 ⟨1,1,1|H′[{I−PG}H′′ ]2|0,0,0⟩= 0 ⟨0,0,0|H′′ [{I−PG}H′]2|0,0,0⟩= 0 ⟨1,1,1|H′′ [{I−PG}H′]2|0,0,0⟩= 0 ⟨1,1,1|H′[{I−PG}H′′ ]2|0,0,0⟩=⟨0,0,0|H′[{I−PG}H′′ ]2|1,1,1⟩=−2iτ(¯ J2)2{ξ2,1ζ1,2ζ1,3+ξ2,2ζ1,3ζ1,1+ξ2,3ζ1,1ζ1,3} ⟨0,0,0|H′′ [{I−PG}H′]2|1,1,1⟩=⟨1,1,1|H′′ [{I−PG}H′]2|0,0,0⟩= 0 ⟨0,0,0|H′[{I−PG}H′′ ]2|0,0,0⟩=⟨1,1,1|H′[{I−PG}H′′ ]2|1,1,1⟩= 0 ⟨1,1,1|H′′ [{I−PG}H′]2|1,1,1⟩=⟨0,0,0|H′′ [{I−PG}H′]2|0,0,0⟩ = 2τ2¯ J2{ζ1,1ζ1,2ξ2,1ξ2,2+ζ1,2ζ1,3ξ2,2ξ2,3+ζ1,3ζ1,1ξ2,3ξ2,1}.(56) As a result, to third order in the boundary interaction strengths, we obtain δ(3) H=PG(H′+H′′ )(I−PG)(H′+H′′ )(I−PG)(H′+H′′ )PG 16( ¯ J1)2 =−3τ3 16 ¯ J2 1 [−iξ2,1ξ2,2ξ2,3]Q +iτ(¯ J2)2 8( ¯ J1)2{ξ2,1ζ1,2ζ1,3+ξ2,2ζ1,3ζ1,1+ξ2,3ζ1,1ζ1,2}Q +τ2¯ J2 (¯ J1)2{ζ1,1ζ1,2ξ2,1ξ2,2+ζ1,2ζ1,3ξ2,2ξ2,3+ζ1,3ζ1,1ξ2,3ξ2,1},(57) with Q=−iη1η2η3being the operator that sends |0,0,0⟩into |1,1,1⟩, and vice versa. We now note that, by construction, over any of the eigenstates of HK,f,1in the large- ¯ J1limit, the average value of ξ1,λ is equal to 0, for any λ. At the same time, the field ξj,λ at j= 1 has fully disappeared from the residual boundary interaction terms. This suggests to account for the updated boundary conditions by simply shortening the lead chains by one site, so that now the boundary conditions at the inner boundary become ξ1,λ = 0, ∀λ= 1,2,3 which, in the field-theoretical language, implies that the boundary conditions for the unfolded fields turn from antiperiodic to periodic. While this is relevant to recover the dimension 1/8 spin field of the Ising model, in our specific case it is not of the utmost importance. At variance, it is interesting to write the residual boundary interaction in the continuum limit as HB,res ≈ −A[−iξ1(0)ξ2(0)ξ3(0)]Q+iB{ξ1(0)ζ2(0)ζ3(0) + ξ2(0)ζ3(0)ζ1(0) + ξ3(0)ζ1(0)ζ2(0)}Q +C{ζ1(0)ζ2(0)ξ1(0)ξ2(0) + ζ2(0)ζ3(0)ξ2(0)ξ3(0) + ζ3(0)ζ1(0)ξ3(0)ξ1(0)}.(58) Taking into account the residual boundary operators at finite J2and τand going through a systematic Schrieffer-Wolff procedure involving boundary couplings and operators and eventually resorting to the continuum fields, we obtain that the leading boundary perturbation in lattice fermion coordinates is given by