Renormalization group analysis for bosonization coefficients in half-odd-integer Kitaev spin chains

Chao Xu; Jianxun Li; Wang Yang

arxiv: 2605.03582 · v2 · pith:OX44KKU5new · submitted 2026-05-05 · ❄️ cond-mat.str-el

Renormalization group analysis for bosonization coefficients in half-odd-integer Kitaev spin chains

Jianxun Li , Chao Xu , Wang Yang This is my paper

Pith reviewed 2026-05-20 23:55 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords bosonizationrenormalization groupKitaev-Gamma chainKitaev-Heisenberg-Gamma chainspin-S chainssymmetry breakingDMRG

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The pith

Breaking of emergent continuous symmetries in bosonization formulas for Kitaev-Gamma chains scales as 1/S in the large-S limit

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies renormalization group analysis to bosonization formulas in spin-S Kitaev-Gamma and Kitaev-Heisenberg-Gamma chains for half-odd-integer S in the region with K negative, Gamma positive and J positive. It establishes that effects tied to the breaking of emergent continuous symmetries scale as 1/S in the large-S limit and finds qualitative agreement with DMRG numerics on Kitaev-Gamma chains. Symmetry analysis for the Kitaev-Heisenberg-Gamma case identifies ten independent bosonization coefficients, five of which the RG flow predicts to be independent of the Heisenberg coupling to linear order. The results are positioned as input for mapping out magnetic ordering in two-dimensional Kitaev models through quasi-one-dimensional ladders.

Core claim

Based on a renormalization group (RG) analysis, we study the bosonization formulas in spin-S Kitaev-Gamma and Kitaev-Heisenberg-Gamma chains in the (K<0,Γ>0,J>0) parameter region, where S is a half-odd integer. We find that the effects associated with the breaking of emergent continuous symmetries in bosonization formulas scale as 1/S in the large-S limit, which is in qualitative agreement with DMRG numerical results for Kitaev-Gamma chains. In Kitaev-Heisenberg-Gamma chains, symmetry analysis reveals ten independent bosonization coefficients, five of which are predicted by the RG analysis to have no dependence on the Heisenberg coupling up to linear order.

What carries the argument

Renormalization group flow applied to the bosonization mapping that tracks the breaking of emergent continuous symmetries

If this is right

The 1/S scaling becomes systematically smaller for larger half-odd-integer S, improving the accuracy of the continuous-symmetry bosonization formulas.
Five of the ten independent bosonization coefficients in Kitaev-Heisenberg-Gamma chains remain independent of the Heisenberg coupling to linear order in the RG flow.
The RG-derived coefficients supply concrete starting values for quasi-one-dimensional ladders used to infer ordering tendencies in two-dimensional Kitaev models.
Qualitative match with existing DMRG data on Kitaev-Gamma chains supports the validity of the RG treatment of symmetry breaking.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The 1/S scaling supplies a practical error estimate that can be used to extrapolate bosonization results from finite-S ladders to the classical large-S regime.
Independence of selected coefficients from J may allow simplified effective models when studying the competition between Kitaev and Heisenberg terms.
The same RG procedure could be applied to other one-dimensional spin chains with emergent continuous symmetries to obtain analogous scaling predictions.

Load-bearing premise

The bosonization mapping and the renormalization group flow remain valid in the (K<0, Γ>0, J>0) parameter region for half-odd-integer S, with the emergent continuous symmetries and their breaking captured accurately by the effective field theory.

What would settle it

Direct numerical extraction of the bosonization coefficients for successive half-odd-integer values of S to test whether the size of the symmetry-breaking corrections decreases proportionally to 1/S.

Figures

Figures reproduced from arXiv: 2605.03582 by Chao Xu, Jianxun Li, Wang Yang.

**Figure 2.** Figure 2: FIG. 2: Bond structures (a) before and (b) after the six view at source ↗

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 3.** Figure 3: FIG. 3: Vertex of (a) interaction and (b) scaling field. view at source ↗

**Figure 4.** Figure 4: FIG. 4: Diagrams for the renormalizations of scaling fields. view at source ↗

**Figure 5.** Figure 5: (a) shows s x i (j) (i = 0, 1, 2) as functions of rL (r = 3j) on a log-log scale, in which DMRG numerics are performed for ϕ = 0.8π on systems of L = 144 sites under periodic boundary conditions, where K = cos(ϕ), Γ = sin(ϕ). In view at source ↗

read the original abstract

Based on a renormalization group (RG) analysis, we study the bosonization formulas in spin-$S$ Kitaev-Gamma and Kitaev-Heisenberg-Gamma chains in the $(K<0,\Gamma>0,J>0)$ parameter region, where $S$ is a half-odd integer. We find that the effects associated with the breaking of emergent continuous symmetries in bosonization formulas scale as $1/S$ in the large-$S$ limit, which is in qualitative agreement with DMRG numerical results for Kitaev-Gamma chains. In Kitaev-Heisenberg-Gamma chains, symmetry analysis reveals ten independent bosonization coefficients, five of which are predicted by the RG analysis to have no dependence on the Heisenberg coupling up to linear order. Our work may offer valuable input for determining magnetic ordering tendencies in two-dimensional Kitaev spin models within a quasi-one-dimensional approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

RG analysis shows symmetry-breaking corrections scale as 1/S and five of ten bosonization coefficients stay independent of J to linear order in these Kitaev chains.

read the letter

The key points are the 1/S scaling for effects from broken emergent symmetries and the prediction that half the coefficients in the Kitaev-Heisenberg-Gamma case carry no linear J dependence. The paper takes the usual bosonization ansatz for half-odd-integer S chains in the (K<0, Γ>0, J>0) window, classifies operators allowed by the lattice symmetries plus the half-odd constraint, and runs a standard perturbative RG flow around the fixed-point Hamiltonian to extract the leading corrections. That produces the ten-coefficient count and the specific independence result, with a qualitative DMRG check for the pure Kitaev-Gamma case that lines up with the scaling claim. The symmetry counting and the linear-order independence both follow directly from the operator analysis rather than from fitting choices. This supplies concrete analytical constraints that could feed into quasi-1D modeling of 2D Kitaev systems, which is the stated motivation. The derivations use established techniques but the concrete predictions on coefficient behavior appear new relative to the cited literature. The weakest assumption—that the bosonization map remains valid in the chosen parameter region—is stated explicitly and is not contradicted by the calculations shown. The DMRG comparison stays qualitative with no error estimates or quantitative fits, and the large-S expansion is applied to half-odd integers where S=1/2 or 3/2 are the practical values, so higher-order terms could matter in real materials. No internal inconsistencies show up in the operator classification or the flow equations. This is aimed at condensed-matter theorists who build effective models for Kitaev magnets or use 1D chains to approximate 2D behavior. A reader working on bosonization or RG in frustrated spin chains would find the coefficient constraints directly usable. It deserves a serious referee because the claims are specific, the methods are standard but applied cleanly, and the results are in principle checkable against numerics.

Referee Report

0 major / 3 minor

Summary. The manuscript applies renormalization group analysis to bosonization formulas for half-odd-integer spin-S Kitaev-Gamma and Kitaev-Heisenberg-Gamma chains in the (K<0, Γ>0, J>0) regime. It concludes that effects from the breaking of emergent continuous symmetries scale as 1/S in the large-S limit, in qualitative agreement with DMRG results for the Kitaev-Gamma case. Symmetry analysis for the Kitaev-Heisenberg-Gamma model identifies ten independent bosonization coefficients, five of which are predicted to be independent of the Heisenberg coupling J to linear order in the RG flow. The work suggests this may inform magnetic ordering studies in two-dimensional Kitaev models via quasi-one-dimensional approaches.

Significance. If the RG analysis and bosonization mapping hold, the results provide a systematic understanding of how microscopic parameters enter effective field theory coefficients in these frustrated chains, with the 1/S scaling offering a concrete bridge to the large-S limit. The prediction of J-independent coefficients constitutes a falsifiable claim that can guide future numerical work. The symmetry classification and perturbative RG flow around the fixed-point Hamiltonian are standard but applied here to a relevant model class.

minor comments (3)

[Abstract and DMRG comparison section] The abstract states qualitative agreement with DMRG but does not specify which observables (e.g., specific correlation functions or ordering tendencies) are compared or the range of S values examined; this should be clarified in the main text or a dedicated comparison section.
[Symmetry analysis for Kitaev-Heisenberg-Gamma chains] While the symmetry count yielding ten independent coefficients follows from lattice symmetries and the half-odd-integer constraint, an explicit listing of the allowed operators or the basis used for the classification would improve reproducibility.
[Introduction or methods] The manuscript should include a brief statement on the validity range of the bosonization mapping in the (K<0, Γ>0, J>0) region, perhaps with a reference to prior works establishing its applicability for half-odd-integer S.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately captures our RG analysis of bosonization coefficients in half-odd-integer Kitaev-Gamma and Kitaev-Heisenberg-Gamma chains, including the 1/S scaling of emergent symmetry-breaking effects and the identification of five J-independent coefficients to linear order in the flow. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation chain begins with lattice symmetries and the half-odd-integer constraint to enumerate ten independent bosonization coefficients, then applies a standard perturbative RG flow to the symmetry-allowed operators around the fixed-point Hamiltonian. The claimed independence of five coefficients from the Heisenberg coupling to linear order follows directly from the structure of the RG beta functions and the operator classification, without any reduction to fitted inputs or self-referential definitions. The 1/S scaling of emergent symmetry breaking is obtained as a perturbative correction in the large-S limit and is compared qualitatively to external DMRG data rather than used as an input. No self-citation is load-bearing for the central claims, and the bosonization mapping is treated as an established starting point whose validity is stated as an assumption rather than derived internally. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities listed. The analysis implicitly assumes standard bosonization and RG applicability in the given parameter regime.

axioms (2)

domain assumption Bosonization formulas remain applicable for half-odd-integer S in the (K<0, Γ>0, J>0) region
Central to the RG analysis described in the abstract.
domain assumption Emergent continuous symmetries and their breaking are correctly captured by the effective theory
Required for the 1/S scaling claim.

pith-pipeline@v0.9.0 · 5682 in / 1394 out tokens · 27985 ms · 2026-05-20T23:55:14.449873+00:00 · methodology

Review history (3 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

RG analysis predicts that C1/C2 = 1 - (ln bs / 0.1π) (ΔΓ/Γ) / (S+1) ... five of which are predicted by the RG analysis to have no dependence on the Heisenberg coupling up to linear order
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

low energy physics ... described by the SU(2)1 WZW model ... Luttinger liquid theory

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

[1]

T: (S x i , Sy i , Sz i )→(−S x i ,−S y i ,−S z i )

work page
[2]

R I I: (S x i , Sy i , Sz i )→(−S z 10−i,−S y 10−i,−S x 10−i)

work page
[3]

R aTa : (Sx i , Sy i , Sz i )→(S z i+1, Sx i+1, Sy i+1)

work page
[4]

R(ˆx, π) : (S x i , Sy i , Sz i )→(S x i ,−S y i ,−S z i )

work page
[5]

R(ˆy, π) : (S x i , Sy i , Sz i )→(−S x i , Sy i ,−S z i )

work page
[6]

R(ˆz, π) : (S x i , Sy i , Sz i )→(−S x i ,−S y i , Sz i ).(35) We note that as discussed in Ref. 66,G′ KΓ is a nonsym- morphic group in the sense of the following short exact sequence 1→ ⟨T 3a⟩ →G ′ KΓ →O h →1,(36) in which⟨T 3a⟩is the translational group generated by spatial translation⟨T 3a⟩of three lattice sites, andO h is the full octahedra group whi...

work page
[7]

4: Diagrams for the renormalizations of scaling fields

RG flows for uniform components FIG. 4: Diagrams for the renormalizations of scaling fields. We start with analyzing how the uniform components of the scaling fields get renormalized by the diagram in Fig. 4 (a) when the cutoff is lowered from Λ0/bto Λ 0/(b+ ∆b), whereb <Λ 0/Λs and ∆b≪1. Calculations show that Fig. 4 (a) contributes the following term δΓ ...

work page
[8]

RG flows for staggered components For the staggered components of the scaling fields, Fig. 4 (a) contributes the following term δΓ 1 t λ(S) s,jlδαγ lnb· Z dτ X n hα s,l(τ, n)S γ s,i(τ, n),(57) in whichλ (S) s,jl is given by λ(S) s,jl =n cΛs jl.(58) The expression of Λ s jl in Eq. (58) is given by 1 t Λs jl lnb=− a 6 3X m=1 e−i 2π 3 (m+ 1 2 )(j−l) × Z Λ/b ...

work page
[9]

|K| −Γ Γ +O |K| −Γ Γ 2 D1 D2 = 1 + πlnb s S+ 1 (λu 0 −λ u

work page
[10]

(77) Using Eq

|K| −Γ Γ +O |K| −Γ Γ 2 . (77) Using Eq. (54,63) and retaining only the leading order terms, we obtain Eq. (12). In particular, notice from Eq. (77) that the deviations ofC 1/C2 andD 1/D2 away from 1 scale as 1/Sin the large-Slimit. Therefore, in the semiclassical limitS≫1, bothC 1/C2 andD 1/D2 approach 1. D. DMRG numerics Spin correlation functions in the...

work page
[11]

Plugging Eqs

J Γ ση =b πlnb s S+ 1 √ 2 3 (λη 0 −λ η 1)∆Γ Γ δη =−b πlnb s S+ 1 2 3(λη 0 −λ η 1)∆Γ −3J Γ νη =b 1− πlnb s S+ 1 (λη 0 + 2λη 1) 2 3 ∆Γ Γ −2 J Γ ρη =σ η,(117) in whichη=D, Con the left correspond toη= u,s on the right; andO( ∆2 Γ Γ2 ),O( ∆ΓJ Γ2 ),O( J2 Γ2 ) terms are ne- glected. Plugging Eqs. (54,63) into Eq. (117), we obtain Eqs. (21,22). Detailed derivati...

work page
[12]

The site with 2S−1 electrons likewise has maximal spinS− 1

work page
[13]

(A7) becomes H(2) eff,ij = P H2 1,ijP −U 2S+ 1 2 .(A11) Because the hopping is diagonal in the color index, only virtual processes with the same coloracontribute

Thus the intermediate-state energy is E(ij) int =−2U S− 1 2 S+ 1 2 .(A9) Therefore, E(ij) 0 −E (ij) int =−U 2S+ 1 2 ,(A10) and Eq. (A7) becomes H(2) eff,ij = P H2 1,ijP −U 2S+ 1 2 .(A11) Because the hopping is diagonal in the color index, only virtual processes with the same coloracontribute. The two equivalent processesi→j→iandj→i→jgive the same contribu...

work page
[14]

4 (a) In this appendix, we evaluate the diagram in Fig

Diagram in Fig. 4 (a) In this appendix, we evaluate the diagram in Fig. 4 (a). We focus on the uniform component of the scaling field, and the treatment for the staggered component is exactly similar. The calculation for RG flow equations of staggered components is exactly similar. We first rewrite the lattice interaction and the external-field coupling i...

work page
[15]

4 (b) Next we consider the one-loop correction to the scaling field generated by theδ Γ term in Fig

Diagram in Fig. 4 (b) Next we consider the one-loop correction to the scaling field generated by theδ Γ term in Fig. 4 (b). We again focus on the uniform component of the scaling fields. The analysis for the staggered component is exactly similar. 21 The contributionF loop,b is given by Floop,b = δΓ 9Lβ X m,m′ X ⃗k,⃗ p,⃗ q,⃗k′,⃗ q′ X ω1,ω2,ω3,ω4,ω5,ω6,ω7 ...

work page
[16]

Explicit RG flow equations The RG flow equations of the scaling fields for spin-SKitaev-Gamma chain are given by (η=u,s) dhx η,1 dlnb = (1−2Sδ Γλη 1)hx η,1 −2Sδ Γλη 0hx η,2 −2Sδ Γλη 1hx η,3, dhx η,2 dlnb = (1−2Sδ Γλη 1)hx η,2 −2Sδ Γλη 0hx η,1 −2Sδ Γλη 1hx η,3, dhx η,3 dlnb =h x η,3,(B26) dhz η,1 dlnb =h z η,1, dhz η,2 dlnb = (1−2Sδ Γλη 1)hz η,2 −2Sδ Γλη 0...

work page
[17]

0 2 3(λη 0 + 2λη 1)   .(C26) One also has A(J) 1 +A (J) 2 =   1 0 0 0−1 0 0 0−2   ,(C27) hence (λη 0 +λ η 1)(A(J) 1 +A (J) 2 ) + 2λη 1A(J) 3 =   λη 0 −λ η 1 0 0 0−(λ η 0 −λ η

work page
[18]

C” forη= s and “D

0 0 0−2(λ η 0 + 2λη 1)   .(C28) Therefore M η 2 =b×   1− 2Slnb s t h λη 0+5λη 1 3 δΓ + (λη 0 −λ η 1)δ J i 0 2Slnb s t √ 2 3 (λη 0 −λ η 1)δ Γ 0 1− 2Slnb s t [(λη 0 +λ η 1)δ Γ −(λ η 0 −λ η 1)δ J] 0 2Slnb s t √ 2 3 (λη 0 −λ η 1)δ Γ 0 1− 2Slnb s t (λη 0 + 2λη 1) 2 3 δΓ −2δ J   . (C29) 26 The bosonization coefficients are givenin Eq. (17). Sinc...

work page
[19]

Explicit forms of bosonization formulas The nonsymmorphic bosonization formulas in theOU 6 frame are given by S′′x 1+3n = (λ D + 3 4 δD)J x + √ 3 4 δDJ y − 1 2 ρDJ z + (−)1+n (λC + 3 4 δC)N x + √ 3 4 δCN y − 1 2 ρCN z , S′′y 1+3n = √ 3 4 δDJ x + (λD + 1 4 δD)J y + √ 3 2 ρDJ z + (−)1+n √ 3 4 δCN x + (λC + 1 4 δC)N y + √ 3 2 ρCN z , S′′z 1+3n =− 1 2 σDJ x +...

work page
[20]

Magnetostriction in the $J$-$K$-$\Gamma$ model: Application of the numerical linked cluster expansion

Reducing to theJ= 0case Notice that the bosonization fieldsJ α, N α (α=x, y, z) are defined in theOU 6 frame. Let’s rotateJ α, N α to the U6 frame, by definingJ α,N α as (J x J y J z) = (J x J y J z)O, (N x N y N z) = (N x N y N z)O,(D4) where the matrixOis defined in Eq. (20). Then we have (S′x 1+3n S′y 1+3n S′z 1+3n) = (J x J y J z)OD1O−1 −(−) n(N x N y...

work page internal anchor Pith review Pith/arXiv arXiv 2006

[1] [1]

T: (S x i , Sy i , Sz i )→(−S x i ,−S y i ,−S z i )

work page

[2] [2]

R I I: (S x i , Sy i , Sz i )→(−S z 10−i,−S y 10−i,−S x 10−i)

work page

[3] [3]

R aTa : (Sx i , Sy i , Sz i )→(S z i+1, Sx i+1, Sy i+1)

work page

[4] [4]

R(ˆx, π) : (S x i , Sy i , Sz i )→(S x i ,−S y i ,−S z i )

work page

[5] [5]

R(ˆy, π) : (S x i , Sy i , Sz i )→(−S x i , Sy i ,−S z i )

work page

[6] [6]

R(ˆz, π) : (S x i , Sy i , Sz i )→(−S x i ,−S y i , Sz i ).(35) We note that as discussed in Ref. 66,G′ KΓ is a nonsym- morphic group in the sense of the following short exact sequence 1→ ⟨T 3a⟩ →G ′ KΓ →O h →1,(36) in which⟨T 3a⟩is the translational group generated by spatial translation⟨T 3a⟩of three lattice sites, andO h is the full octahedra group whi...

work page

[7] [7]

4: Diagrams for the renormalizations of scaling fields

RG flows for uniform components FIG. 4: Diagrams for the renormalizations of scaling fields. We start with analyzing how the uniform components of the scaling fields get renormalized by the diagram in Fig. 4 (a) when the cutoff is lowered from Λ0/bto Λ 0/(b+ ∆b), whereb <Λ 0/Λs and ∆b≪1. Calculations show that Fig. 4 (a) contributes the following term δΓ ...

work page

[8] [8]

RG flows for staggered components For the staggered components of the scaling fields, Fig. 4 (a) contributes the following term δΓ 1 t λ(S) s,jlδαγ lnb· Z dτ X n hα s,l(τ, n)S γ s,i(τ, n),(57) in whichλ (S) s,jl is given by λ(S) s,jl =n cΛs jl.(58) The expression of Λ s jl in Eq. (58) is given by 1 t Λs jl lnb=− a 6 3X m=1 e−i 2π 3 (m+ 1 2 )(j−l) × Z Λ/b ...

work page

[9] [9]

|K| −Γ Γ +O |K| −Γ Γ 2 D1 D2 = 1 + πlnb s S+ 1 (λu 0 −λ u

work page

[10] [10]

(77) Using Eq

|K| −Γ Γ +O |K| −Γ Γ 2 . (77) Using Eq. (54,63) and retaining only the leading order terms, we obtain Eq. (12). In particular, notice from Eq. (77) that the deviations ofC 1/C2 andD 1/D2 away from 1 scale as 1/Sin the large-Slimit. Therefore, in the semiclassical limitS≫1, bothC 1/C2 andD 1/D2 approach 1. D. DMRG numerics Spin correlation functions in the...

work page

[11] [11]

Plugging Eqs

J Γ ση =b πlnb s S+ 1 √ 2 3 (λη 0 −λ η 1)∆Γ Γ δη =−b πlnb s S+ 1 2 3(λη 0 −λ η 1)∆Γ −3J Γ νη =b 1− πlnb s S+ 1 (λη 0 + 2λη 1) 2 3 ∆Γ Γ −2 J Γ ρη =σ η,(117) in whichη=D, Con the left correspond toη= u,s on the right; andO( ∆2 Γ Γ2 ),O( ∆ΓJ Γ2 ),O( J2 Γ2 ) terms are ne- glected. Plugging Eqs. (54,63) into Eq. (117), we obtain Eqs. (21,22). Detailed derivati...

work page

[12] [12]

The site with 2S−1 electrons likewise has maximal spinS− 1

work page

[13] [13]

(A7) becomes H(2) eff,ij = P H2 1,ijP −U 2S+ 1 2 .(A11) Because the hopping is diagonal in the color index, only virtual processes with the same coloracontribute

Thus the intermediate-state energy is E(ij) int =−2U S− 1 2 S+ 1 2 .(A9) Therefore, E(ij) 0 −E (ij) int =−U 2S+ 1 2 ,(A10) and Eq. (A7) becomes H(2) eff,ij = P H2 1,ijP −U 2S+ 1 2 .(A11) Because the hopping is diagonal in the color index, only virtual processes with the same coloracontribute. The two equivalent processesi→j→iandj→i→jgive the same contribu...

work page

[14] [14]

4 (a) In this appendix, we evaluate the diagram in Fig

Diagram in Fig. 4 (a) In this appendix, we evaluate the diagram in Fig. 4 (a). We focus on the uniform component of the scaling field, and the treatment for the staggered component is exactly similar. The calculation for RG flow equations of staggered components is exactly similar. We first rewrite the lattice interaction and the external-field coupling i...

work page

[15] [15]

4 (b) Next we consider the one-loop correction to the scaling field generated by theδ Γ term in Fig

Diagram in Fig. 4 (b) Next we consider the one-loop correction to the scaling field generated by theδ Γ term in Fig. 4 (b). We again focus on the uniform component of the scaling fields. The analysis for the staggered component is exactly similar. 21 The contributionF loop,b is given by Floop,b = δΓ 9Lβ X m,m′ X ⃗k,⃗ p,⃗ q,⃗k′,⃗ q′ X ω1,ω2,ω3,ω4,ω5,ω6,ω7 ...

work page

[16] [16]

Explicit RG flow equations The RG flow equations of the scaling fields for spin-SKitaev-Gamma chain are given by (η=u,s) dhx η,1 dlnb = (1−2Sδ Γλη 1)hx η,1 −2Sδ Γλη 0hx η,2 −2Sδ Γλη 1hx η,3, dhx η,2 dlnb = (1−2Sδ Γλη 1)hx η,2 −2Sδ Γλη 0hx η,1 −2Sδ Γλη 1hx η,3, dhx η,3 dlnb =h x η,3,(B26) dhz η,1 dlnb =h z η,1, dhz η,2 dlnb = (1−2Sδ Γλη 1)hz η,2 −2Sδ Γλη 0...

work page

[17] [17]

0 2 3(λη 0 + 2λη 1)   .(C26) One also has A(J) 1 +A (J) 2 =   1 0 0 0−1 0 0 0−2   ,(C27) hence (λη 0 +λ η 1)(A(J) 1 +A (J) 2 ) + 2λη 1A(J) 3 =   λη 0 −λ η 1 0 0 0−(λ η 0 −λ η

work page

[18] [18]

C” forη= s and “D

0 0 0−2(λ η 0 + 2λη 1)   .(C28) Therefore M η 2 =b×   1− 2Slnb s t h λη 0+5λη 1 3 δΓ + (λη 0 −λ η 1)δ J i 0 2Slnb s t √ 2 3 (λη 0 −λ η 1)δ Γ 0 1− 2Slnb s t [(λη 0 +λ η 1)δ Γ −(λ η 0 −λ η 1)δ J] 0 2Slnb s t √ 2 3 (λη 0 −λ η 1)δ Γ 0 1− 2Slnb s t (λη 0 + 2λη 1) 2 3 δΓ −2δ J   . (C29) 26 The bosonization coefficients are givenin Eq. (17). Sinc...

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[19] [19]

Explicit forms of bosonization formulas The nonsymmorphic bosonization formulas in theOU 6 frame are given by S′′x 1+3n = (λ D + 3 4 δD)J x + √ 3 4 δDJ y − 1 2 ρDJ z + (−)1+n (λC + 3 4 δC)N x + √ 3 4 δCN y − 1 2 ρCN z , S′′y 1+3n = √ 3 4 δDJ x + (λD + 1 4 δD)J y + √ 3 2 ρDJ z + (−)1+n √ 3 4 δCN x + (λC + 1 4 δC)N y + √ 3 2 ρCN z , S′′z 1+3n =− 1 2 σDJ x +...

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[20] [20]

Magnetostriction in the $J$-$K$-$\Gamma$ model: Application of the numerical linked cluster expansion

Reducing to theJ= 0case Notice that the bosonization fieldsJ α, N α (α=x, y, z) are defined in theOU 6 frame. Let’s rotateJ α, N α to the U6 frame, by definingJ α,N α as (J x J y J z) = (J x J y J z)O, (N x N y N z) = (N x N y N z)O,(D4) where the matrixOis defined in Eq. (20). Then we have (S′x 1+3n S′y 1+3n S′z 1+3n) = (J x J y J z)OD1O−1 −(−) n(N x N y...

work page internal anchor Pith review Pith/arXiv arXiv 2006