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On the finite size corrections of anti-ferromagnetic anomalous dimensions in ${\cal N}=4$ SYM

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Giovanni Feverati, Davide Fioravanti, Paolo Grinza, Marco Rossi

High Energy Physics - Theory · Submitted 20 Feb 2006 · 46 references

Pith's reading: Finite-size corrections of order 1/L to anti-ferromagnetic anomalous dimensions in N=4 SYM are derived from Bethe Ansatz NLIEs, including pi^2/(2L) in the SO(6) sector.

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arXiv IDhep-th/0602189 (version 2)
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CitationDavide Fioravanti, Giovanni Feverati, Marco Rossi, Paolo Grinza. "On the finite size corrections of anti-ferromagnetic anomalous dimensions in ${\cal N}=4$ SYM." arXiv:hep-th/0602189 (2006). https://pith.science/paper/VLDNTG2P
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Claim paper
The paper

The paper claims

Finite-size corrections of order 1/L to anti-ferromagnetic anomalous dimensions in N=4 SYM are derived from Bethe Ansatz NLIEs, including pi^2/(2L) in the SO(6) sector.

The reading

Composite operators in N=4 super Yang-Mills theory get quantum corrections to their scaling dimensions. In the planar limit, these dimensions can be computed as energies of an integrable spin chain. Most work focuses on the leading term when the chain length L is large. This paper studies the first correction, of order 1/L, for the operator with the largest anomalous dimension, the so-called anti-ferromagnetic state.

The authors use a technique from integrable models, the non-linear integral equation (NLIE). Starting from the Bethe equations that determine the spin chain spectrum, they rewrite the energy as an integral involving a counting function Z(x) that satisfies a single or coupled non-linear integral equation. This equation is equivalent to the original Bethe equations, but is easier to analyze at large L. Using a standard derivative lemma, they extract the 1/L correction analytically.

For the one-loop SU(2) sector, they recover the known value pi^2/(6L). For the multi-loop SU(2) sector, using the conjectured all-loop Bethe ansatz of Beisert, Dippel and Staudacher, they derive a 1/L coefficient written in terms of Bessel functions. For the SO(6) sector at one loop, they obtain pi^2/(2L), consistent with a three-dimensional central charge. Numerical solution of the NLIEs supports the formulas. The multi-loop results inherit the uncertainty of the underlying Bethe ansatz, which is known to break down at wrapping order g^(2L).

Extended reading notes

Core claim

Equations (5.8), (5.15) and (5.23) state that as L goes to infinity the anti-ferromagnetic energy is E = 2L ln 2 - pi H + pi^2/(6L) + o(1/L) for the Heisenberg chain; E = (8*pi/sqrt(lambda)) L integral J0 J1/(k(e^k+1)) dk + 4i/(L sqrt(lambda)) J1(-i sqrt(lambda)/2) J0(i sqrt(lambda)/2) pi^2/6 + o(1/L) for the multi-loop BDS chain; and E = 2L(pi/2 + ln 2) + pi^2/(2L) + o(1/L) for the SO(6) chain. If true, these are the leading finite-size corrections to the largest anomalous dimensions of the corresponding scalar operators in N=4 SYM.

Load-bearing premise

Section 3 builds all multi-loop SU(2) results on the asymptotic Bethe Ansatz of Beisert, Dippel and Staudacher (Eq. 3.1), which the authors themselves state breaks down at order g^(2L). The 1/L correction in (5.15) is a prediction for N=4 SYM only if this Bethe ansatz remains valid at the relevant finite L; the NLIE method would still be correct for whatever the true Bethe equations are.

Original abstract

Non-linear integral equations derived from Bethe Ansatz are used to evaluate finite size corrections to the highest (i.e. {\it anti-ferromagnetic}) and immediately lower anomalous dimensions of scalar operators in ${\cal N}=4$ SYM. In specific, multi-loop corrections are computed in the SU(2) operator subspace, whereas in the general SO(6) case only one loop calculations have been finalised. In these cases, the leading finite size corrections are given by means of explicit formul\ae and compared with the exact numerical evaluation. In addition, the method here proposed is quite general and especially suitable for numerical evaluations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,640 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.90 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.72 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Non-linear integral equations derived from Bethe Ansatz are used to evaluate finite size corrections to the highest (i.e. {\it anti-ferromagnetic}) and immediately lower anomalous dimensions of scalar operators in ${\cal N}=4$ SYM. In specific, multi-loop corrections are computed in the SU(2) operator subspace, whereas in the general SO(6) case only one loop calculations have been finalised. In these cases, the leading finite size corrections are given by means of explicit formul\ae and compared with the exact numerical evaluation. In addition, the method here proposed is quite general and especially suitable for numerical evaluations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Non-linear integral equations derived from Bethe Ansatz are used to evaluate finite size corrections to the highest (i.e. {\it anti-ferromagnetic}) and immediately lower anomalous dimensions of scalar operators in ${\cal N}=4$ SYM. method 0.52 0.88 0.32 asserted 1.00
s1 In specific, multi-loop corrections are computed in the SU(2) operator subspace, whereas in the general SO(6) case only one loop calculations have been finalised. limitation 0.73 0.75 0.10 asserted 1.00
s2 In these cases, the leading finite size corrections are given by means of explicit formul\ae and compared with the exact numerical evaluation. result 0.93 0.90 0.22 asserted 1.00
s3 In addition, the method here proposed is quite general and especially suitable for numerical evaluations. interpretation 0.43 0.70 0.11 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.87 · evidence from proof or derivation 0.64.

What was examined, and what was not

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Plot of ∆E(L)L− π 2 6 versus L for the state I = (−1, 0, 1, 2) of the Heisenberg spin chain. by I = (−1, 0, 1, 2) (see also
Figure 2. Plot of the counting function Z(x) versus x for the Heisenberg spin chain with L = 12 sites. The position of the four holes quantised by I = (−1, 0, 1, 2) is indicated by the small crosses. The finite size corrections to the anti-ferromagnetic vacuum of the Heisenberg chain have been extensively studied in condensed matter literature [44, 36]. Taking into account the first logarithmic corrections we have E = 2L ln 2 + π 2 6L  1 + 3 8 1 ln3 L + k2 ln ln L ln4 L + k3 ln4 L  + . . . , (6.3) where the numerical values of the constants k2 and k3 have been obtained by a fit
Figure 3. Plot of ∆E(L, λ) L versus L for the state I = (−1 , 0 , 1 , 2) of the many-loops spin chain with λ = 50. However, in close analogy with the Heisenberg chain, the proposed functional form 32
Figure 4. Plot of ∆E(L) L − π 2 2 versus L for the anti-ferromagnetic state of the SO(6) spin chain. loop correction in the SU(2) scalar sub-sector: this effort was at least useful to warm up and check the entire machinery. These NLIEs, equipped with the quantisation conditions for the holes, are totally equivalent to the Bethe equations we started with. But of course they are much more effective for both the analytic and numeric computations. For simplicity’s sake, we limited our analysis to the case of real roots, although the introduced formalism can be easily extended to allow for the p

Figures reproduced from arXiv:hep-th/0602189 by the authors.

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References
How the literature uses it

3 arXiv papers cite this in 4 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 2 sentences without a confident label

Uses its method or tool 1 paper
  • The non-linear integral equation approach developed in the context of $=4$ SYM in e.g. [ref] would be useful.

    Giant Spinons · Conclusions and Discussions · 0.94

Cites it as background 1 paper
  • After solving for $ρ_c$, one obtains a non-linear integral equation[Such type of integral equations first appeared as alternative formulation of the Thermodinamic Bethe Ansatz without strings [ref] , and most recently in supersymmetric gauge theories [ref] .

    Semi-classical analysis of the inner product of Bethe states · One-dimensional effective theory in the semiclassical limit · 0.85

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