Three-body local correlation function in the Lieb-Liniger model: bosonization approach

H. Smith; M. B. Zvonarev; Vadim V. Cheianov

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Three-body local correlation function in the Lieb-Liniger model: bosonization approach

Vadim V. Cheianov, H. Smith, M. B. Zvonarev

Other Condensed Matter · Submitted 20 Feb 2006 · 14 references

Pith's verdict: A clever and likely correct exact formula for g3 in the Lieb-Liniger model, but the proof of exactness leans on an unproven universality assumption that a referee should press.

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arXiv IDcond-mat/0602468 (version 1)

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CitationH. Smith, M. B. Zvonarev, Vadim V. Cheianov. "Three-body local correlation function in the Lieb-Liniger model: bosonization approach." arXiv:cond-mat/0602468 (2006). https://pith.science/paper/LFA4Z3M6

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Quick take on cond-mat/0602468. The paper delivers something genuinely new: an explicit closed-form expression, Eq. (1.12), for the three-body local correlator g3(γ) in the Lieb-Liniger model, expressed through moments of the Lieb-equation distribution. That is a real advance; prior exact results for local correlators in sine-Gordon did not transfer. The combination of q-boson lattice regularization, algebraic Bethe Ansatz, and contour-independent integral identities is original and worth studying on its own. The intermediate identities—the commutator sums (5.37)-(5.40), the Hellmann-Feynman manipulation, the O(δ^4) expansions—are carried out in commendable detail. The formula contains no fitted parameters. Credit where due.

Where the paper is soft: the claim is exact, but the step from the exact contour identity (5.21) to the bosonized evaluation (5.31) retains only the leading terms of the currents (4.35)-(4.36). The authors write 'after some algebra,' but they do not prove that the h.o.t. operators contribute nothing to the contour integral in the limit (5.30). Since W^(m) is an integral over a macroscopic interval, the usual irrelevant-operator power counting does not automatically apply to the product in the integrand. That is a genuine gap in the argument, not a quibble with Luttinger-liquid universality. The same happens later: (6.23)-(6.25) are asserted after 'rather lengthy algebra,' and the final passage to (1.12) is merely sketched. An algebraic slip there would corrupt the formula, and there is no independent numerical check of (1.12). Minor: (2.63) for |m|>2 is only checked numerically, but the proof needs only |m|=1,2, so this is not a real issue.

On balance, the derivation is coherent and the result is probably correct. The paper deserves serious peer review, but the referee should insist on a direct treatment of the irrelevant-operator corrections to (5.31)—or, failing that, a high-precision numerical test of (1.12) for intermediate γ. I'd bring it to the reading group and cite it, with a caveat.

Referee report

Significance

If correct, Eq. (1.12) would be a rare exact expression for a local correlation function in an interacting one-dimensional Bose gas, and it is directly relevant to the three-body recombination experiment of Ref. [1]. The paper's strategy is appealing: it uses conserved currents and contour identities rather than form-factor expansions, and it explicitly determines all non-universal coefficients (α_m, β_m, Kv) from exact ground-state averages of the same model, so there are no fitted parameters. The treatment of the q-boson lattice and the derivation of the intermediate identities are careful and well organized. The main weakness is that the exactness claim is not fully supported at two load-bearing points: the treatment of subleading operators in the contour identity (5.31) and the omitted final algebra in Section 6.2.

Major objections (2)

[§5.2, Eq. (5.31)] The exact contour integral G^{(n,m)}_Γ is evaluated by substituting the bosonized expressions (4.35)–(4.36) and keeping only the leading terms, while dropping the 'h.o.t.' terms. The separation-of-arguments argument in the limit (5.30) does not by itself control the contribution of irrelevant operators: W^{(m)}(0) is an integral of j^{(m)}_τ over a macroscopic segment, so the product entering Eq. (5.21) contains local operators separated by distances of order l at best, and after integration an irrelevant operator can contribute at order l^0. Since Eq. (5.31) is the key input for Eq. (5.40), and Eq. (5.40) is the input for Eq. (1.12), the exactness of the main formula depends on a quantitative estimate, or an exact elimination, of the h.o.t. contributions. The authors explicitly state in §4.4 that the low-energy description by (4.10) is an assumption; for the claimed exact result this assumption must be upgraded to a proof that the omitted operators do not contribute to the particular contour integrals used.
[§6.2, Eqs. (6.23)–(6.25) and Eq. (1.12)] The two decisive algebraic steps of the final section are not shown. First, the continuum limit of the q-boson local operator combination leading to Eq. (6.23) is stated to follow from 'rather lengthy algebra'. Second, the substitution of expansions (6.7)–(6.8) into Eq. (6.24), the use of Eq. (6.22), and the extraction of the leading δ^4 term are summarized as 'after some algebra'. These steps convert the lattice identity (5.40) into the final closed-form expression, so an algebraic error in either step would change the published formula. Please provide the intermediate expansions, at least in an appendix, or include a computer-algebra verification of the algebraic identities connecting Eqs. (6.24), (6.25), and (1.12).

Minor objections (5)

[§1, Eq. (1.12)] The left-hand side uses n without a prior definition; the density was defined as D in Eq. (1.8). Please define n = D or use D consistently throughout.
[§2.4, Eq. (2.63)] The text honestly states that Eq. (2.63) for |m| > 2 was only checked numerically with Mathematica; since the main calculation uses only m = ±1, ±2, this does not affect Eq. (1.12), but the numerical check should be clearly separated from the analytic results used later.
[§6.1, Eqs. (6.5)–(6.6)] The same symbols ρ and Λ are used for the lattice Bethe-ansatz distribution and its continuum counterpart; the authors note this, but a brief notational remark at the first occurrence would improve readability.
[§6.1, Eq. (6.22)] Equation (6.22) uses an ellipsis for higher-order terms; please state explicitly the order of the omitted terms so that the δ^4 counting in Eq. (6.25) can be followed.
[Figure 2] The contour Γ' and the segment Υ are described in the text, but the figure captions are terse; adding the definitions of l, L, and ǫ directly in the captions would help the reader.

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The paper

The paper claims

An exact formula for the three-body local correlation function in the Lieb-Liniger model is derived by combining a q-boson lattice regularization, bosonization, and contour-independent integral identities.

The reading

Atoms trapped in a very thin wire can move only along one direction. When three atoms meet at nearly the same point, they can form a molecule and the gas loses atoms, an effect called three-body recombination. Physicists want to know how often three atoms meet, which is quantified by the three-body correlation function. The Lieb-Liniger model describes such a one-dimensional gas of bosons that interact only when they touch. Although this model is exactly solvable for its energy levels, calculating correlation functions remained hard.

The authors develop a new route to such calculations. They first put the model on a lattice with a special algebraic structure (the q-boson model), which avoids short-distance singularities. Then they describe the long-distance behavior by an effective free-boson theory, the method known as bosonization. The key trick is a set of contour integrals that do not change when the integration path is deformed. These integrals link short-distance quantities, like the three-body correlation, to the long-distance behavior that the effective theory can compute. In this way the exact short-distance result is obtained without using the approximate effective theory at short distances.

The main result is an explicit formula for the three-body correlation as a simple combination of moments of the quasi-momentum distribution, which is the solution of the standard Lieb integral equation. The formula has no free parameters. It is directly comparable to experiments on reduced three-body recombination in one-dimensional gases.

Extended reading notes

Core claim

The load-bearing result is Eq. (1.12): in the thermodynamic limit at finite density, g3(γ)/n^3 = (3/2) γ ǫ'_4 - (5 ǫ_4)/γ^2 + (1 + γ/2) ǫ'_2 - (2 ǫ_2)/γ - (3 ǫ_2 ǫ'_2)/γ + (9 ǫ_2^2)/γ^2, where ǫ_m are the moments (1.13) of the quasi-momentum distribution σ(z) defined by the Lieb equation (1.14)-(1.15), and ǫ'_m = dǫ_m/dγ. If the paper is correct, this is the exact ground-state expectation value of the local three-body operator ψ†(x)^3 ψ(x)^3.

Load-bearing premise

The derivation assumes that the low-energy sector of the q-boson lattice model is governed by the free boson (Luttinger liquid) Hamiltonian (4.10), and that the microscopic conserved currents j(m) admit the bosonized operator expansions (4.35)-(4.36) with coefficients αm, βm exactly fixed by the zero-frequency response functions (4.48)-(4.49). If irrelevant operators contribute nonvanishing corrections to the contour integrals in Section 5 at arbitrarily large l, the exact identity (5.40) linking the short-distance four-boson average to the bosonized data would fail, and Eq. (1.12) would no longer be exact. This enters in Section 4.4 first paragraph, where the authors say 'we shall exploit an assumption that the low-energy physics... is described by the Hamiltonian (4.10)', and the assumption is used in Sections 5.2-5.3 and 6.2.

Original abstract

We develop a method for the calculation of vacuum expectation values of local operators in the Lieb-Liniger model. This method is based on a set of new identities obtained using integrability and effective theory (``bosonization'') description. We use this method to get an explicit expression for the three-body local correlation function, measured in a recent experiment [1].

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We develop a method for the calculation of vacuum expectation values of local operators in the Lieb-Liniger model. This method is based on a set of new identities obtained using integrability and effective theory (``bosonization'') description. We use this method to get an explicit expression for the three-body local correlation function, measured in a recent experiment [1].

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We develop a method for the calculation of vacuum expectation values of local operators in the Lieb-Liniger model.	contribution 0.99	0.95	0.14	asserted 1.00
s1	This method is based on a set of new identities obtained using integrability and effective theory (``bosonization'') description.	method 0.97	0.89	0.08	asserted 1.00
s2	We use this method to get an explicit expression for the three-body local correlation function, measured in a recent experiment [1].	result 0.95	0.89	0.72	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.92 · evidence from proof or derivation 1.00.

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Figures

**Figure 1.** In (a) we show the integration contour Γ of Eq. (5.8). It c

**Figure 2.** In (a) we show the integration contour Γ used in section 5.2

Figures reproduced from arXiv:cond-mat/0602468 by the authors.

References

Works cited (14)

[1]

Tolra B L, O’Hara K M, Huckans J H, Phillips W D, Rolston S L a nd Porto J V 2004 Observation of Reduced Three-Body Recombination in a Correlated 1D Degenerate Bose Gas Phys Rev Lett 92 190401

View on Pith 2004
[2]

Cheianov V V, Smith H and Zvonarev M B 2005 Exact results fo r three-body correlations in a degenerate one-dimensional Bose gas e-print cond-mat/0506609

View on Pith Pith review arXiv 2005
[3]

Smirnov F A 1992 Form Factors in Completely Integrable Models of Quantum Field Theory (World Scientiﬁc: Singapore)

View on Pith 1992
[4]

Lukyanov S and Zamolodchikov Alexander 1997 Exact expec tation values of local ﬁelds in the quantum sine-Gordon model Nucl Phys. B 493 [FS] 571–587

View on Pith 1997
[5]

Lieb E H and Liniger W 1963 Exact Analysis of an Interactin g Bose Gas. I. The General Solution and the Ground State Phys Rev 130 1605–1616

View on Pith 1963
[6]

Amico L and Korepin V E 2004 Universality of the one-dimen sional Bose gas with delta interaction Annals of Physics 314 496–507

View on Pith 2004
[7]

Bogoliubov N M, Bullough R K and Pang G D 1993 Exact solutio n of a q-boson hopping model Phys Rev B 47 11495–11498

View on Pith 1993
[8]

Bogoliubov N M, Izergin A G and Kitanine N A 1998 Correlati on functions for a strongly correlated boson system Nucl Phys B 516 [FS] 501–528

View on Pith 1998
[9]

Korepin V E, Bogoliubov N M and Izergin A G 1993 Quantum Inverse Scattering Method and Correlation Functions (Cambridge: Cambridge University Press)

View on Pith 1993
[10]

Weinberg S 1995 The quantum theory of ﬁelds (Cambridge: Cambridge Univer- sity Press)

View on Pith 1995
[11]

Olshanii M and Dunjko V 2003 Short-Distance Correlatio n Properties of the Lieb-Liniger System and Momentum Distributions of Trapped One-Dimensional Atomic Gases Phys. Rev. Lett. 91 090401

View on Pith 2003
[12]

Giamarchi T 2004 Quantum Physics in One Dimension (Oxford University Press)

View on Pith 2004
[13]

Gogolin A O, Nersesyan A A and Tsvelik A M 1993 Bosonization and Strongly Correlated Systems (Cambridge: Cambridge University Press)

View on Pith 1993
[14]

Haldane F D M 1981 Demonstration of the “Luttinger Liqui d” character of Bethe- ansatz soluble models of 1-D quantum ﬂuids Phys Lett A 81 153–155 40

View on Pith 1981

How the literature uses it

3 arXiv papers cite this in 12 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 · 5 sentences without a confident label

Uses its method or tool 1 paper

Here we review the (ABA) solution, our exposition follows that of [ref] .

Quantum quenches and Generalized Gibbs Ensemble in a Bethe Ansatz solvable lattice model of interacting bosons · The model and its Bethe Ansatz solution · 0.80

Cites it as background 3 papers

The model can serve as a lattice regularization of the Lieb-Liniger model [ref] ; however, in the present work we focus on the lattice model only.

Quantum quenches and Generalized Gibbs Ensemble in a Bethe Ansatz solvable lattice model of interacting bosons · The model and its Bethe Ansatz solution · 0.83
It can serve as a lattice regularization of the Lieb-Liniger model [ref] , and it has connections to combinatorics [ref] and the theory of symmetric functions [ref] .

Integrability breaking in the one dimensional Bose gas: Atomic losses and energy loss · The $q$-boson model and its scaling limit · 0.86
Concerning the higher-body cases small-coupling and large-coupling expansions were performed in [ref] , whereas the exact ground state value of the three-body correlation was calculated in [ref] .

Local correlations in the 1D Bose gas from a scaling limit of the XXZ chain · Introduction · 0.89

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Tolra B L, O’Hara K M, Huckans J H, Phillips W D, Rolston S L a nd Porto J V 2004 Observation of Reduced Three-Body Recombination in a Correlated 1D Degenerate Bose Gas Phys Rev Lett 92 190401

View on Pith 2004

[2] [2]

Cheianov V V, Smith H and Zvonarev M B 2005 Exact results fo r three-body correlations in a degenerate one-dimensional Bose gas e-print cond-mat/0506609

View on Pith Pith review arXiv 2005

[3] [3]

Smirnov F A 1992 Form Factors in Completely Integrable Models of Quantum Field Theory (World Scientiﬁc: Singapore)

View on Pith 1992

[4] [4]

Lukyanov S and Zamolodchikov Alexander 1997 Exact expec tation values of local ﬁelds in the quantum sine-Gordon model Nucl Phys. B 493 [FS] 571–587

View on Pith 1997

[5] [5]

Lieb E H and Liniger W 1963 Exact Analysis of an Interactin g Bose Gas. I. The General Solution and the Ground State Phys Rev 130 1605–1616

View on Pith 1963

[6] [6]

Amico L and Korepin V E 2004 Universality of the one-dimen sional Bose gas with delta interaction Annals of Physics 314 496–507

View on Pith 2004

[7] [7]

Bogoliubov N M, Bullough R K and Pang G D 1993 Exact solutio n of a q-boson hopping model Phys Rev B 47 11495–11498

View on Pith 1993

[8] [8]

Bogoliubov N M, Izergin A G and Kitanine N A 1998 Correlati on functions for a strongly correlated boson system Nucl Phys B 516 [FS] 501–528

View on Pith 1998

[9] [9]

Korepin V E, Bogoliubov N M and Izergin A G 1993 Quantum Inverse Scattering Method and Correlation Functions (Cambridge: Cambridge University Press)

View on Pith 1993

[10] [10]

Weinberg S 1995 The quantum theory of ﬁelds (Cambridge: Cambridge Univer- sity Press)

View on Pith 1995

[11] [11]

Olshanii M and Dunjko V 2003 Short-Distance Correlatio n Properties of the Lieb-Liniger System and Momentum Distributions of Trapped One-Dimensional Atomic Gases Phys. Rev. Lett. 91 090401

View on Pith 2003

[12] [12]

Giamarchi T 2004 Quantum Physics in One Dimension (Oxford University Press)

View on Pith 2004

[13] [13]

Gogolin A O, Nersesyan A A and Tsvelik A M 1993 Bosonization and Strongly Correlated Systems (Cambridge: Cambridge University Press)

View on Pith 1993

[14] [14]

Haldane F D M 1981 Demonstration of the “Luttinger Liqui d” character of Bethe- ansatz soluble models of 1-D quantum ﬂuids Phys Lett A 81 153–155 40

View on Pith 1981