Correlations in Low-Dimensional Quantum Gases

We present a systematic procedure to extract the perturbative series for the ground state energy density in the Lieb-Liniger and Gaudin-Yang models, starting from the Bethe ansatz solution. This makes it possible to calculate explicitly the coefficients of these series and to study their large order behavior. We find that both series diverge factorially and are not Borel summable. In the case of the Gaudin-Yang model, the first Borel singularity is determined by the non-perturbative energy gap. This provides a new perspective on the Cooper instability.

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“…More recently, precise numerical calculations have made it possible to guess the analytic form of some additional coefficients [13][14][15].…”

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confidence: 99%

Exact Perturbative Results for the Lieb–Liniger and Gaudin–Yang Models

Mariño

Reis

2019

J Stat Phys

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“…However, the first three terms were known analytically in this limit for a long time [13][14][15], until a recent work [16] (see also Ref. [17]) which contains conjectures for the analytical form of the first six terms in the ground state energy and the general structure of the following ones based on the double extrapolation of numerical solution of the discrete Bethe ansatz equations.…”

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confidence: 99%

“…. , a 8 which had at least 46 correct digits were sufficient to use the integer relation algorithm [18,19] that recognizes the approximate number as a certain rational combination of basis vectors that are 1, ζfunctions [16,17], but also the combinations of ζ-functions, contrary to the conjecture of Ref. [16].…”

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confidence: 99%

Conjectures about the ground-state energy of the Lieb-Liniger model at weak repulsion

Ristivojević

2019

Phys. Rev. B

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In this paper we develop an alternative description to solve the problem of ground state energy of the Lieb-Liniger model that describes one-dimensional bosons with contact repulsion. For this integrable model we express the Lieb integral equation in the representation of Chebyshev polynomials. The latter form is convenient to efficiently obtain very precise numerical results in the singular limit of weak interaction. Such highly precise data enables us to use the integer relation algorithm to discover the analytical form of the coefficients in the expansion of the ground state energy for small interaction parameter. We obtained the first nine terms of the expansion using quite moderate numerical efforts. The detailed knowledge of behavior of the ground state energy on the interaction immediately leads to the exact perturbative results for the excitation spectrum.Since its introduction in 1963, the Lieb-Liniger model of one-dimensional bosons with contact interaction [1, 2] continues to fascinate the scientific community. This model is remarkable in many respects and, moreover, describes the physics of realistic systems. The attractive case has deep connections with classical two-dimensional systems, in particular with a surface growth described by the Kardar-Parisi-Zhang equation [3,4]. It could be nowadays directly realized in experiments with cold gases [5][6][7], enabling us to better understand the correlation effects in one-dimensional many-body systems. Rather importantly, the Lieb-Liniger model is integrable and admits an exact solution in terms of Bethe ansatz [1,2]. It thus serves as a benchmark for the effective theories which unavoidably contain various levels of approximations [8,9]. The known results for the above model form a cornerstone for quantum one-dimensional physics of interacting particles [10,11].The explicit analytical expressions for various physical quantities of interest in integrable models are often difficult to extract from the exact solution and one is typically restricted to study special cases. The relevant information about the system's wave function and the corresponding energy of the Lieb-Liniger model is encoded into, so called, the Lieb inte-gral equation [1]. Despite its simple form, the ground state energy is only known in the limiting cases. Using the systematic procedure of Ref. [12], at strong repulsion one can generate a power series expansion to an arbitrary order in the inverse coupling strength. The other limit of weak coupling, on the other hand, is more difficult to treat since it is singular [1]. However, the first three terms were known analytically in this limit for a long time [13][14][15], until a recent work [16] (see also Ref. [17]) which contains conjectures for the analytical form of the first six terms in the ground state energy and the general structure of the following ones based on the double extrapolation of numerical solution of the discrete Bethe ansatz equations.In this paper we study the Lieb-Liniger model at weak repulsion. We develop an algorit...

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“…The second-order term was inferred in [44] and checked wih more rigor in [45,46]. Coefficients of order 3 to 5 were inferred from a very accurate numerical study [47,48], and involve the zeta function evaluated at odd arguments, ζ(3) and ζ(5). Very recently, further coefficients have been explicitly obtained using two independent methods.…”

Section: B Weak-coupling Expansionmentioning

confidence: 99%

Conjectures about the structure of strong- and weak-coupling expansions of a few ground-state observables in the Lieb-Liniger and Yang-Gaudin models

Lang

2019

SciPost Phys.

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In this paper, we apply experimental number theory to two integrable quantum models in one dimension, the Lieb-Liniger Bose gas and the Yang-Gaudin Fermi gas with contact interactions. We identify patterns in weak-and strong-coupling series expansions of the ground-state energy, local correlation functions and pressure. Based on the most accurate data available in the literature, we make a few conjectures about their mathematical structure and extrapolate to higher orders. 2

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Correlations in Low-Dimensional Quantum Gases

Cited by 14 publications

References 350 publications

Exact Perturbative Results for the Lieb–Liniger and Gaudin–Yang Models

Exact Perturbative Results for the Lieb–Liniger and Gaudin–Yang Models

Conjectures about the ground-state energy of the Lieb-Liniger model at weak repulsion

Conjectures about the structure of strong- and weak-coupling expansions of a few ground-state observables in the Lieb-Liniger and Yang-Gaudin models

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