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Analytic thermodynamic properties of the Lieb-Liniger gas
by M. L. Kerr, G. De Rosi, K. V. Kheruntsyan
Submission summary
| Authors (as registered SciPost users): | Karen Kheruntsyan |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2404.06092v2 (pdf) |
| Date submitted: | 2024-04-22 09:58 |
| Submitted by: | Kheruntsyan, Karen |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics -- thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.
Current status:
Reports on this Submission
Strengths
1. Comprehensive review of the approximate analytic results for the thermodynamic quantities of the Lieb-Liniger model.
2. New and improved expansions in the thermal quasicondensate regime (regime II), the degenerate nearly ideal Bose gas regime (regime III) and the non-degenerate nearly ideal Bose gas regime (regime IV).
Report
The submitted paper provides a thorough review of the approximate analytic results of the relevant thermodynamic quantities (pressure, entropy, chemical potential, energy, specific heat, isothermal compressibility, and pair correlation function) for the repulsive Lieb-Liniger (LL) model. While in principle all the thermodynamic quantities of the Lieb-Liniger model can be obtained using the Thermodynamic Bethe Ansatz (TBA) formalism it is also useful to have simple analytical formulae valid in different regimes which can be easily compared with experimental data.
The analytic results reported cover the six different regimes of the LL model (quasicondensate, nearly ideal Bose gas, and strongly interacting each of them with two subregimes) and are presented as expansions in the dimensionless strength $\gamma$ and reduced temperature $\tau$. Numerical comparisons with the TBA results highlighting the area of applicability of the various approximations and the crossover boundaries of the different regimes are also presented. The authors also use these results to investigate the density profiles in a trapped system using the local density approximation and propose a new method of thermometry for the LL model.
Probably the most interesting and important result is represented by the expansions derived for the thermal quasicondensate regime (regime II) which shows that the temperature contributions to the pressure and pair correlation functions are not negligible as previously was believed (this also can be used to distinguish the I and II regimes). In addition, the newly derived expansions for regimes III (degenerate nearly ideal Bose gas regime) and IV (non-degenerate nearly ideal Bose gas regime) contain additional $\gamma$ dependent terms compared with the results derived using the Hartree Fock approximation.
This is a clearly written and well-organized review that contains both previously known and new important original results reasons for which I recommend its publication in SciPost Physics Core.
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