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Power-law scalings in weakly-interacting Bose gases at quantum criticality

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Abstract

Weakly interacting quantum systems in low dimensions have been investigated for a long time, but there still remain a number of open questions and a lack of explicit expressions of physical properties of such systems. In this work, we find power-law scalings of thermodynamic observables in low-dimensional interacting Bose gases at quantum criticality. We present a physical picture for these systems with the repulsive interaction strength approaching zero; namely, the competition between the kinetic and interaction energy scales gives rise to power-law scalings with respect to the interaction strength in characteristic thermodynamic observables. This prediction is supported by exact Bethe ansatz solutions in one dimension, demonstrating a simple 1/3-power-law scaling of the critical entropy per particle. Our method also yields results in agreement with a non-perturbative renormalization-group computation in two dimensions. These results provide a new perspective for understanding many-body phenomena induced by weak interactions in quantum gases.

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References

  1. T. D. Lee, K. Huang, and C. N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties, Phys. Rev. 106(6), 1135 (1957)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  2. C. N. Yang and C. P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction, J. Math. Phys. 10(7), 1115 (1969)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71(3), 463 (1999)

    Article  ADS  Google Scholar 

  4. N. Prokof’ev, O. Ruebenacker, and B. Svistunov, Critical point of a weakly interacting two-dimensional Bose gas, Phys. Rev. Lett. 87(27), 270402 (2001)

    Article  Google Scholar 

  5. N. Prokof’ev and B. Svistunov, Two-dimensional weakly interacting Bose gas in the fluctuation region, Phys. Rev. A 66(4), 043608 (2002)

    Article  ADS  Google Scholar 

  6. S. Floerchinger and C. Wetterich, Nonperturbative thermodynamics of an interacting Bose gas, Phys. Rev. A 79(6), 063602 (2009)

    Article  ADS  Google Scholar 

  7. Y. Z. Jiang, Y. Y. Chen, and X. W. Guan, Understanding many-body physics in one dimension from the Lieb-Liniger model, Chin. Phys. B 24(5), 050311 (2015)

    Article  ADS  Google Scholar 

  8. C. Chin, Ultracold atomic gases going strong, Natl. Sci. Rev. 3(2), 168 (2016)

    Article  Google Scholar 

  9. E. Bettelheim, The Whitham approach to the c → 0 limit of the Lieb—Liniger model and generalized hydrodynamics, J. Phys. A Math. Theor. 53(20), 205204 (2020)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  10. A. Posazhennikova, Colloquium: Weakly interacting, dilute Bose gases in 2D, Rev. Mod. Phys. 78(4), 1111 (2006)

    Article  ADS  Google Scholar 

  11. X. W. Guan, M. T. Batchelor, and C. Lee, Fermi gases in one dimension: From Bethe ansatz to experiments, Rev. Mod. Phys. 85(4), 1633 (2013)

    Article  ADS  Google Scholar 

  12. M. Gaudin, Un systeme a une dimension de fermions en interaction, Phys. Lett. A 24(1), 55 (1967)

    Article  MathSciNet  ADS  Google Scholar 

  13. C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive δ-function interaction, Phys. Rev. Lett. 19(23), 1312 (1967)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. M. Takahashi, Ground state energy of the one-dimensional electron system with short-range interaction (I), Prog. Theor. Phys. 44, 348 (1970)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. T. Iida and M. Wadati, Exact analysis of a one-dimensional attractive S-function Fermi gas with arbitrary spin polarization, J. Low Temp. Phys. 148(3–4), 417 (2007)

    Article  MATH  ADS  Google Scholar 

  16. X. W. Guan, Polaron, molecule and pairing in one-dimensional spin-1/2 Fermi gas with an attractive delta-function interaction, Front. Phys. 7(1), 8 (2012)

    Article  ADS  Google Scholar 

  17. X. W. Guan and Z. Q. Ma, One-dimensional multicomponent fermions with δ-function interaction in strong-and weak-coupling limits: Two-component Fermi gas, Phys. Rev. A 85(3), 033632 (2012)

    Article  ADS  Google Scholar 

  18. X. W. Guan, Z. Q. Ma, and B. Wilson, One-dimensional multicomponent fermions with δ-function interaction in strong- and weak-coupling limits: κ-component Fermi gas, Phys. Rev. A 85(3), 033633 (2012)

    Article  ADS  Google Scholar 

  19. C. A. Tracy and H. Widom, On the ground state energy of the δ-function Bose gas, J. Phys. A Math. Theor. 49(29), 294001 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. C. A. Tracy and H. Widom, On the ground state energy of the δ-function Fermi gas, J. Math. Phys. 57(10), 103301 (2016)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  21. S. Prolhac, Ground state energy of the δ-Bose and Fermi gas at weak coupling from double extrapolation, J. Phys. A Math. Theor. 50(14), 144001 (2017)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  22. S. Sachdev, Quantum Phase Transitions, 2nd Ed., Cambridge University Press, 2011

  23. G. G. Batrouni, R. T. Scalettar, and G. T. Zimanyi, Quantum critical phenomena in one-dimensional Bose systems, Phys. Rev. Lett. 65(14), 1765 (1990)

    Article  ADS  Google Scholar 

  24. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80(3), 885 (2008)

    Article  ADS  Google Scholar 

  25. M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, One dimensional bosons: From condensed matter systems to ultracold gases, Rev. Mod. Phys. 83(4), 1405 (2011)

    Article  ADS  Google Scholar 

  26. C. Chin, in: Universal Themes of Bose—Einstein Condensation, edited by D. Snoke, N. Proukakis, and P. Littlewood, Cambridge University Press, 2017, Chapter 9, pp 175–195

  27. T. L. Ho, Universal thermodynamics of degenerate quantum gases in the unitarity limit, Phys. Rev. Lett. 92(9), 090402 (2004)

    Article  ADS  Google Scholar 

  28. T. L. Ho and Q. Zhou, Obtaining the phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases, Nat. Phys. 6(2), 131 (2010)

    Article  Google Scholar 

  29. S. Pilati, S. Giorgini, and N. Prokof’ev, Critical temperature of interacting Bose gases in two and three dimensions, Phys. Rev. Lett. 100(14), 140405 (2008)

    Article  ADS  Google Scholar 

  30. A. Rançon and N. Dupuis, Universal thermodynamics of a two-dimensional Bose gas, Phys. Rev. A 85(6), 063607 (2012)

    Article  ADS  Google Scholar 

  31. J. Kinast, A. Turlapov, J. E. Thomas, Q. Chen, J. Stajic, and K. Levin, Heat capacity of a strongly interacting Fermi gas, Science 307(5713), 1296 (2005)

    Article  ADS  Google Scholar 

  32. L. Luo and J. E. Thomas, Thermodynamic measurements in a strongly interacting Fermi gas, J. Low Temp. Phys. 154(1–2), 1 (2009)

    Article  ADS  Google Scholar 

  33. S. Nascimbène, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon, Exploring the thermodynamics of a universal Fermi gas, Nature 463(7284), 1057 (2010)

    Article  ADS  Google Scholar 

  34. N. Navon, S. Nascimbene, F. Chevy, and C. Salomon, The equation of state of a low-temperature Fermi gas with tunable interactions, Science 328(5979), 729 (2010)

    Article  MATH  ADS  Google Scholar 

  35. C. L. Hung, X. Zhang, N. Gemelke, and C. Chin, Observation of scale invariance and universality in two-dimensional Bose gases, Nature 470(7333), 236 (2011)

    Article  ADS  Google Scholar 

  36. T. Yefsah, R. Desbuquois, L. Chomaz, K. J. Gunter, and J. Dalibard, Exploring the thermodynamics of a two-dimensional Bose gas, Phys. Rev. Lett. 107(13), 130401 (2011)

    Article  ADS  Google Scholar 

  37. M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335(6068), 563 (2012)

    Article  ADS  Google Scholar 

  38. X. Zhang, C. L. Hung, S. K. Tung, and C. Chin, Observation of quantum criticality with ultracold atoms in optical lattices, Science 335(6072), 1070 (2012)

    Article  ADS  Google Scholar 

  39. L. C. Ha, C. L. Hung, X. Zhang, U. Eismann, S. K. Tung, and C. Chin, Strongly interacting two-dimensional Bose gases, Phys. Rev. Lett. 110(14), 145302 (2013)

    Article  ADS  Google Scholar 

  40. A. Vogler, R. Labouvie, F. Stubenrauch, G. Barontini, V. Guarrera, and H. Ott, Thermodynamics of strongly correlated one-dimensional Bose gases, Phys. Rev. A 88, 031603(R) (2013)

    Article  ADS  Google Scholar 

  41. B. Yang, Y. Y. Chen, Y. G. Zheng, H. Sun, H. N. Dai, X. W. Guan, Z. S. Yuan, and J. W. Pan, Quantum criticality and the Tomonaga-Luttinger liquid in one-dimensional Bose gases, Phys. Rev. Lett. 119(16), 165701 (2017)

    Article  ADS  Google Scholar 

  42. X. Zhang, Y. Y. Chen, L. X. Liu, Y. J. Deng, and X. W. Guan, Interaction-induced particle—hole symmetry breaking and fractional exclusion statistics, Natl. Sci. Rev., nwac027 (2022)

  43. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B 40(1), 546 (1989)

    Article  ADS  Google Scholar 

  44. A. Khare, Fractional Statistics and Quantum Theory, World Scientific Publishing Co. Pte. Ltd., 5 Toh Tuck Link, Singapore 596224, 2005, Chapter 5, 2nd Ed

    Book  Google Scholar 

  45. P. Grüter, D. Ceperley, and F. Laloe, Critical temperature of Bose—Einstein condensation of hard-sphere gases, Phys. Rev. Lett. 79(19), 3549 (1997)

    Article  ADS  Google Scholar 

  46. E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas (I): The general solution and the ground state, Phys. Rev. 130(4), 1605 (1963)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  47. X. W. Guan and M. T. Batchelor, Polylogs, thermodynamics and scaling functions of one-dimensional quantum many-body systems, J. Phys. A Math. Theor. 44(10), 102001 (2011)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  48. L. Tonks, The complete equation of state of one, two and three-dimensional gases of hard elastic spheres, Phys. Rev. 50(10), 955 (1936)

    Article  MATH  ADS  Google Scholar 

  49. M. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension, J. Math. Phys. 1(6), 516 (1960)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  50. B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch, Tonks—Girardeau gas of ultracold atoms in an optical lattice, Nature 429(6989), 277 (2004)

    Article  ADS  Google Scholar 

  51. T. Kinoshita, T. Wenger, and D. S. Weiss, Observation of a one-dimensional Tonks-Girardeau gas, Science 305(5687), 1125 (2004)

    Article  ADS  Google Scholar 

  52. Supporting information is available as supplementary materials.

  53. K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47(4), 773 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  54. K. Costello, Renormalization and Effective Field Theory, American Mathematical Society, Providence, Rhode Island, 2011

    Book  MATH  Google Scholar 

  55. G. Passarino, Veltman, renormalizability, calculability, Acta Phys. Pol. B 52(6), 533 (2021)

    Article  MathSciNet  ADS  Google Scholar 

  56. B. Wolf, Y. Tsui, D. Jaiswal-Nagar, U. Tutsch, A. Honecker, K. Remović-Langer, G. Hofmann, A. Prokofiev, W. Assmus, G. Donath, and M. Lang, Magnetocaloric effect and magnetic cooling near a field-induced quantum-critical point, Proc. Natl. Acad. Sci. USA 108(17), 6862 (2011)

    Article  ADS  Google Scholar 

  57. Y. Y. Chen, G. Watanabe, Y. C. Yu, X. W. Guan, and A. del Campo, An interaction-driven many-particle quantum heat engine and its universal behavior, npj Quantum Inf. 5, 88 (2019)

    Article  ADS  Google Scholar 

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Acknowledgements

This work was supported by the National Key Research and Development Program of China under Grant No. 2018YFA0305601, the National Natural Science Foundation of China under Grant No. 11874073, the Chinese Academy of Sciences Strategic Priority Research Program under Grant No. XDB35020100, and the Hefei National Laboratory and the Scientific and Technological Innovation 2030 under Grant No. 2021ZD0301903. X. W. Guan is supported by the National Natural Science Foundation of China under key Grant No. 12134015, and under Grants No. 11874393 and No. 12121004. Y. Y. Chen is supported by the National Natural Science Foundation of China under Grant No. 12104372.

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  1. These authors contributed equally to this work.

Authors and Affiliations

  1. International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China

    Ming-Cheng Liang, Zhi-Xing Lin & Xibo Zhang

  2. Collaborative Innovation Center of Quantum Matter, Beijing, 100871, China

    Ming-Cheng Liang & Xibo Zhang

  3. State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan, 430071, China

    Yang-Yang Chen & Xi-Wen Guan

  4. Institute of Modern Physics, Northwest University, Xi’an, 710127, China

    Yang-Yang Chen

  5. Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra, ACT, 0200, Australia

    Xi-Wen Guan

  6. Beijing Academy of Quantum Information Sciences, Beijing, 100193, China

    Xibo Zhang

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  1. Ming-Cheng Liang
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Correspondence to Xibo Zhang.

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Liang, MC., Lin, ZX., Chen, YY. et al. Power-law scalings in weakly-interacting Bose gases at quantum criticality. Front. Phys. 17, 61501 (2022). https://doi.org/10.1007/s11467-022-1186-x

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