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本文通过数值求解有限温度下一维均匀费米Gaudin-Yang模型的热力学Bethe-ansatz方程, 研究了此模型的基本性质,得到了在给定的温度或给定的相互作用下, 化学势、相互作用、粒子密度和熵的相互变化图像. 对结果分析发现, 在给定温度和相互作用下, 熵随着化学势的变化有一个量子临界区域.
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关键词:
- 热力学Bethe-ansatz方程 /
- 有限温度 /
- Gaudin-Yang模型 /
- 量子临界性质
The one-dimensional system interacting via a delta-function interparticle interaction is a very important one in cold atomic systems and has fundamental importance in many-body physics. In one dimension, due to the geometric confinement induced quantum correlations and quantum fluctuations, there may exist a number of unusual phenomena, such as spin-charge separation, effective fermionization and quantum criticality. This paper studies the basic properties of a uniform one-dimensional Gaudin-Yang model for fermions by solving the thermodynamic Bethe-ansatz equations by a numerical method. Numerically, we use the many-variable Newton’s method to solve the coupled equations. We analyze the physical properties, including density, interaction, temperature and entropy at a given temperature and a given interaction, separately. We know that a lot of researches are limited to zero temperature. However, we cannot reach the absolute zero temperature in the real cold atomic experiment. So it is important to deal with the finite temperature problems. We study the density and entropy as a function of the chemical potential, temperature and interaction and, then give the phase diagrams, respectively. We found that there is a quantum critical zone in the phase diagram of entropy, including the high temperature zone with thermal fluctuations and the Luttinger liquid zone with quantum fluctuations. For a given temperature and low chemical potential, the thermal fluctuations are the main factor in the entropy. With the increase of chemical potential, the system enters the quantum critical zone where the competitive effect between the thermal fluctuations and the quantum fluctuations exists. When the chemical potential is large enough, the quantum fluctuations become the main factor in the system’s entropy, and we get the Luttinger liquid phase. Our results can be further used in the finite temperature density-functional theory and to analyze the collective phenomena at a finite temperature.-
Keywords:
- thermodynamic Bethe-ansatz equations /
- finite temperature /
- Gaudin-Yang model /
- quantum criticality
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[47] Olshanii M 1998 Phys. Rev. Lett. 81 938
[48] Dunjko V, Lorent V, Olshanii M 2001 Phys. Rev. Lett. 86 5413
[49] Hu J H, Wang J J, Gao X L, Okumura M, Igarashi R, Yamada S, Machida M 2010 Phys. Rev. B 82 014202
[50] Wei F X,Gao X L 2014 Journal of Zhejiang Normal University (Nat. Sci.) 37 54 (in Chinese) [卫福霞, 高先龙 2014 浙江师范大学学报(自然科学版) 37 54]
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[1] Anderson M H, Ensher J R, Matthews M R, Wieman C E, Cornell E A 1995 Science 269 198
[2] Feshbach H 1958 Ann. Phys. 5 357
[3] Tiesinga E, Verhaar B J, Stoof H T C 1993 Phys. Rev. A 47 4114
[4] DeMarco B, Jin D S 1999 Science 285 1703
[5] Köhl M, Moritz H, Stöferle T, Gnter K, Esslinger T 2005 Phys. Rev. Lett. 94 080403
[6] Geng T, Yan S B, Wang Y H, Yang H J, Zhang T C, Wang J M 2005 Acta Phys. Sin. 54 5104 (in Chinese) [耿涛, 闫树斌, 王彦华, 杨海菁, 张天才, 王军民 2005 物理学报 54 5104]
[7] Cazalilla M A, Citro R, Giamarchi T, Orignac E, Rigol M 2011 Rev. Mod. Phys. 83 1405
[8] Guan X W, Batchelor M T, Lee C H 2013 Rev. Mod. Phys. 85 1633
[9] Gao X L 2010 Phys. Rev. B 81 104306
[10] Li W, Gao X L, Kollath C, Polini M 2008 Phys. Rev. B 78 195109
[11] Xu Z J, Liu X Y 2011 Acta Phys. Sin. 60 120305 (in Chinese) [徐志君, 刘夏吟 2011 物理学报 60 120305]
[12] Astrakharchik G E, Blume D, Giorgini S, Pitaevskii L P 2004 Phys. Rev. Lett. 93 050402
[13] van Amerongen A H, van Es J J P, Wicke P, Kheruntsyan K V, van Druten N J 2008 Phys. Rev. Lett. 100 090402
[14] Moritz H, Stöferle T, Köhl M, Esslinger T 2003 Phys. Rev. Lett. 91 250402
[15] Yang C N 1967 Phys. Rev. Lett. 19 1312
[16] Gaudin M 1967 Phys. Lett. A 24 55
[17] Menotti C, Stringari S 2002 Phys. Rev. A 66 043610
[18] Gao X L, Asgari R 2008 Phys. Rev. A 77 033604
[19] Gao X L, Polini M, Asgari R, Tosi M P 2006 Phys. Rev. A 73 033609
[20] Guan L M, Chen S, Wang Y P, Ma Z Q 2009 Phys. Rev. Lett. 102 160402
[21] Takahashi M 1971 Prog. Theor. Phys. 46 1388
[22] Hu H, Gao X L, Liu X J 2014 Phys. Rev. A 90 013622
[23] Lee J Y, Guan X W, Sakai K, Batchelor M T 2012 Phys. Rev. B 85 085414
[24] Chen Y Y, Jiang Y Z, Guan X W, Zhou Q 2014 Nat. Commun. 5
[25] Hoffman M D, Javernick P, Loheac A C, Porter W J, Anderson E R, Drut J E 2014 arXiv:1410.7370vl
[26] Zhao E, Guan X W, Liu W V, Batchelor M T, Oshikawa M 2009 Phys. Rev. Lett. 103 140404
[27] Batchelor M T, Foerster A, Guan X W, Kuhn C C N 2010 J. Stat. Mech. P12014
[28] Klmper A, Pâţu O I 2011 Phys. Rev. A 84 051604
[29] Kheruntsyan K V, Gangardt D M, Drummond P D, Shlyapnikov G V 2003 Phys. Rev. Lett. 91 040403
[30] Kheruntsyan K V, Gangardt D M, Drummond P D, Shlyapnikov G V 2005 Phys. Rev. A 71 053615
[31] Takahashi M 1972 Prog. Theor. Phys. 47 69
[32] Usuki T, Kawakami N, Okiji A 1989 Phys. Lett. A 135 476
[33] Usuki T, Kawakami N, Okiji A 1990 J.Phys. Soc. Jpn. 59 1357
[34] Suzuki M 1985 Phys. Rev. B 31 2957
[35] Suzuki M, Inoue M 1987 Prog. Theor. Phys. 78 787
[36] Jttner G, Klmper A, Suzuki J 1998 Nucl. Phys. B 522 471
[37] Klmper A, Bariev R Z 1996 Nucl. Phys. B 458 623
[38] Takahashi M, Shiroishi M 2002 Phys. Rev. B 65 165104
[39] Khatami E, Rigol M 2011 Phys. Rev. A 84 053611
[40] Wolak M J, Rousseau V G, Miniatura C, Grémaud B, Scalettar R T, Batrouni G G 2010 Phys. Rev. A 82 013614
[41] Snyder A, Tanabe I, De Silva T 2011 Phys. Rev. A 83 063632
[42] Carmelo J M P, Gu S J, Sampaio M J 2014 J. Phys. 47 255004
[43] Carmelo J M P, Gu S J, Sacramento P D 2013 Ann. Phys. 339 484
[44] Chen F, Ying H P, Xu T F, Li W Z 1994 Acta Phys. Sin. 43 1672 (in Chinese) [陈锋, 应和平, 徐铁锋, 李文铸 1994 物理学报 43 1672]
[45] Gao X L, Chen A H, Tokatly I V, Kurth S 2012 Phys. Rev. B 86 235139
[46] Campo V L 2014 arXiv:1407.6726vl
[47] Olshanii M 1998 Phys. Rev. Lett. 81 938
[48] Dunjko V, Lorent V, Olshanii M 2001 Phys. Rev. Lett. 86 5413
[49] Hu J H, Wang J J, Gao X L, Okumura M, Igarashi R, Yamada S, Machida M 2010 Phys. Rev. B 82 014202
[50] Wei F X,Gao X L 2014 Journal of Zhejiang Normal University (Nat. Sci.) 37 54 (in Chinese) [卫福霞, 高先龙 2014 浙江师范大学学报(自然科学版) 37 54]
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