Dynamical structure factor and a new method to measure the pairing gap in two-dimensional attractive Fermi-Hubbard model

doi:10.1088/1674-1056/adfbd9

中国物理B ›› 2025, Vol. 34 ›› Issue (11): 117102-117102.doi: 10.1088/1674-1056/adfbd9

Dynamical structure factor and a new method to measure the pairing gap in two-dimensional attractive Fermi-Hubbard model

Huaisong Zhao(赵怀松)¹, Feng Yuan(袁峰)^1,†, Tianxing Ma(马天星)^2,‡, and Peng Zou(邹鹏)^1,¶

1 Centre for Theoretical and Computational Physics, College of Physics, Qingdao University, Qingdao 266071, China;
2 School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China

收稿日期:2025-08-08 修回日期:2025-08-08 接受日期:2025-08-15 出版日期:2025-10-30 发布日期:2025-10-30
通讯作者: Feng Yuan, Tianxing Ma, Peng Zou E-mail:yuan@qdu.edu.cn;txma@bnu.edu.cn;phy.zoupeng@gmail.com

Dynamical structure factor and a new method to measure the pairing gap in two-dimensional attractive Fermi-Hubbard model

Huaisong Zhao(赵怀松)¹, Feng Yuan(袁峰)^1,†, Tianxing Ma(马天星)^2,‡, and Peng Zou(邹鹏)^1,¶

1 Centre for Theoretical and Computational Physics, College of Physics, Qingdao University, Qingdao 266071, China;
2 School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China

Received:2025-08-08 Revised:2025-08-08 Accepted:2025-08-15 Online:2025-10-30 Published:2025-10-30
Contact: Feng Yuan, Tianxing Ma, Peng Zou E-mail:yuan@qdu.edu.cn;txma@bnu.edu.cn;phy.zoupeng@gmail.com

摘要/Abstract

摘要： The measurement of the pairing gap is crucial for investigating the physical properties of superconductors or superfluids. We propose a strategy to measure the pairing gap through the dynamical excitations. With the random phase approximation (RPA), we study the dynamical excitations of a two-dimensional attractive Fermi-Hubbard model by calculating its dynamical structure factor. Two distinct collective modes emerge: a Goldstone phonon mode at transferred momentum ${\bm q}=\left[0,0\right]$ and a roton mode at ${\bm q}=\left[\pi,\pi\right]$. The roton mode exhibits a sharp molecular peak in the low-energy regime. Notably, the area under the roton molecular peak scales with the square of the pairing gap, which holds even in three-dimensional and spin-orbit coupled (SOC) optical lattices. This finding suggests an experimental approach to measure the pairing gap in lattice systems by analyzing the dynamical structure factor at ${\bm q}=\left[\pi,\pi\right]$.

关键词: attractive Fermi-Hubbard model, superfluid, dynamical excitations, roton mode, pairing gap measurement

Abstract: The measurement of the pairing gap is crucial for investigating the physical properties of superconductors or superfluids. We propose a strategy to measure the pairing gap through the dynamical excitations. With the random phase approximation (RPA), we study the dynamical excitations of a two-dimensional attractive Fermi-Hubbard model by calculating its dynamical structure factor. Two distinct collective modes emerge: a Goldstone phonon mode at transferred momentum ${\bm q}=\left[0,0\right]$ and a roton mode at ${\bm q}=\left[\pi,\pi\right]$. The roton mode exhibits a sharp molecular peak in the low-energy regime. Notably, the area under the roton molecular peak scales with the square of the pairing gap, which holds even in three-dimensional and spin-orbit coupled (SOC) optical lattices. This finding suggests an experimental approach to measure the pairing gap in lattice systems by analyzing the dynamical structure factor at ${\bm q}=\left[\pi,\pi\right]$.

Key words: attractive Fermi-Hubbard model, superfluid, dynamical excitations, roton mode, pairing gap measurement

中图分类号: (Lattice fermion models (Hubbard model, etc.))

71.10.Fd

03.75.Kk (Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow) 67.85.-d (Ultracold gases, trapped gases)

Huaisong Zhao(赵怀松), Feng Yuan(袁峰), Tianxing Ma(马天星), and Peng Zou(邹鹏). Dynamical structure factor and a new method to measure the pairing gap in two-dimensional attractive Fermi-Hubbard model[J]. 中国物理B, 2025, 34(11): 117102-117102.

参考文献

[1] Feld M, Fröhlich B, Vogt E, Koschorreck M and Köhl M 2011 Nature 480 75
[2] Stewart J T, Gaebler J P and Jin D S 2008 Nature 454 744
[3] Fröhlich B, Feld M, Vogt E, Koschorreck M, Zwerger W and Köhl M 2011 Phys. Rev. Lett. 106 105301
[4] Chin C, Bartenstein M, Altmeyer A, Riedl S, Jochim S, Denschlag J H and Grimm R 2004 Science 305 1128
[5] Sommer A T, Cheuk L W, Ku M J H, Bakr W S and Zwierlein M W 2012 Phys. Rev. Lett. 108 045302
[6] Zhai H 2015 Rep. Prog. Phys. 78 026001
[7] Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S and Zwierlein M W 2012 Phys. Rev. Lett. 109 095302
[8] Wang P, Yu Z Q, Fu Z, Miao J, Huang L, Chai S, Zhai H and Zhang J 2012 Phys. Rev. Lett. 109 095301
[9] Wang Z Y, Cheng X C, Wang B Z, Zhang J Y, Lu Y H, Yi C R, Niu S, Deng Y, Liu X J, Chen S and Pan J W 2021 Science 372 271
[10] Wu F, Guo G C, Zhang W and Yi W 2013 Phys. Rev. Lett. 110 110401
[11] Han R, Yuan F and Zhao H 2023 New J. Phys. 25 023011
[12] Veeravalli G, Kuhnle E, Dyke P and Vale C J 2008 Phys. Rev. Lett. 101 250403
[13] Hoinka S, Dyke P, Lingham M G, Kinnunen J J, Bruun G M and Vale C J 2017 Nat. Phys. 13 943
[14] Dyke P, Musolino S, Kurkjian H, Ahmed-Braun D J M, Pennings A, Herrera I, Hoinka S, Kokkelmans S J J M F, Colussi V E and Vale C J 2024 Phys. Rev. Lett. 132 223402
[15] Biss H, Sobirey L, Luick N, Bohlen M, Kinnunen J J, Bruun G M, Lompe T and Moritz H 2022 Phys. Rev. Lett. 128 100401
[16] Senaratne R, Cavazos-Cavazos D, Wang S, He F, Chang Y T, Kafle A, Pu H, Guan X W and Hulet R G 2022 Science 376 1305
[17] Li X, Luo X, Wang S, Xie K, Liu X P, Hu H, Chen Y A, Yao X C and Pan J W 2022 Science 375 528
[18] Pagano G, Mancini M, Cappellini G, Lombardi P, Schöfer F, Hu H, Liu X J, Catani J, Sias C, Inguscio M and Fallani L 2014 Nat. Phys. 10 198
[19] Combescot R, Giorgini S and Stringari S 2006 Europhys. Lett. 75 695
[20] Combescot R, KaganMY and Stringari S 2006 Phys. Rev. A 74 042717
[21] Zou P, Kuhnle E D, Vale C J and Hu H 2010 Phys. Rev. A 82 061605(R)
[22] Zou P, Dalfovo F, Sharma R, Liu X J and Hu H 2016 New J. Phys. 18 113044
[23] Zou P, Hu H and Liu X J 2018 Phys. Rev. A 98 011602(R)
[24] Hu H, Zou P, and Liu X J 2018 Phys. Rev. A 97 023615
[25] Zou P, Zhao H, He L, Liu X J and Hu H 2021 Phys. Rev. A 103 053310
[26] Kuhnle E D, Hu H, Liu X J, Dyke P, Mark M, Drummond P D, Hannaford P and Vale C J 2010 Phys. Rev. Lett. 105 070402
[27] Watabe S and Nikuni T 2010 Phys. Rev. A 82 033622
[28] Sobirey L, Biss H, Luick N, Bohlen M, Moritz H and Lompe T 2022 Phys. Rev. Lett. 129 083601
[29] Vitali E, Shi H, Qin M and Zhang S 2017 Phys. Rev. A 96 061601(R)
[30] Zhao H, Gao X, Liang W, Zou P and Yuan F 2020 New J. Phys. 22 093012
[31] Gao Z, He L, Zhao H, Peng S G and Zou P 2023 Phys. Rev. A 107 013304
[32] Bloch I, Dalibard J and Zwerger W 2008 Rev. Mod. Phys. 80 885
[33] Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y, Chen S, Liu X J and Pan J W 2016 Science 354 83
[34] Greiner M, Mandel O, Esslinger T, Hänsch T W and Bloch I 2002 Nature 415 39
[35] Spielman I B, Phillips W D and Porto J V 2008 Phys. Rev. Lett. 100 120402
[36] Thomas C K, Barter T H, Leung T H, Okano M, Jo G B, Guzman J, Kimchi I, Vishwanath A and Stamper-Kurn D M 2017 Phys. Rev. Lett. 119 100402
[37] Jördens R, Strohmaier N, Günter K, Moritz H and Esslinger T 2008 Nature 455 204
[38] Schneider U, Hackermüller L, Will S, Best T, Bloch I, Costi T A, Helmes R W, Rasch D and Rosch A 2008 Science 322 1520
[39] Greif D, Uehlinger T, Jotzu G, Tarruell L and Esslinger T 2013 Science 340 1307
[40] Hart R A, Duarte P M, Yang T L, Liu X, Paiva T, Khatami E, Scalettar R T, Trivedi N, Huse D A and Hulet R G 2015 Nature 519 211
[41] Parsons M F, Mazurenko A, Chiu C S, Ji G, Greif D and Greiner M 2016 Science 353 1253
[42] Cheuk L W, Nichols M A, Lawrence K R, Okan M, Zhang H, Khatami E, Trivedi N, Paiva T, Rigol M and Zwierlein M W 2016 Science 353 1260
[43] Koepsell J, Bourgund D, Sompet P, Hirthe S, Bohrdt A,Wang Y, Grusdt F, Demler E, Salomon G, Gross C and Bloch I 2021 Science 374 82
[44] Boll M, Hilker T A, Salomon G, Omran A, Nespolo J, Pollet L, Bloch I and Gross C 2016 Science 353 1257
[45] Brown P T, Mitra D, Guardado-Sanchez E, Schauß P, Kondov S S, Khatami E, Paiva T, Trivedi N, Huse D A and Bakr W S 2017 Science 357 1385
[46] Arovas D P, Berg E, Kivelson S A and Raghu S 2022 Annual Review of Condensed Matter Physics 13 239
[47] Scalettar R T, Loh E Y, Gubernatis J E, Moreo A, White S R, Scalapino D J, Sugar R L and Dagotto E 1989 Phys. Rev. Lett. 62 1407
[48] Kyung B, Allen S and Tremblay A M S 2001 Phys. Rev. B 64 075116
[49] Honerkamp C and Hofstetter W 2004 Phys. Rev. Lett. 92 170403
[50] Mondaini R, Nikoliø’c P and Rigol M 2015 Phys. Rev. A 92 013601
[51] Cocchi E, Miller L A, Drewes J H, Koschorreck M, Pertot D, Brennecke F and Köhl M 2016 Phys. Rev. Lett. 116 175301
[52] Strohmaier N, Takasu Y, Günter K, Jördens R, Köhl M, Moritz H and Esslinger T 2007 Phys. Rev. Lett. 99 220601
[53] Ho A F, Cazalilla M A and Giamarchi T 2009 Phys. Rev. A 79 033620
[54] Moreo A and Scalapino D J 2007 Phys. Rev. Lett. 98 216402
[55] Gukelberger J, Lienert S, Kozik E, Pollet L and Troyer M 2016 Phys. Rev. B 94 075157
[56] Paiva T, dos Santos R R, Scalettar R T and Denteneer P J H 2004 Phys. Rev. B 69 184501
[57] Shenoy V B 2008 Phys. Rev. B 78 134503
[58] Mitra D, Brown P T, Guardado-Sanchez E, Kondov S S, Devakul T, Huse D A, Schauß P and Bakr W S 2018 Nat. Phys. 14 173
[59] Brown P T, Guardado-Sanchez E, Spar B M, Huang EW, Devereaux T P and Bakr W S 2020 Nat. Phys. 16 26
[60] Hackermüller L, Schneider U, Moreno-Cardoner M, Kitagawa T, Best T, Will S, Demler E, Altman E, Bloch I and Paredes B 2010 Science 327 1621
[61] Gall M, Chan C F, Wurz N and Köhl M 2020 Phys. Rev. Lett. 124 010403
[62] Schneider U, Hackermüller L, Ronzheimer J P, Will S, Braun S, Best T, Bloch I, Demler E, Mandt S, Rasch D and Rosch A 2012 Nat. Phys. 8 213
[63] Hartke T, Oreg B, Turnbaugh C, Jia N and Zwierlein M 2023 Science 381 82
[64] Vitali E, Kelly P, Lopez A, Bertaina G and Galli D E 2020 Phys. Rev. A 102 053324
[65] Zhao H, Han R, Qin L, Yuan F and Zou P 2024 Phys. Rev. A 110 063326
[66] Liu X J, Hu H, Minguzzi A and Tosi M P 2004 Phys. Rev. A 69 043605
[67] He L 2016 Ann. Phys. 373 470
[68] Ganesh R, Paramekanti A and Burkov A A 2009 Phys. Rev. A 80 043612
[69] Han R, Yuan F and Zhao H 2022 Europhys. Lett. 139 25001
[70] Nocera A, Patel N D, Fernandez-Baca J, Dagotto E and Alvarez G 2016 Phys. Rev. B 94 205145
[71] Zhang S 1990 Phys. Rev. Lett. 65 120
[72] Qin M, Schäfer T, Andergassen S, Corboz P and Gull E 2022 Annual Review of Condensed Matter Physics 13 275
[73] Belkhir L and Randeria M 1994 Phys. Rev. B 49 6829
[74] Schäfer T,Wentzell N, Šimkovic F, He Y Y, Hille C, Klett M, Eckhardt C J, Arzhang B, Harkov V, Le Régent F M, A. Kirsch, Wang Y, Kim A J, Kozik E, Stepanov E A, Kauch A, Andergassen S, Hansmann P, Rohe D, Vilk Y M, LeBlanc J P F, Zhang S, Tremblay A M S, Ferrero M, Parcollet O and Georges A 2021 Phys. Rev. X 11 011058
[75] Hafermann H, van Loon E G C P, Katsnelson M I, Lichtenstein A I and Parcollet O 2014 Phys. Rev. B 90 235105
[76] Feng S, Kuang L and Zhao H 2015 Physica C 517 5
[77] Anderson P W, Lee P A, Randeria M, Rice T M, Trivedi N and Zhang F C 2004 J. Phys.: Condens. Matter 16 R755
[78] Crameri F 2018 Geosci. Model Dev. 11 2541

Dynamical structure factor and a new method to measure the pairing gap in two-dimensional attractive Fermi-Hubbard model

Dynamical structure factor and a new method to measure the pairing gap in two-dimensional attractive Fermi-Hubbard model

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