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与伽利略不变性的超流体不同, 具有洛伦兹不变性的超流体中除了声子模之外, 还存在希格斯振幅模(Higgs amplitude mode). 在二维情况下, 由于存在十分剧烈的衰变成声子模的过程, 希格斯模是否仍然是一个能产生尖锐线性响应的激发子成为了一个问题. 近年来的进展最终对这一持续数十年的争论做出了肯定的回答, 证实了希格斯的可观测性. 在这里, 我们回顾一系列的数值方面的工作; 它们以二维超流体(superfluid)到莫特绝缘体(Mott insulator) 量子相变点(SF-MI QCP) 附近的具有洛伦兹不变性的超流体为对象, 成功探测到了希格斯模的线性响应信号. 特别是, 我们介绍了一种如何使用平衡态系统的蒙特卡罗算法计算强关联系统的延迟响应函数(retarded response function)的方法. 该方法主要包含两个核心步骤: 即通过路径积分表示下的蠕虫算法这一高效的蒙特卡罗算法计算平衡态系统的虚时间关联函数, 然后利用数值解析延拓方法从虚时间关联函数中获得实时间(实频率)的响应函数. 将该数值方法应用于二维SF-MI QCP附近的玻色-哈伯德模型(Bose-Hubbard Model), 结果表明尽管在超流相中, 希格斯模衰变过程非常剧烈, 但是在动能算符相对应的延迟响应函数的虚部中, 仍然可以观测到希格斯模所对应的尖锐的共振峰. 进一步的研究表明, 在莫特绝缘相, 甚至常流体相中, 也可能存在类似的共振峰信号. 由于可以在光晶格中超冷原子系统等凝聚态中观测到SF-MI QCP, 因此希格斯共振峰有望通过实验进行直接探测. 此外我们指出, 同样的希格斯共振峰还存在于所有和SF-MI QCP具有相同普适类((2+1)维相对论性U(1)临界性)的量子临界系统中.In additional to the phonon (massless Goldstone mode) in Galilean invariant superfluid, there is another type of mode known as the Higgs amplitude mode in superfluid with emergent Lorentz invariance. In two dimensions, due to the strong decay into phonons, whether this Higgs mode is a detectable excitation with sharp linear response has been controversial for decades. Recent progress gives a positive answer to this question. Here, we review a series of numerical studies of the linear response of a two-dimensional Lorentz invariant superfluid near the superfluid-Mott insulator quantum critical point (SF-MI QCP). Particularly, we introduce a numerical procedure to unbiasedly calculate the linear response properties of strongly correlated systems. The numerical procedure contains two crucial steps, i.e., one is to use a highly efficient quantum Monte Carlo method, the worm algorithm in the imaginary-time path-integral representation, to calculate the imaginary time correlation functions for the system in equilibrium; and then, the other is, based on the imaginary time correlation functions, to use the numerical analytical continuation method for obtaining the real-time (real-frequency) linear response function. Applying this numerical procedure to the two-dimensional Bose Hubbard model near SF-MI QCP, it is found that despite strong damping, the Higgs boson survives as a prominent resonance peak in the kinetic energy response function. Further investigations also suggest a similar but less prominent resonance peak near SF-MI QCP on the MI side, and even on the normal liquid side. Since SF-MI quantum criticality can be realized by ultracold aotms in optical lattice, the Higgs resonance peak can be directly observed in experiment. In addition, we point out that the same Higgs resonance peak exists in all quantum critical systems with the same universality, namely (2 + 1)-dimensional relativistic U(1) criticality, as SF-MI QCP.
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Keywords:
- linear response /
- quantum criticality /
- Higgs mode /
- Monte Carlo
[1] Sachdev S 2011 Quantum Phase Transitions (2nd Ed.) (Cambridge: Cambridge University Press)
[2] Goldstone J 1961 Nuovo. Cim. 19 154
[3] Weinberg S 1996 The quantum theory of fields(Vol. 2) (Cambridge: Cambridge University Press)
[4] Anderson P W 1963 Phys. Rev 130 439
[5] Higgs P W 1964 Phys. Rev. Lett. 13 508
[6] Pekker D, Varma C M 2015 Annu. Rev. Condens. Matter Phys. 6 269
[7] Sooryakumar R, Klein M V 1980 Phys. Rev. Lett. 45 660
[8] Sooryakumar R, Klein M V 1981 Phys. Rev. B 23 3213
[9] Jaksch D, Bruder C, Cirac J I, Gardiner C W, Zoller P 1998 Phys. Rev. Lett. 81 3108
[10] Fisher M P A, Weichman P B, Grinstein G, Fisher D S 1989 Phys. Rev. B 40 546
[11] Capogrosso-Sansone B, Söyler S G, Prokof'ev N V, Svistunov B V 2008 Phys. Rev. A 77 015602
[12] Bissbort U, Götze S, Li Y, Heinze J, Krauser J S, Weinberg M, Becker C, Sengstock K, Hofstetter W 2011 Phys. Rev. Lett. 106 205303
[13] Regg Ch, Normand B, Matsumoto M, Furrer A, McMorrow D F, Krämer K W, Gdel H U, Gvasaliya S N, Mutka H, Boehm M 2008 Phys. Rev. Lett. 100 205701
[14] Chubukov A V, Sachdev S, Ye J 1994 Phys. Rev. B 49 11919
[15] Sachdev S 1999 Phys. Rev. B 59 14054
[16] Zwerger W 2004 Phys. Rev. Lett. 92 027203
[17] Podolsky D, Auerbach A, Arovas D P 2011 Phys. Rev. B 84 174522
[18] Podolsky D, Sachdev S 2012 Phys. Rev. B 86 054508
[19] Katan Y T, Podolsky D 2015 Phys. Rev. B 91 075132
[20] Endres M, Fukuhara T, Pekker D, Cheneau M, Schauß P, Gross C, Demler E, Kuhr S, Bloch I 2012 Nature 487 454
[21] Pollet L, Prokof'ev N 2012 Phys. Rev. Lett. 109 010401
[22] Gazit S, Podolsky D, Auerbach A 2013 Phys. Rev. Lett. 110 140401
[23] Chen K, Liu L, Deng Y, Pollet L, Prokof'ev N 2013 Phys. Rev. Lett. 110 170403
[24] Gazit S, Podolsky D, Auerbach A, Arovas D P 2013 Phys. Rev. B 88 235108
[25] Rancon A, Dupuis N 2014 Phys. Rev. B 89 180501
[26] Burovski E, Machta J, Prokof'ev N V, Svistunov B V 2006 Phys. Rev. B 74 132502
[27] Campostrini M, Hasenbusch M, Pelissetto A, Vicari E 2006 Phys. Rev. B 74 144506
[28] Prokof'ev N V, Svistunov B V, Tupitsyn I S 1998 Sov. Phys.-JETP 87 310
[29] Prokof'ev N V, Svistunov B V, Tupitsyn I S 1998 Phys. Lett. A 238 253
[30] Caër L D 2010 Understanding Quantum Phase Transitions (Boca Raton: Taylor & Francis)
[31] Mishchenko A S, Prokof'ev N V, Sakamoto A, Svistunov B V 2000 Phys. Rev. B 62 6317
[32] Silver R N, Sivia D S, Gubernatis J E 1990 Phys. Rev. B 41 2380
[33] Jarrell M, Gubernatis J E 1996 Phys. Rep 269 133
[34] Prokof'ev N V, Svistunov B V 2013 Jetp Lett. 97 649
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[1] Sachdev S 2011 Quantum Phase Transitions (2nd Ed.) (Cambridge: Cambridge University Press)
[2] Goldstone J 1961 Nuovo. Cim. 19 154
[3] Weinberg S 1996 The quantum theory of fields(Vol. 2) (Cambridge: Cambridge University Press)
[4] Anderson P W 1963 Phys. Rev 130 439
[5] Higgs P W 1964 Phys. Rev. Lett. 13 508
[6] Pekker D, Varma C M 2015 Annu. Rev. Condens. Matter Phys. 6 269
[7] Sooryakumar R, Klein M V 1980 Phys. Rev. Lett. 45 660
[8] Sooryakumar R, Klein M V 1981 Phys. Rev. B 23 3213
[9] Jaksch D, Bruder C, Cirac J I, Gardiner C W, Zoller P 1998 Phys. Rev. Lett. 81 3108
[10] Fisher M P A, Weichman P B, Grinstein G, Fisher D S 1989 Phys. Rev. B 40 546
[11] Capogrosso-Sansone B, Söyler S G, Prokof'ev N V, Svistunov B V 2008 Phys. Rev. A 77 015602
[12] Bissbort U, Götze S, Li Y, Heinze J, Krauser J S, Weinberg M, Becker C, Sengstock K, Hofstetter W 2011 Phys. Rev. Lett. 106 205303
[13] Regg Ch, Normand B, Matsumoto M, Furrer A, McMorrow D F, Krämer K W, Gdel H U, Gvasaliya S N, Mutka H, Boehm M 2008 Phys. Rev. Lett. 100 205701
[14] Chubukov A V, Sachdev S, Ye J 1994 Phys. Rev. B 49 11919
[15] Sachdev S 1999 Phys. Rev. B 59 14054
[16] Zwerger W 2004 Phys. Rev. Lett. 92 027203
[17] Podolsky D, Auerbach A, Arovas D P 2011 Phys. Rev. B 84 174522
[18] Podolsky D, Sachdev S 2012 Phys. Rev. B 86 054508
[19] Katan Y T, Podolsky D 2015 Phys. Rev. B 91 075132
[20] Endres M, Fukuhara T, Pekker D, Cheneau M, Schauß P, Gross C, Demler E, Kuhr S, Bloch I 2012 Nature 487 454
[21] Pollet L, Prokof'ev N 2012 Phys. Rev. Lett. 109 010401
[22] Gazit S, Podolsky D, Auerbach A 2013 Phys. Rev. Lett. 110 140401
[23] Chen K, Liu L, Deng Y, Pollet L, Prokof'ev N 2013 Phys. Rev. Lett. 110 170403
[24] Gazit S, Podolsky D, Auerbach A, Arovas D P 2013 Phys. Rev. B 88 235108
[25] Rancon A, Dupuis N 2014 Phys. Rev. B 89 180501
[26] Burovski E, Machta J, Prokof'ev N V, Svistunov B V 2006 Phys. Rev. B 74 132502
[27] Campostrini M, Hasenbusch M, Pelissetto A, Vicari E 2006 Phys. Rev. B 74 144506
[28] Prokof'ev N V, Svistunov B V, Tupitsyn I S 1998 Sov. Phys.-JETP 87 310
[29] Prokof'ev N V, Svistunov B V, Tupitsyn I S 1998 Phys. Lett. A 238 253
[30] Caër L D 2010 Understanding Quantum Phase Transitions (Boca Raton: Taylor & Francis)
[31] Mishchenko A S, Prokof'ev N V, Sakamoto A, Svistunov B V 2000 Phys. Rev. B 62 6317
[32] Silver R N, Sivia D S, Gubernatis J E 1990 Phys. Rev. B 41 2380
[33] Jarrell M, Gubernatis J E 1996 Phys. Rep 269 133
[34] Prokof'ev N V, Svistunov B V 2013 Jetp Lett. 97 649
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