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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
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[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
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[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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