Analytical approach to quantum phase transitions of ultracold Bose gases in bipartite optical lattices using the generalized Green’s function method
Zhi Lin1,2(), Jun Zhang2, Ying Jiang2,3,4
1. Department of Physics, State Key Laboratory of Surface Physics and Laboratory of Advanced Materials, Fudan University, Shanghai 200433, China 2. Department of Physics, Shanghai University, Shanghai 200444, China 3. Qian Weichang College, Shanghai University, Shanghai 200444, China 4. Key Lab for Astrophysics, Shanghai 200234, China
In order to investigate the quantum phase transitions and the time-of-flight absorption pictures analytically in a systematic way for ultracold Bose gases in bipartite optical lattices, we present a generalized Green’s function method. Utilizing this method, we study the quantum phase transitions of ultracold Bose gases in two types of bipartite optical lattices, i.e., a hexagonal lattice with normal Bose–Hubbard interaction and a d-dimensional hypercubic optical lattice with extended Bose–Hubbard interaction. Furthermore, the time-of-flight absorption pictures of ultracold Bose gases in these two types of lattices are also calculated analytically. In hexagonal lattice, the time-of-flight interference patterns of ultracold Bose gases obtained by our analytical method are in good qualitative agreement with the experimental results of Soltan-Panahi, et al. [Nat. Phys. 7, 434 (2011)]. In square optical lattice, the emergence of peaks at (±,±) in the time-of-flight absorption pictures, which is believed to be a sort of evidence of the existence of a supersolid phase, is clearly seen when the system enters the compressible phase from charge-density-wave phase.
. [J]. Frontiers of Physics, 2018, 13(4): 136401.
Zhi Lin, Jun Zhang, Ying Jiang. Analytical approach to quantum phase transitions of ultracold Bose gases in bipartite optical lattices using the generalized Green’s function method. Front. Phys. , 2018, 13(4): 136401.
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415(6867), 39 (2002) https://doi.org/10.1038/415039a
2
M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), and U. Sen, Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond, Adv. Phys. 56(2), 243 (2007) https://doi.org/10.1080/00018730701223200
3
I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80(3), 885 (2008); and the references therein. https://doi.org/10.1103/RevModPhys.80.885
4
S. Sachdev, Quantum Phase Transitions, Cambridge: Cambridge University Press, 1999
5
M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluidinsulator transition, Phys. Rev. B 40(1), 546 (1989) https://doi.org/10.1103/PhysRevB.40.546
6
D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold bosonic atoms in optical lattices, Phys. Rev. Lett. 81(15), 3108 (1998) https://doi.org/10.1103/PhysRevLett.81.3108
7
J. K. Freericks and H. Monien, Strong-coupling expansions for the pure and disordered Bose–Hubbard model, Phys. Rev. B 53(5), 2691 (1996) https://doi.org/10.1103/PhysRevB.53.2691
8
B. Capogrosso-Sansone, N. V. Prokof’ev, and B. V. Svistunov, Phase diagram and thermodynamics of the threedimensional Bose-Hubbard model, Phys. Rev. B 75(13), 134302 (2007) https://doi.org/10.1103/PhysRevB.75.134302
Z. Lin, J. Zhang, and Y. Jiang, Quantum phase transitions of ultracold Bose systems in nonrectangular optical lattices, Phys. Rev. A 85(2), 023619 (2012) https://doi.org/10.1103/PhysRevA.85.023619
11
N. Teichmann, D. Hinrichs, and M. Holthaus, Reference data for phase diagrams of triangular and hexagonal bosonic lattices, Europhys. Lett. 91(1), 10004 (2010) https://doi.org/10.1209/0295-5075/91/10004
12
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 3rd Ed., Clarendon Press, 1996
13
H. Kleinert and V. Schulte-Frohlinde, Critical Properties of ϕ4-Theories, World Scientific, 2001 https://doi.org/10.1142/4733
14
V. A. Kashurnikov, N. V. Prokof’ev, and B. V. Svistunov, Revealing the superfluid–Mott-insulator transition in an optical lattice, Phys. Rev. A 66, 031601(R) (2002)
15
A. Hoffmann and A. Pelster, Visibility of cold atomic gases in optical lattices for finite temperatures, Phys. Rev. A 79(5), 053623 (2009) https://doi.org/10.1103/PhysRevA.79.053623
16
Z. Lin, J. Zhang, and Y. Jiang, Visibility of ultracold Bose system in triangular optical lattices, Phys. Rev. A 86(3), 033625 (2012) https://doi.org/10.1103/PhysRevA.86.033625
M. Ohliger, Diploma thesis, Free University of Berlin, 2008
19
C. Becker, P. Soltan-Panahi, J. Kronjäger, S. Dörscher, K. Bongs, and K. Sengstock, Ultracold quantum gases in triangular optical lattices, New J. Phys. 12(6), 065025 (2010) https://doi.org/10.1088/1367-2630/12/6/065025
20
T. D. Graß, F. E. A. dos Santos, and A. Pelster, Excitation spectra of bosons in optical lattices from the Schwinger–Keldysh calculation, Phys. Rev. A 84(1), 013613 (2011) https://doi.org/10.1103/PhysRevA.84.013613
21
T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72(12), 126401 (2009) https://doi.org/10.1088/0034-4885/72/12/126401
22
C. Trefzger, C. Menotti, B. Capogrosso-Sansone, and M. Lewenstein, Ultracold dipolar gases in optical lattices, J. Phys. At. Mol. Opt. Phys. 44(19), 193001 (2011) https://doi.org/10.1088/0953-4075/44/19/193001
23
A. Lauer, D. Muth, and M. Fleischhauer, Transportinduced melting of crystals of Rydberg dressed atoms in a one-dimensional lattice, New J. Phys. 14(9), 095009 (2012) https://doi.org/10.1088/1367-2630/14/9/095009
24
P. Schauß, M. Cheneau, M. Endres, T. Fukuhara, S. Hild, A. Omran, T. Pohl, C. Gross, S. Kuhr, and I. Bloch, Observation of spatially ordered structures in a two-dimensional Rydberg gas, Nature 491(7422), 87 (2012) https://doi.org/10.1038/nature11596
25
A. Safavi-Naini, S. G. Soyler, G. Pupillo, H. R. Sadeghpour, and B. Capogrosso-Sansone, Quantum phases of dipolar bosons in bilayer geometry, New J. Phys. 15(1), 013036 (2013) https://doi.org/10.1088/1367-2630/15/1/013036
26
E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, Phase diagram of two-component bosons on an optical lattice, New J. Phys. 5, 113 (2003) https://doi.org/10.1088/1367-2630/5/1/113
27
P. Soltan-Panahi, D. Lühmann, J. Struck, P. Windpassinger, and K. Sengstock, Quantum phase transition to unconventional multi-orbital superfluidity in optical lattices, Nat. Phys. 8, 71 (2012)
28
A. Eckardt, P. Hauke, P. Soltan-Panahi, C. Becker, K. Sengstock, and M. Lewenstein, Frustrated quantum antiferromagnetism with ultracold bosons in a triangular lattice, Europhys. Lett. 89(1), 10010 (2010) https://doi.org/10.1209/0295-5075/89/10010
29
S. Pielawa, E. Berg, and S. Sachdev, Frustrated quantum Ising spins simulated by spinless bosons in a tilted lattice: From a quantum liquid to antiferromagnetic order, Phys. Rev. B 86(18), 184435 (2012) https://doi.org/10.1103/PhysRevB.86.184435
30
J. Ye, K. Zhang, Y. Li, Y. Chen, and W. Zhang, Optical Bragg, atomic Bragg and cavity QED detections of quantum phases and excitation spectra of ultracold atoms in bipartite and frustrated optical lattices, Ann. Phys. 328, 103 (2013) https://doi.org/10.1016/j.aop.2012.09.006
31
S. Peil, J. V. Porto, B. Laburthe Tolra, J. M. Obrecht, B. E. King, M. Subbotin, S. L. Rolston, and W. D. Phillips, Patterned loading of a Bose-Einstein condensate into an optical lattice, Phys. Rev. A 67, 051603(R) (2003)
32
J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V. Porto, Lattice of double wells for manipulating pairs of cold atoms, Phys. Rev. A 73(3), 033605 (2006) https://doi.org/10.1103/PhysRevA.73.033605
33
S. Fölling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera, T. Müller, and I. Bloch, Direct observation of second-order atom tunnelling, Nature 448(7157), 1029 (2007) https://doi.org/10.1038/nature06112
34
P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger, M. Moreno-Cardoner, S. Fölling, and I. Bloch, Counting atoms using interaction blockade in an optical superlattice, Phys. Rev. Lett. 101(9), 090404 (2008) https://doi.org/10.1103/PhysRevLett.101.090404
35
G. B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M. Stamper-Kurn, Ultracold atoms in a tunable optical Kagome lattice, Phys. Rev. Lett. 108(4), 045305 (2012) https://doi.org/10.1103/PhysRevLett.108.045305
36
T. Wang, X. F. Zhang, S. Eggert, and A. Pelster, Generalized effective-potential Landau theory for bosonic quadratic superlattices, Phys. Rev. A 87(6), 063615 (2013) https://doi.org/10.1103/PhysRevA.87.063615
37
M. Ohliger and A. Pelster, M. Ohliger, A. Pelster, and J. World, Green’s Function Approach to the Bose- Hubbard Model, World Journal of Condensed Matter Physics 3, 125 (2013), arXiv: 0810.4399
38
P. Soltan-Panahi, J. Struck, P. Hauke, A. Bick, W. Plenkers, G. Meineke, C. Becker, P. Windpassinger, M. Lewenstein, and K. Sengstock, Multi-component quantum gases in spin-dependent hexagonal lattices, Nat. Phys. 7(5), 434 (2011)
39
M. Iskin and J. K. Freericks, Strong-coupling perturbation theory for the extended Bose-Hubbard model, Phys. Rev. A 79(5), 053634 (2009) https://doi.org/10.1103/PhysRevA.79.053634
40
M. Iskin and J. K. Freericks, Momentum distribution of the insulating phases of the extended Bose-Hubbard model, Phys. Rev. A 80(6), 063610 (2009) https://doi.org/10.1103/PhysRevA.80.063610
41
D. van Oosten, P. van der Straten, and H. T. C. Stoof, Quantum phases in an optical lattice, Phys. Rev. A 63(5), 053601 (2001) https://doi.org/10.1103/PhysRevA.63.053601
42
B. Bradlyn, F. E. A. dos Santos, and A. Pelster, Effective action approach for quantum phase transitions in bosonic lattices, Phys. Rev. A 79(1), 013615 (2009) https://doi.org/10.1103/PhysRevA.79.013615
43
M. Peskin and D. Schröder, An Introduction to Quantum Field Theory, Westview Press, Boulder, 1995
44
D. L. Kovrizhin, G. V. Pai, and S. Sinha, Density wave and supersolid phases of correlated bosons in an optical lattice, Europhys. Lett. 72(2), 162 (2005) https://doi.org/10.1209/epl/i2005-10231-y
45
M. Köhl, H. Moritz, T. Stöferle, K. Günter, and T. Esslinger, Fermionic atoms in a three dimensional optical lattice: Observing Fermi surfaces, dynamics, and interactions, Phys. Rev. Lett. 94(8), 080403 (2005) https://doi.org/10.1103/PhysRevLett.94.080403
46
C. Becker, P. Soltan-Panahi, J. Kronjäger, S. Dörscher, K. Bongs, and K. Sengstock, Ultracold quantum gases in triangular optical lattices, New J. Phys. 12(6), 065025 (2010) https://doi.org/10.1088/1367-2630/12/6/065025
47
M. Köhl, H. Moritz, T. Stöferle, C. Schori, and T. Esslinger, Superfluid to Mott insulator transition in one, two, and three dimensions, J. Low Temp. Phys. 138(3–4), 635 (2005) https://doi.org/10.1007/s10909-005-2273-4
48
I. B. Spielman, W. D. Phillips, and J. V. Porto, Mottinsulator transition in a two-dimensional atomic Bose gas, Phys. Rev. Lett. 98(8), 080404 (2007) https://doi.org/10.1103/PhysRevLett.98.080404
49
M. Iskin, Route to supersolidity for the extended Bose- Hubbard model, Phys. Rev. A 83, 051606(R) (2011)
O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.S. Lühmann, B. A. Malomed, T. Sowiński, and J. Zakrzewski, Non-standard Hubbard models in optical lattices: A review, Rep. Prog. Phys. 78(6), 066001 (2015) https://doi.org/10.1088/0034-4885/78/6/066001
52
M. Lewenstein, A. Sanpera, and V. Ahufinger, Ultracold Atoms in Optical Lattices: Simulating Quantum Many- Body Systems, Oxford: Oxford University Press, 2012, pp 182–183 https://doi.org/10.1093/acprof:oso/9780199573127.001.0001
53
K. Góral, K. Rza¸żewski, and T. Pfau, Bose–Einstein condensation with magnetic dipole-dipole forces, Phys. Rev. A 61, 051601(R) (2000)
54
K. Góral and L. Santos, Ground state and elementary excitations of single and binary Bose-Einstein condensates of trapped dipolar gases, Phys. Rev. A 66(2), 023613 (2002) https://doi.org/10.1103/PhysRevA.66.023613
T. Sowiński, O. Dutta, P. Hauke, L. Tagliacozzo, and M. Lewenstein, Dipolar molecules in optical lattices, Phys. Rev. Lett. 108(11), 115301 (2012) https://doi.org/10.1103/PhysRevLett.108.115301
57
S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Extended Bose–Hubbard models with ultracold magnetic atoms, Science 352(6282), 201 (2016) https://doi.org/10.1126/science.aac9812
58
Disgusting ghost peaks are well known in cubic lattice systems [15]. Our method shows the existence of ghost peaks in square lattice when J/U>(J/U)c, but no ghost peak in triangular [16] and hexagonal lattice for arbitrary J/U. Thus, the existence of disgusting ghost peaks is not only due to the divergence of re-summed Green’ function, but also depends on the lattice structure or some unknown reasons. At the critical point (V ˜0= V ˜0c ), the ground state of the system is neither localized phases (MI or CDW) nor compressible phases (SS or SF), but it includes characteristic fingerprints of the physical properties of both localized and compressible phases. At V ˜0c , some tiny satellite peaks appear in ‘SS’ phase but not in ‘SF’ phase. The appearance of those tiny peaks can be deemed to be an evidence of ‘SS’ phase, since it coincides with the feature of ‘SS’ phase. In the case of J/U>(J/U)c, our theory may not be exactly solid, but it is available for triangular [16] and hexagonal systems. The above-mentioned argument indicates that when J/U>(J/U)c, if these satellite peaks appear in SS phase, these are real peaks; but they should be taken as ghost peaks in SF phases if existing, since there is no such peaks at the critical point where our theory is valid and it also does not coincide with the features of SF phase.
59
V. W. Scarola, E. Demler, and S. Das Sarma, Searching for a supersolid in cold-atom optical lattices, Phys. Rev. A 73, 051601(R) (2006)
