Strong-coupling perturbation theory for the extended Bose-Hubbard ...

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M. ISKIN AND J. K. FREERICKS PHYSICAL REVIEW A 79, 053634 2009 2.7 Mott(2) ∞ dimensions (zV=0.4U) m-f ext s-c the power-series expansion. Therefore, we conclude that, even in infinite dimensions, the agreement of the third-order strong-coupling perturbation theory with the exact meanfield theory is quite good. µ/U 1.8 0.9 CDW(2,1) Mott(1) CDW(1,0) 0 0 0.04 0.08 0.12 and V j dip =zVn A for jB sublattice. This leads to two coupled equations for A and B . Since A , B →0 + near the phase boundary, Eq. 19 can be satisfied only if 1 z 2 t 2 = n A +1 Un A + zVn B − − n B +1 Un B + zVn A − − x = dt/U FIG. 3. Color online Chemical potential in units of U versus x=dt/U phase diagram for a d→-dimensional hypercubic lattice. Here the nearest-neighbor repulsion scales inversely with d such that zV=0.4U. The dotted lines correspond to phase boundaries for the Mott insulator to superfluid and CDW insulator to supersolid states as determined from the third-order strongcoupling perturbation theory s-c. The circles correspond to the extrapolation fit ext discussed in the text. The red solid lines correspond to phase boundaries for the Mott insulator to superfluid and CDW insulator to supersolid states as determined from the meanfield theory m-f which becomes exact for d→. n A Un A −1 + zVn B − n B Un B −1 + zVn A − , 22 which gives a quartic equation for . Since a simple closedform analytic solution for is not possible, we solve Eq. 22 with MATHEMATICA for each of the CDW lobes separately. In the d→ limit, we also checked that our strongcoupling perturbation results for the CDW lobes agree with this exact solution when the latter is expanded out to third order in t, providing again an independent check of the algebra. In Fig. 3, the results of the third-order strong-coupling perturbation theory dotted lines is compared to those of the exact mean-field theory red solid lines and of the extrapolation technique circles for an infinite d→-dimensional hypercubic lattice when zV=0.4U. Notice that in infinite dimensions, both t and V must scale inversely with d such that dt and dV are finite. The extrapolated solutions are indistinguishable from the exact ones for the Mott lobes, and they are within 5% of each other for the tips of the CDW lobes. It turns out that this minor disagreement around the tips of the CDW lobes is due to the large coefficient of the Ot 4 term in V. CONCLUSIONS We analyzed the zero-temperature phase diagram of the extended BH model with on-site and nearest-neighbor boson-boson repulsions in d1-dimensional hypercubic lattices. We used the many-body version of Rayleigh- Schrödinger perturbation theory in the kinetic-energy term with respect to the ground state of the system when the kinetic-energy term is absent. This technique was previously used to discuss the phase diagram of the on-site BH model 7,8, and its extrapolated results showed excellent agreement with the recent quantum Monte Carlo simulations 25,26. Here, we generalized this method to the extended BH model, hoping to develop an analytical approach which could be as accurate as the numerical ones. We derived analytical expressions for the phase boundaries between the incompressible Mott or CDW insulators and compressible superfluid or supersolid phases up to third order in the hopping t. However, we remark that the strong-coupling perturbation theory developed here cannot be used to calculate the phase boundary between two compressible phases, e.g., the supersolid-to-superfluid transition. We also proposed a chemical-potential extrapolation technique based on the scaling theory to extrapolate our thirdorder power-series expansion into a functional form that is appropriate for the Mott or CDW lobes. We believe some of our results could potentially be observed with ultracold dipolar Bose gases loaded into optical lattices 27,29. This is motivated by the recent success in observing superfluid–to–Mott insulator transition with ultracold pointlike Bose gases loaded into optical lattices. Such lattices are created by the intersection of laser fields, and they are nondissipative periodic potential-energy surfaces for the atoms. An ultracold dipolar Bose gas can be realized in many ways with optical lattices. For instance, heteronuclear molecules which have permanent electric dipole moments, Rydberg atoms which have very large induced electric dipole moment, or chromiumlike atoms which have large intrinsic magnetic moment can be used to generate sufficiently strong long-ranged dipole-dipole interactions. This work can be extended in several ways if desired. For instance, our current results for the CDW phase are not directly applicable to the one-dimensional case. We are currently working on this problem and will report our results elsewhere. In addition, it turns out that the coefficient of the Ot 4 term in the power series is generally small for the Mott lobes, but it can become very large for the CDW lobes when zVU. Therefore, inclusion of the Ot 4 is necessary to improve the accuracy of the phase boundaries near the tips of the CDW lobes when zVU. One can also examine frustration effects by performing these calculations on nonbipartite lattices. In addition, one can include the next-nearestneighbor repulsion term to the current model, which would lead to additional striped CDW phases. Lastly, one can also 053634-8
STRONG-COUPLING PERTURBATION THEORY FOR THE… examine how the momentum distribution changes with the hopping in the CDW phase or in the Mott phase when there is a nearest-neighbor repulsion. This last calculation could have direct relevance for experiments on these systems and would generalize recent results for the V=0 case 30. PHYSICAL REVIEW A 79, 053634 2009 ACKNOWLEDGMENTS We would like to thank E. Tiesinga for many useful discussions. J.K.F. acknowledges support under USARO Grant No. W911NF0710576 with funds from the DARPA OLE Program. 1 D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 1998. 2 M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature London 415, 392002. 3 I. B. Spielman, W. D. Phillips, and J. V. Porto, Phys. Rev. Lett. 98, 080404 2007. 4 I. B. Spielman, W. D. Phillips, and J. V. Porto, Phys. Rev. Lett. 100, 120402 2008. 5 F. Gerbier, S. Trotzky, S. Folling, U. Schnorrberger, J. D. Thompson, A. Widera, I. Bloch, L. Pollet, M. Troyer, B. Capogrosso-Sansone, N. V. Prokofev, and B. V. Svistunov, Phys. Rev. Lett. 101, 155303 2008. 6 M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 1989. 7 J. K. Freericks and H. Monien, Europhys. Lett. 26, 545 1994. 8 J. K. Freericks and H. Monien, Phys. Rev. B 53, 2691 1996. 9 P. Buonsante, V. Penna, and A. Vezzani, Phys. Rev. B 70, 184520 2004. 10 S. Ospelkaus, A. Pe’er, K.-K. Ni, J. J. Zirbel, B. Neyenhuis, S. Kotochigova, P. S. Julienne, J. Ye, and D. S. Jin, Nat. Phys. 4, 622 2008. 11 K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Science 322, 231 2008. 12 C. Bruder, Rosario Fazio, and Gerd Schön, Phys. Rev. B 47, 342 1993. 13 Parhat Niyaz, R. T. Scalettar, C. Y. Fong, and G. G. Batrouni, Phys. Rev. B 50, 362 1994. 14 G. G. Batrouni, R. T. Scalettar, G. T. Zimanyi, and A. P. Kampf, Phys. Rev. Lett. 74, 2527 1995. 15 Anne van Otterlo, Karl-Heinz Wagenblast, Reinhard Baltin, C. Bruder, Rosario Fazio, and Gerd Schön, Phys. Rev. B 52, 16176 1995. 16 Till D. Kühner, Steven R. White, and H. Monien, Phys. Rev. B 61, 12474 2000. 17 Pinaki Sengupta, Leonid P. Pryadko, Fabien Alet, Matthias Troyer, and Guido Schmid, Phys. Rev. Lett. 94, 207202 2005. 18 D. L. Kovrizhin, G. Venketeswara Pai, and S. Sinha, Europhys. Lett. 72, 162 2005. 19 S. R. Hassan, L. de Medici, and A.-M. S. Tremblay, Phys. Rev. B 76, 144420 2007. 20 Yu-Chun Chen, Roger G. Melko, Steffan Wessel, and Ying-Jer Kao, Phys. Rev. B 77, 014524 2008. 21 A. J. Leggett, Phys. Rev. Lett. 25, 1543 1970. 22 E. Kim and M. H. W. Chan, Science 305, 1941 2004. 23 See Ref. 20 and references therein for the effects of nextnearest-neighbor hopping-interaction terms on the phase diagram, and see Ref. 19 and references therein for a related model on triangular lattices. 24 L. D. Landau and L. M. Lifshitz, Quantum Mechanics Butterworth-Heinemann, Oxford, 1981. 25 B. Capogrosso-Sansone, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. B 75, 134302 2007. 26 B. Capogrosso-Sansone, S. G. Söyler, N. Prokof’ev, and B. Svistunov, Phys. Rev. A 77, 015602 2008. 27 C. Trefzger, C. Menotti, and M. Lewenstein, Phys. Rev. A 78, 043604 2008. 28 D. van Oosten, P. van der Straten, and H. T. C. Stoof, Phys. Rev. A 63, 053601 2001. 29 K. Goral, L. Santos, and M. Lewenstein, Phys. Rev. Lett. 88, 170406 2002. 30 J. K. Freericks, H. R. Krishnamurthy, Yasuyuki Kato, Naoki Kawashima, and Nandini Trivedi, e-print arXiv:0902.3435, Phys. Rev. A to be published. 053634-9
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