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PHYSICAL REVIEW LETTERS
PRL 113, 155303 (2014)
Strong Interaction Effects and Criticality of Bosons in Shaken Optical Lattices Wei Zheng,1 Boyang Liu,1 Jiao Miao,1 Cheng Chin,2 and Hui Zhai1,* 1
Institute for Advanced Study, Tsinghua University, Beijing 100084, China James Franck Institute, Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA (Received 5 March 2014; revised manuscript received 31 August 2014; published 10 October 2014) 2
We study the quantum phase transitions and identify a tricritical point between a normal Bose superfluid, a superfluid that breaks additional Z2 Ising symmetry, and a Mott insulator in a recent shaken optical lattice experiment. We show that near the transition between normal and Z2 symmetry breaking superfluids, bosons can condense into a momentum state with high or even locally maximum kinetic energies due to the interaction effect. We present a general low-energy effective field theory that treats both the superfluid transition and the Ising transition in a uniform framework. Using the perturbative renormalization group method, we find that the critical behavior of the quantum phase transition belongs to a universality class different from that of a dilute Bose gas. DOI: 10.1103/PhysRevLett.113.155303
PACS numbers: 67.85.-d, 03.75.Lm, 05.30.Rt, 64.70.Tg
Critical phenomena lie in the center of modern manybody physics because different systems can develop a universal behavior near the phase transition. Two of the most paradigmatic phase transitions are the Ising transition and the superfluid transition, across which a discrete Z2 symmetry and a Uð1Þ gauge symmetry are spontaneously broken, respectively. If a system can exhibit phase transitions of both types, their interplay can lead to a novel type of critical phenomena. Such a system is not known in real materials, to the best of our knowledge, but has been recently demonstrated in cold atom systems. So far there are at least three approaches to realize such a transition in cold atom experiments: (a) Bose condensates with spin-orbit coupling induced by Raman transitions, where the transition is driven by changing the Raman coupling strength [1,2]; (b) Bose condensates in an optical lattice with staggered magnetic field, where the transition is driven by changing the ratio of two hopping amplitudes along two different spatial directions [3]; and (c) Bose condensates in a shaking optical lattice, where the transition is driven by tuning the shaking frequency [4] or shaking amplitude [5]. These systems share the following common feature. As one changes a tunable parameter f, their single particle dispersion along one direction, denoted by ϵðkx Þ, gradually evolves from a single minimum at kx ¼ 0 (normal superfluid regime, labeled by SF) to double degenerate minima at k . In the latter case, generally one should assume the condensate wave function as a superposition of φðkþ Þ and φðk− Þ, and it is up to the interaction between bosons to determine the superposition coefficients [6–10]. When interaction favors a condensate wave function to be either purely φðkþ Þ or purely φðk− Þ, the superfluid will break the Z2 symmetry and is labeled by Z2 SF. In this Letter we investigate the strong interaction effect in this transition. We show that transition between SF and 0031-9007=14=113(15)=155303(5)
Z2 SF strongly depends on the interaction between particles, and this dependence exhibits opposite behavior in the shallow lattice and the deep lattice regimes. More importantly, in contrast to previous works [6–10] on a uniform system of model (a), our work focuses on model (c) in which the Mott insulator will emerge with strong interactions. We identify a quantum tricritical point between Mott insulator (MI), SF, and Z2 SF phases. For normal SF to MI transition, the upper critical dimensions are dc ¼ 3 and dc ¼ 2 for the case with and without particle-hole symmetry, respectively. Here we show that at this tricritical point, dc is changed to dc ¼ 7=2 and dc ¼ 5=2, respectively. That is to say, the importance of interactions changes from marginal to relevant for three dimensions with particle-hole symmetry, or for two dimensions without particle-hole symmetry, which can yield a critical phenomena different from the normal SF to MI transition. This has been overlooked in previous studies of lattice modes with spin-orbit coupling or staggered magnetic flux [11–20]. Shaken lattice model.—Consider a d-dimensional lattice (d ¼ 1, 2, or 3), and along one spatial direction (say, xˆ ), the relative phase θ between two counterpropagating lasers is time-periodically modulated, which will result in a time-dependent lattice potential [21,22], HðtÞ ¼
θðtÞ kˆ 2x ; þ Vcos2 k0 x þ 2 2m
ð1Þ
where θðtÞ ¼ f cos ðωtÞ, f is the shaking amplitude, and Δx ¼ f=ð2k0 Þ is the maximum lattice displacement. By employing the Bessel function expansion, the lattice potential can be expressed as VðtÞ ¼ VJ0 ðfÞcos2 ðk0 xÞ − VJ1 ðfÞ sin ð2k0 xÞ cos ðωtÞ þ …. The first term gives rise to a static lattice potential with a static
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PHYSICAL REVIEW LETTERS
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band structure ελ ðkx Þ and the Bloch wave function φλ;kx ðxÞ. (λ is the band index and kx is the quasimomentum.) Denoting Δ as the separation between the bottom of s band and top of p band, we consider the experimental situation ω ≳ Δ [5], as shown in Fig. 1(a). This near resonance condition make the situation quite different from previous shaking lattice proposals and experiments [4,22,23]. The calculation of single particle band structure [5] can be therefore simplified by keeping only s and px bands [24], and the time-dependent potential can also been simplified as VðtÞ ¼ −VJ1 ðfÞ sin ð2k0 xÞ cos ðωtÞ, which couples s and px bands. In the two-band bases, and upon a rotating wave approximation, it is straightforward to show that the eigenenergies are given by ϵ ðkx Þ ¼ Akx =2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ2kx =4 þ jΩkx j2 ;
ð2Þ
where Ωkx ¼ −VJ1 ðfÞhφp;kx j sin ð2k0 xÞjφs;kx i=2, Akx ¼ εp ðkx Þ þ εs ðkx Þ þ ω, Δkx ¼ εp ðkx Þ − εs ðkx Þ − ω. Two eigenwave functions are denoted by φþ;kx ðxÞ and φ−;kx ðxÞ, respectively. Experimentally, as shaking is adiabatically turned on, bosons will remain in the ϵþ ðkx Þ band and the lifetime can be as long as 1 s [5]. We show in Fig. 1(b) that across a critical shaking amplitude f 0c , ϵþ ðkx Þ exhibits a transition from single minimum at zero momentum to double minima at finite momentum kmin . For f > f 0c , we can assume the condensate wave function to be a linear superposition as ΨðxÞ ¼ sin αψ þ;kmin ðxÞ þ cos αψ þ;−kmin ðxÞ. Whereas in this case the interaction energy is always minimized by choosing α equaling to zero or π=2. That is to say, the condensate will break the Z2 symmetry across f 0c . Such a transition, as well as domain wall formation in the symmetry breaking phase, has been observed in a recent experiment [5].
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Interaction shift of SF to Z2 SF transition.—For Bose condensates in a single momentum state, the mean-field interaction energy can be simplified as [25] sp pp 2 2 ϵint ðkx Þ ¼ U ss ð3Þ kx ns;kx þ 4U kx ns;kx np;kx þ U kx np;kx ; R 0 where Uλkxλ ¼ g dxjφλ0 ;kx j2 jφλ;kx j2 , nλ;kx is the boson number, g is the interaction constant, λ and λ0 denote s or px. With Eqs. (2) and (3), the total energy of the condensate is written as ϵðkx Þ ¼ ϵþ ðkx Þ þ ϵint ðkx Þ. The condensate momentum kc and the critical point f c can be determined by minimizing ϵðkx Þ with respect to kx . Our findings are summarized in Fig. 2. For deep lattices in the tight binding region [Figs. 2(a) and 2(c)], the 0 dominant contribution to Uλkxλ is the on-site interaction of localized Wannier states. Since the Wannier wave function for the px band is more extended, and because the mixing between the s and px band is stronger for smaller momentum, the repulsive interaction energy has a minimum at kx ¼ 0. Therefore, in the shaded area of Fig. 2(b), even when the kinetic energy already exhibits double minimum, the total energy still possesses a single minimum at zero momentum. In another word, in this regime, bosons are condensed into the local maximum of single particle kinetic energy, as shown in Fig. 2(c). In contrast, for shallow lattices where the Bloch wave function behaves like plane waves, around kx ≈ 0, for the s band the Bloch wave function is nearly a constant, whereas for the px band the Bloch wave function behaves like sinð2k0 xÞ, which has a stronger spatial modulation. Thus,
4 4 3
3.8
2 3.6
1 0
3.4
−1
(b)
(a)
−2 −1
0
3.2 1 −1
0
1
FIG. 1 (color online). Band structure of a shaken lattice. (a) Band structure before shaking. The red solid line on the top is the dressed s band with energy shifted by an energy ℏω. (b) The solid line is the dispersion for small shaking amplitude f < f 0c and the dashed line is the dispersion for large shaking amplitude f > f 0c . Energy is plotted in units of lattice recoil energy Er ¼ ℏ2 k20 =ð2mÞ.
FIG. 2 (color online). Interaction shifts of SF-Z2 SF quantum critical point. (a),(c) Deep lattice with V=Er ¼ 16 and ℏω=Er ¼ 7.1; (b),(d) Shallow lattice with V=Er ¼ 4 and ℏω=Er ¼ 4.4. (a) and (b) Condensate momentum kc as a function of f. Blue dashed line is for noninteracting and the red solid line is for the interacting case with gn=Er ¼ 1. (c) and (d) Interaction (gn)-shaking amplitude(f) phase diagram for a fixed frequency ω. Gray shaded areas show regions where atoms condense to states with finite kinetic energy.
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Eðjϕj; kx Þ ¼ ðk4x þ ak2x Þjϕj2 þ rjϕj2 þ ðα þ βk2x Þjϕj4 : ð7Þ The ground state is determined by ∂E=∂kx ¼ 0 and ∂E=∂jϕj ¼ 0, which give rise to three different phases: ϕ ¼ 0 as the MI phase; ϕ ≠ 0 and kx ¼ 0 as the SF phase; and ϕ ≠ 0 and kx ≠ 0 as the Z2 SF phase. The phase diagram is shown in Fig. 3 in terms of f and g. We find all transitions are second order and all three phases meet at a tricritical point at a ¼ r ¼ 0, around which the interaction effect is the strongest. Hereafter, we are concerned about the critical exponent nearby the quantum tricritical point. Our discussion can be divided into two cases: Case A.—No particle-hole symmetry. K 1 ≠ 0 and the K 2 term becomes irrelevant [26]. In this case, it is straightforward to show that the scaling dimension of ϕ is dim½ϕ ¼ −ð2d þ 7Þ=4 and the critical dimension is 5=2 [25]. For a physical system with d ¼ 2, the scaling dimensions of r, a, and α are dim ¼ 2, dim ¼ 1, and dim½α ¼ 1=2, respectively, and all these three terms are relevant. This is different from the conventional Bose Hubbard model with quadratic 2
Z2 SF 1
dd rdτfK 1 ϕ ∂ τ ϕ þ K 2 j∂ τ ϕj2 þ E½ϕ ; ϕg;
Shaking amplitude
Z
[26]. The α and β term come from interaction between bosons. The parameter α is positive for repulsive interactions. In the microscopic model discussed above, parameter β can be either positive or negative. Here, for simplicity, we consider the case with positive β. The parameter a and r are two main control parameters in this system; a controls single particle dispersion. As f changes across f 0c , a changes from positive to negative. r controls interaction strength g. As g increases, r changes from negative to positive, which drives a transition from SF to MI. This effective action basically includes all low-energy terms that are allowed by symmetry of this system and consistent with the microscopic model. Assuming ϕ ¼ jϕjeikx x , we can rewrite E as
0 c
the repulsive interaction is enhanced as the px component increases, and the interaction energy displays a local maximum at zero momentum. Therefore, opposite to the tight binding region, f c decreases as repulsive interaction increases. In the shaded area of Fig. 2(d), bosons are condensed into a finite momentum state which is not the kinetic energy minimum state. This phenomenon is quite intriguing since it invalidates the conventional notion that bosons always condense into the single particle energy minimum. This happens when the self-energy correction due to interactions has strong momentum dependence. So far, this effect has been proposed by Li et al. in the spin-orbit coupled 87 Rb system [system (a) [10]]. Comparing these two systems, in the spin-orbit coupled system this shift is relatively weak because the interaction constants between different spin states of 87 Rb atom are very close [10]. While in our system, this shift is due to the difference of interaction sp constants between different bands, such as Uss kx , U kx , and pp Ukx , and their difference is very large because of the different behaviors of Bloch wave functions. Therefore, this effect is much more profound and is much easier to observe experimentally. Second, in our system the phenomena are richer because the interaction shift can be either positive or negative depending on lattice depth. Tricritical point between SF, Z2 SF, and MI.—Since our system is a lattice system, it will enter a MI phase for sufficiently strong interactions. Here we shall explore a general low-energy effective theory to describe both the SFMI transition and the Ising transition. This effective field theory should capture two key ingredients from microscopic physics as discussed above: (i) the kinetic energy expanded in terms of small kx is given by ak2x þ bk4x, where a can change sign and b > 0; and (ii) the interaction term contains both a constant term and a term proportional to k2x . Nearby the SF-MI transition point, the partition function can be expanded in terms of order parameter ϕ as [26,27] Z Z ¼ D½ϕ ; ϕ expf−S½ϕ ; ϕg; ð4Þ S¼
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PHYSICAL REVIEW LETTERS
0 c
PRL 113, 155303 (2014)
ð5Þ
and (by setting b ¼ 1) E ¼ j∂ 2x ϕj2 þ aj∂ x ϕj2 þ T þ rjϕj2 þ αjϕj4 þ βjϕ∂ x ϕj2 ; ð6Þ where ϕ is the order parameter, T ¼ j∂ y ϕj2 for d ¼ 2 and T ¼ j∂ y ϕj2 þ j∂ z ϕj2 for d ¼ 3. The K 1 and K 2 terms describe the first-order and second-order derivative term for the temporal dynamics of the ϕ field. When the particle and hole excitation of a MI state have the same energy, which is referred to as the particle-hole symmetric case, K 1 vanishes
0
MI
SF 1
2
2
1
0
Interaction
1 c
2 c
FIG. 3 (color online). Schematic of the tricritical point of interacting bosons in a shaken lattice. The three phases are the Mott insulator phase (MI), normal superfluid (SF) phase, and the superfluid phase that breaks Z2 symmetry (Z2 SF). gc defines the critical interaction strength for the normal superfluid to Mott insulator transition.
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dispersion, where dim½α ¼ 0 and the α term is marginal in two dimensions. The scaling dimension of β is dim½β ¼ −1=2, and the β term is irrelevant. That means although the momentum-dependent interaction plays an important role at the mean-field level to shift the critical value, it does not play a significant role for fluctuations beyond the mean field. The one-loop renormalizaiton group (RG) equations are derived as [25] da ¼ a; dl
dr dα α2 ¼ 2r; ¼ ϵα − I ðaÞ; dl dl 1þr 2
ð8Þ
where ϵ ¼ 5=2 − d ¼ 1=2, I 2 is a function of a defined in the Supplemental Material [25]. In addition to the Gaussian fixed point at ða; r; αÞ ¼ ð0; 0; 0Þ, these RG equations give another non-Gaussian fixed point at ða;r;αÞ¼(0;0;ϵ= I 2 ð0Þ). The flow diagram is shown in Fig. 4(a). However, since in this case r does not receive any correction from interaction, the critical exponent of superfluid transition still remains as its mean-field value ν ¼ 1=2, as in the usual Bose gas case [27,28]. Case B.—Particle-hole symmetry. K 1 ¼ 0 [26]. In this case, the scaling dimension of ϕ is dim½ϕ ¼ −ð5 þ 2dÞ=4 and the critical dimension is 7=2. In this case, for a system in two dimensions, ϵ ¼ 7=2 − d ¼ 3=2. Since ϵ > 1 it is not accurate to treat the system by means of perturbative expansion. For a system with d ¼ 3, ϵ ¼ 1=2, and the scaling dimensions of different terms are dim ¼ 2, dim ¼ 1, dim½α ¼ 1=2, and dim½β ¼ −1=2, respectively. These are all identical to case A. However, in this case, the one-loop RG equations read da ¼ a; dl
dr 2αI 3 ðaÞ ffi; ¼ 2r þ pffiffiffiffiffiffiffiffiffiffi dl 1þr
dα 5I 3 ðaÞα2 ¼ ϵα − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; dl 2 ð1 þ rÞ3
More importantly, the critical exponent of superfluid transition ν ¼ 1=ð2 − 2ϵ=5Þ ≃ 1=2 þ ϵ=10 ¼ 11=20 is now different from the mean-field value [25]. This is also different from conventional bosons with k2 dispersion with K 1 ¼ 0, where interaction is marginal for d ¼ 3 [27]. In this sense, it represents a new type of critical behavior. Conclusion and experimental detection.—We have shown that interaction between particles can significantly shift the location of SF to Z2 SF transition. Bosons can condense to momentum state with high or locally maximum kinetic energy near the quantum critical point. This can be observed in experiment by measuring the transition point with different interaction strength. We have also shown that there exists a quantum tricritical point between the MI, SF, and Z2 SF phases, nearby which interactions dictate the universal behavior and yield new universal critical exponents. Previously, for the conventional Bose Hubbard model, the critical exponent ν ¼ 1=2 has been measured using in situ density measurements inside a harmonic trap [29]. In the system of shaken lattice, one can tune the interaction to the vicinity of the Mott transition, tune the chemical potential to the particle-hole symmetric point, and tune the band dispersion by shaking to the vicinity of the Ising transition. Thus, with the same in situ method, this new critical phenomenon predicted here can be experimentally verified. We thank Xiaoliang Qi, Fa Wang, Hong Yao, and Dimo I. Uzunov for helpful discussions. This work is supported by the Tsinghua University Initiative Scientific Research Program (H. Z.), NSFC Grants No. 11004118 (H. Z.) and No. 11174176 (H. Z.), and NKBRSFC under Grant No. 2011CB921500 (H. Z.), NSF MRSEC (DMR0820054) (C. C.), NSF Grant No. PHY- 0747907 (C. C.), and ARO-MURI Grant No. W911NF-14-1-0003 (C. C.).
ð9Þ *
where ϵ ¼ 7=2 − d ¼ 1=2 and I 3 ðaÞ is also defined in the Supplemental Material [25]. The key difference is that the r term now receives correction from interaction. The new non-Gaussian fixed point is located at ða;r;αÞ¼½0;−2ϵ=5; 2ϵ=(5I 3 ð0Þ), and the flow diagram is shown in Fig. 4(b).
(a)
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(b)
FIG. 4. Renormalization group flow diagram of (a) case A (no particle-hole symmetry) and (b) case B (particle-hole symmetry).
[email protected] [1] Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Nature (London) 471, 83 (2011). [2] S. C. Ji, J. Y. Zhang, L. Zhang, Z. D. Du, W. Zheng, Y. J. Deng, H. Zhai, S. Chen, and J. W. Pan, Nat. Phys. 10, 314 (2014). [3] M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011). [4] J. Struck, C. Ölschläger, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger, and K. Sengstock, Science 333, 996 (2011). [5] C. V. Parker, L. C. Ha, and C. Chin, Nat. Phys. 9, 769 (2013). [6] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A 78, 023616 (2008). [7] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett. 105, 160403 (2010). [8] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011). [9] C. J. Wu, I. Mondragon-Shem, and X. F. Zhou, Chin. Phys. Lett. 28, 097102 (2011).
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[10] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 108, 225301 (2012). [11] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi, Phys. Rev. Lett. 109, 085302 (2012). [12] J. Radic, A. Di Ciolo, K. Sun, and V. Galitski, Phys. Rev. Lett. 109, 085303 (2012). [13] Z. Cai, X. Zhou, and C. Wu, Phys. Rev. A 85, 061605(R) (2012). [14] S. Mandal, K. Saha, and K. Sengupta, Phys. Rev. B 86, 155101 (2012). [15] J. Yao and S. Zhang, Phys. Rev. A 90, 023608 (2014). [16] J. Zhao, S. Hu, J. Chang, P. Zhang, and X. Wang, Phys. Rev. A 89, 043611 (2014). [17] J. Zhao, S. Hu, J. Chang, F. Zheng, P. Zhang, and X. Wang, Phys. Rev. B 90, 085117 (2014). [18] M. Piraud, Z. Cai, I. P. McCulloch, and U. Schollwock, Phys. Rev. A 89, 063618 (2014). [19] S. Peotta, L. Mazza, E. Vicari, M. Polini, R. Fazio, and D. Rossini, J. Stat. Mech. 09 (2014) P09005. [20] Z. Xu, W. S. Cole, and S. Zhang, Phys. Rev. A 89, 051604 (R) (2014).
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[21] N. Gemelke, E. Sarajlic, Y. Bidel, S. Hong, and S. Chu, Phys. Rev. Lett. 95, 170404 (2005). [22] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. 99, 220403 (2007). [23] A. Eckardt, C. Weiss, and M. Holthaus, Phys. Rev. Lett. 95, 260404 (2005). [24] We also verified numerically that qualitatively our results are not changed by including higher bands. [25] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.113.155303 for a detailed derivation of mean-field interaction energy, counting of the scaling dimension, and the derivation of the renormalization group equations. [26] S. Sachdev, Quantum Phase Transition, 2nd ed. (Cambridge University Press, Cambridge, England, 2011), Chap. 9. [27] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989). [28] D. I. Uzunov, Phys. Lett. 87A, 11 (1981). [29] X. Zhang, C. L. Hung, S. K. Tung, and C. Chin, Science 335, 1070 (2012).
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Strong Interaction Effects and Criticality of Bosons in Shaken Optical Lattices Wei Zheng,1 Boyang Liu,1 Jiao Miao,1 Cheng Chin,2 and Hui Zhai1,* 1
Institute for Advanced Study, Tsinghua University, Beijing 100084, China James Franck Institute, Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA (Received 5 March 2014; revised manuscript received 31 August 2014; published 10 October 2014) 2
We study the quantum phase transitions and identify a tricritical point between a normal Bose superfluid, a superfluid that breaks additional Z2 Ising symmetry, and a Mott insulator in a recent shaken optical lattice experiment. We show that near the transition between normal and Z2 symmetry breaking superfluids, bosons can condense into a momentum state with high or even locally maximum kinetic energies due to the interaction effect. We present a general low-energy effective field theory that treats both the superfluid transition and the Ising transition in a uniform framework. Using the perturbative renormalization group method, we find that the critical behavior of the quantum phase transition belongs to a universality class different from that of a dilute Bose gas. DOI: 10.1103/PhysRevLett.113.155303
PACS numbers: 67.85.-d, 03.75.Lm, 05.30.Rt, 64.70.Tg
Critical phenomena lie in the center of modern manybody physics because different systems can develop a universal behavior near the phase transition. Two of the most paradigmatic phase transitions are the Ising transition and the superfluid transition, across which a discrete Z2 symmetry and a Uð1Þ gauge symmetry are spontaneously broken, respectively. If a system can exhibit phase transitions of both types, their interplay can lead to a novel type of critical phenomena. Such a system is not known in real materials, to the best of our knowledge, but has been recently demonstrated in cold atom systems. So far there are at least three approaches to realize such a transition in cold atom experiments: (a) Bose condensates with spin-orbit coupling induced by Raman transitions, where the transition is driven by changing the Raman coupling strength [1,2]; (b) Bose condensates in an optical lattice with staggered magnetic field, where the transition is driven by changing the ratio of two hopping amplitudes along two different spatial directions [3]; and (c) Bose condensates in a shaking optical lattice, where the transition is driven by tuning the shaking frequency [4] or shaking amplitude [5]. These systems share the following common feature. As one changes a tunable parameter f, their single particle dispersion along one direction, denoted by ϵðkx Þ, gradually evolves from a single minimum at kx ¼ 0 (normal superfluid regime, labeled by SF) to double degenerate minima at k . In the latter case, generally one should assume the condensate wave function as a superposition of φðkþ Þ and φðk− Þ, and it is up to the interaction between bosons to determine the superposition coefficients [6–10]. When interaction favors a condensate wave function to be either purely φðkþ Þ or purely φðk− Þ, the superfluid will break the Z2 symmetry and is labeled by Z2 SF. In this Letter we investigate the strong interaction effect in this transition. We show that transition between SF and 0031-9007=14=113(15)=155303(5)
Z2 SF strongly depends on the interaction between particles, and this dependence exhibits opposite behavior in the shallow lattice and the deep lattice regimes. More importantly, in contrast to previous works [6–10] on a uniform system of model (a), our work focuses on model (c) in which the Mott insulator will emerge with strong interactions. We identify a quantum tricritical point between Mott insulator (MI), SF, and Z2 SF phases. For normal SF to MI transition, the upper critical dimensions are dc ¼ 3 and dc ¼ 2 for the case with and without particle-hole symmetry, respectively. Here we show that at this tricritical point, dc is changed to dc ¼ 7=2 and dc ¼ 5=2, respectively. That is to say, the importance of interactions changes from marginal to relevant for three dimensions with particle-hole symmetry, or for two dimensions without particle-hole symmetry, which can yield a critical phenomena different from the normal SF to MI transition. This has been overlooked in previous studies of lattice modes with spin-orbit coupling or staggered magnetic flux [11–20]. Shaken lattice model.—Consider a d-dimensional lattice (d ¼ 1, 2, or 3), and along one spatial direction (say, xˆ ), the relative phase θ between two counterpropagating lasers is time-periodically modulated, which will result in a time-dependent lattice potential [21,22], HðtÞ ¼
θðtÞ kˆ 2x ; þ Vcos2 k0 x þ 2 2m
ð1Þ
where θðtÞ ¼ f cos ðωtÞ, f is the shaking amplitude, and Δx ¼ f=ð2k0 Þ is the maximum lattice displacement. By employing the Bessel function expansion, the lattice potential can be expressed as VðtÞ ¼ VJ0 ðfÞcos2 ðk0 xÞ − VJ1 ðfÞ sin ð2k0 xÞ cos ðωtÞ þ …. The first term gives rise to a static lattice potential with a static
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© 2014 American Physical Society
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band structure ελ ðkx Þ and the Bloch wave function φλ;kx ðxÞ. (λ is the band index and kx is the quasimomentum.) Denoting Δ as the separation between the bottom of s band and top of p band, we consider the experimental situation ω ≳ Δ [5], as shown in Fig. 1(a). This near resonance condition make the situation quite different from previous shaking lattice proposals and experiments [4,22,23]. The calculation of single particle band structure [5] can be therefore simplified by keeping only s and px bands [24], and the time-dependent potential can also been simplified as VðtÞ ¼ −VJ1 ðfÞ sin ð2k0 xÞ cos ðωtÞ, which couples s and px bands. In the two-band bases, and upon a rotating wave approximation, it is straightforward to show that the eigenenergies are given by ϵ ðkx Þ ¼ Akx =2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ2kx =4 þ jΩkx j2 ;
ð2Þ
where Ωkx ¼ −VJ1 ðfÞhφp;kx j sin ð2k0 xÞjφs;kx i=2, Akx ¼ εp ðkx Þ þ εs ðkx Þ þ ω, Δkx ¼ εp ðkx Þ − εs ðkx Þ − ω. Two eigenwave functions are denoted by φþ;kx ðxÞ and φ−;kx ðxÞ, respectively. Experimentally, as shaking is adiabatically turned on, bosons will remain in the ϵþ ðkx Þ band and the lifetime can be as long as 1 s [5]. We show in Fig. 1(b) that across a critical shaking amplitude f 0c , ϵþ ðkx Þ exhibits a transition from single minimum at zero momentum to double minima at finite momentum kmin . For f > f 0c , we can assume the condensate wave function to be a linear superposition as ΨðxÞ ¼ sin αψ þ;kmin ðxÞ þ cos αψ þ;−kmin ðxÞ. Whereas in this case the interaction energy is always minimized by choosing α equaling to zero or π=2. That is to say, the condensate will break the Z2 symmetry across f 0c . Such a transition, as well as domain wall formation in the symmetry breaking phase, has been observed in a recent experiment [5].
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Interaction shift of SF to Z2 SF transition.—For Bose condensates in a single momentum state, the mean-field interaction energy can be simplified as [25] sp pp 2 2 ϵint ðkx Þ ¼ U ss ð3Þ kx ns;kx þ 4U kx ns;kx np;kx þ U kx np;kx ; R 0 where Uλkxλ ¼ g dxjφλ0 ;kx j2 jφλ;kx j2 , nλ;kx is the boson number, g is the interaction constant, λ and λ0 denote s or px. With Eqs. (2) and (3), the total energy of the condensate is written as ϵðkx Þ ¼ ϵþ ðkx Þ þ ϵint ðkx Þ. The condensate momentum kc and the critical point f c can be determined by minimizing ϵðkx Þ with respect to kx . Our findings are summarized in Fig. 2. For deep lattices in the tight binding region [Figs. 2(a) and 2(c)], the 0 dominant contribution to Uλkxλ is the on-site interaction of localized Wannier states. Since the Wannier wave function for the px band is more extended, and because the mixing between the s and px band is stronger for smaller momentum, the repulsive interaction energy has a minimum at kx ¼ 0. Therefore, in the shaded area of Fig. 2(b), even when the kinetic energy already exhibits double minimum, the total energy still possesses a single minimum at zero momentum. In another word, in this regime, bosons are condensed into the local maximum of single particle kinetic energy, as shown in Fig. 2(c). In contrast, for shallow lattices where the Bloch wave function behaves like plane waves, around kx ≈ 0, for the s band the Bloch wave function is nearly a constant, whereas for the px band the Bloch wave function behaves like sinð2k0 xÞ, which has a stronger spatial modulation. Thus,
4 4 3
3.8
2 3.6
1 0
3.4
−1
(b)
(a)
−2 −1
0
3.2 1 −1
0
1
FIG. 1 (color online). Band structure of a shaken lattice. (a) Band structure before shaking. The red solid line on the top is the dressed s band with energy shifted by an energy ℏω. (b) The solid line is the dispersion for small shaking amplitude f < f 0c and the dashed line is the dispersion for large shaking amplitude f > f 0c . Energy is plotted in units of lattice recoil energy Er ¼ ℏ2 k20 =ð2mÞ.
FIG. 2 (color online). Interaction shifts of SF-Z2 SF quantum critical point. (a),(c) Deep lattice with V=Er ¼ 16 and ℏω=Er ¼ 7.1; (b),(d) Shallow lattice with V=Er ¼ 4 and ℏω=Er ¼ 4.4. (a) and (b) Condensate momentum kc as a function of f. Blue dashed line is for noninteracting and the red solid line is for the interacting case with gn=Er ¼ 1. (c) and (d) Interaction (gn)-shaking amplitude(f) phase diagram for a fixed frequency ω. Gray shaded areas show regions where atoms condense to states with finite kinetic energy.
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Eðjϕj; kx Þ ¼ ðk4x þ ak2x Þjϕj2 þ rjϕj2 þ ðα þ βk2x Þjϕj4 : ð7Þ The ground state is determined by ∂E=∂kx ¼ 0 and ∂E=∂jϕj ¼ 0, which give rise to three different phases: ϕ ¼ 0 as the MI phase; ϕ ≠ 0 and kx ¼ 0 as the SF phase; and ϕ ≠ 0 and kx ≠ 0 as the Z2 SF phase. The phase diagram is shown in Fig. 3 in terms of f and g. We find all transitions are second order and all three phases meet at a tricritical point at a ¼ r ¼ 0, around which the interaction effect is the strongest. Hereafter, we are concerned about the critical exponent nearby the quantum tricritical point. Our discussion can be divided into two cases: Case A.—No particle-hole symmetry. K 1 ≠ 0 and the K 2 term becomes irrelevant [26]. In this case, it is straightforward to show that the scaling dimension of ϕ is dim½ϕ ¼ −ð2d þ 7Þ=4 and the critical dimension is 5=2 [25]. For a physical system with d ¼ 2, the scaling dimensions of r, a, and α are dim ¼ 2, dim ¼ 1, and dim½α ¼ 1=2, respectively, and all these three terms are relevant. This is different from the conventional Bose Hubbard model with quadratic 2
Z2 SF 1
dd rdτfK 1 ϕ ∂ τ ϕ þ K 2 j∂ τ ϕj2 þ E½ϕ ; ϕg;
Shaking amplitude
Z
[26]. The α and β term come from interaction between bosons. The parameter α is positive for repulsive interactions. In the microscopic model discussed above, parameter β can be either positive or negative. Here, for simplicity, we consider the case with positive β. The parameter a and r are two main control parameters in this system; a controls single particle dispersion. As f changes across f 0c , a changes from positive to negative. r controls interaction strength g. As g increases, r changes from negative to positive, which drives a transition from SF to MI. This effective action basically includes all low-energy terms that are allowed by symmetry of this system and consistent with the microscopic model. Assuming ϕ ¼ jϕjeikx x , we can rewrite E as
0 c
the repulsive interaction is enhanced as the px component increases, and the interaction energy displays a local maximum at zero momentum. Therefore, opposite to the tight binding region, f c decreases as repulsive interaction increases. In the shaded area of Fig. 2(d), bosons are condensed into a finite momentum state which is not the kinetic energy minimum state. This phenomenon is quite intriguing since it invalidates the conventional notion that bosons always condense into the single particle energy minimum. This happens when the self-energy correction due to interactions has strong momentum dependence. So far, this effect has been proposed by Li et al. in the spin-orbit coupled 87 Rb system [system (a) [10]]. Comparing these two systems, in the spin-orbit coupled system this shift is relatively weak because the interaction constants between different spin states of 87 Rb atom are very close [10]. While in our system, this shift is due to the difference of interaction sp constants between different bands, such as Uss kx , U kx , and pp Ukx , and their difference is very large because of the different behaviors of Bloch wave functions. Therefore, this effect is much more profound and is much easier to observe experimentally. Second, in our system the phenomena are richer because the interaction shift can be either positive or negative depending on lattice depth. Tricritical point between SF, Z2 SF, and MI.—Since our system is a lattice system, it will enter a MI phase for sufficiently strong interactions. Here we shall explore a general low-energy effective theory to describe both the SFMI transition and the Ising transition. This effective field theory should capture two key ingredients from microscopic physics as discussed above: (i) the kinetic energy expanded in terms of small kx is given by ak2x þ bk4x, where a can change sign and b > 0; and (ii) the interaction term contains both a constant term and a term proportional to k2x . Nearby the SF-MI transition point, the partition function can be expanded in terms of order parameter ϕ as [26,27] Z Z ¼ D½ϕ ; ϕ expf−S½ϕ ; ϕg; ð4Þ S¼
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0 c
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ð5Þ
and (by setting b ¼ 1) E ¼ j∂ 2x ϕj2 þ aj∂ x ϕj2 þ T þ rjϕj2 þ αjϕj4 þ βjϕ∂ x ϕj2 ; ð6Þ where ϕ is the order parameter, T ¼ j∂ y ϕj2 for d ¼ 2 and T ¼ j∂ y ϕj2 þ j∂ z ϕj2 for d ¼ 3. The K 1 and K 2 terms describe the first-order and second-order derivative term for the temporal dynamics of the ϕ field. When the particle and hole excitation of a MI state have the same energy, which is referred to as the particle-hole symmetric case, K 1 vanishes
0
MI
SF 1
2
2
1
0
Interaction
1 c
2 c
FIG. 3 (color online). Schematic of the tricritical point of interacting bosons in a shaken lattice. The three phases are the Mott insulator phase (MI), normal superfluid (SF) phase, and the superfluid phase that breaks Z2 symmetry (Z2 SF). gc defines the critical interaction strength for the normal superfluid to Mott insulator transition.
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dispersion, where dim½α ¼ 0 and the α term is marginal in two dimensions. The scaling dimension of β is dim½β ¼ −1=2, and the β term is irrelevant. That means although the momentum-dependent interaction plays an important role at the mean-field level to shift the critical value, it does not play a significant role for fluctuations beyond the mean field. The one-loop renormalizaiton group (RG) equations are derived as [25] da ¼ a; dl
dr dα α2 ¼ 2r; ¼ ϵα − I ðaÞ; dl dl 1þr 2
ð8Þ
where ϵ ¼ 5=2 − d ¼ 1=2, I 2 is a function of a defined in the Supplemental Material [25]. In addition to the Gaussian fixed point at ða; r; αÞ ¼ ð0; 0; 0Þ, these RG equations give another non-Gaussian fixed point at ða;r;αÞ¼(0;0;ϵ= I 2 ð0Þ). The flow diagram is shown in Fig. 4(a). However, since in this case r does not receive any correction from interaction, the critical exponent of superfluid transition still remains as its mean-field value ν ¼ 1=2, as in the usual Bose gas case [27,28]. Case B.—Particle-hole symmetry. K 1 ¼ 0 [26]. In this case, the scaling dimension of ϕ is dim½ϕ ¼ −ð5 þ 2dÞ=4 and the critical dimension is 7=2. In this case, for a system in two dimensions, ϵ ¼ 7=2 − d ¼ 3=2. Since ϵ > 1 it is not accurate to treat the system by means of perturbative expansion. For a system with d ¼ 3, ϵ ¼ 1=2, and the scaling dimensions of different terms are dim ¼ 2, dim ¼ 1, dim½α ¼ 1=2, and dim½β ¼ −1=2, respectively. These are all identical to case A. However, in this case, the one-loop RG equations read da ¼ a; dl
dr 2αI 3 ðaÞ ffi; ¼ 2r þ pffiffiffiffiffiffiffiffiffiffi dl 1þr
dα 5I 3 ðaÞα2 ¼ ϵα − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; dl 2 ð1 þ rÞ3
More importantly, the critical exponent of superfluid transition ν ¼ 1=ð2 − 2ϵ=5Þ ≃ 1=2 þ ϵ=10 ¼ 11=20 is now different from the mean-field value [25]. This is also different from conventional bosons with k2 dispersion with K 1 ¼ 0, where interaction is marginal for d ¼ 3 [27]. In this sense, it represents a new type of critical behavior. Conclusion and experimental detection.—We have shown that interaction between particles can significantly shift the location of SF to Z2 SF transition. Bosons can condense to momentum state with high or locally maximum kinetic energy near the quantum critical point. This can be observed in experiment by measuring the transition point with different interaction strength. We have also shown that there exists a quantum tricritical point between the MI, SF, and Z2 SF phases, nearby which interactions dictate the universal behavior and yield new universal critical exponents. Previously, for the conventional Bose Hubbard model, the critical exponent ν ¼ 1=2 has been measured using in situ density measurements inside a harmonic trap [29]. In the system of shaken lattice, one can tune the interaction to the vicinity of the Mott transition, tune the chemical potential to the particle-hole symmetric point, and tune the band dispersion by shaking to the vicinity of the Ising transition. Thus, with the same in situ method, this new critical phenomenon predicted here can be experimentally verified. We thank Xiaoliang Qi, Fa Wang, Hong Yao, and Dimo I. Uzunov for helpful discussions. This work is supported by the Tsinghua University Initiative Scientific Research Program (H. Z.), NSFC Grants No. 11004118 (H. Z.) and No. 11174176 (H. Z.), and NKBRSFC under Grant No. 2011CB921500 (H. Z.), NSF MRSEC (DMR0820054) (C. C.), NSF Grant No. PHY- 0747907 (C. C.), and ARO-MURI Grant No. W911NF-14-1-0003 (C. C.).
ð9Þ *
where ϵ ¼ 7=2 − d ¼ 1=2 and I 3 ðaÞ is also defined in the Supplemental Material [25]. The key difference is that the r term now receives correction from interaction. The new non-Gaussian fixed point is located at ða;r;αÞ¼½0;−2ϵ=5; 2ϵ=(5I 3 ð0Þ), and the flow diagram is shown in Fig. 4(b).
(a)
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(b)
FIG. 4. Renormalization group flow diagram of (a) case A (no particle-hole symmetry) and (b) case B (particle-hole symmetry).
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