Self-Consistent Approach for Mapping Interacting Systems in Continuous

ISSN: 0256-307X 中国物理快报 Chinese Physics Letters

Volume 29 Number 8 August 2012 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/cpl http://cpl.iphy.ac.cn

C HINESE P HYSICAL S OCIETY

CHIN. PHYS. LETT. Vol. 29, No. 8 (2012) 083701 Self-Consistent Approach for Mapping Interacting Systems in Continuous Space to Lattice Models * WU Biao(吴飙)1**, XU Yong(徐勇)2, DONG Lin(董霖)3, SHI Jun-Ren(施均仁)1 1International Center for Quantum Materials, Peking University, Beijing 100871 2Institute of Physics, Chinese Academy of Sciences, Beijing 100190 3Department of Physics and Astronomy, and Rice Quantum Institute, Rice University, Houston, Texas 77251-1892, USA

(Received 23 February 2012) We propose a general variational principle for mapping the interacting systems in continuous space to lattice models. Based on the principle, we derive a set of self-consistent nonlinear equations for the Wannier functions (or, equivalently for the Bloch functions). These equations show that the Wannier functions can be strongly influenced by the interaction and be significantly different from their non-interacting counterparts. Theapproach is demonstrated with interacting bosons in an optical lattice, and illustrated quantitatively by a simple model of interacting bosons in a double well potential. It is shown that the so-determined lattice model parameters can be significantly different from their non-interacting values.

PACS: 37.10.Jk, 71.10.Fd DOI: 10.1088/0256-307X/29/8/083701

Although the real condensed matter systems al- linear equations for the Wannier functions (or, equiv- ways live in the continuous space, it is not uncom- alently, the corresponding Bloch functions), with the mon to use lattice models to describe them, for ex- coefficients of the equations expressed in the correla- ample, many systems can be well described by the tion functions of the lattice model. As a result, these Hubbard model.[1,2] Traditionally, the lattice model is nonlinear equations have to be solved self-consistently often intuitively motivated, and the choice of Wannier with the lattice model. Even though the approach is functions[3] used in the mapping from the continuous general, we demonstrate it with interacting boson in space to the lattice is casual. This lack of rigor in an optical lattice, and eventually illustrated it quanti- choosing Wannier functions was noticed by Kohn and tatively using a simple model of interacting bosons in a he proposed a variational approach to pick the “best” double well potential. It is shown that so-determined Wannier function. However, his approach relies on lattice model parameters can be significantly different the choice of the trial localization function and the re- from their non-interacting values. sultant Wannier functions can not be regarded as the To illustrate our approach, we consider the map- best even in principle.[4] ping of an interacting bosonic cold-atom system in the This issue is becoming more urgent with the de- presence of the periodic potential to a single-band lat- velopments in ultra-cold atom systems. For the ultra- tice model. The extensions to the fermionic systems cold atom systems, the experiments have reached and/or the multi-band lattice models are straightfor- the high precision to clearly demonstrate the vari- ward. The Hamiltonian in the continuous space can ation of effective lattice model parameters, such as in general be written as the strength of the on-site repulsion with the number of atoms per site.[5,6] In the traditional electron ∫︁ 2 [︁ ~ 2 ]︁ systems, there have also been efforts to combine the 퐻ˆ = 푑푟 휓ˆ†(푟) − ∇ + 푉 (푟) 휓ˆ(푟) 2푚 first-principles density functional calculations with the 1 ∫︁ [︁ ]︁ strong-correlation techniques, and the latter is usually + 푑푟푑푟′ 휓ˆ†(푟)휓ˆ†(푟′)푈(|푟 − 푟′|)휓ˆ(푟′)휓ˆ(푟) , based on the lattice models.[7] These developments are 2 calling for the more rigorous theoretical basis for prop- (1) erly mapping an interacting system from the continuous space to the lattice space. This issue has been where 푚 is the mass of the atom, 푉 (푟) is a peri- [7−15] the focus of many theoretical efforts. However, odic potential and 푈(|푟|) is the interaction between the theory is still unsatisfactory because there exists two atoms. Under the single-band approximation, the no generally accepted criteria for what the best set of bosonic field operator 휓(푟) can be expanded as Wannier functions is. In this Letter we propose a general variational ˆ ∑︁ principle for mapping an interacting system in contin- 휓(푟) = 푎ˆ푗푊푗(푟), (2) uous space to a lattice model: the choice of an incom- 푗 plete set of Wannier functions minimizes the calcu- lated ground state energy of the lattice model. Based where 푊푗(푟) = 푊 (푟 − 푟푗) is the Wannier function at on the principle, we derive a set of self-consistent non- site 푗 and 푎ˆ푗 is the associated annihilation operator.

*Supported by the National Natural Science Foundation of China under Grant No 10825417. **Corresponding author. Email: [email protected] © 2012 Chinese Physical Society and IOP Publishing Ltd 083701-1 CHIN. PHYS. LETT. Vol. 29, No. 8 (2012) 083701

The corresponding single-band lattice model reads 푗 푗 푗 푗 3 4 ∫︁ 3 4 ∑︁ † † ′[︁ * ′ ∑︁ † ∑︁ † † + ⟨푎ˆ 푎ˆ 푎ˆ푗 푎ˆ푗 ⟩ 푑푟 푊 (푟 + 푟푗 − 푟푗 ) 퐻ˆ = − 퐽 푎ˆ 푎ˆ + 푈 푎ˆ 푎ˆ 푎ˆ 푎ˆ . 푗1 푗2 3 4 2 1 푠푏 푗1푗2 푗1 푗2 푗1푗4푗2푗3 푗1 푗2 푗3 푗4 푗1푗2 푗1푗2 푗1푗2 ]︁ (3) ′ ′ × 푊 (푟 + 푟푗2 − 푟푗4 )푈(|푟 − 푟|) 푊 (푟 + 푟푗2 − 푟푗3 ). The parameters are given by (8) ∫︁ * It is clear that the above equation depends on the 퐽푗 푗 = − 푑푟푊 (푟)퐻0푊푗 (푟), (4) 1 2 푗1 2 solution of the single-band lattice model (3). Before ∫︁ solving Eq. (3), one can employ various approxima- 1 ′[︁ * 푈푗 푗 푗 푗 = 푑푟푑푟 푊 (푟)푊푗 (푟) tions, for example, by choosing to keep only the near- 1 4 2 3 2 푗1 4 ]︁ est neighbor tunneling 퐽 and the on-site interaction × 푈(|푟 − 푟′|)푊 * (푟′)푊 (푟′) , 푈 in the single-band Hamiltonian (3). Various many- 푗2 푗3 (5) body techniques[16−19] can be employed to solve the resulting simple lattice model. The solution, no mat- 2 ~ 2 ter what approximations are employed to get it, is still where 퐻0 ≡ − 2푚 ∇ + 푉 (푟). a valid trial wave-function of a functional of the Wan- Further approximations for 퐻ˆ푠푏 are often needed. For instance, to obtain the well-known Bose-Hubbard nier functions. To implement the variational principle model, one keeps only two terms, the nearest neigh- Eq. (6), it is essential to keep all terms in Eq. (8), even bor tunneling and the on-site interaction. Various ex- when the corresponding terms in Eq. (3) (e.g., off-site act or approximated many-body techniques, such as interaction) had been ignored in solving the lattice direct diagonalization, Gutzwiller projection,[16] den- model. For instance, if the interaction terms are ig- sity matrix renormalization group (DMRG),[17] time- nored when solving Eq. (3), Eq. (8) becomes the usual evolving block decimation (TEBD),[18] or quantum Hartree–Fock approximation. It is important to note Monte Carlo,[19] can be employed to solve the result- that Wannier functions obtained in this way not only ing simple lattice model. It is important to observe depend on the approximations applied to Eq. (3), but that the solution, no matter what approximations are also depend on the many-body technique employed employed to get it, can be considered as a trial wave- to solve the resulting lattice model. We call such an function of a functional of the Wannier functions. approach a self-consistent single-band approximation. As an example, we consider the case of a deep The ground state |퐺푡⟩ in the single-band ap- Mott-insulator regime, where the ground state has the proximation can be formally written as |퐺푡⟩ = form |푛 , 푛 , . . . , 푛 ⟩ with 푛 being the average num- 퐹 (ˆ푎†)|vaccum⟩, where 퐹 is a certain function. The 0 0 0 0 푗 ber of particles per site. In this case, we have Wannier function in Eq. (2) is usually pre-determined. Here the Wannier function is not known a priori. We ⟨푎ˆ† 푎ˆ ⟩ = 푛 훿 , (9) 푗1 푗2 0 푗1,푗2 look for the Wannier functions that minimize the sys- ⟨푎ˆ† 푎ˆ† 푎ˆ 푎ˆ ⟩ = 푛2훿 훿 + 푛2훿 훿 tem’s single-band ground state energy, 푗1 푗2 푗3 푗4 0 푗1,푗3 푗2,푗4 0 푗1,푗4 푗2,푗3 − (푛2 + 푛 )훿 훿 훿 . ˆ ˆ 0 0 푗1,푗2 푗2,푗3 푗3,푗4 퐸퐺 = ⟨퐺푡|퐻|퐺푡⟩ = ⟨퐺푡|퐻푠푏|퐺푡⟩, (6) (10) As a result, Eq. (8) is simplified and has the form for where 퐻ˆ푠푏 is the usual single-band lattice Hamiltonian We achieve the minimization of the ground state 푈(|푟|) = 푔0훿(푟), energy 퐸퐺 by varying the Wannier function under the 휇0 ∑︁ 2 orthonormal constraints, ℎ = ∫︀ 푑푟푊 *(푟)푊 (푟−푟 ) = 푊 (푟) = 퐻0푊 (푟) + 푔0푛0 |푊 (푟 − 푟푗)| 푊 (푟) 푗 푗 푁0 푟 ̸=0 훿0,푗. According to the Feynman–Hellman theorem, we 푗 2 have + 푔0(푛0 − 1)|푊 (푟)| 푊 (푟). (11) ∑︀ 훿(퐸퐺 − 푗 휇푗ℎ푗) When the off-site terms, which are often very small, 훿푊 *(푟) are ignored, the above equation has the form of the familiar Gross–Pitaevskii equation.[20,21] Therefore, it ˆ 훿퐻푠푏 ∑︁ 휇푗훿ℎ푗 = ⟨퐺 | |퐺 ⟩ − = 0, (7) is very clear that the Wannier function is greatly in- 푡 훿푊 *(푟) 푡 훿푊 *(푟) 푗 fluenced by the interaction whenever 푛0 ≥ 2. A periodic system can be described alternatively where 휇’s are Lagrangian multipliers. After straight- with Bloch functions. If we place the system in a box forward computation, we obtain a nonlinear equation of 푁 lattice sites, the Wannier functions are related for the Wannier functions to Bloch functions as 1 ∑︁ ∑︁ −푖푘·푟푛 휇 푊 (푟 − 푟 ) 푊 (푟 − 푟푛) = √ 푒 Ψ 푘(푟), (12) 푗 푗 푁 푗 푘 ∑︁ † where Ψ푘 is a Bloch function with Bloch wave num- = ⟨푎ˆ푗 푎ˆ푗2 ⟩퐻0푊 (푟 + 푟푗1 − 푟푗2 ) 1 ber 푘 and is normalized to one. In terms of Bloch 푗1,푗2 083701-2 CHIN. PHYS. LETT. Vol. 29, No. 8 (2012) 083701

* functions, the nonlinear equation (8) becomes a set of 휓−푘 = 휓푘. We have also ignored the scattering pro- nonlinear equations cesses with nonzero 퐾. Finally, we use a one-dimensional double-well po- ∑︁ ∫︁ 휈˜ Ψ (푟) = 퐻 Ψ (푟) + 푃 푑푟′ tential under a periodic boundary condition to illus- 푘 푘 0 푘 푘1푘푘3푘4 trate our theory. The two Wannier functions for the ⟨푘1푘푘3푘4⟩ [︁ ]︁ left well and the right well are related to the ground · Ψ * (푟′)Ψ (푟′)푈(|푟′ − 푟|) Ψ (푟), state and the first excited state for the double-well 푘1 푘3 푘4 (13) potential as follows: √ √ where ⟨푘1푘푘3푘4⟩ stands for summation with the con- 2 2 푊푙 = (Ψ0 + Ψ1), 푊푟 = (Ψ0 − Ψ1), (18) straint 푘1 + 푘 = 푘3 + 푘4 + 퐾, and 푃푘1푘푘3푘4 = 2 2 ˆ† ˆ† ˆ ˆ ˆ† ˆ ˆ 1 ∑︀ −푖푘·푟푛 ⟨푏 푏 푏푘 푏푘 ⟩/⟨푏 푏푘⟩; 푏푘 = √ 푎ˆ푛푒 , and 푘1 푘 3 4 푘 푁 푛 where Ψ0 and Ψ1 are chosen such that they are both 1 ∑︀ −푖푘·푟푛 ˆ† ˆ 휈˜푘 = ( 푁 푛 휇푛푒 )/⟨푏푘푏푘⟩. 퐾 is a reciprocal lat- positive in the left well. These two Wannier functions tice. In deriving Eq. (13), we have used the fact that satisfy the nonlinear equation (8). The corresponding the Bloch functions of different wave numbers 푘 are lattice model is orthogonal to each other automatically and, therefore, [︁ ]︁ one can have an equation for each different Bloch wave † † 퐻ˆ2 = − 퐽 + 2(푁0 − 1)푈3 (ˆ푎 푎ˆ푟 +푎 ˆ 푎ˆ푙)+ number 푘. 푙 푟 † † † † † † We now apply Eq. (13) in the superfluid regime, + 푈2(ˆ푎푟푎ˆ푟푎ˆ푙푎ˆ푙 + 4ˆ푎푙 푎ˆ푙푎ˆ푟푎ˆ푟 +푎 ˆ푙 푎ˆ푙 푎ˆ푟푎ˆ푟) where the Bogoliubov mean-field theory can be + 푈(ˆ푎†푎ˆ†푎ˆ 푎ˆ +푎 ˆ†푎ˆ†푎ˆ 푎ˆ ), (19) applied.[22] For the simple interaction, 푈(|푟|) = 푙 푙 푙 푙 푟 푟 푟 푟 푔0훿(푟), the mean-field Hamiltonian is where 푈2 = 푈푙푙푟푟, 푈3 = 푈푙푟푟푟, and 푈 = 푈푙푙푙푙. [︁ ]︁ We choose the double-well potential as a part of ˆ 2 ∑︁ ˆ† ˆ 퐻푏ℎ = 휖0풩0 + 푈0풩0 + 휖푘 + 4풩0푈푘 푏푘푏푘 the one dimensional optical lattice created experimen- 푘̸=0 tally in Ref. [23]. Thus, the double well potential is ∑︁ (︁ )︁ 2 + 풩 푈 ˆ푏† ˆ푏† + ˆ푏 ˆ푏 , (14) given by 푉 (푥) = 푉0 sin (푘퐿푥), where 푘퐿 is the wave 0 푘 푘 −푘 푘 −푘 number of the laser that creates the potential. Due 푘̸=0 to the lateral confinement, the interaction strength 푔0 is given by 푔 = (4휋 2푎 /푚)(푚휔 /(2휋 )) = 2 휔 푎 , where 풩0 is the number of atoms in the state 휓0, 0 ~ 푠 ⊥ ~ ~ ⊥ 푠 ∫︀ * ∫︀ 2 2 where 푎푠 is the s-wave scattering length and 휔⊥ the 휖푘 = 푑푟휓푘퐻0휓푘, and 푈푘 = (푔0/2) 푑푟|휓푘| |휓0| . Following the standard procedure,[22] we find perpendicular confinement frequency. In our numerical calculations, the Hamiltonian in Eq. (19) is di- 푃 = 푣2 (훿 훿 + 훿 훿 ) agonalized directly and Eq. (8) is solved with the 푘1푘푘3푘4 푘1 푘1,푘4 푘,푘3 푘1,푘3 푘,푘4 * nonlinear equation solver in MATLAB. We use 푐 = 푢 푢푘 푣푘 2 푘 3 3 휋푚푔0/( 푘퐿) as the dimensionless interaction param- + 훿−푘1,푘훿−푘3,푘4 ~ 푣푘 eter.

− 2풩0훿푘1,0훿푘,0훿푘3,0, (15) The numerical results for the dependences of 퐽 √ and 푈 on the well depth and interaction strength are √︀ 2 2 2 2 where ℰ푘 = 휂푘 − 4풩0 푈푘; 푢0 = 푣0 = 풩0, 푢푘̸=0 = shown in terms of the ratios 퐽/퐽0 and 푈/푈0 in Fig. 1. 2 2 2 2 2 2 퐽0 and 푈0 are the tunneling parameter and on-site in- 1 + 푣푘̸=0; and 푣푘̸=0 = [휂푘/(휂푘 − 4풩0 푈푘) − 1]/2, with 0 0 teraction obtained with single-particle Wannier func- 휂푘 = 휖푘 − 휖0 + 4풩0푈푘 − 2풩0푈0. This leads to a set of simplified nonlinear equations for Bloch functions, tion. These numerical results have confirmed our in- ∑︁ tuitive understanding that the Wannier function is 휈˜ Ψ = 퐻 Ψ + 푔 풩 |Ψ |2Ψ + 2푔 푣2 |Ψ |2Ψ + 0 0 0 0 0 0 0 0 0 푘′ 푘′ 0 broadened as the interaction strength 푐 increases. 푘′̸=0 This is evidently illustrated by the decrease of 푈/푈0 ∑︁ 2 푔0 푢푘′푣푘′|Ψ푘′| Ψ0, (16) with the increase of 푐 shown in the right column of 푘′̸=0 Fig. 1. Although the broadening of the Wannier function always leads to the decrease of 푈/푈0, it does not and for 푘 ̸= 0 always lead to the increase of 퐽/퐽0 as one would intuitively expect. This is demonstrated such that 퐽/퐽0 푢푘 2 휈˜푘Ψ푘 = 퐻0Ψ푘 + 푔0(2 + )풩0|Ψ0| Ψ푘+ generally decreases as 푐 increases in the right col- 푣푘 umn. This is also supported by the fact that 퐽/퐽0 ∑︁ [︁ 푢푘 2 ]︁ 2 푔0 푢푘′ 푣푘′ + 2푣푘′ |Ψ푘′ | Ψ푘. can decrease with 푈/푈0 as 푣 increase in Figs. 1(a2) 푣푘 푘′̸=0 (17) and 1(a3). This rich behavior of 퐽/퐽0 is due to the fact that 퐽 depends not only on the size of the Wan- Since 푢푘 and 푣푘 themselves depend on 휓푘, the above nier function tail but also on the detailed oscillatory two equations have to be solved self-consistently. Note structure of the tail. Nevertheless, one can still have that in the above derivation we have assumed that the some intuitive understanding of the behavior of 퐽/퐽0. lattice potential is symmetric, 푉 (푟) = 푉 (−푟), so that For the right column, one can roughly understand the 083701-3 CHIN. PHYS. LETT. Vol. 29, No. 8 (2012) 083701 decrease of 퐽/퐽0 with increasing 푐 as the result of the much with 푐 when there is only one particle per well effective reduction of band width, which becomes nar- on average. As shown in Figs. 1(a1) and 1(b1), the rower as the increased interaction enhances the band- change of 퐽/퐽0 and 푈/푈0 caused by the interaction energy at the 푘 = 0 state more quickly than 푘 = 휋 is less than 1%. However, the change becomes signif- state. icant when there is more than one particle per well. We also notice that 퐽/퐽0 and 푈/푈0 does not vary This is evident in the lower four panels of Fig. 1.

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Fig. 1. The tunneling parameter 퐽 and the on-site interaction 푈 as functions of the double-well depth and the interaction strength. For (a1, a2, a3), 푐 = 2.5; for (b1, b2, b3), 푣 = 5.0. The left vertical axis is for the solid line while the right one is for the dashed line. The results for 푐 < 0.1 are not accurate due to the numerical difficulties 2 2 caused by the intrinsic structure problem in the system. The energy unit is the recoil energy 퐸푟 = ~ 푘퐿/2푚. The inset in (a1) shows how 퐽 and 푈 change with 푣.

We expect that the behaviors of 퐽/퐽0 and 푈/푈0 re- B. W. acknowledges the hospitality of the Aspen vealed in Fig. 1 are not limited to the double-well sys- Center for Physics, where part of the work was carried tems, and should hold generally for ultracold bosons out. in optical lattices. This is to be confirmed by future work. Currently in typical experiments, ultracold atoms References are also trapped by a harmonic potential.[20,21] Con- [1] Ashcroft N W and Mermin N D Solid State Phys. (Sanders, sequently, the wells are not identical to each other. Philadelphia 1976) Also a random potential can be added to make the [2] Jaksch D et al 1998 Phys. Rev. Lett. 81 3108 wells non-identical.[24] Nevertheless, single-band ap- [3] Wannier G H 1937 Phys. Rev. 52 191 proximation can still be applied as long as the dif- [4] Kohn W 1973 Phys. Rev. B 7 4388 [5] Campbell G K et al 2006 Science 313 649 ference between the site energies is smaller than the [6] Will S et al 2010 Nature 465 197 energy gaps between the ground state and the first [7] Marzari N and Vanderbilt D 1997 Phys. Rev. B 56 12847 excited state in the wells. For simplicity, we consider [8] Li J et al 2006 New J. Phys. 8 154 a one-dimensional potential of 푁 wells, which are not [9] Hazzard K R A and Mueller E J 2010 Phys. Rev. A 81 031602(R) identical. In this case, the Wannier functions for the [10] Liang Z et al 2009 arXiv:0903.4058 lowest “band” can be defined as [11] Schneider P et al 2009 Phys. Rev. A 80 013404 Phys. Rev. Lett. 78 푁−1 [12] Esry B D et al 1997 3594 1 ∑︁ 푖 2푘푗휋 [13] Masiello D et al 2005 Phys. Rev. A 72 063624 푊푗(푥) = √ 푒 푁 휓푘, 푗 = 0, 1, . . . , 푁 − 1, [14] Büchler H P 2010 Phys. Rev. Lett. 104 090402 푁 푘=0 [15] Hazzard K R A and Mueller E J 2010 Phys. Rev. A 81 (20) 031602 [16] Rokhsar D S and Kotliar B G 1991 Phys. Rev. B 44 10328 where 휓푘’s are the lowest 푁 eigenstates. Our varia- [17] White S R 1992 Phys. Rev. Lett. 69 2863 tional approach can be easily adopted to this case with [18] Vidal G 2004 Phys. Rev. Lett. 93 040502 just one modification. Since the Wannier functions at [19] Prokofév N V et al 1998 J. Exp. Theor. Phys. 87 310 different sites have different shapes, the constraint is [20] Morsch O and Oberthaler M 2006 Rev. Mod. Phys. 78 179 now ℎ = ∫︀ 푑푥푊 *(푥)푊 (푥) = 훿 . As a result, [21] Yukalov V 2009 Laser Phys. 19 1 푛,푚 푛 푚 푛,푚 [22] Pethick C J and Smith H Bose-Einstein Condensation in one obtains a set of nonlinear equations for the Wan- Dilute Gases (Cambridge 2002) nier functions. Since everything is straightforward, we [23] Paredes B et al 2004 Nature 429 277 shall not write out the equations here. [24] Zhou S Q and Ceperley D M 2009 arXiv:0907.5053 083701-4 Chinese Physics Letters Volume 29 Number 8 August 2012

GENERAL 080201 Explicit Multisymplectic Fourier Pseudospectral Scheme for the Klein–Gordon–Zakharov Equations CAI Jia-Xiang, LIANG Hua 080202 The Generalized Wronskian Solution to a Negative KdV-mKdV Equation LIU Yu-Qing, CHEN Deng-Yuan, HU Chao 080203 DNA Dynamics Studied Using the Homogeneous Balance Method E. M. E. Zayed, A. H. Arnous 080301 Quantum Gate Implementations in the Separated Ion-Traps by Fast Laser Pulses ZHANG Miao, WEI Lian-Fu 080302 Approximate Analytical Solution of the Yukawa Potential with Arbitrary Angular Momenta M. Hamzavi, M. Movahedi, K.-E. Thylwe, A. A. Rajabi 080303 Controlling Excitation Inversion of a Cooper Pair Box Interacting with a Nanomechanical Resonator C. Valverde, H. C. B. de Oliveira, A. T. Avelar, B. Baseia 080304 Mocking up a Dephasing Channel with a Minimal-Sized Environment WU Zhen 080401 Proposed Test of the Equivalence Principle with Rotating Cold Polar Molecules HU Zhong-Kun, KE Yi, DENG Xiao-Bing, ZHOU Ze-Bing, LUO Jun 080402 Conserved Quantities in f(R) Gravity via Noether Symmetry M. Farasat Shamir, Adil Jhangeer, Akhlaq Ahmad Bhatti 080403 Casimir–Polder-Like Force for an Atom in Hartle–Hawking Vacuum outside a Schwarzschild Black Hole ZHANG Jia-Lin, YU Hong-Wei 080404 LRS Bianchi Type-II Inﬂationary Universe with Massless Scalar Field and Time Varying Λ Raj Bali, Swati 080501 Ground State Energy of 1D Attractive δ-Function Interacting Fermi Gas WANG Ya-Hui, MA Zhong-Qi 080502 Modiﬁed Static Floor Field and Exit Choice for Pedestrian Evacuation XU Yan, HUANG Hai-Jun, YONG Gui 080503 Dynamics of a Cortical Neural Network Based on a Simple Model QU Jing-Yi, WANG Ru-Bin 080504 A Statistical Model for Predicting Thermal Chemical Reaction Rate: Application to Bimolecule Reactions LI Wang-Yao, LIN Zheng-Zhe, XU Jian-Jun, NING Xi-Jing 080701 Improvement on AMS Measurement of 236U at CIAE WANG Xiang-Gao, HE Ming, DONG Ke-Jun, JIANG Shan

THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS 081301 Photoproduction of J/ψ in pp and PbPb Collisions at Leading Order ZHU Jia-Qing, LI Yun-De

NUCLEAR PHYSICS 082501 New Measurements for 8He Excited States XIAO Jun, YE Yan-Lin, CAO Zhong-Xin, JIANG Dong-Xing, ZHENG Tao, HUA Hui, LI Zhi-Huan, LI Xiang-Qing, GE Yu-Cheng, LOU Jian-Ling, PANG Dan-Yang, LV Lin-Hui, LI Qi-Te, QIAO Rui, YOU Hai-Bo, CHEN Rui-Jiu, H. Sakurai, H. Otsu, M. Nishimura, S. Sakaguchi, H. Baba, Y. Togano, K. Yoneda, LI Chen, WANG Shuo, WANG He, LI Kuo-Ang, Y. Nakayama, Y. Kondo, S. Deguchi, Y. Satou, K. H. Tshoo ATOMIC AND MOLECULAR PHYSICS 083301 Temporal Electronic Structures of Nonresonant Raman Excited Virtual States of P-Hydroxybenzoic Acid FANG Chao, SUN Li-Feng

083302 Transition Spectrum Intensities and Absorption Coeﬃcients of Hydride BH2 Free Radical Molecule WU Dong-Lan, TAN Bin, ZHANG Xin-Qin, XIE An-Dong 083701 Self-Consistent Approach for Mapping Interacting Systems in Continuous Space to Lattice Models WU Biao, XU Yong, DONG Lin, SHI Jun-Ren

FUNDAMENTAL AREAS OF PHENOMENOLOGY(INCLUDING APPLICATIONS) 084201 Self-Similar Vortex Solitons for the Distributed-Coefficient Nonlinear SchrödingerEquation ZHAO Bi, DAI Chao-Qing, ZHANG Jie-Fang 084202 Advanced Aperture Synthesis by Wave-Front Combination in Generalized Phase-Shifting Interferometry XU Xian-Feng, CAI Lü-Zhong,LU Guang-Can, HAN Guo-Xia, TIAN Yan-Jie, ZHANG Qian, JIAO Zhi-Yong 084203 All-Optical Time Slot Interchange Using a Cascaded Optical Buffer FENG Zhen, SHENG Xin-Zhi, WU Chong-Qing, LI Zheng-Yong, MAO Ya-Ya 084204 Fabrication and Characterization of a 2 × 2 Microfiber Knot Resonator Coupler A. A. Jasim, A. Z. Zulkifli, M. Z. Muhammad, H. Ahmad, S. W. Harun 084205 A Compact Low Noise Single Frequency Linearly Polarized DBR Fiber Laser at 1550 nm ZHANG Wei-Nan, LI Can, MO Shu-Pei, YANG Chang-Sheng, FENG Zhou-Ming, XU Shan-Hui, SHEN Shao-Xiong, PENG Ming-Ying, ZHANG Qin-Yuan, YANG Zhong-Min 084206 A 1342-nm Passively Mode-Locked Picosecond Oscillator with a Semiconductor Saturable Absorber Mirror NIU Gang, YAN Ying, MA Yun-Feng, SHI Zhao-Hui, HUANG Yu-Tao, WANG Lu-Lu, SONG Ya-Li, FAN Zhong-Wei, Mikhail Grishin, Donatas Joksas, Zenonas Kuprionis 084207 Effect of Oxygen Vacancy on the Band Gap and Nanosecond Laser-Induced Damage Threshold of Ta2O5 Films XU Cheng, YANG Shuai, WANG Ji-Fei, NIU Ji-Nan, MA Hao, QIANG Ying-Huai, LIU Jiong-Tian, LI Da-Wei, TAO Chun-Xian 084208 Single-Fundamental-Mode 850 nm Surface Relief VCSEL WEI Si-Min, XU Chen, DENG Jun, ZHU Yan-Xu, MAO Ming-Ming, XIE Yi-Yang, XU Kun, CAO Tian, LIU Jiu-Cheng 084209 Quantum Information Transfer Based on Frequency Modes in Circuit QED WANG Chao-Quan 084210 Single-Mode Propagation and Highly Efficient Second Harmonic Generation in Periodically Poled MgO-Doped LiNbO3 on Insulator Rib Waveguide ZHOU Yu-Jie, FENG Li-Qun, SUN Jun-Qiang 084211 Two-Dimensional Nonlinear Laser Spectra Calculations with Numerical Path Integral LIANG Xian-Ting 084212 Two-Dimensional Optical Lattice Solitons in Photovoltaic-Photorefractive Crystals GUO Jian-Bang, LU Ke-Qing, NIU Ping-Juan, YU Li-Yuan, XING Hai-Ying 084213 Characteristics Improvement of L-Band Superfluorescent Fiber Source Using Unpumped Erbium-Doped Fiber WANG Xiu-Lin, HUANG Wen-Cai, CAI Zhi-Ping 084214 Radially and Azimuthally Polarized Beams Generated by a Composite Spiral Zone Plate HUA Yi-Lei, WANG Zi-Qiang, LI Hai-Liang, GAO Nan, DU Yu-Chan 084215 Refractive Index Profiles of Copper Ion Exchange Glass Planar Waveguides XIA Hong-Yun, TENG Chuan-Xin, ZHAO Xiao-Wei, ZHENG Jie 084216 A Lens Assisted Phase Microscope Based on Ptychography PAN Xing-Chen, LIN Qiang, LIU Cheng, ZHU Jian-Qiang 084217 Swept Frequency Measurement of Electrooptic Phase Modulators Using Dispersive Fibers ZHANG Shang-Jian, ZHANG Xiao-Xia, LIU Yong 084601 Group Invariant Solutions of the Two-Dimensional Elastodynamics Problem in the Polar Coordinate System LI Hou-Guo, HUANG Ke-Fu 084701 Global Stability Analysis of Flow Past Two Side-by-Side Circular Cylinders at Low Reynolds Numbers by a POD-Galerkin Spectral Method ZHANG Wei, CHEN Cheng 084702 Flow Visualization of Supersonic Flow over a Finite Cylinder WANG Deng-Pan, ZHAO Yu-Xin, XIA Zhi-Xun, WANG Qing-Hua, LUO Zhen-Bing 084703 Flow Patterns in the Sedimentation of a Capsule-Shaped Particle NIE De-Ming, LIN Jian-Zhong, ZHANG Kai 084704 A Variant of the Classical Von KármánFlow for a Couple Stress Fluid A. A. Farooq, A. M. Siddiqui, M. A. Rana, T. Haroon 084705 On the Exact Solution for Axisymmetric Flow and Heat Transfer over a Nonlinear Radially Stretching Sheet Azeem Shahzad, Ramzan Ali, Masood Khan 084706 Dynamic Analysis of the Smooth-and-Discontinuous Oscillator under Constant Excitation TIAN Rui-Lan, WU Qi-Liang, LIU Zhong-Jia, YANG Xin-Wei 084707 Applications of the CE/SE Scheme to Incompressible Viscous Flows in Two-Sided Lid-Driven Square Cavities YANG Duo-Xing, ZHANG De-Liang

PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES 085201 Mode Conversion and Whistler Wave Generation on an Alfv´enResonance Layer in High Beta Plasmas LUAN Qi-Bin, SHI Yi-Peng, WANG Xiao-Gang 085202 K-Shell Spectra from CH Foam Tamped Ti Layers Irradiated with Nanosecond Laser Pulses ZHAO Yang, ZHU Tuo, WEI Min-Xi, XIONG Gang, SONG Tian-Ming, HU Zhi-Min, HUANG Cheng-Wu, SHANG Wan-Li, YANG Guo-Hong, ZHANG Ji-Yan, YANG Jia-Min 085203 Combining Eﬀects between LHW and IBW Injections on EAST DUAN Wen-Xue, MA Zhi-Wei, WU Bin

CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMAL PROPERTIES

086201 Pressure Eﬀects on the Magnetic Phase Transition of Mn3SnC1−xNx (x = 0, 0.5) HU Jing-Yu, WEN Yong-Chun, YAO Yuan, WANG Cong, ZHAO Qing, JIN Chang-Qing, YU Ri-Cheng 086402 Scaling of the Leading Response in Linear Quench Dynamics in the Quantum Ising Model YU Wing-Chi, WANG Li-Gang, GU Shi-Jian, LIN Hai-Qing 086801 Sacriﬁce-Template Synthesis of CdTe Nanorod Arrays in Glycol via a Solvothermal Process DENG Yuan, LIU Jing, WANG Yao, LIANG Li-Xing 086802 Electric-Field Switching of Exciton Spin Splitting in Asymmetrical Coupled Quantum Dots LI Xiao-Jing

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES 087201 Unipolar Resistive Switching Effects Based on Al/ZnO/P++-Si Diodes for Nonvolatile Memory Applications SHI Wei, TAI Qiang, XIA Xian-Hai, YI Ming-Dong, XIE Ling-Hai, FAN Qu-Li, WANG Lian-Hui, WEI Ang, HUANG Wei 087202 Spin Filtering in a Rashba Electron Waveguide Induced by Edge Disorder XIAO Xian-Bo, LI Fei, LIU Nian-Hua 087203 Determination of Channel Temperature in AlGaN/GaN HEMTs by Pulsed I–V Characteristics WANG Jian-Hui, WANG Xin-Hua, PANG Lei, CHEN Xiao-Juan, JIN Zhi, LIU Xin-Yu 087204 Forward Current Transport Mechanism and Schottky Barrier Characteristics of a Ni/Au Contact on n-GaN YAN Da-Wei, ZHU Zhao-Min, CHENG Jian-Min, GU Xiao-Feng, LU Hai 087301 Energy Band Structure of the Electron Gas in Periodic Quantum Wells MAO Sheng-Hong, MA Yu-Ting, XUE Ju-Kui 087302 Numerical Simulation of a P+ a-SiC:H/N+ Poly-Si Solar Cell with High Efficiency and Fill Factor SHAO Qing-Yi, CHEN A-Qing, ZHU Kai-Gui, ZHANG Juan 087303 Double-Peak N-Shaped Negative Differential Resistance in a Quantum Dot Field Effect Transistor XU Xiao-Na, WANG Xiao-Dong, LI Yue-Qiang, CHEN Yan-Ling, JI An, ZENG Yi-Ping, YANG Fu-Hua 087304 Optical Investigation of Sm3+ Doped Zinc-Lead-Phosphate Glass Raja J. Amjad, M. R. Sahar, S. K. Ghoshal, M. R. Dousti, S. Riaz, B. A. Tahir 087401 Growth and Characterization of High-Quality Single Crystals of Ni- and Zn-Doped Bi2Sr2Ca(Cu2−xMx)O8+δ (M = Ni or Zn) High-Temperature Superconductors LIU Shan-Yu, ZHANG Wen-Tao, ZHAO Lin, LIU Hai-Yun, WU Yue, LIU Guo-Dong, DONG Xiao-Li, ZHOU Xing-Jiang 087501 Tailoring the Microstructure of NiZn Ferrite for Power Field Applications TANG Xiao-Li, SU Hua, ZHANG Huai-Wu 087701 Structural and Electrical Characteristics of Amorphous ErAlO Gate Dielectric Films ZHU Yan-Yan, FANG Ze-Bo, TAN Yong-Sheng

087801 Observation of Two-Photon Induced Excited-State Absorption Phenomena in C60 and C70 Derivatives OUYANG Xin-Hua, LU Liang, GE Zi-Yi 087802 Eﬀects of Annealing on Thermoluminescence Peak Positions and Trap Depths of Synthetic and Natural Quartz by Means of the Various Heating Rate Method H¨useyinToktami¸s,A. Necmeddin Yazici

CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY 088101 Eﬀects of the V/III Ratio of a Low-Temperature GaN Buﬀer Layer on the Structural and Optical Properties of a-GaN Films Grown on r-Plane Sapphire Substrates by MOCVD TIAN Yu, DAI Jiang-Nan, XIONG Hui, ZHENG Guang, RYU My, FANG Yan-Yan, CHEN Chang-Qing

088102 Improvement of the Quality of a GaN Epilayer by Employing a SiNx Interlayer YANG De-Chao, LIANG Hong-Wei, SONG Shi-Wei, LIU Yang, SHEN Ren-Sheng, LUO Ying-Min, ZHAO Hai-Feng, DU Guo-Tong

088103 Superhydrophilic and Wetting Behavior of TiO2 Films and their Surface Morphologies WANG Wei, ZHANG Da-Wei, TAO Chun-Xian, WANG Qi, WANG Wen-Na, HUANG Yuan-Shen, NI Zheng-Ji, ZHUANG Song-Lin, LI Hai-Xia, MEI Ting 088104 Effect Mechanism of a Direct Current on the Solidification of Immiscible Alloys JIANG Hong-Xiang, ZHAO Jiu-Zhou 088105 Cathode-Control Alloying at an Au-ZnSe Nanowire Contact via in Situ Joule Heating ZENG Ya-Ping, WANG Yan-Guo, QU Bai-Hua, YU Hong-Chun 088401 Single and Double Superconducting Coplanar Waveguide Resonators ZHAO Na, LIU Jian-She, LI Hao, LI Tie-Fu, CHEN Wei 088501 Low Power and High Sensitivity MOSFET-Based Pressure Sensor ZHANG Zhao-Hua, REN Tian-Ling, ZHANG Yan-Hong, HAN Rui-Rui, LIU Li-Tian 088502 Modeling, Simulation and Analysis of Thermal Resistance in Multi-finger AlGaN/GaN HEMTs on SiC Substrates WANG Jian-Hui, WANG Xin-Hua, PANG Lei, CHEN Xiao-Juan, LIU Xin-Yu 088701 Experimental Study of Entropy Production in Cells under Alternating Electric Field DING Chang-Jiang, LUO Liao-Fu 088702 Modulation of Amyloid-β Conformation by Charge State of N-Terminal Disordered Region XI Wen-Hui, LI Wen-Fei, WANG Wei 088703 Weak Field-Induced Evolution of Spiral Wave in Small-World Networks of Hodgkin–Huxley Neurons WANG Ya-Min, LIU Yong, WANG Jing, LIU Yu-Rong 088801 Polymer Photovoltaic Cells Based on Ultraviolet-Ozone-Treated Vanadium-Doped Indium Oxide Anodes GUO Xiao-Yang, LUO Jin-Song, CHEN Hong, LIU Xing-Yuan 088901 Cooperative Enhancement of Cost Threshold in the Spatial N-Person Snowdrift Game LU Dong-Liang, ZHANG Hong-Bin, GE Juan, XU Chen 088902 Power-Law Exponent for Exponential Growth Network WANG Li-Na, CHEN Bin, ZANG Chen-Rui

GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS 089401 Anomalous Resistivity Associated with Secondary Islands in the Reconnection Region ZHOU Meng, SU Wei, DENG Xiao-Hua, YAO Ming 089402 Spatial Evolution of Electrostatic Solitary Waves along Plasma Sheet Boundary Layer Adjacent to the Magnetic Reconnection X-Line LI Shi-You, ZHANG Shi-Feng, DENG Xiao-Hua, CAI Hong 089601 Cross-Shock Electrostatic Potential and Ion Reﬂection in Quasi-Parallel Supercritical Collisionless Shocks SU Yan-Qing, LU Quan-Ming