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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(Ç飙)1**, XU Yong(M])2, DONG Lin(Â霖)3, SHI Jun-Ren(施均;)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 num- ber of atoms per site.[5;6] In the traditional electron Z 2 h ~ 2 i systems, there have also been efforts to combine the H^ = dr ^†(r) − r + V (r) ^(r) 2m first-principles density functional calculations with the 1 Z h i strong-correlation techniques, and the latter is usually + drdr0 ^†(r) ^†(r0)U(jr − r0j) ^(r0) ^(r) ; 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 contin- uous space to the lattice space. This issue has been where m is the mass of the atom, V (r) is a peri- [7−15] the focus of many theoretical efforts. However, odic potential and U(jrj) 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 (r) can be expanded as Wannier functions is. In this Letter we propose a general variational ^ X principle for mapping an interacting system in contin- (r) = a^jWj(r); (2) uous space to a lattice model: the choice of an incom- j plete set of Wannier functions minimizes the calcu- lated ground state energy of the lattice model. Based where Wj(r) = W (r − rj) is the Wannier function at on the principle, we derive a set of self-consistent non- site j and a^j 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 j j j j 3 4 Z 3 4 X † † 0h * 0 X † X † † + ha^ a^ a^j a^j i dr W (r + rj − rj ) H^ = − J a^ a^ + U a^ a^ a^ a^ : j1 j2 3 4 2 1 sb j1j2 j1 j2 j1j4j2j3 j1 j2 j3 j4 j1j2 j1j2 j1j2 i (3) 0 0 × W (r + rj2 − rj4 )U(jr − rj) W (r + rj2 − rj3 ): The parameters are given by (8) Z * It is clear that the above equation depends on the Jj j = − drW (r)H0Wj (r); (4) 1 2 j1 2 solution of the single-band lattice model (3). Before Z solving Eq. (3), one can employ various approxima- 1 0h * Uj j j j = drdr W (r)Wj (r) tions, for example, by choosing to keep only the near- 1 4 2 3 2 j1 4 i est neighbor tunneling J and the on-site interaction × U(jr − r0j)W * (r0)W (r0) ; U in the single-band Hamiltonian (3). Various many- j2 j3 (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 H0 ≡ − 2m r + V (r): a valid trial wave-function of a functional of the Wan- Further approximations for H^sb 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 jGti in the single-band ap- Mott-insulator regime, where the ground state has the proximation can be formally written as jGti = form jn ; n ; : : : ; n i with n being the average num- F (^a†)jvaccumi; where F is a certain function. The 0 0 0 0 j 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 ha^† a^ i = n 훿 ; (9) j1 j2 0 j1;j2 look for the Wannier functions that minimize the sys- ha^† a^† a^ a^ i = n2훿 훿 + n2훿 훿 tem’s single-band ground state energy, j1 j2 j3 j4 0 j1;j3 j2;j4 0 j1;j4 j2;j3 − (n2 + n )훿 훿 훿 : ^ ^ 0 0 j1;j2 j2;j3 j3;j4 EG = hGtjHjGti = hGtjHsbjGti; (6) (10) As a result, Eq. (8) is simplified and has the form for where H^sb is the usual single-band lattice Hamiltonian We achieve the minimization of the ground state U(jrj) = g0훿(r), energy EG by varying the Wannier function under the 휇0 X 2 orthonormal constraints, h = R drW *(r)W (r−r ) = W (r) = H0W (r) + g0n0 jW (r − rj)j W (r) j j N0 r 6=0 훿0;j: According to the Feynman–Hellman theorem, we j 2 have + g0(n0 − 1)jW (r)j W (r): (11) P 훿(EG − j 휇jhj) When the off-site terms, which are often very small, 훿W *(r) are ignored, the above equation has the form of the familiar Gross–Pitaevskii equation.[20;21] Therefore, it ^ 훿Hsb X 휇j훿hj = hG j jG i − = 0; (7) is very clear that the Wannier function is greatly in- t 훿W *(r) t 훿W *(r) j fluenced by the interaction whenever n0 ≥ 2.
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