Lie algebraic similarity transformed Hamiltonians for lattice model systems

Abstract

We present a class of Lie algebraic similarity transformations generated by exponentials of twobody on-site hermitian operators whose Hausdorff series can be summed exactly without truncation. The correlators are defined over the entire lattice and include the Gutzwiller factor n i↑ n i↓ , and twosite products of density (n i↑ + n i↓ ) and spin (n i↑ − n i↓ ) operators. The resulting non-hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with polynomial computational cost. The proposed similarity transformation generates locally weighted orbital transformations of the reference determinant. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the 1D and 2D repulsive Hubbard model where it yields accurate results for small and medium size interaction strengths.Introduction.-Hamiltonian similarity transformations are ubiquitous in many areas of physics, including electronic structure and condensed matter theories, and have been applied in a myriad of contexts [1][2][3][4][5][6]. JastrowGutzwiller correlation factors are also very popular as variational wave functions in quantum Monte Carlo and other applications [7][8][9][10][11][12][13][14][15]. Non-variational solutions have also been discussed in the literature. Tsuneyuki [16] presented a Hilbert space Jastrow method based on a Gutzwiller factor i n i↑ n i↓ and applied it to the 1D Hubbard model, minimizing its energy variance as in the transcorrelated method [17][18][19]. Neuscamman et al. [20] proposed many-body Jastrow correlators, diagonal in the lattice basis, and truncated them to a subset of sites matching a given pattern; these authors compared projective solutions with those obtained stochastically via Monte Carlo.Here, we consider Hamiltonian transformations of the form e −J He J based on hermitian correlators J built from general two-body products of on-site operators over the entire lattice. The transformations here are generated by density (charge), spin, and Gutzwiller factor correlators, including density-spin crossed terms. Similar Jastrowtype correlators have been extensively discussed in the literature but almost always in a variational context [10]. Our transformed Hamiltonian is non-hermitian but can be solved in mean-field via projective equations similar in spirit to those of coupled cluster theory [20,21]. In this sense, the model is an extension that fits under the generalized coupled cluster label [22][23][24]. The fundamental difference is that traditional coupled cluster is formulated with particle-hole excitations out of a reference determinant via a non-hermitian cluster operator; the present model is constructed with on-site hermitian correlators.The main result of this paper is the realization that the Hausdorff series resulting from the non-unitary similarity transformation e −J He J can be analytically summed. This result follows from Lie algebraic arguments [25] after recognizing that bot...

Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations

“…The superscript (TDL) denotes thermodynamic limit results from Eq. (23-24) for both the energy and J parameter, and an absent superscript refers Table II refer to energies obtained with restricted/unrestricted Hartree-Fock reference states with a general two-body correlator 73 , which includes all possible density-density correlations…”

“…The transcorrelated method has been quite widely applied in combination with explicitly correlated methods in quantum chemistry [79][80][81][82] , with approximations being employed to terminate the commutator series arising from the evaluation of e −τĤ eτ 83,84 . The explicit similarity transformation of the Hubbard Hamiltonian(1) with a Gutzwiller(2) 72,85 or more general correlator 73,74 , which can be obtained without approximations due to a terminating commutator series, has been solved on a mean-field level 72 . In the present work, we will not restrict ourselves to a mean-field solution, but solve for the exact ground state ofH with the FCIQMC method 75,86 .…”

Correlated Band Structure of a Transition Metal Oxide ZnO Obtained from a Many-Body Wave Function Theory

“…Moreover, the TC method is deterministic, i.e., free from the statistical error, unlike the QMC methods. Accurate calculations for the Hubbard model [35,36] and molecular systems [37][38][39] were also reported using the TC method or other theories that have a close relationship with the TC method. However, insofar as solid-state calculations are concerned, the TC method has so far been applied only to weakly correlated systems.…”

Iterative diagonalization of the non-Hermitian transcorrelated Hamiltonian using a plane-wave basis set: Application to sp-electron systems with deep core states

“…For solid-states calculations, the TC method is shown to reproduce accurate band structures [15][16][17] and optical absorption spectra [18]. The TC method and a related theory have been successfully applied also to the Hubbard model [19,20]. It is noteworthy that the TC method has a close relationship with some QMC methods in the sense that both methods employ the Jastrow correlation factor.…”