Strong parameter renormalization from optimum lattice model orbitals

Which is the best single-particle basis to express a Hubbard-like lattice model? A rigorous variational answer to this question leads to equations the solution of which depends in a self-consistent manner on the lattice ground state. Contrary to naive expectations, for arbitrary small interactions, the optimized orbitals differ from the noninteracting ones, leading also to substantial changes in the model parameters as shown analytically and in an explicit numerical solution for a simple double-well one-dimensional case. At strong coupling, we obtain the direct exchange interaction with a very large renormalization with important consequences for the explanation of ferromagnetism with model Hamiltonians. Moreover, in the case of two atoms and two fermions we show that the optimization equations are closely related to reduced density-matrix functional theory, thus establishing an unsuspected correspondence between continuum and lattice approaches.

Figure 1

(a) Caticha potential, $V_{\mathrm{Ca}}(x;a,b)$, along with the effective potentials, $v_0$ and $v_1$, in the limit of vanishing interaction, $\alpha \to 0$, and for $\alpha =6$. For $\alpha \to 0$, only $v_1\ne 0$. Other parameters: $t_0=0.2$. (b) Noninteracting and optimized Wannier states for $\alpha \to 0$ and $\alpha =6$. (c) Ground-state density (red lines) compared with the noninteracting density (black line). The solid and dotted lines show the density calculated with optimized and noninteracting orbitals, respectively.

Figure 2

(a) Renormalization of the hopping amplitude and of $U-V$ as a function of $\alpha$ shown as a ratio between the matrix element (M.E.) computed with optimized orbitals (OO) and noninteracting orbitals (NIO). (b) Direct and superexchange couplings as a function of $\alpha$. (c) Energy of the singlet and of the triplet as a function of $\alpha$. In both panels (b) and (c) solid and dotted lines indicate quantities calculated using optimized and noninteracting orbitals. Energies are in units of $I_0$. In these units the noninteracting parameters are $t_{\text{NIO}}=0.1,{(U-V)}_{\mathrm{NIO}}=0.31.$