Ground States of the Hubbard Model

Abstract

A first step toward understanding a quantum Hamiltonian is to search for its ground state. In the absence of an exact solution, a judicial choice of a family of variational states Ψ^γ, where γ are variational parameters, can be fruitful. The variational theorem states that

$$\frac{{\langle {{\Psi }^{\gamma }}|\mathcal{H}|{{\Psi }^{\gamma }}\rangle }}{{\langle {{\Psi }^{\gamma }}|{{\Psi }^{\gamma }}\rangle }} = {{E}^{\gamma }} \geqslant {{E}_{0}},$$

(4.1)

where E ₀ is the exact ground state energy. A systematic improvement of the ground state energy and wave function can be achieved by minimizing the left-hand side of (4.1) with respect to ever larger families of variational states. The variational approach is conceptually straightforward and avoids the mathematically subtle convergence problems that plague perturbation theories and asymptotic expansions.