Polaron on a Lattice with a Trap in the Su–Schrieffer–Heeger Approximation

Abstract

The wave function and energy of the ground state of a lattice described by the Su–Schrieffer–Heeger Hamiltonian with a trap were found using three different methods. In the first of these methods, the energy of the ground state was calculated by numerical minimization of the energy functional. In the second, an auxiliary Hamiltonian was constructed, the diagonalization of which determined the energy of the ground state and the wave functions. And in the third, the wave function is completely restored analytically from the asymptotics of the wave function. All the three approaches produced results that agreed with an accuracy of 10^–15.

Appendix

Let us write the initial functional to be minimized in the form

$$\begin{gathered} E\{ y,\Psi \} = \frac{1}{2}\sum {y_{j}^{2}} + \left\langle {\Psi \left| {{{H}^{{\text{e}}}}} \right|\Psi } \right\rangle , \\ H_{{j,j}}^{{\text{e}}} = 2 - U{{\delta }_{{j,0}}},\,\,\,\,H_{{j,j + 1}}^{{\text{e}}} = H_{{j + 1,j}}^{{\text{e}}} = (\alpha {{y}_{j}} - 1){{\Psi }_{j}}{{\Psi }_{{j + 1}}}. \\ \end{gathered} $$

Let us minimize this functional as follows. Initially, we choose some initial set of coordinates y_j, e.g., an exponentially decreasing function: y_j ~ exp(–| j|). Using these y_j values, we minimize the functional in $\Psi .$ If the wave function $\Psi$ is the ground state of the Hamiltonian H^e, then we obtain the minimum energy. This problem can be solved differently and, e.g., the energy in y_j can begin to be minimized using a gradient method. Then, at each step, the functional should be minimized over all the values of $\Psi .$ This means that, in the course of this process, the system is always in the ground state of the Hamiltonian H^e. The minimum is reached if ${{y}_{j}} = - 2\alpha {{\Psi }_{{j + 1}}}{{\Psi }_{j}}.$ And then the wave function is the ground state of the Hamiltonian H^e_. It may seem that the obtained coincidence of the minimum of the functional and the ground state of the corresponding Hamiltonian is fully accidental. But this is not the case. One can give a wealth of examples of such a correspondence. For example, let us consider the Hamiltonian with an arbitrary diagonal nonlinearity:

$$H_{{j,j}}^{{\text{e}}} = 2 - f({{\Psi }_{j}}),\,\,\,\,H_{{j,j + 1}}^{{\text{e}}} = H_{{j,j + 1}}^{{\text{e}}} = - 1.$$

If, e.g.,  f(Ψ) = Ψ², then this a discrete variant of a nonlinear Schrödinger equation. It turns out that, for any function f(Ψ), one can match a functional the minimum of which corresponds to the ground state of the considered Hamiltonian.