# Can I use imaginary time propagation for many-body problems?

There are various ways to numerically find the ground state energy and wavefunction of a many-body Hamiltonian. You can diagonalize the Hamiltonian and pick out the lowest eigenstate, or you use Lancoz.

My proposal is that can I use imaginary time propagation for many-body problems?

Make it simple, I multiply a trial wavefunction $$|\psi_0\rangle = \sum_{i\in\text{eigen}} w^0_i |i\rangle$$ by an operator $\exp(-\hat H \tau)$. Then we have $$|\psi_\tau\rangle = \sum_{i\in\text{eigen}} w^\tau_i |i\rangle$$ with $w_i^\tau = w^0_i\exp(-E_i\tau)$.

For a sufficiently large $\tau$, we have $\exp(-\hat H \tau)\approx w_g^\tau \hat P_g$. The trial function will be projected to the ground state!

Choosing a complete set of basis states, we can numerically calculate the operator by taking Taylor expansion of the exponential and iterate to the $n$-th order, finally we obtain a matrix. Now multiply the matrix on a trial function written in terms of the basis we chose, then normalize it, and we get the ground state wavefunction.

Will it be accurate, stable and fast?

• Let's still concentrate on simple methods with a minmum number of subroutines. Some people are using operator $(\hat H - E)^n$ to find the ground state. I just modified it to make the procedure simple and clear. I realized that I don't need to write down the matrix at all, recording the resulting vector will suffice. – Machine Oct 1 '12 at 6:56