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?
Best Answer
This is the basis of a pretty common set of techniques to find ground state properties. The hard part is writing down the matrix and multiplying it against trial wavefunctions in a large many-body basis. The projection intuition itself is not enough, but it turns out we can use:
Projector Quantum Monte Carlo (there's a lot of literature on this, but see for example http://arxiv.org/abs/0807.0682 section IV) to sample efficiently the effect of hitting a trial state with high powers of the hamiltonian matrix.
Time-evolved Block Decimation in imaginary time. This technique is closely related to DMRG, and again it's just a matter of having a good trial state (a Matrix Product State) and an efficient way of applying the thermal evolution operator (technical, but all the details are in http://arxiv.org/abs/quant-ph/0301063)