# 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?

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)

• 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. Oct 1, 2012 at 6:56
• I guess I'm not sure what you mean -- if you are interested in a vector in the full many-body Hilbert space, even without the explicit matrix, you're still just proposing the Power Method, of which Lanczos is an improvement (and one that doesn't cost any more!) but it's still restricted to very small systems.
– wsc
Oct 1, 2012 at 12:53

Any arbitrary time-dependent wave function is already a sum of decaying terms if you use an imaginary time. The real problem is to find such a wave function analytically. It can be done by the perturbation theory.

Many years ago (1985) we used this idea for the Green function to estimate the ground state energy in quarkonuim (confining potential). The more perturbative terms you use, the better is accuracy, but it was hard to advance further than the third order (analytically). And we had to use non-linear approximations for the obtained series in order to better approximate the decaying exponential at large times.