  A statistical method is derived for the calculation of thermodynamic
properties of many-body systems at low temperatures. This method is based on
the self-healing diffusion Monte Carlo method for complex functions [F. A.
Reboredo J. Chem. Phys. 136, 204101 (2012)] and some ideas of the correlation
function Monte Carlo approach [D. M. Ceperley and B. Bernu, J. Chem. Phys. 89,
6316 (1988)]. In order to allow the evolution in imaginary time to describe the
density matrix, we remove the fixed-node restriction using complex
antisymmetric guiding wave functions. In the process we obtain a parallel
algorithm that optimizes a small subspace of the many-body Hilbert space to
have maximum overlap with the subspace spanned by the lowest-energy eigenstates
of a many-body Hamiltonian. We show in a model system that the partition
function is progressively maximized within this subspace. We show that the
subspace spanned by the small basis systematically converges towards the
subspace spanned by the lowest energy eigenstates. Possible applications of
this method to calculate the thermodynamic properties of many-body systems near
the ground state are discussed. The resulting basis can be also used to
accelerate the calculation of the ground or excited states with Quantum Monte
Carlo.

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