Memory efficient Fock-space recursion scheme for many-fermion correlation functions

Green’s function is the standard paradigm to study collective

properties of many-body quantum systems. The density of states, electronic

conductivity can be directly computed from one and two body excitations

(few-body Green’s functions) in the many-particle ground state. To compute

this few-body Green’s function exactly for lattice fermion systems, one needs

to calculate many-body Green’s functions in vacuum. To compute many-

body Green’s functions in vacuum, we need to invert the resolvent operator

defined as [ω + iη − H]⁻¹

in the relevant particle number sectors of the Fock

space, where ω, η , and H are the frequency, regulator and Hubbard Hamil-

tonian respectively. However, exponential Hilbert space growth limits such

calculations to small system sizes. In our work, we have developed a new

algorithm that allows a mapping between the Hilbert space of an L size pe-

riodic chain of spinless fermions to an abstract one-dimensional Fock-space

lattice with open boundary conditions. At half-filling, our method allows

O(1/L) suppression in RAM usage in the exact computation of many-body

correlations in a vacuum, as compared to brute force methods. We named

this method as Fock-space recursive Green’s functions (F-RGF) method [1].

We derive exact relations for computing few-body Green’s functions from

the above-mentioned resolvent calculations. While our relations are general

for arbitrary p-body excitations in many-body ground states, for demon-

stration purposes, we compute two-hole excitation spectra in partially-filled

many-body ground states for the Hubbard model that are relevant to spec-

troscopies that measures local few-body density of states.