Imaginary-time Green's and correlation functions

Andro Sabashvili, Mats Granath, Hugo U. R. Strand +1 more

Rihito Sakurai, Wataru Mizukami, Hiroshi Shinaoka

Quantitative descriptions of strongly correlated materials pose a considerable challenge in condensed matter physics and chemistry. A promising approach to address this problem is quantum embedding methods. In particular, the dynamical mean-field theory (DMFT) maps the original system to an effective quantum impurity model comprising correlated orbitals embedded in an electron bath. The biggest bottleneck in DMFT calculations is numerically solving the quantum impurity model, i.e., computing the Green's function. Past studies have proposed theoretical methods to compute the Green's function of a quantum impurity model in polynomial time using a quantum computer. So far, however, efficient methods for computing the imaginary-time Green's functions have not been established despite the advantages of the imaginary-time formulation. We propose a quantum-classical hybrid algorithm for computing imaginary-time Green's functions on quantum devices with limited hardware resources by applying the variational quantum simulation. Using a quantum circuit simulator, we verified this algorithm by computing Green's functions for a dimer model as well as a four-site impurity model obtained by DMFT calculations of the single-band Hubbard model, although our method can be applied to general imaginary-time correlation functions.

G. Yue

Ben Yu-Kuang Hu

Using a simple contour deformation method it is shown that there is a very general relationship between imaginary-time correlation functions and real-time commutators. This method is used to show easy derivations of the equivalence between (1) two dissimilar forms of the Kubo formula for the electrical conductivity tensor σμν(ω), and (2) the imaginary-time-ordered Matsubara and the real-time retarded Green’s functions. In deriving example (1), the infra- red divergent paramagnetic and diamagnetic parts of σμν(ω) arise naturally from the imaginary-time formulation, and can be explicitly combined to give a single simple regular expression at ω→0.

Norman J. M. Horing

Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.