Formal Framework for Multi-Particle Self-Consistent Field Methods with Tensor-Compressed Correlation

Jiawei Yan, Jonas B. Profe, Yuta Murakami +1 more

We develop a nonequilibrium multi-orbital extension of the two-particle self-consistent theory and apply it to the bilayer Hubbard model as a minimal platform to investigate correlation effects in the presence of interlayer interactions and tunneling. The method determines vertex corrections in the spin and charge channels self-consistently at the two-particle level, thereby avoiding the spurious finite-temperature phase transitions that limit dynamical mean-field theory in two dimensions. We derive the spectral self-energy and implement the framework directly on the real-frequency axis within the Keldysh nonequilibrium Green's function formalism, enabling the treatment of both equilibrium and non-equilibrium steady states without relying on numerical analytic continuation. As an application, we demonstrate that a pseudogap can emerge in the bilayer Hubbard model when spin, charge, or excitonic fluctuations become sufficiently strong. Instabilities in different channels are also evaluated in an unbiased manner across the parameter space. Remarkably, we find that the excitonic susceptibility grows with increasing interlayer bias, before it gets suppressed at large biases by the charge imbalance between the layers. This work establishes a versatile and computationally efficient framework for investigating correlated multi-orbital systems under nonequilibrium conditions.

Per‐Olov Löwdin

The independent-particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked-cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.

Shouhong Wang, Tian Ma

Robert A. Kromhout

A self-consistent field Hamiltonian is constructed, for arbitrary pair interactions, which yields (in principle) an exact value for the energy of a chosen state of a system of $N$ particles, as well as estimates for other states. The method for extending the formalism to interactions involving arbitrary numbers of particles is indicated and is straightforward. The method is carried out within the bound-state framework, and while motivated by the recent work of Bethe, results in a Hamiltonian somewhat different from his. An argument is presented which indicates that the independent-particle eigenfunctions of this Hamiltonian are "good" representations of the original $N$-body system. A discussion of the practicality of the computations is included.

Airi Kawasaki, Osamu Sugino